Muhammad Yaseen

A Comprehensive Statistical Analysis of the Holy Quran

Muhammad Yaseen — Tue, 05 May 2026 04:00:00 GMT

library(knitr)
read_chunk("100Analysis.R")

1 Semantic Analysis of Quran

2 Surah Level Analysis

2.1 Surah Level Statistics

2.1.1 Number of Ayats, Words, and Characters

2.1.2 Top 10 longest Surahs by Ayat count

3 Summary by revelation type (Meccan vs Medinan)

4 Word Level Analysis

Citation

BibTeX citation:

@online{yaseen2026,
  author = {Yaseen, Muhammad},
  title = {A {Comprehensive} {Statistical} {Analysis} of the {Holy}
    {Quran}},
  date = {2026-05-05},
  url = {https://myaseen208.com/blogs/2025-06-28_Quran/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2026. “A Comprehensive Statistical Analysis of the Holy Quran.” May 5. https://myaseen208.com/blogs/2025-06-28_Quran/.

Batting Performance of Babar Azam and Virat Kohli

Muhammad Yaseen — Fri, 04 Apr 2025 04:00:00 GMT

Test Batting Performance of Babar Azam

Figure 1: Babar Azam

Overall Performance

Mat: Number of Matches played

Inns: Number of Innings played

NO: Number of NOT OUT

Runs: Runs scored

HS: Highest Runs Scored

Ave: Average

BF: Balls Faced

SR: Strike Rate

Hundreds: Number of Hundreds Scored

Fifties: Number of Fifties Scored

Fours: Number of Fours hitted

Sixes: Number of Sixes hitted

Performance by Years

Test Batting Performance of Babar Azam over the years

Performance versus different oppositions

Test Batting Performance of Babar Azam over the years by each inning

Performance by Years and Innings

Test Batting Performance of Babar Azam versus different oppositions

Test Batting Performance of Virat Kohli

Figure 2: Virat Kohli

Overall Performance

Mat: Number of Matches played

Inns: Number of Innings played

NO: Number of NOT OUT

Runs: Runs scored

HS: Highest Runs Scored

Ave: Average

BF: Balls Faced

SR: Strike Rate

Hundreds: Number of Hundreds Scored

Fifties: Number of Fifties Scored

Fours: Number of Fours hitted

Sixes: Number of Sixes hitted

Performance by Years

Test Batting Performance of Virat Kohli over the years

Performance versus different oppositions

Test Batting Performance of Virat Kohli over the years by each inning

Performance by Years and Innings

Test Batting Performance of Virat Kohli versus different oppositions

Citation

BibTeX citation:

@online{yaseen2025,
  author = {Yaseen, Muhammad},
  title = {Batting {Performance} of {Babar} {Azam} and {Virat} {Kohli}},
  date = {2025-04-04},
  url = {https://myaseen208.com/blogs/2025-04-04_Cricket/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2025. “Batting Performance of Babar Azam and Virat Kohli.” April 4. https://myaseen208.com/blogs/2025-04-04_Cricket/.

Pakistan Population Census

Muhammad Yaseen — Tue, 15 Aug 2023 04:00:00 GMT

Introduction

The Pakistan Bureau of Statistics (PBS) holds the pivotal responsibility of orchestrating, collating, and disseminating the outcomes of the population census within the nation. On the significant date of August 5, 2023, the Council of Common Interests (CCI) accorded its approval for the findings derived from the Census of 2023, a comprehensive endeavor led by the PBS to capture the demographic landscape of Pakistan. To facilitate widespread access to this crucial data, the PBS published the conclusive results of the Census 2023 on its official website in a PDF format, albeit one that is not machine-readable.

However, this presentation in PDF form poses a significant challenge to researchers and analysts aiming to delve into the insights embedded within this valuable data. The intricacies of extracting, processing, and interpreting information from a non-machine-readable format like PDF hinder the swift utilization of this data for in-depth analysis. Recognizing this impediment, a strategic solution has been crafted in the form of an R programming language package termed PakPC2023.

The PakPC2023 R package has been meticulously developed to surmount the obstacles presented by the non-machine-readable PDF format. This specialized software acts as a bridge, effectively converting the cumbersome PDF-based Census 2023 data into a format that is amenable for analysis within the R software environment. Researchers, analysts, and data enthusiasts can seamlessly install the PakPC2023 package from the Comprehensive R Archive Network (CRAN), a repository that serves as a hub for R packages, ensuring easy accessibility to this innovative tool.

By implementing the PakPC2023 package, the door is opened for researchers to harness the power of the Census 2023 data in an efficient and effective manner. This strategic initiative not only streamlines the process of accessing and utilizing the census data but also empowers researchers to unlock valuable insights and trends that lie within, contributing to informed decision-making and comprehensive socio-economic understanding.

This dashboard allows you to explore datasets from the Pakistan Population Census of 2017 and 2023. Choose a dataset from the sidebar to generate tables and graphs.

Province Level Analysis

Figure 1: Population by Regions

The visual representation presented in Figure 1 provides a comprehensive overview of the population distribution across various regions of Pakistan. This data has been meticulously collected from the Pakistan Population Census of 2023, serving as an accurate reflection of the demographic landscape of the country.

A closer examination of the graph reveals that the province of Punjab emerges as the most populous region, boasting an impressive population count of approximately 128 million individuals.

Following closely, the province of Sindh contributes significantly to the nation’s population with approximately 56 million residents.

Turning our attention towards the remaining regions, Khyber Pakhtunkhwa stands as a substantial population center with around 41 million inhabitants, further emphasizing its role in the overall demographic makeup of Pakistan. On the other hand, Balochistan, while having a comparably smaller population, still contributes notably with around 15 million people.

It is noteworthy that the data showcased in Figure 1 not only underscores the population sizes of these regions but also underscores the diversity and complexity of Pakistan’s demographic distribution. This information serves as a valuable resource for policymakers, researchers, and anyone interested in gaining insights into the population dynamics of Pakistan.

Figure 2: Population by Regions & Area

Figure 2 illustrates the distribution of Pakistan’s population across various regions and areas, providing insights drawn from the Pakistan Population Census 2023. The data underscores a significant urban-rural divide within the country, shedding light on the demographic landscape.

Notably, the data reveals that a substantial portion of Pakistan’s population resides in rural areas, accounting for a staggering figure of more than 147.75 million individuals. In contrast, the urban population stands at slightly over 93.75 million. This distribution underscores the predominance of rural living conditions in Pakistan.”),

Examining the regional disparities, the province of Punjab emerges as the leader in terms of rural population, boasting an impressive count of over 75.71 million inhabitants in rural settings. Sindh, on the other hand, leads the pack in urban population, hosting a significant urban community of over 29.92 million individuals.

The data presented within Figure 2 serves to accentuate the pronounced urban-rural divide prevalent within Pakistan. The statistics reaffirm the prevalence of rural living conditions, with Punjab standing out as the province with the highest rural population, surpassing the 75.71 million mark.

The second spot in terms of rural population is claimed by Sindh, hosting more than 25.77 million residents in rural areas. In contrast, Balochistan registers as the province with the lowest rural population, counting slightly over 4.61 million individuals.

Shifting the focus to urban populations, Punjab once again takes the lead, this time in the urban category, as it accommodates a substantial urban populace exceeding 52.00 million. Sindh follows closely, contributing to the urban landscape with over 29.90 million inhabitants. In stark contrast, Khyber Pakhtunkhwa and Balochistan exhibit the smallest urban populations, each recording figures slightly surpassing 6.10 million and 4.60 million individuals, respectively.

In conclusion, Figure 2 casts a spotlight on the demographic distribution within Pakistan, drawing attention to the prevailing urban-rural divide. The robust presentation of data underscores the dominance of rural populations, particularly in provinces like Punjab and Sindh, while also providing valuable insights into the urban landscape across various regions of the country.

Figure 3: Average Annual Population Growth Rate by Regions

Figure 4: Average Annual Population Growth Rate by Regions & Area

Division Level Analysis

Figure 5: Population by Regions & Division

Figure 6: Average Annual Population Growth Rate by Regions & Division

District Level Analysis

Figure 7: Population by Division & District for Punjab

Figure 8: Population by Division & District for Sindh

Figure 9: Population by Division & District for KP

Figure 10: Population by Division & District for Balochistan

Figure 11: Average Annual Population Growth Rate by Division & District for Punjab

Figure 12: Average Annual Population Growth Rate by Division & District for Sindh

Figure 13: Average Annual Population Growth Rate by Division & District for KP

Figure 14: Average Annual Population Growth Rate by Division & District for Balochistan

Citation

BibTeX citation:

@online{yaseen2023,
  author = {Yaseen, Muhammad},
  title = {Pakistan {Population} {Census}},
  date = {2023-08-15},
  url = {https://myaseen208.com/blogs/2025-04-04_PakPC/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2023. “Pakistan Population Census.” August 15. https://myaseen208.com/blogs/2025-04-04_PakPC/.

An Introduction to R Programming Language

Muhammad Yaseen — Tue, 18 Apr 2023 04:00:00 GMT

Introduction

R is a free, open-source programming language and software environment for statistical computing, bioinformatics, visualization and general computing. R provides a wide variety of statistical and graphical techniques, and is highly extensible. The latest version of R can be obtained from https://cran.r-project.org/bin/windows/base/.

R R for Windows (32/64 bit)

RStudio RStudio

Manual An Introduction to R

Ref Card R Reference Card

New York Times New York Times

Nature Article Nature Article

Slides

The Slides may be accessed from Slides.

Press f for Fullscreen view.

Citation

BibTeX citation:

@online{yaseen2023,
  author = {Yaseen, Muhammad},
  title = {An {Introduction} to {R} {Programming} {Language}},
  date = {2023-04-18},
  url = {https://myaseen208.com/blogs/2023-04-18_IntroR/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2023. “An Introduction to R Programming Language.” April 18. https://myaseen208.com/blogs/2023-04-18_IntroR/.

Statistics: The Art & Science of Learning from Data

Muhammad Yaseen — Fri, 20 May 2022 04:00:00 GMT

knitr::read_chunk("2022-05-20_TUF.R")

Introduction

Statistics

Statistics is the science of uncertainty & variability

Statistics turns data into information

Data -> Information -> Knowledge -> Wisdom

Data Driven Decisions (3Ds)

Statistics is the interpretation of Science

Statistics is the Art & Science of learning from data

Variable

Characteristic that may vary from individual to individual

Height, Weight, CGPA etc

Measurement

Process of assigning numbers or labels to objects or states in accordance with logically accepted rules

Figure 1: Measurement

Measurement Scales

Nominal Scale: Obersvations may be classified into mutually exclusive & exhaustive classes or categories

Ordinal Scale: Obersvations may be ranked

Interval Scale: Difference between obersvations is meaningful

Ratio Scale: Ratio between obersvations is meaningful & true zero point

Figure 2: Measurement Scales

Exploring Data with Graphs & Numerical Summaries

Graphs

Numerical Summaries

Example

The following data shows the ID, Gender (Male, Female), Age, Race (Mexican American, Non-Hispanic Black, Non-Hispanic White, Other Hispanic, Other/Mixed), BMI, and BMI.Cat (Under Weight, Normal Weight, Over Weight, Obese) from the National Health and Nutrition Examination Survey (NHANES). The survey is conducted by the National Center for Health Statistics (NCHS), and data are publicly available at: https://www.cdc.gov/nchs/nhanes.htm . NHANES data are reported in well over one thousand peer-reviewed journal publications every year.

Descriptive Statistics

Number of Observations

Measures of Central Tendency

Measures of Central Dispersion

Measures of Skewness

Measures of Kurtosis

Example

The following data shows the ID, Gender (Male, Female), Age, Race (Mexican American, Non-Hispanic Black, Non-Hispanic White, Other Hispanic, Other/Mixed), BMI, and BMI.Cat (Under Weight, Normal Weight, Over Weight, Obese) from the National Health and Nutrition Examination Survey (NHANES). The survey is conducted by the National Center for Health Statistics (NCHS), and data are publicly available at: https://www.cdc.gov/nchs/nhanes.htm . NHANES data are reported in well over one thousand peer-reviewed journal publications every year.

Correlation Analysis

Example

The following data shows the ID, Gender (Male, Female), Age, Race (Mexican American, Non-Hispanic Black, Non-Hispanic White, Other Hispanic, Other/Mixed), BMI, and BMI.Cat (Under Weight, Normal Weight, Over Weight, Obese) from the National Health and Nutrition Examination Survey (NHANES). The survey is conducted by the National Center for Health Statistics (NCHS), and data are publicly available at: https://www.cdc.gov/nchs/nhanes.htm . NHANES data are reported in well over one thousand peer-reviewed journal publications every year.

An Introduction to Linear Models

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Figure 3: Population Regression Function

Example

The following data shows the ID, Gender (Male, Female), Age, Race (Mexican American, Non-Hispanic Black, Non-Hispanic White, Other Hispanic, Other/Mixed), BMI, and BMI.Cat (Under Weight, Normal Weight, Over Weight, Obese) from the National Health and Nutrition Examination Survey (NHANES). The survey is conducted by the National Center for Health Statistics (NCHS), and data are publicly available at: https://www.cdc.gov/nchs/nhanes.htm . NHANES data are reported in well over one thousand peer-reviewed journal publications every year.

Analysis of Variance (ANOVA)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 4: Analysis of Variance

Example

The following data shows the ID, Gender (Male, Female), Age, Race (Mexican American, Non-Hispanic Black, Non-Hispanic White, Other Hispanic, Other/Mixed), BMI, and BMI.Cat (Under Weight, Normal Weight, Over Weight, Obese) from the National Health and Nutrition Examination Survey (NHANES). The survey is conducted by the National Center for Health Statistics (NCHS), and data are publicly available at: https://www.cdc.gov/nchs/nhanes.htm . NHANES data are reported in well over one thousand peer-reviewed journal publications every year.

Analysis of Covariance (ANCOVA)

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 5: Analysis of Covariance

Example

The following data shows the ID, Gender (Male, Female), Age, Race (Mexican American, Non-Hispanic Black, Non-Hispanic White, Other Hispanic, Other/Mixed), BMI, and BMI.Cat (Under Weight, Normal Weight, Over Weight, Obese) from the National Health and Nutrition Examination Survey (NHANES). The survey is conducted by the National Center for Health Statistics (NCHS), and data are publicly available at: https://www.cdc.gov/nchs/nhanes.htm . NHANES data are reported in well over one thousand peer-reviewed journal publications every year.

R Software

R is a free, open-source programming language and software environment for statistical computing, bioinformatics, visualization and general computing. R provides a wide variety of statistical and graphical techniques, and is highly extensible. The latest version of R can be obtained from https://cran.r-project.org/bin/.

RStudio

RStudio is a powerful integrated development environment (IDE) for R, and it can be downloaded from https://www.rstudio.com/products/rstudio/download/.

Figure 6: An Introduction to R

R Short Reference Card

Figure 7: R Short Reference Card

2009 New York Times Article on R

Figure 8: 2009 New York Times Article on R

2015 Nature Article on R

Figure 9: 2015 Nature Article on R

Citation

BibTeX citation:

@online{yaseen2022,
  author = {Yaseen, Muhammad and Yaseen, Muhammad},
  title = {Statistics: {The} {Art} \& {Science} of {Learning} from
    {Data}},
  date = {2022-05-20},
  url = {https://myaseen208.com/blogs/2022-05-20_TUF/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad, and Muhammad Yaseen. 2022. “Statistics: The Art & Science of Learning from Data.” May 20. https://myaseen208.com/blogs/2022-05-20_TUF/.

Design & Analysis of Field Experiments using R

Muhammad Yaseen — Fri, 06 Mar 2020 05:00:00 GMT

Introduction

R is a free, open-source programming language and software environment for statistical computing, bioinformatics, visualization and general computing. R provides a wide variety of statistical and graphical techniques, and is highly extensible. The latest version of R can be obtained from https://cran.r-project.org/bin/windows/base/.

R R for Windows (32/64 bit)

RStudio RStudio

Manual An Introduction to R

Ref Card R Reference Card

New York Times New York Times

Nature Article Nature Article

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Figure 1

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

# Fertilizer (Kg/acre)
# Production (Bushels/acre)
Fertilizer <- c(100, 200, 400, 500)
Production <- c( 70,  70,  80, 100)
df1        <- data.frame(Fertilizer, Production)
df1

  Fertilizer Production
1        100         70
2        200         70
3        400         80
4        500        100

library(ggplot2)
p1 <- 
  ggplot(data = df1, mapping = aes(x = Fertilizer, y = Production)) + 
  geom_point() +
  labs(x = "Fertilizer", y = "Production") +
  theme_bw()
print(p1)

fm1 <- lm(formula = Production ~ Fertilizer, data = df1)
summary(fm1)


Call:
lm(formula = Production ~ Fertilizer, data = df1)

Residuals:
 1  2  3  4 
 4 -3 -7  6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 59.00000    7.95299   7.419   0.0177 *
Fertilizer   0.07000    0.02345   2.985   0.0963 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.416 on 2 degrees of freedom
Multiple R-squared:  0.8167,    Adjusted R-squared:  0.725 
F-statistic: 8.909 on 1 and 2 DF,  p-value: 0.0963

anova(fm1)

Analysis of Variance Table

Response: Production
           Df Sum Sq Mean Sq F value Pr(>F)  
Fertilizer  1    490     490  8.9091 0.0963 .
Residuals   2    110      55                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Fertilizer	Yield
0.3	10
0.6	15
0.9	30
1.2	35
1.5	25
1.8	30
2.1	50
2.4	45

Fert  <- c(0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4)
Yield <- c(10, 15, 30, 35, 25, 30, 50, 45)
df2   <- data.frame(Fert, Yield)
df2

  Fert Yield
1  0.3    10
2  0.6    15
3  0.9    30
4  1.2    35
5  1.5    25
6  1.8    30
7  2.1    50
8  2.4    45

library(ggplot2)
p2 <- 
  ggplot(data = df2, mapping = aes(x = Fert, y = Yield)) + 
  geom_point() +
  labs(x = "Fertilizer", y = "Yield") +
  theme_bw()
print(p2)

fm2 <- lm(formula = Yield ~ Fert, data = df2)
summary(fm2)


Call:
lm(formula = Yield ~ Fert, data = df2)

Residuals:
   Min     1Q Median     3Q    Max 
-7.441 -4.018 -2.441  7.351  7.798 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    8.036      5.504   1.460   0.1946   
Fert          16.270      3.633   4.478   0.0042 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.064 on 6 degrees of freedom
Multiple R-squared:  0.7697,    Adjusted R-squared:  0.7313 
F-statistic: 20.05 on 1 and 6 DF,  p-value: 0.004202

anova(fm2)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value   Pr(>F)   
Fert       1 1000.6  1000.6  20.052 0.004202 **
Residuals  6  299.4    49.9                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Weekly Income ($)	Weekly Expenditures ($)
80	70
100	65
120	90
140	95
160	110
180	115
200	120
220	140
240	155
260	150

# Weekly Income ($) of a Family
# Weekly Expenditures ($) of a Family
Income       <- seq(from = 80, to = 260, by = 20)
Expenditures <- c(70, 65, 90, 95, 110, 115, 120, 140, 155, 150)
df3          <- data.frame(Income, Expenditures)
df3

   Income Expenditures
1      80           70
2     100           65
3     120           90
4     140           95
5     160          110
6     180          115
7     200          120
8     220          140
9     240          155
10    260          150

library(ggplot2)
p3 <- 
  ggplot(data = df3, mapping = aes(x = Income, y = Expenditures)) + 
  geom_point() +
  scale_x_continuous(labels = scales::dollar) +
  scale_y_continuous(labels = scales::dollar) +
  labs(x = "Weekly income, $", y = "Weekly consumption expenditures, $") +
  theme_bw()
print(p3)

fm3 <- lm(formula = Expenditures ~ Income, data = df3)
summary(fm3)


Call:
lm(formula = Expenditures ~ Income, data = df3)

Residuals:
    Min      1Q  Median      3Q     Max 
-10.364  -4.977   1.409   4.364   8.364 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 24.45455    6.41382   3.813  0.00514 ** 
Income       0.50909    0.03574  14.243 5.75e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.493 on 8 degrees of freedom
Multiple R-squared:  0.9621,    Adjusted R-squared:  0.9573 
F-statistic: 202.9 on 1 and 8 DF,  p-value: 5.753e-07

anova(fm3)

Analysis of Variance Table

Response: Expenditures
          Df Sum Sq Mean Sq F value    Pr(>F)    
Income     1 8552.7  8552.7  202.87 5.753e-07 ***
Residuals  8  337.3    42.2                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Fertilizer (Kg)	Rainfall (mm)	Yield (Kg)
100	10	40
200	20	50
300	10	50
400	30	70
500	20	65
600	20	65
700	30	80

# Fertilizer (Kg)
# Rainfall (mm)
# Yield (Kg)
Fertilizer <- seq(from = 100, to = 700, by = 100)
Rainfall   <- c(10, 20, 10, 30, 20, 20, 30)
Yield      <- c(40, 50, 50, 70, 65, 65, 80)
df4        <- data.frame(Fertilizer, Rainfall, Yield)
df4

  Fertilizer Rainfall Yield
1        100       10    40
2        200       20    50
3        300       10    50
4        400       30    70
5        500       20    65
6        600       20    65
7        700       30    80

library(ggplot2)
ggplot(data = df4, mapping = aes(x = Fertilizer, y = Rainfall, color = Yield)) +
  geom_point(alpha = 0.8, size = 2) +
  scale_color_viridis_c() +
  theme_minimal()

fm4 <- lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)
summary(fm4)


Call:
lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)

Residuals:
      1       2       3       4       5       6       7 
-0.2381 -2.3810  2.1429  1.6667  1.1905 -2.6190  0.2381 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 28.095238   2.491482  11.277 0.000352 ***
Fertilizer   0.038095   0.005832   6.532 0.002838 ** 
Rainfall     0.833333   0.154303   5.401 0.005690 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.315 on 4 degrees of freedom
Multiple R-squared:  0.9814,    Adjusted R-squared:  0.972 
F-statistic: 105.3 on 2 and 4 DF,  p-value: 0.0003472

anova(fm4)

Analysis of Variance Table

Response: Yield
           Df Sum Sq Mean Sq F value    Pr(>F)    
Fertilizer  1 972.32  972.32 181.500 0.0001756 ***
Rainfall    1 156.25  156.25  29.167 0.0056899 ** 
Residuals   4  21.43    5.36                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(lm.beta)
lm.beta(fm4)


Call:
lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)

Standardized Coefficients::
(Intercept)  Fertilizer    Rainfall 
         NA   0.5944301   0.4914732

Polynomial Regression Analysis

Quantifying non-linear dependency of a normal response on quantitative explanatory variable(s)

Example

An experiment was conducted to evaluate the effects of different levels of nitrogen. Three levels of nitrogen: 0, 10 and 20 grams per plot were used in the experiment. Each treatment was replicated twice and data is given below:

Nitrogen	Yield
0	5
0	7
10	15
10	17
20	9
20	11

Nitrogen <- c(0, 0, 10, 10, 20, 20)
Yield    <- c(5, 7, 15, 17,  9, 11)
df5      <- data.frame(Nitrogen, Yield)
df5

  Nitrogen Yield
1        0     5
2        0     7
3       10    15
4       10    17
5       20     9
6       20    11

library(ggplot2)
p5 <- 
  ggplot(data = df5, mapping = aes(x = Nitrogen, y = Yield)) + 
  geom_point() +
  labs(x = "Nitrogen", y = "Yield") +
  theme_bw()
print(p5)

fm5 <- lm(formula = Yield ~ Nitrogen + I(Nitrogen^2), data = df5)
# fm5 <- lm(formula = Yield ~ poly(x = Nitrogen, degree = 2, raw = TRUE), data = df5)
summary(fm5)


Call:
lm(formula = Yield ~ Nitrogen + I(Nitrogen^2), data = df5)

Residuals:
 1  2  3  4  5  6 
-1  1 -1  1 -1  1 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.00000    1.00000   6.000  0.00927 **
Nitrogen       1.80000    0.25495   7.060  0.00584 **
I(Nitrogen^2) -0.08000    0.01225  -6.532  0.00729 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.414 on 3 degrees of freedom
Multiple R-squared:  0.9441,    Adjusted R-squared:  0.9068 
F-statistic: 25.33 on 2 and 3 DF,  p-value: 0.01322

anova(fm5)

Analysis of Variance Table

Response: Yield
              Df Sum Sq Mean Sq F value   Pr(>F)   
Nitrogen       1 16.000  16.000   8.000 0.066276 . 
I(Nitrogen^2)  1 85.333  85.333  42.667 0.007292 **
Residuals      3  6.000   2.000                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Analysis of Variance (ANOVA)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 2

Example

Yield	Variety
5	V1
6	V1
7	V1
15	V2
16	V2
17	V2

Yield   <- c(5, 6, 7, 15, 16, 17)
Variety <- gl(n = 2, k = 3, length = 2*3, labels = c("V1", "V2"))
df6     <- data.frame(Yield, Variety)
df6

  Yield Variety
1     5      V1
2     6      V1
3     7      V1
4    15      V2
5    16      V2
6    17      V2

library(ggplot2)
p6 <- 
  ggplot(data = df6, mapping = aes(x = Variety, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Variety", y = "Yield") +
  theme_bw()
print(p6)

fm6 <- lm(formula = Yield ~ Variety, data = df6)
summary(fm6)


Call:
lm(formula = Yield ~ Variety, data = df6)

Residuals:
         1          2          3          4          5          6 
-1.000e+00 -4.774e-15  1.000e+00 -1.000e+00 -1.110e-16  1.000e+00 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   6.0000     0.5774   10.39 0.000484 ***
VarietyV2    10.0000     0.8165   12.25 0.000255 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1 on 4 degrees of freedom
Multiple R-squared:  0.974, Adjusted R-squared:  0.9675 
F-statistic:   150 on 1 and 4 DF,  p-value: 0.0002552

anova(fm6)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value    Pr(>F)    
Variety    1    150     150     150 0.0002552 ***
Residuals  4      4       1                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Yield	Variety
5	V1
7	V1
15	V2
17	V2
17	V3
19	V3

Yield   <- c(5, 7, 15, 17, 17, 19)
Variety <- gl(n = 3, k = 2, length = 3*2, labels = c("V1", "V2", "V3"))
df7     <- data.frame(Yield, Variety)
df7

  Yield Variety
1     5      V1
2     7      V1
3    15      V2
4    17      V2
5    17      V3
6    19      V3

library(ggplot2)
p7 <- 
  ggplot(data = df7, mapping = aes(x = Variety, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Variety", y = "Yield") +
  theme_bw()
print(p7)

fm7 <- lm(formula = Yield ~ Variety, data = df7)
summary(fm7)


Call:
lm(formula = Yield ~ Variety, data = df7)

Residuals:
 1  2  3  4  5  6 
-1  1 -1  1 -1  1 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.000      1.000   6.000  0.00927 **
VarietyV2     10.000      1.414   7.071  0.00582 **
VarietyV3     12.000      1.414   8.485  0.00344 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.414 on 3 degrees of freedom
Multiple R-squared:  0.965, Adjusted R-squared:  0.9416 
F-statistic: 41.33 on 2 and 3 DF,  p-value: 0.006553

anova(fm7)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value   Pr(>F)   
Variety    2 165.33  82.667  41.333 0.006553 **
Residuals  3   6.00   2.000                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Analysis of Covariance (ANCOVA)

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 3

Example

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Yield    <- c(51, 52, 53, 54, 56, 57, 55, 56, 58, 59, 62, 63)
Fert     <- rep(x = c(80, 90, 100), each = 2)
Variety  <- gl(n = 2, k = 6, length = 2*6, labels = c("V1", "V2"), ordered = FALSE)
df8         <- data.frame(Yield, Fert, Variety)
df8

   Yield Fert Variety
1     51   80      V1
2     52   80      V1
3     53   90      V1
4     54   90      V1
5     56  100      V1
6     57  100      V1
7     55   80      V2
8     56   80      V2
9     58   90      V2
10    59   90      V2
11    62  100      V2
12    63  100      V2

Same intercepts but different slopes

fm8 <- lm(formula = Yield ~ Fert + Variety, data = df8)
summary(fm8)


Call:
lm(formula = Yield ~ Fert + Variety, data = df8)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.8333 -0.8333  0.1667  0.1667  1.1667 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 26.83333    2.54557   10.54 2.30e-06 ***
Fert         0.30000    0.02805   10.69 2.04e-06 ***
VarietyV2    5.00000    0.45812   10.91 1.72e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7935 on 9 degrees of freedom
Multiple R-squared:  0.9629,    Adjusted R-squared:  0.9546 
F-statistic: 116.7 on 2 and 9 DF,  p-value: 3.657e-07

anova(fm8)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value    Pr(>F)    
Fert       1 72.000   72.00  114.35 2.042e-06 ***
Variety    1 75.000   75.00  119.12 1.720e-06 ***
Residuals  9  5.667    0.63                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(ggplot2)
p8 <- 
  ggplot(data = df8, mapping = aes(x = Fert, y = Yield, color = Variety)) + 
  geom_point() +
  geom_line(mapping = aes(y = predict(fm8))) +
  labs(x = "Fert", y = "Yield") +
  theme_bw()
print(p8)

Different intercepts and different slopes

fm9 <- lm(formula = Yield ~ Fert * Variety, data = df8)
summary(fm9)


Call:
lm(formula = Yield ~ Fert * Variety, data = df8)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.83333 -0.33333 -0.08333  0.66667  0.66667 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)    31.33333    3.05903  10.243 7.09e-06 ***
Fert            0.25000    0.03385   7.385 7.73e-05 ***
VarietyV2      -4.00000    4.32612  -0.925   0.3822    
Fert:VarietyV2  0.10000    0.04787   2.089   0.0701 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.677 on 8 degrees of freedom
Multiple R-squared:  0.976, Adjusted R-squared:  0.967 
F-statistic: 108.4 on 3 and 8 DF,  p-value: 8.11e-07

anova(fm9)

Analysis of Variance Table

Response: Yield
             Df Sum Sq Mean Sq  F value    Pr(>F)    
Fert          1 72.000  72.000 157.0909 1.538e-06 ***
Variety       1 75.000  75.000 163.6364 1.315e-06 ***
Fert:Variety  1  2.000   2.000   4.3636   0.07013 .  
Residuals     8  3.667   0.458                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(ggplot2)
p9 <- 
  ggplot(data = df8, mapping = aes(x = Fert, y = Yield, color = Variety)) + 
  geom_point() +
  geom_line(mapping = aes(y = predict(fm9))) +
  labs(x = "Fert", y = "Yield") +
  theme_bw()
print(p9)

Correlation Analysis

Linear Relationship between Quantitative Variables

Simple Correlation Analysis

Linear Relationship between Two Quantitative Variables

Example

Sparrow Wing length (cm)	Sparrow Tail length (cm)
10.4	7.4
10.8	7.6
11.1	7.9
10.2	7.2
10.3	7.4
10.2	7.1
10.7	7.4
10.5	7.2
10.8	7.8
11.2	7.7
10.6	7.8
11.4	8.3

WingLength  <- c(10.4, 10.8, 11.1, 10.2, 10.3, 10.2, 10.7, 10.5, 10.8, 11.2, 10.6, 11.4)
TailLength  <- c(7.4, 7.6, 7.9, 7.2, 7.4, 7.1, 7.4, 7.2, 7.8, 7.7, 7.8, 8.3)
df10        <- data.frame(WingLength, TailLength)
df10

   WingLength TailLength
1        10.4        7.4
2        10.8        7.6
3        11.1        7.9
4        10.2        7.2
5        10.3        7.4
6        10.2        7.1
7        10.7        7.4
8        10.5        7.2
9        10.8        7.8
10       11.2        7.7
11       10.6        7.8
12       11.4        8.3

library(ggplot2)
p10 <- 
  ggplot(data = df10, mapping = aes(x= WingLength, y = TailLength)) + 
  geom_point() +
  labs(x = "Sparrow Wing Length (cm)", y = "Sparrow Tail Length (cm)") +
  theme_bw()
print(p10)

cor(df10$WingLength, df10$TailLength)

[1] 0.8703546

cor.test(df10$WingLength, df10$TailLength)


    Pearson's product-moment correlation

data:  df10$WingLength and df10$TailLength
t = 5.5893, df = 10, p-value = 0.0002311
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5923111 0.9631599
sample estimates:
      cor 
0.8703546

Partial Correlation Analysis

Linear Relationship between Quantitative Variables while holding/keeping all other constants

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

LeafArea    <- c(72, 174, 116, 78, 134, 95, 113, 98, 148, 42)
LeafMoist   <- c(75, 81, 83, 79, 81, 80, 81, 74, 78, 58)
TotShLength <- c(307, 529, 632, 527, 442, 525, 481, 710, 422, 345)
df11        <- data.frame(LeafArea, LeafMoist, TotShLength)
df11

   LeafArea LeafMoist TotShLength
1        72        75         307
2       174        81         529
3       116        83         632
4        78        79         527
5       134        81         442
6        95        80         525
7       113        81         481
8        98        74         710
9       148        78         422
10       42        58         345

library(ppcor)
pcor(df11)

$estimate
                LeafArea LeafMoist  TotShLength
LeafArea     1.000000000 0.6509096 -0.007012236
LeafMoist    0.650909618 1.0000000  0.324334659
TotShLength -0.007012236 0.3243347  1.000000000

$p.value
              LeafArea  LeafMoist TotShLength
LeafArea    0.00000000 0.05760462   0.9857154
LeafMoist   0.05760462 0.00000000   0.3944839
TotShLength 0.98571537 0.39448385   0.0000000

$statistic
               LeafArea LeafMoist TotShLength
LeafArea     0.00000000  2.268502 -0.01855309
LeafMoist    2.26850175  0.000000  0.90714704
TotShLength -0.01855309  0.907147  0.00000000

$n
[1] 10

$gp
[1] 1

$method
[1] "pearson"

Multiple Correlation Analysis

Linear Relationship between a Quantitative Variable and set of other Quantitative Variables

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

fm12 <- lm(formula = TotShLength ~ LeafArea + LeafMoist, data = df11)
sqrt(summary(fm12)$r.squared)

[1] 0.421277

Completely Randomized Design (CRD)

Used when experimental material is homogeneous

Example

The following table shows some of the results of an experiment on the effects of applications of sulphur in reducing scab disease of potatoes. The object in applying sulphur is to increase the acidity of the soil, since scab does not thrive in very acid soil. In addition to untreated plots which serve as a control, 3 amounts of dressing were compared—300, 600, and 900 lb. per acre. Both a fall and a spring application of each amount was tested, so that in all there were seven distinct treatments. The sulphur was spread by hand on the surface of the soil, and then diced into a depth of about 4 inches. The quantity to be analyzed is the “scab index”. That is roughly speaking, the percentage of the surface area of the potato that is infected with scab. It is obtained by examining 100 potatoes at random from each plot, grading each potato on a scale from 0 to 100% infected, and taking the average.

ScabIndex <-
      c(
        12, 10,  24, 29, 
        30, 18,  32, 26, 
         9,  9,  16,  4,
        30,  7,  21,  9,
        16, 10,  18, 18,
        18, 24,  12, 19,
        10,  4,   4,  5,
        17,  7,  16, 17
      )
Trt  <- factor(
          x = rep(x = 1:7, times = c(8, 4, 4, 4, 4, 4, 4))
        , labels = c("O", "F3", "S3", "F6", "S6", "F9", "S9")
        )

df13 <- data.frame(Trt, ScabIndex)
df13

   Trt ScabIndex
1    O        12
2    O        10
3    O        24
4    O        29
5    O        30
6    O        18
7    O        32
8    O        26
9   F3         9
10  F3         9
11  F3        16
12  F3         4
13  S3        30
14  S3         7
15  S3        21
16  S3         9
17  F6        16
18  F6        10
19  F6        18
20  F6        18
21  S6        18
22  S6        24
23  S6        12
24  S6        19
25  F9        10
26  F9         4
27  F9         4
28  F9         5
29  S9        17
30  S9         7
31  S9        16
32  S9        17

library(ggplot2)
p13 <- 
  ggplot(data = df13, mapping = aes(x = Trt, y = ScabIndex)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Scab Index") +
  theme_bw()
print(p13)

fm13 <- lm(formula = ScabIndex ~ Trt, data = df13)
summary(fm13)


Call:
lm(formula = ScabIndex ~ Trt, data = df13)

Residuals:
    Min      1Q  Median      3Q     Max 
-12.625  -4.844   0.625   3.594  13.250 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   22.625      2.369   9.549 8.08e-10 ***
TrtF3        -13.125      4.104  -3.198 0.003734 ** 
TrtS3         -5.875      4.104  -1.432 0.164666    
TrtF6         -7.125      4.104  -1.736 0.094858 .  
TrtS6         -4.375      4.104  -1.066 0.296601    
TrtF9        -16.875      4.104  -4.112 0.000372 ***
TrtS9         -8.375      4.104  -2.041 0.051977 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.702 on 25 degrees of freedom
Multiple R-squared:  0.4641,    Adjusted R-squared:  0.3355 
F-statistic: 3.608 on 6 and 25 DF,  p-value: 0.01026

anova(fm13)

Analysis of Variance Table

Response: ScabIndex
          Df  Sum Sq Mean Sq F value  Pr(>F)  
Trt        6  972.34 162.057  3.6081 0.01026 *
Residuals 25 1122.88  44.915                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Randomized Complete Block Design (RCBD)

Used when experimental material is heterogenous in one direction

Example

Yield : Yield of barley, SoilType : Soil Type, and Trt : 5 sources and a control

Yield <- c(
             32.1,  35.6,   41.9,   35.4
          ,  30.1,  31.5,   37.1,   30.8
          ,  25.4,  27.4,   33.8,   31.1
          ,  24.1,  33.0,   35.6,   31.4
          ,  26.1,  31.0,   33.8,   31.9
          ,  23.2,  24.8,   26.7,   26.7
          )
SoilType <- gl(n = 4, k = 6, length = 4*6
               , labels = c("I", "II", "III", "IV"), ordered = FALSE)
Trt      <- gl(n = 6, k = 1, length = 4*6
               , labels = c("(NH4)2SO4", "NH4NO3", "CO(NH2)2", "Ca(NO3)2", "NaNO3", "Control"), ordered = FALSE)

df14 <- data.frame(Yield, SoilType, Trt)
df14

   Yield SoilType       Trt
1   32.1        I (NH4)2SO4
2   35.6        I    NH4NO3
3   41.9        I  CO(NH2)2
4   35.4        I  Ca(NO3)2
5   30.1        I     NaNO3
6   31.5        I   Control
7   37.1       II (NH4)2SO4
8   30.8       II    NH4NO3
9   25.4       II  CO(NH2)2
10  27.4       II  Ca(NO3)2
11  33.8       II     NaNO3
12  31.1       II   Control
13  24.1      III (NH4)2SO4
14  33.0      III    NH4NO3
15  35.6      III  CO(NH2)2
16  31.4      III  Ca(NO3)2
17  26.1      III     NaNO3
18  31.0      III   Control
19  33.8       IV (NH4)2SO4
20  31.9       IV    NH4NO3
21  23.2       IV  CO(NH2)2
22  24.8       IV  Ca(NO3)2
23  26.7       IV     NaNO3
24  26.7       IV   Control

library(ggplot2)
p14 <- 
  ggplot(data = df14, mapping = aes(x = Trt, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Yield") +
  theme_bw()
print(p14)

fm14 <- lm(formula = Yield ~ SoilType + Trt, data = df14)
summary(fm14)


Call:
lm(formula = Yield ~ SoilType + Trt, data = df14)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.0208 -2.4229  0.0792  2.1354  6.7958 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   35.354      2.826  12.511 2.44e-09 ***
SoilTypeII    -3.500      2.664  -1.314   0.2087    
SoilTypeIII   -4.233      2.664  -1.589   0.1329    
SoilTypeIV    -6.583      2.664  -2.471   0.0259 *  
TrtNH4NO3      1.050      3.263   0.322   0.7521    
TrtCO(NH2)2   -0.250      3.263  -0.077   0.9399    
TrtCa(NO3)2   -2.025      3.263  -0.621   0.5442    
TrtNaNO3      -2.600      3.263  -0.797   0.4380    
TrtControl    -1.700      3.263  -0.521   0.6100    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.615 on 15 degrees of freedom
Multiple R-squared:  0.3512,    Adjusted R-squared:  0.005215 
F-statistic: 1.015 on 8 and 15 DF,  p-value: 0.4653

anova(fm14)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value Pr(>F)
SoilType   3 133.62  44.539  2.0915 0.1443
Trt        5  39.31   7.862  0.3692 0.8618
Residuals 15 319.43  21.295

Latin Square Design

Used when experimental material is heterogenous in two perpendicular directions

Example

The following table shows the field layout and yield of a 5×5 Latin square experiment on the effect of spacing on yield of millet plants. Five levels of spacing were used. The data on yield (grams/plot) was recorded and is given below:

Yield <- c(
           257, 230, 279, 287, 202,
           245, 283, 245, 280, 260,
           182, 252, 280, 246, 250,
           203, 204, 227, 193, 259,
           231, 271, 266, 334, 338
           )
Row <- factor(x = rep(x = 1:5, each = 5))
Col <- factor(x = rep(x = 1:5, times = 5))
Trt <- c(
          "B", "E", "A", "C", "D"
        , "D", "A", "E", "B", "C"
        , "E", "B", "C", "D", "A"
        , "A", "C", "D", "E", "B"
        , "C", "D", "B", "A", "E"
         )

df15 <- data.frame(Yield, Row, Col, Trt)
df15

   Yield Row Col Trt
1    257   1   1   B
2    230   1   2   E
3    279   1   3   A
4    287   1   4   C
5    202   1   5   D
6    245   2   1   D
7    283   2   2   A
8    245   2   3   E
9    280   2   4   B
10   260   2   5   C
11   182   3   1   E
12   252   3   2   B
13   280   3   3   C
14   246   3   4   D
15   250   3   5   A
16   203   4   1   A
17   204   4   2   C
18   227   4   3   D
19   193   4   4   E
20   259   4   5   B
21   231   5   1   C
22   271   5   2   D
23   266   5   3   B
24   334   5   4   A
25   338   5   5   E

library(ggplot2)
p15 <- 
  ggplot(data = df15, mapping = aes(x = Trt, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Yield") +
  theme_bw()
print(p15)

fm15 <- lm(formula = Yield ~ Row + Col + Trt, data = df15)
summary(fm15)


Call:
lm(formula = Yield ~ Row + Col + Trt, data = df15)

Residuals:
   Min     1Q Median     3Q    Max 
-44.68 -12.48   1.12  16.52  54.92 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   240.08      23.43  10.247 2.75e-07 ***
Row2           11.60      20.55   0.565   0.5828    
Row3           -9.00      20.55  -0.438   0.6692    
Row4          -33.80      20.55  -1.645   0.1259    
Row5           37.00      20.55   1.801   0.0969 .  
Col2           24.40      20.55   1.187   0.2580    
Col3           35.80      20.55   1.742   0.1070    
Col4           44.40      20.55   2.161   0.0516 .  
Col5           38.20      20.55   1.859   0.0877 .  
TrtB           -7.00      20.55  -0.341   0.7393    
TrtC          -17.40      20.55  -0.847   0.4137    
TrtD          -31.60      20.55  -1.538   0.1500    
TrtE          -32.20      20.55  -1.567   0.1431    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 32.49 on 12 degrees of freedom
Multiple R-squared:  0.6536,    Adjusted R-squared:  0.3073 
F-statistic: 1.887 on 12 and 12 DF,  p-value: 0.1426

anova(fm15)

Analysis of Variance Table

Response: Yield
          Df  Sum Sq Mean Sq F value  Pr(>F)  
Row        4 13601.4  3400.3  3.2212 0.05164 .
Col        4  6146.2  1536.5  1.4556 0.27581  
Trt        4  4156.6  1039.1  0.9844 0.45229  
Residuals 12 12667.3  1055.6                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Factorial Experiment under RCBD

Three strains of perennial ryegrass were grown as swards at each of the fertilizer levels. The three strains were S23, Kent & a control. The fertilizer levels were heavy & average. The experiment was laid out under randomized complete block design. The mid summer dry matter yields, in units of 10 lb/acre were as follows. Analyze the data & construct a table of means with appropriate standard errors. Draw any graphs you deem fit to explain the results & write a short report on the conclusions to be drawn from this experiment.

DMY <- 
  c(
    299,    318,    284,    279,
    247,    202,    171,    183,
    403,    439,    355,    324,
    222,    170,    192,    176,
    382,    353,    383,    310,
    233,    216,    200,    143
    )
Blk  <- gl(n = 4, k = 1, length = 24, labels = c("I", "II", "III", "IV"))
Vart <- gl(n = 3, k = 8, length = 24, labels = c("S23", "Control", "Kent"))
Manu <- gl(n = 2, k = 4, length = 24, labels = c("Heavy", "Average"))

df1 <- data.frame(DMY, Blk, Vart, Manu)
df1

   DMY Blk    Vart    Manu
1  299   I     S23   Heavy
2  318  II     S23   Heavy
3  284 III     S23   Heavy
4  279  IV     S23   Heavy
5  247   I     S23 Average
6  202  II     S23 Average
7  171 III     S23 Average
8  183  IV     S23 Average
9  403   I Control   Heavy
10 439  II Control   Heavy
11 355 III Control   Heavy
12 324  IV Control   Heavy
13 222   I Control Average
14 170  II Control Average
15 192 III Control Average
16 176  IV Control Average
17 382   I    Kent   Heavy
18 353  II    Kent   Heavy
19 383 III    Kent   Heavy
20 310  IV    Kent   Heavy
21 233   I    Kent Average
22 216  II    Kent Average
23 200 III    Kent Average
24 143  IV    Kent Average

fm1 <- lm(formula = DMY ~ Blk + Vart * Manu, data = df1)
anova(fm1)

Analysis of Variance Table

Response: DMY
          Df Sum Sq Mean Sq  F value    Pr(>F)    
Blk        3  12814    4271   7.1611  0.003294 ** 
Vart       2   6196    3098   5.1935  0.019324 *  
Manu       1 131128  131128 219.8375 2.287e-10 ***
Vart:Manu  2   9590    4795   8.0389  0.004239 ** 
Residuals 15   8947     596                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(emmeans)
emmip(fm1, Vart ~ Manu)

emmip(fm1, Manu ~ Vart)

emmeans(fm1, ~ Vart)

 Vart    emmean   SE df lower.CL upper.CL
 S23        248 8.63 15      229      266
 Control    285 8.63 15      267      304
 Kent       278 8.63 15      259      296

Results are averaged over the levels of: Blk, Manu 
Confidence level used: 0.95

emmeans(fm1, ~ Manu)

 Manu    emmean   SE df lower.CL upper.CL
 Heavy      344 7.05 15      329      359
 Average    196 7.05 15      181      211

Results are averaged over the levels of: Blk, Vart 
Confidence level used: 0.95

emmeans(fm1, ~ Manu|Vart)

Vart = S23:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      295 12.2 15      269      321
 Average    201 12.2 15      175      227

Vart = Control:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      380 12.2 15      354      406
 Average    190 12.2 15      164      216

Vart = Kent:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      357 12.2 15      331      383
 Average    198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95

emmeans(fm1, ~ Vart|Manu)

Manu = Heavy:
 Vart    emmean   SE df lower.CL upper.CL
 S23        295 12.2 15      269      321
 Control    380 12.2 15      354      406
 Kent       357 12.2 15      331      383

Manu = Average:
 Vart    emmean   SE df lower.CL upper.CL
 S23        201 12.2 15      175      227
 Control    190 12.2 15      164      216
 Kent       198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95

emmeans(fm1, pairwise ~ Vart)

$emmeans
 Vart    emmean   SE df lower.CL upper.CL
 S23        248 8.63 15      229      266
 Control    285 8.63 15      267      304
 Kent       278 8.63 15      259      296

Results are averaged over the levels of: Blk, Manu 
Confidence level used: 0.95 

$contrasts
 contrast       estimate   SE df t.ratio p.value
 S23 - Control    -37.25 12.2 15  -3.050  0.0208
 S23 - Kent       -29.62 12.2 15  -2.426  0.0689
 Control - Kent     7.62 12.2 15   0.624  0.8092

Results are averaged over the levels of: Blk, Manu 
P value adjustment: tukey method for comparing a family of 3 estimates

emmeans(fm1, pairwise ~ Manu)

$emmeans
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      344 7.05 15      329      359
 Average    196 7.05 15      181      211

Results are averaged over the levels of: Blk, Vart 
Confidence level used: 0.95 

$contrasts
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average      148 9.97 15  14.827 <0.0001

Results are averaged over the levels of: Blk, Vart

emmeans(fm1, pairwise ~ Vart|Manu)

$emmeans
Manu = Heavy:
 Vart    emmean   SE df lower.CL upper.CL
 S23        295 12.2 15      269      321
 Control    380 12.2 15      354      406
 Kent       357 12.2 15      331      383

Manu = Average:
 Vart    emmean   SE df lower.CL upper.CL
 S23        201 12.2 15      175      227
 Control    190 12.2 15      164      216
 Kent       198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95 

$contrasts
Manu = Heavy:
 contrast       estimate   SE df t.ratio p.value
 S23 - Control    -85.25 17.3 15  -4.936  0.0005
 S23 - Kent       -62.00 17.3 15  -3.590  0.0071
 Control - Kent    23.25 17.3 15   1.346  0.3927

Manu = Average:
 contrast       estimate   SE df t.ratio p.value
 S23 - Control     10.75 17.3 15   0.622  0.8102
 S23 - Kent         2.75 17.3 15   0.159  0.9861
 Control - Kent    -8.00 17.3 15  -0.463  0.8893

Results are averaged over the levels of: Blk 
P value adjustment: tukey method for comparing a family of 3 estimates

emmeans(fm1, pairwise ~ Manu|Vart)

$emmeans
Vart = S23:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      295 12.2 15      269      321
 Average    201 12.2 15      175      227

Vart = Control:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      380 12.2 15      354      406
 Average    190 12.2 15      164      216

Vart = Kent:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      357 12.2 15      331      383
 Average    198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95 

$contrasts
Vart = S23:
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average     94.2 17.3 15   5.458 <0.0001

Vart = Control:
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average    190.2 17.3 15  11.016 <0.0001

Vart = Kent:
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average    159.0 17.3 15   9.207 <0.0001

Results are averaged over the levels of: Blk

Stability Analysis

library(stability)
library(help = stability)
data(ge_data)

Individual Analysis of Variance for each Location

Yield.indiv_anova <-
         indiv_anova(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
          )
Yield.indiv_anova

$`Analysis of Variance for FSD`
            Df   Sum Sq Mean Sq F value Pr(>F)    
Gen         59 45524231  771597   32.62 <2e-16 ***
Residuals   60  1419180   23653                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for FSDR`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 14019781  237623   2.724 7.91e-05 ***
Residuals   60  5234860   87248                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Okara`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 26802597  454281   3.538 1.22e-06 ***
Residuals   60  7703642  128394                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Sargodha`
            Df   Sum Sq Mean Sq F value  Pr(>F)    
Gen         59 38381469  650533   4.254 4.3e-08 ***
Residuals   60  9175301  152922                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Gujranwala`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 41863914  709558   4.687 6.55e-09 ***
Residuals   60  9082956  151383                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for KotNaina`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 26692263  452411     4.3 3.51e-08 ***
Residuals   60  6313095  105218                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for KSK`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 17890629  303231   3.472 1.68e-06 ***
Residuals   60  5239575   87326                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Khanewal`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 15125214  256360   2.451 0.000345 ***
Residuals   60  6274965  104583                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Sahiwal`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 26109918  442541   4.095 8.82e-08 ***
Residuals   60  6484679  108078                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Bhakkar`
            Df   Sum Sq Mean Sq F value  Pr(>F)   
Gen         59 10870598  184247   1.862 0.00885 **
Residuals   60  5936717   98945                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Bahawalnagar`
            Df  Sum Sq Mean Sq F value Pr(>F)  
Gen         59 9718292  164717   1.716 0.0196 *
Residuals   60 5760765   96013                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

attr(,"subheadings")
 [1] "Analysis of Variance for FSD"         
 [2] "Analysis of Variance for FSDR"        
 [3] "Analysis of Variance for Okara"       
 [4] "Analysis of Variance for Sargodha"    
 [5] "Analysis of Variance for Gujranwala"  
 [6] "Analysis of Variance for KotNaina"    
 [7] "Analysis of Variance for KSK"         
 [8] "Analysis of Variance for Khanewal"    
 [9] "Analysis of Variance for Sahiwal"     
[10] "Analysis of Variance for Bhakkar"     
[11] "Analysis of Variance for Bahawalnagar"

Combined Analysis of Variance

YieldANOVA <-
     add_anova(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )
YieldANOVA

$anova
Analysis of Variance Table

Response: .data[[Y]]
           Df    Sum Sq  Mean Sq F value    Pr(>F)    
Env        10 263940496 26394050 68.0551 1.995e-08 ***
Rep(Env)   11   4266167   387833                      
Gen        59  71662431  1214617 12.2482 < 2.2e-16 ***
Gen:Env   590 201336476   341248  3.4411 < 2.2e-16 ***
Residuals 649  64359568    99167                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Additive Main Effects and Multiplicative Interaction (AMMI) Analysis

Yield.ammi <-
     ammi(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )
Yield.ammi

$anova
Analysis of Variance Table

Response: .data[[Y]]
           Df    Sum Sq  Mean Sq F value    Pr(>F)    
Env        10 263940496 26394050 68.0551 1.995e-08 ***
Rep(Env)   11   4266167   387833                      
Gen        59  71662431  1214617 12.2482 < 2.2e-16 ***
Gen:Env   590 201336476   341248  3.4411 < 2.2e-16 ***
Residuals 649  64359568    99167                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$pc.anova
# A tibble: 11 × 9
   PC     SingVal  Percent accumPerc    Df       SS  Mean.Sq  F.value   Pr..F.
                                 
 1 PC1   4.50e+ 3 2.01e+ 1      20.1    68 4.04e+ 7 5.95e+ 5 6.00e+ 0 0       
 2 PC2   4.17e+ 3 1.73e+ 1      37.4    66 3.49e+ 7 5.28e+ 5 5.33e+ 0 0       
 3 PC3   4.05e+ 3 1.63e+ 1      53.7    64 3.27e+ 7 5.12e+ 5 5.16e+ 0 0       
 4 PC4   3.49e+ 3 1.21e+ 1      65.8    62 2.44e+ 7 3.93e+ 5 3.96e+ 0 0       
 5 PC5   3.38e+ 3 1.13e+ 1      77.1    60 2.29e+ 7 3.81e+ 5 3.84e+ 0 0       
 6 PC6   3.11e+ 3 9.63e+ 0      86.7    58 1.94e+ 7 3.34e+ 5 3.37e+ 0 4.02e-14
 7 PC7   2.30e+ 3 5.25e+ 0      92.0    56 1.06e+ 7 1.89e+ 5 1.90e+ 0 1.45e- 4
 8 PC8   2.03e+ 3 4.10e+ 0      96.1    54 8.25e+ 6 1.53e+ 5 1.54e+ 0 9.52e- 3
 9 PC9   1.52e+ 3 2.30e+ 0      98.4    52 4.63e+ 6 8.91e+ 4 8.98e- 1 6.77e- 1
10 PC10  1.28e+ 3 1.63e+ 0     100      50 3.27e+ 6 6.55e+ 4 6.60e- 1 9.66e- 1
11 PC11  2.08e-11 4.32e-28     100      48 8.69e-22 1.81e-23 1.83e-28 1   e+ 0

Additive Main Effects and Multiplicative Interaction (AMMI) Biplot Analysis

ammi_biplot(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )

$aami.biplot


$MeanPC1Plot

Genotype plus Genotypes by Environment (GGE) Interaction Biplot Analysis

gge_biplot(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )

$gge.biplot

Citation

BibTeX citation:

@online{yaseen2020,
  author = {Yaseen, Muhammad},
  title = {Design \& {Analysis} of {Field} {Experiments} Using {R}},
  date = {2020-03-06},
  url = {https://myaseen208.com/blogs/2020-02-06_UAF/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2020. “Design & Analysis of Field Experiments Using R.” March 6. https://myaseen208.com/blogs/2020-02-06_UAF/.

Training Course on Capacity Building of NARS Scientists in Advance Analytical Techniques

Muhammad Yaseen — Mon, 25 Nov 2019 05:00:00 GMT

Introduction

R is a free, open-source programming language and software environment for statistical computing, bioinformatics, visualization and general computing. R provides a wide variety of statistical and graphical techniques, and is highly extensible. The latest version of R can be obtained from https://cran.r-project.org/bin/windows/base/.

R R for Windows (32/64 bit)

RStudio RStudio

Manual An Introduction to R

Ref Card R Reference Card

New York Times New York Times

Nature Article Nature Article

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Figure 1

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

# Fertilizer (Kg/acre)
# Production (Bushels/acre)
Fertilizer <- c(100, 200, 400, 500)
Production <- c( 70,  70,  80, 100)
df1        <- data.frame(Fertilizer, Production)
df1

  Fertilizer Production
1        100         70
2        200         70
3        400         80
4        500        100

library(ggplot2)
p1 <- 
  ggplot(data = df1, mapping = aes(x = Fertilizer, y = Production)) + 
  geom_point() +
  labs(x = "Fertilizer", y = "Production") +
  theme_bw()
print(p1)

fm1 <- lm(formula = Production ~ Fertilizer, data = df1)
summary(fm1)


Call:
lm(formula = Production ~ Fertilizer, data = df1)

Residuals:
 1  2  3  4 
 4 -3 -7  6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 59.00000    7.95299   7.419   0.0177 *
Fertilizer   0.07000    0.02345   2.985   0.0963 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.416 on 2 degrees of freedom
Multiple R-squared:  0.8167,    Adjusted R-squared:  0.725 
F-statistic: 8.909 on 1 and 2 DF,  p-value: 0.0963

anova(fm1)

Analysis of Variance Table

Response: Production
           Df Sum Sq Mean Sq F value Pr(>F)  
Fertilizer  1    490     490  8.9091 0.0963 .
Residuals   2    110      55                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Fertilizer	Yield
0.3	10
0.6	15
0.9	30
1.2	35
1.5	25
1.8	30
2.1	50
2.4	45

Fert  <- c(0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4)
Yield <- c(10, 15, 30, 35, 25, 30, 50, 45)
df2   <- data.frame(Fert, Yield)
df2

  Fert Yield
1  0.3    10
2  0.6    15
3  0.9    30
4  1.2    35
5  1.5    25
6  1.8    30
7  2.1    50
8  2.4    45

library(ggplot2)
p2 <- 
  ggplot(data = df2, mapping = aes(x = Fert, y = Yield)) + 
  geom_point() +
  labs(x = "Fertilizer", y = "Yield") +
  theme_bw()
print(p2)

fm2 <- lm(formula = Yield ~ Fert, data = df2)
summary(fm2)


Call:
lm(formula = Yield ~ Fert, data = df2)

Residuals:
   Min     1Q Median     3Q    Max 
-7.441 -4.018 -2.441  7.351  7.798 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    8.036      5.504   1.460   0.1946   
Fert          16.270      3.633   4.478   0.0042 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.064 on 6 degrees of freedom
Multiple R-squared:  0.7697,    Adjusted R-squared:  0.7313 
F-statistic: 20.05 on 1 and 6 DF,  p-value: 0.004202

anova(fm2)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value   Pr(>F)   
Fert       1 1000.6  1000.6  20.052 0.004202 **
Residuals  6  299.4    49.9                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Weekly Income ($)	Weekly Expenditures ($)
80	70
100	65
120	90
140	95
160	110
180	115
200	120
220	140
240	155
260	150

# Weekly Income ($) of a Family
# Weekly Expenditures ($) of a Family
Income       <- seq(from = 80, to = 260, by = 20)
Expenditures <- c(70, 65, 90, 95, 110, 115, 120, 140, 155, 150)
df3          <- data.frame(Income, Expenditures)
df3

   Income Expenditures
1      80           70
2     100           65
3     120           90
4     140           95
5     160          110
6     180          115
7     200          120
8     220          140
9     240          155
10    260          150

library(ggplot2)
p3 <- 
  ggplot(data = df3, mapping = aes(x = Income, y = Expenditures)) + 
  geom_point() +
  scale_x_continuous(labels = scales::dollar) +
  scale_y_continuous(labels = scales::dollar) +
  labs(x = "Weekly income, $", y = "Weekly consumption expenditures, $") +
  theme_bw()
print(p3)

fm3 <- lm(formula = Expenditures ~ Income, data = df3)
summary(fm3)


Call:
lm(formula = Expenditures ~ Income, data = df3)

Residuals:
    Min      1Q  Median      3Q     Max 
-10.364  -4.977   1.409   4.364   8.364 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 24.45455    6.41382   3.813  0.00514 ** 
Income       0.50909    0.03574  14.243 5.75e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.493 on 8 degrees of freedom
Multiple R-squared:  0.9621,    Adjusted R-squared:  0.9573 
F-statistic: 202.9 on 1 and 8 DF,  p-value: 5.753e-07

anova(fm3)

Analysis of Variance Table

Response: Expenditures
          Df Sum Sq Mean Sq F value    Pr(>F)    
Income     1 8552.7  8552.7  202.87 5.753e-07 ***
Residuals  8  337.3    42.2                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Fertilizer (Kg)	Rainfall (mm)	Yield (Kg)
100	10	40
200	20	50
300	10	50
400	30	70
500	20	65
600	20	65
700	30	80

# Fertilizer (Kg)
# Rainfall (mm)
# Yield (Kg)
Fertilizer <- seq(from = 100, to = 700, by = 100)
Rainfall   <- c(10, 20, 10, 30, 20, 20, 30)
Yield      <- c(40, 50, 50, 70, 65, 65, 80)
df4        <- data.frame(Fertilizer, Rainfall, Yield)
df4

  Fertilizer Rainfall Yield
1        100       10    40
2        200       20    50
3        300       10    50
4        400       30    70
5        500       20    65
6        600       20    65
7        700       30    80

library(ggplot2)
ggplot(data = df4, mapping = aes(x = Fertilizer, y = Rainfall, color = Yield)) +
  geom_point(alpha = 0.8, size = 2) +
  scale_color_viridis_c() +
  theme_minimal()

fm4 <- lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)
summary(fm4)


Call:
lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)

Residuals:
      1       2       3       4       5       6       7 
-0.2381 -2.3810  2.1429  1.6667  1.1905 -2.6190  0.2381 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 28.095238   2.491482  11.277 0.000352 ***
Fertilizer   0.038095   0.005832   6.532 0.002838 ** 
Rainfall     0.833333   0.154303   5.401 0.005690 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.315 on 4 degrees of freedom
Multiple R-squared:  0.9814,    Adjusted R-squared:  0.972 
F-statistic: 105.3 on 2 and 4 DF,  p-value: 0.0003472

anova(fm4)

Analysis of Variance Table

Response: Yield
           Df Sum Sq Mean Sq F value    Pr(>F)    
Fertilizer  1 972.32  972.32 181.500 0.0001756 ***
Rainfall    1 156.25  156.25  29.167 0.0056899 ** 
Residuals   4  21.43    5.36                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(lm.beta)
lm.beta(fm4)


Call:
lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)

Standardized Coefficients::
(Intercept)  Fertilizer    Rainfall 
         NA   0.5944301   0.4914732

Polynomial Regression Analysis

Quantifying non-linear dependency of a normal response on quantitative explanatory variable(s)

Example

An experiment was conducted to evaluate the effects of different levels of nitrogen. Three levels of nitrogen: 0, 10 and 20 grams per plot were used in the experiment. Each treatment was replicated twice and data is given below:

Nitrogen	Yield
0	5
0	7
10	15
10	17
20	9
20	11

Nitrogen <- c(0, 0, 10, 10, 20, 20)
Yield    <- c(5, 7, 15, 17,  9, 11)
df5      <- data.frame(Nitrogen, Yield)
df5

  Nitrogen Yield
1        0     5
2        0     7
3       10    15
4       10    17
5       20     9
6       20    11

library(ggplot2)
p5 <- 
  ggplot(data = df5, mapping = aes(x = Nitrogen, y = Yield)) + 
  geom_point() +
  labs(x = "Nitrogen", y = "Yield") +
  theme_bw()
print(p5)

fm5 <- lm(formula = Yield ~ Nitrogen + I(Nitrogen^2), data = df5)
# fm5 <- lm(formula = Yield ~ poly(x = Nitrogen, degree = 2, raw = TRUE), data = df5)
summary(fm5)


Call:
lm(formula = Yield ~ Nitrogen + I(Nitrogen^2), data = df5)

Residuals:
 1  2  3  4  5  6 
-1  1 -1  1 -1  1 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.00000    1.00000   6.000  0.00927 **
Nitrogen       1.80000    0.25495   7.060  0.00584 **
I(Nitrogen^2) -0.08000    0.01225  -6.532  0.00729 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.414 on 3 degrees of freedom
Multiple R-squared:  0.9441,    Adjusted R-squared:  0.9068 
F-statistic: 25.33 on 2 and 3 DF,  p-value: 0.01322

anova(fm5)

Analysis of Variance Table

Response: Yield
              Df Sum Sq Mean Sq F value   Pr(>F)   
Nitrogen       1 16.000  16.000   8.000 0.066276 . 
I(Nitrogen^2)  1 85.333  85.333  42.667 0.007292 **
Residuals      3  6.000   2.000                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Analysis of Variance (ANOVA)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 2

Example

Yield	Variety
5	V1
6	V1
7	V1
15	V2
16	V2
17	V2

Yield   <- c(5, 6, 7, 15, 16, 17)
Variety <- gl(n = 2, k = 3, length = 2*3, labels = c("V1", "V2"))
df6     <- data.frame(Yield, Variety)
df6

  Yield Variety
1     5      V1
2     6      V1
3     7      V1
4    15      V2
5    16      V2
6    17      V2

library(ggplot2)
p6 <- 
  ggplot(data = df6, mapping = aes(x = Variety, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Variety", y = "Yield") +
  theme_bw()
print(p6)

fm6 <- lm(formula = Yield ~ Variety, data = df6)
summary(fm6)


Call:
lm(formula = Yield ~ Variety, data = df6)

Residuals:
         1          2          3          4          5          6 
-1.000e+00 -4.774e-15  1.000e+00 -1.000e+00 -1.110e-16  1.000e+00 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   6.0000     0.5774   10.39 0.000484 ***
VarietyV2    10.0000     0.8165   12.25 0.000255 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1 on 4 degrees of freedom
Multiple R-squared:  0.974, Adjusted R-squared:  0.9675 
F-statistic:   150 on 1 and 4 DF,  p-value: 0.0002552

anova(fm6)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value    Pr(>F)    
Variety    1    150     150     150 0.0002552 ***
Residuals  4      4       1                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Yield	Variety
5	V1
7	V1
15	V2
17	V2
17	V3
19	V3

Yield   <- c(5, 7, 15, 17, 17, 19)
Variety <- gl(n = 3, k = 2, length = 3*2, labels = c("V1", "V2", "V3"))
df7     <- data.frame(Yield, Variety)
df7

  Yield Variety
1     5      V1
2     7      V1
3    15      V2
4    17      V2
5    17      V3
6    19      V3

library(ggplot2)
p7 <- 
  ggplot(data = df7, mapping = aes(x = Variety, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Variety", y = "Yield") +
  theme_bw()
print(p7)

fm7 <- lm(formula = Yield ~ Variety, data = df7)
summary(fm7)


Call:
lm(formula = Yield ~ Variety, data = df7)

Residuals:
 1  2  3  4  5  6 
-1  1 -1  1 -1  1 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.000      1.000   6.000  0.00927 **
VarietyV2     10.000      1.414   7.071  0.00582 **
VarietyV3     12.000      1.414   8.485  0.00344 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.414 on 3 degrees of freedom
Multiple R-squared:  0.965, Adjusted R-squared:  0.9416 
F-statistic: 41.33 on 2 and 3 DF,  p-value: 0.006553

anova(fm7)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value   Pr(>F)   
Variety    2 165.33  82.667  41.333 0.006553 **
Residuals  3   6.00   2.000                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Analysis of Covariance (ANCOVA)

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 3

Example

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Yield    <- c(51, 52, 53, 54, 56, 57, 55, 56, 58, 59, 62, 63)
Fert     <- rep(x = c(80, 90, 100), each = 2)
Variety  <- gl(n = 2, k = 6, length = 2*6, labels = c("V1", "V2"), ordered = FALSE)
df8         <- data.frame(Yield, Fert, Variety)
df8

   Yield Fert Variety
1     51   80      V1
2     52   80      V1
3     53   90      V1
4     54   90      V1
5     56  100      V1
6     57  100      V1
7     55   80      V2
8     56   80      V2
9     58   90      V2
10    59   90      V2
11    62  100      V2
12    63  100      V2

Same intercepts but different slopes

fm8 <- lm(formula = Yield ~ Fert + Variety, data = df8)
summary(fm8)


Call:
lm(formula = Yield ~ Fert + Variety, data = df8)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.8333 -0.8333  0.1667  0.1667  1.1667 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 26.83333    2.54557   10.54 2.30e-06 ***
Fert         0.30000    0.02805   10.69 2.04e-06 ***
VarietyV2    5.00000    0.45812   10.91 1.72e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7935 on 9 degrees of freedom
Multiple R-squared:  0.9629,    Adjusted R-squared:  0.9546 
F-statistic: 116.7 on 2 and 9 DF,  p-value: 3.657e-07

anova(fm8)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value    Pr(>F)    
Fert       1 72.000   72.00  114.35 2.042e-06 ***
Variety    1 75.000   75.00  119.12 1.720e-06 ***
Residuals  9  5.667    0.63                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(ggplot2)
p8 <- 
  ggplot(data = df8, mapping = aes(x = Fert, y = Yield, color = Variety)) + 
  geom_point() +
  geom_line(mapping = aes(y = predict(fm8))) +
  labs(x = "Fert", y = "Yield") +
  theme_bw()
print(p8)

Different intercepts and different slopes

fm9 <- lm(formula = Yield ~ Fert * Variety, data = df8)
summary(fm9)


Call:
lm(formula = Yield ~ Fert * Variety, data = df8)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.83333 -0.33333 -0.08333  0.66667  0.66667 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)    31.33333    3.05903  10.243 7.09e-06 ***
Fert            0.25000    0.03385   7.385 7.73e-05 ***
VarietyV2      -4.00000    4.32612  -0.925   0.3822    
Fert:VarietyV2  0.10000    0.04787   2.089   0.0701 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.677 on 8 degrees of freedom
Multiple R-squared:  0.976, Adjusted R-squared:  0.967 
F-statistic: 108.4 on 3 and 8 DF,  p-value: 8.11e-07

anova(fm9)

Analysis of Variance Table

Response: Yield
             Df Sum Sq Mean Sq  F value    Pr(>F)    
Fert          1 72.000  72.000 157.0909 1.538e-06 ***
Variety       1 75.000  75.000 163.6364 1.315e-06 ***
Fert:Variety  1  2.000   2.000   4.3636   0.07013 .  
Residuals     8  3.667   0.458                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(ggplot2)
p9 <- 
  ggplot(data = df8, mapping = aes(x = Fert, y = Yield, color = Variety)) + 
  geom_point() +
  geom_line(mapping = aes(y = predict(fm9))) +
  labs(x = "Fert", y = "Yield") +
  theme_bw()
print(p9)

Correlation Analysis

Linear Relationship between Quantitative Variables

Simple Correlation Analysis

Linear Relationship between Two Quantitative Variables

Example

Sparrow Wing length (cm)	Sparrow Tail length (cm)
10.4	7.4
10.8	7.6
11.1	7.9
10.2	7.2
10.3	7.4
10.2	7.1
10.7	7.4
10.5	7.2
10.8	7.8
11.2	7.7
10.6	7.8
11.4	8.3

WingLength  <- c(10.4, 10.8, 11.1, 10.2, 10.3, 10.2, 10.7, 10.5, 10.8, 11.2, 10.6, 11.4)
TailLength  <- c(7.4, 7.6, 7.9, 7.2, 7.4, 7.1, 7.4, 7.2, 7.8, 7.7, 7.8, 8.3)
df10        <- data.frame(WingLength, TailLength)
df10

   WingLength TailLength
1        10.4        7.4
2        10.8        7.6
3        11.1        7.9
4        10.2        7.2
5        10.3        7.4
6        10.2        7.1
7        10.7        7.4
8        10.5        7.2
9        10.8        7.8
10       11.2        7.7
11       10.6        7.8
12       11.4        8.3

library(ggplot2)
p10 <- 
  ggplot(data = df10, mapping = aes(x= WingLength, y = TailLength)) + 
  geom_point() +
  labs(x = "Sparrow Wing Length (cm)", y = "Sparrow Tail Length (cm)") +
  theme_bw()
print(p10)

cor(df10$WingLength, df10$TailLength)

[1] 0.8703546

cor.test(df10$WingLength, df10$TailLength)


    Pearson's product-moment correlation

data:  df10$WingLength and df10$TailLength
t = 5.5893, df = 10, p-value = 0.0002311
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5923111 0.9631599
sample estimates:
      cor 
0.8703546

Partial Correlation Analysis

Linear Relationship between Quantitative Variables while holding/keeping all other constants

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

LeafArea    <- c(72, 174, 116, 78, 134, 95, 113, 98, 148, 42)
LeafMoist   <- c(75, 81, 83, 79, 81, 80, 81, 74, 78, 58)
TotShLength <- c(307, 529, 632, 527, 442, 525, 481, 710, 422, 345)
df11        <- data.frame(LeafArea, LeafMoist, TotShLength)
df11

   LeafArea LeafMoist TotShLength
1        72        75         307
2       174        81         529
3       116        83         632
4        78        79         527
5       134        81         442
6        95        80         525
7       113        81         481
8        98        74         710
9       148        78         422
10       42        58         345

library(ppcor)
pcor(df11)

$estimate
                LeafArea LeafMoist  TotShLength
LeafArea     1.000000000 0.6509096 -0.007012236
LeafMoist    0.650909618 1.0000000  0.324334659
TotShLength -0.007012236 0.3243347  1.000000000

$p.value
              LeafArea  LeafMoist TotShLength
LeafArea    0.00000000 0.05760462   0.9857154
LeafMoist   0.05760462 0.00000000   0.3944839
TotShLength 0.98571537 0.39448385   0.0000000

$statistic
               LeafArea LeafMoist TotShLength
LeafArea     0.00000000  2.268502 -0.01855309
LeafMoist    2.26850175  0.000000  0.90714704
TotShLength -0.01855309  0.907147  0.00000000

$n
[1] 10

$gp
[1] 1

$method
[1] "pearson"

Multiple Correlation Analysis

Linear Relationship between a Quantitative Variable and set of other Quantitative Variables

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

fm12 <- lm(formula = TotShLength ~ LeafArea + LeafMoist, data = df11)
sqrt(summary(fm12)$r.squared)

[1] 0.421277

Completely Randomized Design (CRD)

Used when experimental material is homogeneous

Example

The following table shows some of the results of an experiment on the effects of applications of sulphur in reducing scab disease of potatoes. The object in applying sulphur is to increase the acidity of the soil, since scab does not thrive in very acid soil. In addition to untreated plots which serve as a control, 3 amounts of dressing were compared—300, 600, and 900 lb. per acre. Both a fall and a spring application of each amount was tested, so that in all there were seven distinct treatments. The sulphur was spread by hand on the surface of the soil, and then diced into a depth of about 4 inches. The quantity to be analyzed is the “scab index”. That is roughly speaking, the percentage of the surface area of the potato that is infected with scab. It is obtained by examining 100 potatoes at random from each plot, grading each potato on a scale from 0 to 100% infected, and taking the average.

ScabIndex <-
      c(
        12, 10,  24, 29, 
        30, 18,  32, 26, 
         9,  9,  16,  4,
        30,  7,  21,  9,
        16, 10,  18, 18,
        18, 24,  12, 19,
        10,  4,   4,  5,
        17,  7,  16, 17
      )
Trt  <- factor(
          x = rep(x = 1:7, times = c(8, 4, 4, 4, 4, 4, 4))
        , labels = c("O", "F3", "S3", "F6", "S6", "F9", "S9")
        )

df13 <- data.frame(Trt, ScabIndex)
df13

   Trt ScabIndex
1    O        12
2    O        10
3    O        24
4    O        29
5    O        30
6    O        18
7    O        32
8    O        26
9   F3         9
10  F3         9
11  F3        16
12  F3         4
13  S3        30
14  S3         7
15  S3        21
16  S3         9
17  F6        16
18  F6        10
19  F6        18
20  F6        18
21  S6        18
22  S6        24
23  S6        12
24  S6        19
25  F9        10
26  F9         4
27  F9         4
28  F9         5
29  S9        17
30  S9         7
31  S9        16
32  S9        17

library(ggplot2)
p13 <- 
  ggplot(data = df13, mapping = aes(x = Trt, y = ScabIndex)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Scab Index") +
  theme_bw()
print(p13)

fm13 <- lm(formula = ScabIndex ~ Trt, data = df13)
summary(fm13)


Call:
lm(formula = ScabIndex ~ Trt, data = df13)

Residuals:
    Min      1Q  Median      3Q     Max 
-12.625  -4.844   0.625   3.594  13.250 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   22.625      2.369   9.549 8.08e-10 ***
TrtF3        -13.125      4.104  -3.198 0.003734 ** 
TrtS3         -5.875      4.104  -1.432 0.164666    
TrtF6         -7.125      4.104  -1.736 0.094858 .  
TrtS6         -4.375      4.104  -1.066 0.296601    
TrtF9        -16.875      4.104  -4.112 0.000372 ***
TrtS9         -8.375      4.104  -2.041 0.051977 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.702 on 25 degrees of freedom
Multiple R-squared:  0.4641,    Adjusted R-squared:  0.3355 
F-statistic: 3.608 on 6 and 25 DF,  p-value: 0.01026

anova(fm13)

Analysis of Variance Table

Response: ScabIndex
          Df  Sum Sq Mean Sq F value  Pr(>F)  
Trt        6  972.34 162.057  3.6081 0.01026 *
Residuals 25 1122.88  44.915                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Randomized Complete Block Design (RCBD)

Used when experimental material is heterogenous in one direction

Example

Yield : Yield of barley, SoilType : Soil Type, and Trt : 5 sources and a control

Yield <- c(
             32.1,  35.6,   41.9,   35.4
          ,  30.1,  31.5,   37.1,   30.8
          ,  25.4,  27.4,   33.8,   31.1
          ,  24.1,  33.0,   35.6,   31.4
          ,  26.1,  31.0,   33.8,   31.9
          ,  23.2,  24.8,   26.7,   26.7
          )
SoilType <- gl(n = 4, k = 6, length = 4*6
               , labels = c("I", "II", "III", "IV"), ordered = FALSE)
Trt      <- gl(n = 6, k = 1, length = 4*6
               , labels = c("(NH4)2SO4", "NH4NO3", "CO(NH2)2", "Ca(NO3)2", "NaNO3", "Control"), ordered = FALSE)

df14 <- data.frame(Yield, SoilType, Trt)
df14

   Yield SoilType       Trt
1   32.1        I (NH4)2SO4
2   35.6        I    NH4NO3
3   41.9        I  CO(NH2)2
4   35.4        I  Ca(NO3)2
5   30.1        I     NaNO3
6   31.5        I   Control
7   37.1       II (NH4)2SO4
8   30.8       II    NH4NO3
9   25.4       II  CO(NH2)2
10  27.4       II  Ca(NO3)2
11  33.8       II     NaNO3
12  31.1       II   Control
13  24.1      III (NH4)2SO4
14  33.0      III    NH4NO3
15  35.6      III  CO(NH2)2
16  31.4      III  Ca(NO3)2
17  26.1      III     NaNO3
18  31.0      III   Control
19  33.8       IV (NH4)2SO4
20  31.9       IV    NH4NO3
21  23.2       IV  CO(NH2)2
22  24.8       IV  Ca(NO3)2
23  26.7       IV     NaNO3
24  26.7       IV   Control

library(ggplot2)
p14 <- 
  ggplot(data = df14, mapping = aes(x = Trt, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Yield") +
  theme_bw()
print(p14)

fm14 <- lm(formula = Yield ~ SoilType + Trt, data = df14)
summary(fm14)


Call:
lm(formula = Yield ~ SoilType + Trt, data = df14)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.0208 -2.4229  0.0792  2.1354  6.7958 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   35.354      2.826  12.511 2.44e-09 ***
SoilTypeII    -3.500      2.664  -1.314   0.2087    
SoilTypeIII   -4.233      2.664  -1.589   0.1329    
SoilTypeIV    -6.583      2.664  -2.471   0.0259 *  
TrtNH4NO3      1.050      3.263   0.322   0.7521    
TrtCO(NH2)2   -0.250      3.263  -0.077   0.9399    
TrtCa(NO3)2   -2.025      3.263  -0.621   0.5442    
TrtNaNO3      -2.600      3.263  -0.797   0.4380    
TrtControl    -1.700      3.263  -0.521   0.6100    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.615 on 15 degrees of freedom
Multiple R-squared:  0.3512,    Adjusted R-squared:  0.005215 
F-statistic: 1.015 on 8 and 15 DF,  p-value: 0.4653

anova(fm14)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value Pr(>F)
SoilType   3 133.62  44.539  2.0915 0.1443
Trt        5  39.31   7.862  0.3692 0.8618
Residuals 15 319.43  21.295

Latin Square Design

Used when experimental material is heterogenous in two perpendicular directions

Example

The following table shows the field layout and yield of a 5×5 Latin square experiment on the effect of spacing on yield of millet plants. Five levels of spacing were used. The data on yield (grams/plot) was recorded and is given below:

Yield <- c(
           257, 230, 279, 287, 202,
           245, 283, 245, 280, 260,
           182, 252, 280, 246, 250,
           203, 204, 227, 193, 259,
           231, 271, 266, 334, 338
           )
Row <- factor(x = rep(x = 1:5, each = 5))
Col <- factor(x = rep(x = 1:5, times = 5))
Trt <- c(
          "B", "E", "A", "C", "D"
        , "D", "A", "E", "B", "C"
        , "E", "B", "C", "D", "A"
        , "A", "C", "D", "E", "B"
        , "C", "D", "B", "A", "E"
         )

df15 <- data.frame(Yield, Row, Col, Trt)
df15

   Yield Row Col Trt
1    257   1   1   B
2    230   1   2   E
3    279   1   3   A
4    287   1   4   C
5    202   1   5   D
6    245   2   1   D
7    283   2   2   A
8    245   2   3   E
9    280   2   4   B
10   260   2   5   C
11   182   3   1   E
12   252   3   2   B
13   280   3   3   C
14   246   3   4   D
15   250   3   5   A
16   203   4   1   A
17   204   4   2   C
18   227   4   3   D
19   193   4   4   E
20   259   4   5   B
21   231   5   1   C
22   271   5   2   D
23   266   5   3   B
24   334   5   4   A
25   338   5   5   E

library(ggplot2)
p15 <- 
  ggplot(data = df15, mapping = aes(x = Trt, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Yield") +
  theme_bw()
print(p15)

fm15 <- lm(formula = Yield ~ Row + Col + Trt, data = df15)
summary(fm15)


Call:
lm(formula = Yield ~ Row + Col + Trt, data = df15)

Residuals:
   Min     1Q Median     3Q    Max 
-44.68 -12.48   1.12  16.52  54.92 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   240.08      23.43  10.247 2.75e-07 ***
Row2           11.60      20.55   0.565   0.5828    
Row3           -9.00      20.55  -0.438   0.6692    
Row4          -33.80      20.55  -1.645   0.1259    
Row5           37.00      20.55   1.801   0.0969 .  
Col2           24.40      20.55   1.187   0.2580    
Col3           35.80      20.55   1.742   0.1070    
Col4           44.40      20.55   2.161   0.0516 .  
Col5           38.20      20.55   1.859   0.0877 .  
TrtB           -7.00      20.55  -0.341   0.7393    
TrtC          -17.40      20.55  -0.847   0.4137    
TrtD          -31.60      20.55  -1.538   0.1500    
TrtE          -32.20      20.55  -1.567   0.1431    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 32.49 on 12 degrees of freedom
Multiple R-squared:  0.6536,    Adjusted R-squared:  0.3073 
F-statistic: 1.887 on 12 and 12 DF,  p-value: 0.1426

anova(fm15)

Analysis of Variance Table

Response: Yield
          Df  Sum Sq Mean Sq F value  Pr(>F)  
Row        4 13601.4  3400.3  3.2212 0.05164 .
Col        4  6146.2  1536.5  1.4556 0.27581  
Trt        4  4156.6  1039.1  0.9844 0.45229  
Residuals 12 12667.3  1055.6                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Factorial Experiment under RCBD

Three strains of perennial ryegrass were grown as swards at each of the fertilizer levels. The three strains were S23, Kent & a control. The fertilizer levels were heavy & average. The experiment was laid out under randomized complete block design. The mid summer dry matter yields, in units of 10 lb/acre were as follows. Analyze the data & construct a table of means with appropriate standard errors. Draw any graphs you deem fit to explain the results & write a short report on the conclusions to be drawn from this experiment.

DMY <- 
  c(
    299,    318,    284,    279,
    247,    202,    171,    183,
    403,    439,    355,    324,
    222,    170,    192,    176,
    382,    353,    383,    310,
    233,    216,    200,    143
    )
Blk  <- gl(n = 4, k = 1, length = 24, labels = c("I", "II", "III", "IV"))
Vart <- gl(n = 3, k = 8, length = 24, labels = c("S23", "Control", "Kent"))
Manu <- gl(n = 2, k = 4, length = 24, labels = c("Heavy", "Average"))

df1 <- data.frame(DMY, Blk, Vart, Manu)
df1

   DMY Blk    Vart    Manu
1  299   I     S23   Heavy
2  318  II     S23   Heavy
3  284 III     S23   Heavy
4  279  IV     S23   Heavy
5  247   I     S23 Average
6  202  II     S23 Average
7  171 III     S23 Average
8  183  IV     S23 Average
9  403   I Control   Heavy
10 439  II Control   Heavy
11 355 III Control   Heavy
12 324  IV Control   Heavy
13 222   I Control Average
14 170  II Control Average
15 192 III Control Average
16 176  IV Control Average
17 382   I    Kent   Heavy
18 353  II    Kent   Heavy
19 383 III    Kent   Heavy
20 310  IV    Kent   Heavy
21 233   I    Kent Average
22 216  II    Kent Average
23 200 III    Kent Average
24 143  IV    Kent Average

fm1 <- lm(formula = DMY ~ Blk + Vart * Manu, data = df1)
anova(fm1)

Analysis of Variance Table

Response: DMY
          Df Sum Sq Mean Sq  F value    Pr(>F)    
Blk        3  12814    4271   7.1611  0.003294 ** 
Vart       2   6196    3098   5.1935  0.019324 *  
Manu       1 131128  131128 219.8375 2.287e-10 ***
Vart:Manu  2   9590    4795   8.0389  0.004239 ** 
Residuals 15   8947     596                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(emmeans)
emmip(fm1, Vart ~ Manu)

emmip(fm1, Manu ~ Vart)

emmeans(fm1, ~ Vart)

 Vart    emmean   SE df lower.CL upper.CL
 S23        248 8.63 15      229      266
 Control    285 8.63 15      267      304
 Kent       278 8.63 15      259      296

Results are averaged over the levels of: Blk, Manu 
Confidence level used: 0.95

emmeans(fm1, ~ Manu)

 Manu    emmean   SE df lower.CL upper.CL
 Heavy      344 7.05 15      329      359
 Average    196 7.05 15      181      211

Results are averaged over the levels of: Blk, Vart 
Confidence level used: 0.95

emmeans(fm1, ~ Manu|Vart)

Vart = S23:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      295 12.2 15      269      321
 Average    201 12.2 15      175      227

Vart = Control:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      380 12.2 15      354      406
 Average    190 12.2 15      164      216

Vart = Kent:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      357 12.2 15      331      383
 Average    198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95

emmeans(fm1, ~ Vart|Manu)

Manu = Heavy:
 Vart    emmean   SE df lower.CL upper.CL
 S23        295 12.2 15      269      321
 Control    380 12.2 15      354      406
 Kent       357 12.2 15      331      383

Manu = Average:
 Vart    emmean   SE df lower.CL upper.CL
 S23        201 12.2 15      175      227
 Control    190 12.2 15      164      216
 Kent       198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95

emmeans(fm1, pairwise ~ Vart)

$emmeans
 Vart    emmean   SE df lower.CL upper.CL
 S23        248 8.63 15      229      266
 Control    285 8.63 15      267      304
 Kent       278 8.63 15      259      296

Results are averaged over the levels of: Blk, Manu 
Confidence level used: 0.95 

$contrasts
 contrast       estimate   SE df t.ratio p.value
 S23 - Control    -37.25 12.2 15  -3.050  0.0208
 S23 - Kent       -29.62 12.2 15  -2.426  0.0689
 Control - Kent     7.62 12.2 15   0.624  0.8092

Results are averaged over the levels of: Blk, Manu 
P value adjustment: tukey method for comparing a family of 3 estimates

emmeans(fm1, pairwise ~ Manu)

$emmeans
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      344 7.05 15      329      359
 Average    196 7.05 15      181      211

Results are averaged over the levels of: Blk, Vart 
Confidence level used: 0.95 

$contrasts
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average      148 9.97 15  14.827 <0.0001

Results are averaged over the levels of: Blk, Vart

emmeans(fm1, pairwise ~ Vart|Manu)

$emmeans
Manu = Heavy:
 Vart    emmean   SE df lower.CL upper.CL
 S23        295 12.2 15      269      321
 Control    380 12.2 15      354      406
 Kent       357 12.2 15      331      383

Manu = Average:
 Vart    emmean   SE df lower.CL upper.CL
 S23        201 12.2 15      175      227
 Control    190 12.2 15      164      216
 Kent       198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95 

$contrasts
Manu = Heavy:
 contrast       estimate   SE df t.ratio p.value
 S23 - Control    -85.25 17.3 15  -4.936  0.0005
 S23 - Kent       -62.00 17.3 15  -3.590  0.0071
 Control - Kent    23.25 17.3 15   1.346  0.3927

Manu = Average:
 contrast       estimate   SE df t.ratio p.value
 S23 - Control     10.75 17.3 15   0.622  0.8102
 S23 - Kent         2.75 17.3 15   0.159  0.9861
 Control - Kent    -8.00 17.3 15  -0.463  0.8893

Results are averaged over the levels of: Blk 
P value adjustment: tukey method for comparing a family of 3 estimates

emmeans(fm1, pairwise ~ Manu|Vart)

$emmeans
Vart = S23:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      295 12.2 15      269      321
 Average    201 12.2 15      175      227

Vart = Control:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      380 12.2 15      354      406
 Average    190 12.2 15      164      216

Vart = Kent:
 Manu    emmean   SE df lower.CL upper.CL
 Heavy      357 12.2 15      331      383
 Average    198 12.2 15      172      224

Results are averaged over the levels of: Blk 
Confidence level used: 0.95 

$contrasts
Vart = S23:
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average     94.2 17.3 15   5.458 <0.0001

Vart = Control:
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average    190.2 17.3 15  11.016 <0.0001

Vart = Kent:
 contrast        estimate   SE df t.ratio p.value
 Heavy - Average    159.0 17.3 15   9.207 <0.0001

Results are averaged over the levels of: Blk

Stability Analysis

library(stability)
library(help = stability)
data(ge_data)

Individual Analysis of Variance for each Location

Yield.indiv_anova <-
         indiv_anova(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
          )
Yield.indiv_anova

$`Analysis of Variance for FSD`
            Df   Sum Sq Mean Sq F value Pr(>F)    
Gen         59 45524231  771597   32.62 <2e-16 ***
Residuals   60  1419180   23653                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for FSDR`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 14019781  237623   2.724 7.91e-05 ***
Residuals   60  5234860   87248                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Okara`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 26802597  454281   3.538 1.22e-06 ***
Residuals   60  7703642  128394                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Sargodha`
            Df   Sum Sq Mean Sq F value  Pr(>F)    
Gen         59 38381469  650533   4.254 4.3e-08 ***
Residuals   60  9175301  152922                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Gujranwala`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 41863914  709558   4.687 6.55e-09 ***
Residuals   60  9082956  151383                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for KotNaina`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 26692263  452411     4.3 3.51e-08 ***
Residuals   60  6313095  105218                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for KSK`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 17890629  303231   3.472 1.68e-06 ***
Residuals   60  5239575   87326                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Khanewal`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 15125214  256360   2.451 0.000345 ***
Residuals   60  6274965  104583                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Sahiwal`
            Df   Sum Sq Mean Sq F value   Pr(>F)    
Gen         59 26109918  442541   4.095 8.82e-08 ***
Residuals   60  6484679  108078                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Bhakkar`
            Df   Sum Sq Mean Sq F value  Pr(>F)   
Gen         59 10870598  184247   1.862 0.00885 **
Residuals   60  5936717   98945                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$`Analysis of Variance for Bahawalnagar`
            Df  Sum Sq Mean Sq F value Pr(>F)  
Gen         59 9718292  164717   1.716 0.0196 *
Residuals   60 5760765   96013                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

attr(,"subheadings")
 [1] "Analysis of Variance for FSD"         
 [2] "Analysis of Variance for FSDR"        
 [3] "Analysis of Variance for Okara"       
 [4] "Analysis of Variance for Sargodha"    
 [5] "Analysis of Variance for Gujranwala"  
 [6] "Analysis of Variance for KotNaina"    
 [7] "Analysis of Variance for KSK"         
 [8] "Analysis of Variance for Khanewal"    
 [9] "Analysis of Variance for Sahiwal"     
[10] "Analysis of Variance for Bhakkar"     
[11] "Analysis of Variance for Bahawalnagar"

Combined Analysis of Variance

YieldANOVA <-
     add_anova(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )
YieldANOVA

$anova
Analysis of Variance Table

Response: .data[[Y]]
           Df    Sum Sq  Mean Sq F value    Pr(>F)    
Env        10 263940496 26394050 68.0551 1.995e-08 ***
Rep(Env)   11   4266167   387833                      
Gen        59  71662431  1214617 12.2482 < 2.2e-16 ***
Gen:Env   590 201336476   341248  3.4411 < 2.2e-16 ***
Residuals 649  64359568    99167                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Additive Main Effects and Multiplicative Interaction (AMMI) Analysis

Yield.ammi <-
     ammi(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )
Yield.ammi

$anova
Analysis of Variance Table

Response: .data[[Y]]
           Df    Sum Sq  Mean Sq F value    Pr(>F)    
Env        10 263940496 26394050 68.0551 1.995e-08 ***
Rep(Env)   11   4266167   387833                      
Gen        59  71662431  1214617 12.2482 < 2.2e-16 ***
Gen:Env   590 201336476   341248  3.4411 < 2.2e-16 ***
Residuals 649  64359568    99167                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

$pc.anova
# A tibble: 11 × 9
   PC     SingVal  Percent accumPerc    Df       SS  Mean.Sq  F.value   Pr..F.
                                 
 1 PC1   4.50e+ 3 2.01e+ 1      20.1    68 4.04e+ 7 5.95e+ 5 6.00e+ 0 0       
 2 PC2   4.17e+ 3 1.73e+ 1      37.4    66 3.49e+ 7 5.28e+ 5 5.33e+ 0 0       
 3 PC3   4.05e+ 3 1.63e+ 1      53.7    64 3.27e+ 7 5.12e+ 5 5.16e+ 0 0       
 4 PC4   3.49e+ 3 1.21e+ 1      65.8    62 2.44e+ 7 3.93e+ 5 3.96e+ 0 0       
 5 PC5   3.38e+ 3 1.13e+ 1      77.1    60 2.29e+ 7 3.81e+ 5 3.84e+ 0 0       
 6 PC6   3.11e+ 3 9.63e+ 0      86.7    58 1.94e+ 7 3.34e+ 5 3.37e+ 0 4.02e-14
 7 PC7   2.30e+ 3 5.25e+ 0      92.0    56 1.06e+ 7 1.89e+ 5 1.90e+ 0 1.45e- 4
 8 PC8   2.03e+ 3 4.10e+ 0      96.1    54 8.25e+ 6 1.53e+ 5 1.54e+ 0 9.52e- 3
 9 PC9   1.52e+ 3 2.30e+ 0      98.4    52 4.63e+ 6 8.91e+ 4 8.98e- 1 6.77e- 1
10 PC10  1.28e+ 3 1.63e+ 0     100      50 3.27e+ 6 6.55e+ 4 6.60e- 1 9.66e- 1
11 PC11  2.08e-11 4.32e-28     100      48 8.69e-22 1.81e-23 1.83e-28 1   e+ 0

Additive Main Effects and Multiplicative Interaction (AMMI) Biplot Analysis

ammi_biplot(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )

$aami.biplot


$MeanPC1Plot

Genotype plus Genotypes by Environment (GGE) Interaction Biplot Analysis

gge_biplot(
            .data = ge_data
          , .y    = Yield
          , .rep  = Rep
          , .gen  = Gen
          , .env  = Env
      )

$gge.biplot

Citation

BibTeX citation:

@online{yaseen2019,
  author = {Yaseen, Muhammad},
  title = {Training {Course} on {Capacity} {Building} of {NARS}
    {Scientists} in {Advance} {Analytical} {Techniques}},
  date = {2019-11-25},
  url = {https://myaseen208.com/blogs/2019-11-25_NARC/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2019. “Training Course on Capacity Building of NARS Scientists in Advance Analytical Techniques.” November 25. https://myaseen208.com/blogs/2019-11-25_NARC/.

How to write a Research Paper and Hands on Training of SPSS - A Statistical Tool

Muhammad Yaseen — Thu, 18 Apr 2019 04:00:00 GMT

Introduction

Statistics

Statistics deals with uncertainty & variability

Statistics turns data into information

Data -> Information -> Knowledge -> Wisdom

Statistics is the interpretation of Science

Statistics is the Art & Science of learning from data

Variable

Characteristic that may vary from individual to individual

Measurement

Process of assigning numbers or labels to objects or states in accordance with logically accepted rules

Measurement Scales

Nominal Scale: Obersvations may be classified into mutually exclusive & exhaustive categories

Ordinal Scale: Obersvations may be ranked

Interval Scale: Difference between obersvations is meaningful

Ratio Scale: Ratio between obersvations is meaningful & true zero point

Descriptive Statistics

No of observations

Measures of Central Tendency

Measures of Dispersion

Measures of Skewness

Measures of Kurtosis

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

Analyze > Descriptive Statistics > Descriptives …

Boxwhisker Diagram

Pictorial display of five number summary (Minimum, Q1, Q2, Q3 and Maximum)

Example

Yield	Variety
5	V1
6	V1
7	V1
15	V2
16	V2
17	V2

Graphs > Legacy Dialogs > Scatter/Boxplot …

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

Graphs > Legacy Dialogs > Scatter/Dot …

Analyze > Regression > Linear …

Exercise

Fertilizer	Yield
0.3	10
0.6	15
0.9	30
1.2	35
1.5	25
1.8	30
2.1	50
2.4	45

Exercise

Weekly Income ($)	Weekly Expenditures ($)
80	70
100	65
120	90
140	95
160	110
180	115
200	120
220	140
240	155
260	150

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Fertilizer (Kg)	Rainfall (mm)	Yield (Kg)
100	10	40
200	20	50
300	10	50
400	30	70
500	20	65
600	20	65
700	30	80

Analyze > Regression > Linear …

Correlation Analysis

Linear Relationship between Quantitative Variables

Simple Correlation Analysis

Linear Relationship between Two Quantitative Variables

Example

Sparrow Wing length (cm)	Sparrow Tail length (cm)
10.4	7.4
10.8	7.6
11.1	7.9
10.2	7.2
10.3	7.4
10.2	7.1
10.7	7.4
10.5	7.2
10.8	7.8
11.2	7.7
10.6	7.8
11.4	8.3

Analyze > Correlate > Bivariate …

Partial Correlation Analysis

Linear Relationship between Quantitative Variables while holding/keeping all other constants

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Analyze > Correlate > Partial …

Citation

BibTeX citation:

@online{yaseen2019,
  author = {Yaseen, Muhammad},
  title = {How to Write a {Research} {Paper} and {Hands} on {Training}
    of {SPSS} - {A} {Statistical} {Tool}},
  date = {2019-04-18},
  url = {https://myaseen208.com/blogs/2019-04-18-UOL/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2019. “How to Write a Research Paper and Hands on Training of SPSS - A Statistical Tool.” April 18. https://myaseen208.com/blogs/2019-04-18-UOL/.

Improving Quality in Textile Industry using Six Sigma with R

Muhammad Yaseen — Sat, 12 Jan 2019 05:00:00 GMT

Citation

BibTeX citation:

@online{yaseen2019,
  author = {Yaseen, Muhammad},
  title = {Improving {Quality} in {Textile} {Industry} Using {Six}
    {Sigma} with {R}},
  date = {2019-01-12},
  url = {https://myaseen208.com/blogs/2019-SixSigmawithR/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2019. “Improving Quality in Textile Industry Using Six Sigma with R.” January 12. https://myaseen208.com/blogs/2019-SixSigmawithR/.

Regression Analysis with Python

Muhammad Yaseen — Thu, 20 Sep 2018 04:00:00 GMT

Introduction

In God we trust, all others must bring data. (W. Edwards Deming)

In Data we trust, all others must bring data.

Statistics

Statistics is the science of uncertainty & variability

Statistics turns data into information

Data -> Information -> Knowledge -> Wisdom

Data Driven Decisions (3Ds)

Statistics is the interpretation of Science

Statistics is the Art & Science of learning from data

Variable

Characteristic that may vary from individual to individual

Height, Weight, CGPA etc

Measurement

Process of assigning numbers or labels to objects or states in accordance with logically accepted rules

Measurement Scales

Nominal Scale: Obersvations may be classified into mutually exclusive & exhaustive classes or categories

Ordinal Scale: Obersvations may be ranked

Interval Scale: Difference between obersvations is meaningful

Ratio Scale: Ratio between obersvations is meaningful & true zero point

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Population Regression Function

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

Weekly Income ($)	Weekly Expenditures ($)
80	70
100	65
120	90
140	95
160	110
180	115
200	120
220	140
240	155
260	150

Income = [80, 100, 120, 140, 160, 180, 200, 220, 240, 260]
Expend = [70, 65, 90, 95, 110, 115, 120, 140, 155, 150]

import pandas as pd
df1 = pd.DataFrame(
 {
   "Income": Income
 , "Expend": Expend
 }
 )

print(df1)

   Income  Expend
0      80      70
1     100      65
2     120      90
3     140      95
4     160     110
5     180     115
6     200     120
7     220     140
8     240     155
9     260     150

from matplotlib import pyplot as plt
fig = plt.figure()
plt.scatter(
  df1["Income"]
, df1["Expend"]
, color = "green"
, marker = "o"
)
plt.title("Scatter plot of Weekly Income (\$) and Weekly Expenditures (\$)")
plt.xlabel("Weekly Income (\$)")
plt.ylabel("Weekly Expenditures (\$)")
plt.show()

from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm
Reg1 = ols(formula = "Expend ~ Income", data = df1)
Fit1 = Reg1.fit()
print(Fit1.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                 Expend   R-squared:                       0.962
Model:                            OLS   Adj. R-squared:                  0.957
Method:                 Least Squares   F-statistic:                     202.9
Date:                Wed, 29 Apr 2026   Prob (F-statistic):           5.75e-07
Time:                        14:14:27   Log-Likelihood:                -31.781
No. Observations:                  10   AIC:                             67.56
Df Residuals:                       8   BIC:                             68.17
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     24.4545      6.414      3.813      0.005       9.664      39.245
Income         0.5091      0.036     14.243      0.000       0.427       0.592
==============================================================================
Omnibus:                        1.060   Durbin-Watson:                   2.680
Prob(Omnibus):                  0.589   Jarque-Bera (JB):                0.777
Skew:                          -0.398   Prob(JB):                        0.678
Kurtosis:                       1.891   Cond. No.                         561.
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

print(Fit1.params)

Intercept    24.454545
Income        0.509091
dtype: float64

print(Fit1.fittedvalues)

0     65.181818
1     75.363636
2     85.545455
3     95.727273
4    105.909091
5    116.090909
6    126.272727
7    136.454545
8    146.636364
9    156.818182
dtype: float64

print(Fit1.resid)

0     4.818182
1   -10.363636
2     4.454545
3    -0.727273
4     4.090909
5    -1.090909
6    -6.272727
7     3.545455
8     8.363636
9    -6.818182
dtype: float64

print(Fit1.bse)

Intercept    6.413817
Income       0.035743
dtype: float64

print(Fit1.centered_tss)

8890.0

print(anova_lm(Fit1))

           df       sum_sq      mean_sq           F        PR(>F)
Income    1.0  8552.727273  8552.727273  202.867925  5.752746e-07
Residual  8.0   337.272727    42.159091         NaN           NaN

fig = plt.figure()
plt.scatter(
  df1["Income"]
, df1["Expend"]
, color = "green"
, marker = "o"
)
plt.plot(df1["Income"], Fit1.fittedvalues)
plt.title("Regression plot of Weekly Income (\$) and Weekly Expenditures (\$)")
plt.xlabel("Weekly Income (\$)")
plt.ylabel("Weekly Expenditures (\$)")
plt.show()

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Fertilizer (Kg)	Rainfall (mm)	Yield (Kg)
100	10	40
200	20	50
300	10	50
400	30	70
500	20	65
600	20	65
700	30	80

import numpy as np
Fertilizer = np.arange(100, 800, 100)
Rainfall   = [10, 20, 10, 30, 20, 20, 30]
Yield      = [40, 50, 50, 70, 65, 65, 80]

import pandas as pd
df2 = pd.DataFrame(
 {
   "Fertilizer": Fertilizer
 , "Rainfall": Rainfall
 , "Yield": Yield
 }
 )

print(df2)

   Fertilizer  Rainfall  Yield
0         100        10     40
1         200        20     50
2         300        10     50
3         400        30     70
4         500        20     65
5         600        20     65
6         700        30     80

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(111, projection = "3d")
ax.scatter(
  df2["Fertilizer"]
, df2["Rainfall"]
, df2["Yield"]
, color = "green"
, marker = "o"
, alpha  = 1
)
ax.set_xlabel("Fertilizer")
ax.set_ylabel("Rainfall")
ax.set_zlabel("Yield")
plt.show()

from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm
Reg2 = ols(formula = "Yield ~ Fertilizer + Rainfall", data = df2)
Fit2 = Reg2.fit()
print(Fit2.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                  Yield   R-squared:                       0.981
Model:                            OLS   Adj. R-squared:                  0.972
Method:                 Least Squares   F-statistic:                     105.3
Date:                Wed, 29 Apr 2026   Prob (F-statistic):           0.000347
Time:                        14:14:28   Log-Likelihood:                -13.848
No. Observations:                   7   AIC:                             33.70
Df Residuals:                       4   BIC:                             33.53
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     28.0952      2.491     11.277      0.000      21.178      35.013
Fertilizer     0.0381      0.006      6.532      0.003       0.022       0.054
Rainfall       0.8333      0.154      5.401      0.006       0.405       1.262
==============================================================================
Omnibus:                          nan   Durbin-Watson:                   2.249
Prob(Omnibus):                    nan   Jarque-Bera (JB):                0.705
Skew:                          -0.408   Prob(JB):                        0.703
Kurtosis:                       1.677   Cond. No.                     1.28e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.28e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

print(Fit2.params)

Intercept     28.095238
Fertilizer     0.038095
Rainfall       0.833333
dtype: float64

print(Fit2.fittedvalues)

0    40.238095
1    52.380952
2    47.857143
3    68.333333
4    63.809524
5    67.619048
6    79.761905
dtype: float64

print(Fit2.resid)

0   -0.238095
1   -2.380952
2    2.142857
3    1.666667
4    1.190476
5   -2.619048
6    0.238095
dtype: float64

print(Fit2.bse)

Intercept     2.491482
Fertilizer    0.005832
Rainfall      0.154303
dtype: float64

print(Fit2.centered_tss)

1150.0

print(anova_lm(Fit2))

             df      sum_sq     mean_sq           F    PR(>F)
Fertilizer  1.0  972.321429  972.321429  181.500000  0.000176
Rainfall    1.0  156.250000  156.250000   29.166667  0.005690
Residual    4.0   21.428571    5.357143         NaN       NaN

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot as plt
import numpy as np
import pandas as pd
from matplotlib import cm

fig = plt.figure()
ax = fig.add_subplot(111, projection="3d")

ax.scatter(
    df2["Fertilizer"],
    df2["Rainfall"],
    df2["Yield"],
    color="green",
    marker="o",
    alpha=1
)

ax.set_xlabel("Fertilizer")
ax.set_ylabel("Rainfall")
ax.set_zlabel("Yield")

x_surf = np.arange(100, 720, 20)
y_surf = np.arange(10, 32, 2)
x_surf, y_surf = np.meshgrid(x_surf, y_surf)

exog = pd.DataFrame({
    "Fertilizer": x_surf.ravel(),
    "Rainfall": y_surf.ravel()
})

out = Fit2.predict(exog=exog)

# ✅ Convert to numpy before reshaping
z_surf = out.to_numpy().reshape(x_surf.shape)

ax.plot_surface(
    x_surf,
    y_surf,
    z_surf,
    rstride=1,
    cstride=1,
    color="None",
    alpha=0.4
)

plt.show()

Polynomial Regression Analysis

Quantifying non-linear dependency of a normal response on quantitative explanatory variable(s)

Example

An experiment was conducted to evaluate the effects of different levels of nitrogen. Three levels of nitrogen: 0, 10 and 20 grams per plot were used in the experiment. Each treatment was replicated twice and data is given below:

Nitrogen	Yield
0	5
0	7
10	15
10	17
20	9
20	11

Nitrogen = [0, 0, 10, 10, 20, 20]
Yield    = [5, 7, 15, 17,  9, 11]

import pandas as pd
df3 = pd.DataFrame(
 {
   "Nitrogen": Nitrogen
 , "Yield": Yield
 }
 )

print(df3)

   Nitrogen  Yield
0         0      5
1         0      7
2        10     15
3        10     17
4        20      9
5        20     11

from matplotlib import pyplot as plt
fig = plt.figure()
plt.scatter(
  df3["Nitrogen"]
, df3["Yield"]
, color = "green"
, marker = "o"
)
plt.title("Scatter plot of Nitrogen and Yield")
plt.xlabel("Nitrogen")
plt.ylabel("Yield")
plt.show()

from statsmodels.formula.api import ols
from statsmodels.stats.anova import anova_lm
Reg3 = ols(formula = "Yield ~ Nitrogen + I(Nitrogen**2)", data = df3)
Fit3 = Reg3.fit()
print(Fit3.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                  Yield   R-squared:                       0.944
Model:                            OLS   Adj. R-squared:                  0.907
Method:                 Least Squares   F-statistic:                     25.33
Date:                Wed, 29 Apr 2026   Prob (F-statistic):             0.0132
Time:                        14:14:29   Log-Likelihood:                -8.5136
No. Observations:                   6   AIC:                             23.03
Df Residuals:                       3   BIC:                             22.40
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
====================================================================================
                       coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------
Intercept            6.0000      1.000      6.000      0.009       2.818       9.182
Nitrogen             1.8000      0.255      7.060      0.006       0.989       2.611
I(Nitrogen ** 2)    -0.0800      0.012     -6.532      0.007      -0.119      -0.041
==============================================================================
Omnibus:                          nan   Durbin-Watson:                   3.333
Prob(Omnibus):                    nan   Jarque-Bera (JB):                1.000
Skew:                           0.000   Prob(JB):                        0.607
Kurtosis:                       1.000   Cond. No.                         418.
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

print(Fit3.params)

Intercept           6.00
Nitrogen            1.80
I(Nitrogen ** 2)   -0.08
dtype: float64

print(Fit3.fittedvalues)

0     6.0
1     6.0
2    16.0
3    16.0
4    10.0
5    10.0
dtype: float64

print(Fit3.resid)

0   -1.0
1    1.0
2   -1.0
3    1.0
4   -1.0
5    1.0
dtype: float64

print(Fit3.bse)

Intercept           1.000000
Nitrogen            0.254951
I(Nitrogen ** 2)    0.012247
dtype: float64

print(Fit3.centered_tss)

107.33333333333334

print(anova_lm(Fit3))

                   df     sum_sq    mean_sq          F    PR(>F)
Nitrogen          1.0  16.000000  16.000000   8.000000  0.066276
I(Nitrogen ** 2)  1.0  85.333333  85.333333  42.666667  0.007292
Residual          3.0   6.000000   2.000000        NaN       NaN

fig = plt.figure()
plt.scatter(
  df3["Nitrogen"]
, df3["Yield"]
, color = "green"
, marker = "o"
)
plt.plot(df3["Nitrogen"], Fit3.fittedvalues)
plt.title("Regression plot of Nitrogen and Yield")
plt.xlabel("Nitrogen")
plt.ylabel("Yield")
plt.show()

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad and Yaseen, Muhammad},
  title = {Regression {Analysis} with {Python}},
  date = {2018-09-20},
  url = {https://myaseen208.com/blogs/2018-09-20_RegPython/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad, and Muhammad Yaseen. 2018. “Regression Analysis with Python.” September 20. https://myaseen208.com/blogs/2018-09-20_RegPython/.

Bioinformatics with R

Muhammad Yaseen — Thu, 20 Sep 2018 04:00:00 GMT

Statistics

Statistics is the science of uncertainty & variability

Statistics turns data into information

Data -> Information -> Knowledge -> Wisdom

Data Driven Decisions (3Ds)

Statistics is the interpretation of Science

Statistics is the Art & Science of learning from data

Variable

Characteristic that may vary from individual to individual

Height, Weight, CGPA etc

Measurement

Process of assigning numbers or labels to objects or states in accordance with logically accepted rules

Measurement Scales

Nominal Scale: Obersvations may be classified into mutually exclusive & exhaustive classes or categories

Ordinal Scale: Obersvations may be ranked

Interval Scale: Difference between obersvations is meaningful

Ratio Scale: Ratio between obersvations is meaningful & true zero point

Nominal Data

Example

The following data shows the gender of a sample of twenty students from the University of Agriculture, Faisalabad.

Student	Gender
1	Male
2	Male
3	Female
4	Female
5	Female
6	Female
7	Male
8	Male
9	Male
10	Male
11	Female
12	Female
13	Male
14	Male
15	Female
16	Female
17	Male
18	Male
19	Male
20	Male

library(tidyverse)
df1 <- tibble::tibble(
    Student = seq(from = 1, to = 20, by = 1)
  , Gender  = rep(x = c("Male", "Female", "Male", "Female", "Male", "Female", "Male"), c(2, 4, 4, 2, 2, 2, 4))
  )
df1

# A tibble: 20 × 2
   Student Gender
       
 1       1 Male  
 2       2 Male  
 3       3 Female
 4       4 Female
 5       5 Female
 6       6 Female
 7       7 Male  
 8       8 Male  
 9       9 Male  
10      10 Male  
11      11 Female
12      12 Female
13      13 Male  
14      14 Male  
15      15 Female
16      16 Female
17      17 Male  
18      18 Male  
19      19 Male  
20      20 Male

Frequency Table

df1Freq <- 
      df1 %>%
      dplyr::count(Gender) %>%
      dplyr::rename(f = n) %>%
      dplyr::mutate(
        rf = f/sum(f)
      , pf = rf*100  
      )
df1Freq

# A tibble: 2 × 4
  Gender     f    rf    pf
      
1 Female     8   0.4    40
2 Male      12   0.6    60

Simple Bar Chart

ggplot(
       data = df1
    ,  mapping = aes(x = Gender)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Simple Bar Chart", x = "Gender", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Example

Following data presents the number of nucleotides of gene sequence (A, C, G, T). This is illustrated by the Zyxin gene which plays an important role in cell adhesion (Golub et al, 1999). The accession number (X94991.1) of one of its variants can be found in a data base like NCBI (UniGene). Given data will be used to illustrate the construction of pie chart from the frequency table of four nucleotides.

A	C	G	T
410	789	573	394

Data from the GenBank can also be imported directly by the following code.

# library(ape)
# df2 <- table(read.GenBank(c("X94991.1"),as.character=TRUE))
# df2

Frequency Table

df3 <- tibble(
    G.Sequence   = c("A", "C", "G", "T")
  , Nucleotides  = c(410, 789, 573, 394)
  )
df3

# A tibble: 4 × 2
  G.Sequence Nucleotides
              
1 A                  410
2 C                  789
3 G                  573
4 T                  394

Simple Bar Chart

ggplot(
        data = df3
      , mapping = aes(x = G.Sequence, y = Nucleotides)) +
  geom_bar(stat = "identity") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Simple Bar Chart", x = "Gene Sequence", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Ordinal Data

Example

The following data shows the grades of a sample of twenty students from the University of Agriculture, Faisalabad.

Student	Grade
1	A
2	B
3	B
4	C
5	A
6	D
7	F
8	C
9	B
10	D
11	F
12	A
13	B
14	B
15	C
16	D
17	C
18	B
19	C
20	D

df4 <- tibble::tibble(
    Student = seq(from = 1, to = 20, by = 1)
  , Grade  = c("A", "B", "B", "C", "A", "D", "F", "C", "B", "D", "F", "A", "B", "B", "C", "D", "C",  "B", "C", "D")
  )
df4

# A tibble: 20 × 2
   Student Grade
      
 1       1 A    
 2       2 B    
 3       3 B    
 4       4 C    
 5       5 A    
 6       6 D    
 7       7 F    
 8       8 C    
 9       9 B    
10      10 D    
11      11 F    
12      12 A    
13      13 B    
14      14 B    
15      15 C    
16      16 D    
17      17 C    
18      18 B    
19      19 C    
20      20 D

Frequency Table

df4Freq <- 
      df4 %>%
      dplyr::count(Grade) %>%
      dplyr::rename(f = n) %>%
      dplyr::mutate(
        rf = f/sum(f)
      , pf = rf*100
      , cf = cumsum(f)
      )
df4Freq

# A tibble: 5 × 5
  Grade     f    rf    pf    cf
      
1 A         3  0.15    15     3
2 B         6  0.3     30     9
3 C         5  0.25    25    14
4 D         4  0.2     20    18
5 F         2  0.1     10    20

Simple Bar Chart

ggplot(
       data = df4
    ,  mapping = aes(x = Grade)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Simple Bar Chart", x = "Grades", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Two Way Contingency Table

Example

The following data shows the gender and residental status of a sample of twenty students from the University of Agriculture, Faisalabad.

Student	Gender	RS
1	Male	Boarding
2	Male	Non-Boarding
3	Female	Non-Boarding
4	Female	Boarding
5	Female	Boarding
6	Female	Non-Boarding
7	Male	Non-Boarding
8	Male	Boarding
9	Male	Non-Boarding
10	Male	Non-Boarding
11	Female	Non-Boarding
12	Female	Boarding
13	Male	Non-Boarding
14	Male	Non-Boarding
15	Female	Non-Boarding
16	Female	Non-Boarding
17	Male	Non-Boarding
18	Male	Non-Boarding
19	Male	Non-Boarding
20	Male	Boarding

df5 <- tibble::tibble(
    Student = seq(from = 1, to = 20, by = 1)
  , Gender  = rep(x = c("Male", "Female", "Male", "Female", "Male", "Female", "Male"), c(2, 4, 4, 2, 2, 2, 4))
  , RS      = c("B", "NB", "NB", "B", "B", "NB", "NB", "B", "NB", "NB", "NB", "B", "NB", "NB", "NB", "NB", "NB", "NB", "NB", "B")
  )
df5

# A tibble: 20 × 3
   Student Gender RS   
        
 1       1 Male   B    
 2       2 Male   NB   
 3       3 Female NB   
 4       4 Female B    
 5       5 Female B    
 6       6 Female NB   
 7       7 Male   NB   
 8       8 Male   B    
 9       9 Male   NB   
10      10 Male   NB   
11      11 Female NB   
12      12 Female B    
13      13 Male   NB   
14      14 Male   NB   
15      15 Female NB   
16      16 Female NB   
17      17 Male   NB   
18      18 Male   NB   
19      19 Male   NB   
20      20 Male   B

Cross Tables

df5 %>%
  dplyr::count(Gender) %>%
  dplyr::rename(f = n)

# A tibble: 2 × 2
  Gender     f
    
1 Female     8
2 Male      12

df5 %>%
  dplyr::count(RS) %>%
  dplyr::rename(f = n)

# A tibble: 2 × 2
  RS        f
   
1 B         6
2 NB       14

df5 %>%
  dplyr::count(Gender, RS) %>%
  dplyr::rename(f = n)

# A tibble: 4 × 3
  Gender RS        f
     
1 Female B         3
2 Female NB        5
3 Male   B         3
4 Male   NB        9

library(janitor)
df5CrossTab <- 
      df5 %>%
      janitor::tabyl(dat = ., var1 = Gender, var2 = RS) %>%
      janitor::adorn_totals(dat = ., where = c("row", "col"))
df5CrossTab

 Gender B NB Total
 Female 3  5     8
   Male 3  9    12
  Total 6 14    20

Simple Bar Charts

ggplot(
       data = df5
    ,  mapping = aes(x = Gender)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Simple Bar Chart", x = "Gender", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(
       data = df5
    ,  mapping = aes(x = RS)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Simple Bar Chart", x = "Residental Status", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Multiple Bar Charts

ggplot(
       data = df5
    ,  mapping = aes(x = Gender, fill = RS)) +
  geom_bar(position = "dodge") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Multiple Bar Chart", x = "Gender", fill = "Residental Status", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(
       data = df5
    ,  mapping = aes(x = RS, fill = Gender)) +
  geom_bar(position = "dodge") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Multiple Bar Chart", x = "Residental Status", fill = "Gender", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Component Bar Charts

ggplot(
       data = df5
    ,  mapping = aes(x = Gender, fill = RS)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Component Bar Chart", x = "Gender", fill = "Residental Status", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(
       data = df5
    ,  mapping = aes(x = RS, fill = Gender)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Component Bar Chart", x = "Residental Status", fill = "Gender", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Example

Source: OMB Statistical Policy Working Paper 22. https://www.hhs.gov/sites/default/files/spwp22.pdf Following data set consists of information concerning delinquent children. Recorded variables are Number of Delinquent Children by County and Education Level of Household Head.

Cross Tables

df6 <- tibble(
  Delinquent = gl(n = 4, k = 4, length = 16, labels = c("Alpha", "Beta", "Gamma", "Delta"))
, EduLevel   = gl(n = 4, k = 1, length = 16, labels = c("Low", "Medium", "High", "Very High"))
, Freq       = c(15, 0, 5, 0, 20, 10, 10, 15, 5, 10, 10, 0, 10, 15, 5, 5)
)

df6

# A tibble: 16 × 3
   Delinquent EduLevel   Freq
              
 1 Alpha      Low          15
 2 Alpha      Medium        0
 3 Alpha      High          5
 4 Alpha      Very High     0
 5 Beta       Low          20
 6 Beta       Medium       10
 7 Beta       High         10
 8 Beta       Very High    15
 9 Gamma      Low           5
10 Gamma      Medium       10
11 Gamma      High         10
12 Gamma      Very High     0
13 Delta      Low          10
14 Delta      Medium       15
15 Delta      High          5
16 Delta      Very High     5

df6 %>%
  xtabs(data = ., Freq ~ Delinquent)

Delinquent
Alpha  Beta Gamma Delta 
   20    55    25    35

df6 %>%
  xtabs(data = ., Freq ~ EduLevel)

EduLevel
      Low    Medium      High Very High 
       50        35        30        20

df6 %>%
  xtabs(data = ., Freq ~ Delinquent + EduLevel)

          EduLevel
Delinquent Low Medium High Very High
     Alpha  15      0    5         0
     Beta   20     10   10        15
     Gamma   5     10   10         0
     Delta  10     15    5         5

Multiple Bar Charts

ggplot(
       data = df6
    ,  mapping = aes(x = Delinquent, y = Freq, fill = EduLevel)) +
  geom_bar(stat = "identity", position = "dodge") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Multiple Bar Chart", x = "Delinquent", fill = "Education Level", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(
       data = df6
    ,  mapping = aes(x = EduLevel, y = Freq, fill = Delinquent)) +
  geom_bar(stat = "identity", position = "dodge") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Multiple Bar Chart", x = "Education Level", fill = "Delinquent", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Component Bar Charts

ggplot(
       data = df6
    ,  mapping = aes(x = Delinquent, y = Freq, fill = EduLevel)) +
  geom_bar(stat = "identity") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Component Bar Chart", x = "Delinquent", fill = "Education Level", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(
       data = df6
    ,  mapping = aes(x = EduLevel, y = Freq, fill = Delinquent)) +
  geom_bar(stat = "identity") +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Component Bar Chart", x = "Education Level", fill = "Delinquent", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Count Data

Example

The following data shows the number of notebook a sample of twenty students keeping.

Student	Notebook
1	3
2	1
3	0
4	2
5	2
6	4
7	5
8	1
9	1
10	2
11	3
12	4
13	2
14	5
15	1
16	5
17	4
18	2
19	2
20	3

df7 <- tibble::tibble(
    Student  = seq(from = 1, to = 20, by = 1)
  , Notebook = c(3, 1, 0, 2, 2, 4, 5, 1, 1, 2, 3, 4, 2, 5, 1, 5, 4, 2, 2, 3)
  )
df7

# A tibble: 20 × 2
   Student Notebook
         
 1       1        3
 2       2        1
 3       3        0
 4       4        2
 5       5        2
 6       6        4
 7       7        5
 8       8        1
 9       9        1
10      10        2
11      11        3
12      12        4
13      13        2
14      14        5
15      15        1
16      16        5
17      17        4
18      18        2
19      19        2
20      20        3

Frequency Table

df7Freq <- 
      df7 %>%
      dplyr::count(Notebook) %>%
      dplyr::rename(f = n) %>%
      dplyr::mutate(
        rf = f/sum(f)
      , pf = rf*100
      , cf = cumsum(f)
      )
df7Freq

# A tibble: 6 × 5
  Notebook     f    rf    pf    cf
         
1        0     1  0.05     5     1
2        1     4  0.2     20     5
3        2     6  0.3     30    11
4        3     3  0.15    15    14
5        4     3  0.15    15    17
6        5     3  0.15    15    20

Simple Bar Chart

ggplot(
       data = df7
    ,  mapping = aes(x = Notebook)) +
  geom_bar() +
  scale_y_continuous(expand = c(0, 0)) +
  labs(title = "Simple Bar Chart", x = "Notebooks", y = "Frequency") + 
  theme_bw() + 
  theme(plot.title = element_text(hjust = 0.5))

Continuous Data

Example

The following data is the final plant height (cm) of thirty plants of wheat. 87 91 89 88 89 91 87 92 90 98 95 97 96 100 101 96 98 99 98 100 102 99 101 105 103 107 105 106 107 112

df8 <- tibble::tibble(
  PlantHeight = c(
  87,   91, 89, 88, 89, 91, 87, 92, 90, 98, 95, 97, 96, 100,    101,
  96,   98, 99, 98, 100,    102,    99, 101,    105,    103,    107,    105,    106,    107,    112
  )
)
df8

# A tibble: 30 × 1
   PlantHeight
         
 1          87
 2          91
 3          89
 4          88
 5          89
 6          91
 7          87
 8          92
 9          90
10          98
# ℹ 20 more rows

Frequency Distribution

df9 <- df8 %>% 
  summarize(
    R = max(PlantHeight) - min(PlantHeight)
  , k = floor(1 + 3.3*log10(length(PlantHeight)))
  , h = R/k
    )

df8Freq <- df8 %>% 
  mutate(
          Classes = cut(
                         x              = PlantHeight
                       , breaks         = df9$k
                       , include.lowest = TRUE
                       , right          = FALSE
                       )
          ) %>%
  count(Classes) %>% 
  tidyr::separate(col = Classes, into = c("LB", "UB"), sep = ",", remove = FALSE) %>%
  rename(f = n) %>%
  mutate(
    LB = readr::parse_number(x = LB)
  , UB = readr::parse_number(x = UB)
  , rf = f/sum(f)
  , pf = f/sum(f)*100
  , cf = cumsum(f)
  , MidPoint = (LB + UB)/2
  ) 
  
df8Freq

# A tibble: 5 × 8
  Classes      LB    UB     f    rf    pf    cf MidPoint
                
1 [87,92)      87    92     8 0.267  26.7     8     89.5
2 [92,97)      92    97     4 0.133  13.3    12     94.5
3 [97,102)     97   102    10 0.333  33.3    22     99.5
4 [102,107)   102   107     5 0.167  16.7    27    104. 
5 [107,112]   107   112     3 0.1    10      30    110.

Histogram

ggplot(
       data = df8
     , mapping = aes(x = PlantHeight)) + 
  geom_histogram() +
  scale_y_continuous(expand = c(0, 0)) +
  theme_bw()+
    labs(title = "Histogram for Plant Height", x = "Plant Height", y = "Frequency") + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(data = df8Freq, mapping = aes(x = MidPoint, y = f))  + 
  geom_point() + 
  geom_line() + 
  scale_y_continuous(expand = c(0, 0)) +
  theme_bw()+
    labs(title = "Frequency for Plant Height", x = "Mid Point", y = "Frequency") + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(data = df8Freq, mapping = aes(x = MidPoint, y = cf)) + 
  geom_point()+ 
  geom_line() +
  scale_y_continuous(expand = c(0, 0)) +
  theme_bw()+
    labs(title = "Cummulative Frequency Polygon", x = "Mid Point", y = "Cummulative Frequency") + 
  theme(plot.title = element_text(hjust = 0.5))

Stem and Leaf Plot

stem(x = df8$PlantHeight, scale = 1, width = 80, atom = 1e-08)


  The decimal point is 1 digit(s) to the right of the |

   8 | 77899
   9 | 0112
   9 | 566788899
  10 | 001123
  10 | 55677
  11 | 2

Box Plot

ggplot(data = df8 , aes( y = PlantHeight)) + 
  geom_boxplot()+
  theme_bw()

Example

The golub table contains gene expression values from 3051 genes taken from 38 Leukemia patients. Twenty seven patients are diagnosed as acute lymphoblastic leukemia (ALL) and eleven as acute myeloid leukemia (AML). The golub.gnames table contains information on the gene, including gene index, manufacturing ID, and biological name. Following table presents the gene expression value by their tumor type.

tumortype	genevalue	tumortype	genevalue
ALL	2.10892	ALL	1.78352
ALL	1.52405	ALL	0.45827
ALL	1.96403	ALL	2.18119
ALL	2.33597	ALL	2.31428
ALL	1.85111	ALL	1.99927
ALL	1.99391	ALL	1.36844
ALL	2.06597	ALL	2.37351
ALL	1.81649	ALL	1.83485
ALL	2.17622	AML	0.88941
ALL	1.80861	AML	1.45014
ALL	2.44562	AML	0.42904
ALL	1.90496	AML	0.82667
ALL	2.76610	AML	0.63637
ALL	1.32551	AML	1.02250
ALL	2.59385	AML	0.12758
ALL	1.92776	AML	-0.74333
ALL	1.10546	AML	0.73784
ALL	1.27645	AML	0.49470
ALL	1.83051	AML	1.12058

df10 <- tibble(
   genevalue = c(
              2.10892, 1.52405, 1.96403, 2.33597, 1.85111, 1.99391
            , 2.06597, 1.81649, 2.17622, 1.80861, 2.44562, 1.90496
            , 2.76610, 1.32551, 2.59385, 1.92776, 1.10546, 1.27645
            , 1.83051, 1.78352, 0.45827, 2.18119, 2.31428, 1.99927
            , 1.36844, 2.37351, 1.83485, 0.88941, 1.45014, 0.42904
            , 0.82667, 0.63637, 1.02250, 0.12758, -0.74333, 0.73784
            , 0.49470, 1.12058
            )
  , tumortype = rep(c("ALL","AML"), c(27, 11))
)

df10

# A tibble: 38 × 2
   genevalue tumortype
            
 1      2.11 ALL      
 2      1.52 ALL      
 3      1.96 ALL      
 4      2.34 ALL      
 5      1.85 ALL      
 6      1.99 ALL      
 7      2.07 ALL      
 8      1.82 ALL      
 9      2.18 ALL      
10      1.81 ALL      
# ℹ 28 more rows

Frequency Distribution

df11 <- df10 %>% 
  summarize(
    R = max(genevalue) - min(genevalue)
  , k = floor(1 + 3.3*log10(length(genevalue)))
  , h = R/k
    )

df10Freq <- df10 %>% 
  mutate(
          Classes = cut(
                         x              = genevalue
                       , breaks         = df11$k
                       , include.lowest = TRUE
                       , right          = FALSE
                       )
          ) %>%
  count(Classes) %>% 
  tidyr::separate(col = Classes, into = c("LB", "UB"), sep = ",", remove = FALSE) %>%
  rename(f = n) %>%
  mutate(
    LB = readr::parse_number(x = LB)
  , UB = readr::parse_number(x = UB)
  , rf = f/sum(f)
  , pf = f/sum(f)*100
  , cf = cumsum(f)
  , MidPoint = (LB + UB)/2
  ) 
  
df10Freq

# A tibble: 6 × 8
  Classes             LB     UB     f     rf    pf    cf MidPoint
                         
1 [-0.747,-0.158) -0.747 -0.158     1 0.0263  2.63     1   -0.452
2 [-0.158,0.426)  -0.158  0.426     1 0.0263  2.63     2    0.134
3 [0.426,1.01)     0.426  1.01      7 0.184  18.4      9    0.718
4 [1.01,1.6)       1.01   1.6       8 0.211  21.1     17    1.31 
5 [1.6,2.18)       1.6    2.18     15 0.395  39.5     32    1.89 
6 [2.18,2.77]      2.18   2.77      6 0.158  15.8     38    2.48

Histogram

ggplot(
       data = df10
     , mapping = aes(x = genevalue)) + 
  geom_histogram() +
  scale_y_continuous(expand = c(0, 0)) +
  theme_bw()+
    labs(title = "Histogram for Gene Value", x = "Gene Value", y = "Frequency") + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(data = df10Freq, mapping = aes(x = MidPoint, y = f))  + 
  geom_point() + 
  geom_line() + 
  scale_y_continuous(expand = c(0, 0)) +
  theme_bw()+
    labs(title = "Frequency for Gene Value", x = "Mid Point", y = "Frequency") + 
  theme(plot.title = element_text(hjust = 0.5))

ggplot(data = df10Freq, mapping = aes(x = MidPoint, y = cf)) + 
  geom_point()+ 
  geom_line() +
  scale_y_continuous(expand = c(0, 0)) +
  theme_bw()+
    labs(title = "Cummulative Frequency Polygon", x = "Mid Point", y = "Cummulative Frequency") + 
  theme(plot.title = element_text(hjust = 0.5))

Stem and Leaf Plot

stem(x = df10$genevalue, scale = 1, width = 80, atom = 1e-08)


  The decimal point is at the |

  -0 | 7
  -0 | 
   0 | 14
   0 | 556789
   1 | 011334
   1 | 5588888999
   2 | 00011223344
   2 | 68

Box Plot

ggplot(data = df10 , aes(x = tumortype, y = genevalue)) + 
  geom_boxplot()+
  theme_bw()

Measures of Centeral Tendency

df7 %>%
  summarize(
            n       = length(Notebook)
          , Mean    = mean(Notebook)
          , Median  = median(Notebook)
          , Minimum = min(Notebook) 
          , Maximum = max(Notebook) 
          , Q1      = quantile(x = Notebook, probs = 0.25)
          , Q2      = quantile(x = Notebook, probs = 0.50)
          , Q3      = quantile(x = Notebook, probs = 0.75)
  )

# A tibble: 1 × 8
      n  Mean Median Minimum Maximum    Q1    Q2    Q3
              
1    20   2.6      2       0       5  1.75     2     4

df8 %>%
  summarize(
            n       = length(PlantHeight)
          , Mean    = mean(PlantHeight)
          , Median  = median(PlantHeight)
          , Minimum = min(PlantHeight) 
          , Maximum = max(PlantHeight) 
          , Q1      = quantile(x = PlantHeight, probs = 0.25)
          , Q2      = quantile(x = PlantHeight, probs = 0.50)
          , Q3      = quantile(x = PlantHeight, probs = 0.75)
  )

# A tibble: 1 × 8
      n  Mean Median Minimum Maximum    Q1    Q2    Q3
              
1    30  97.6     98      87     112  91.2    98  102.

df10 %>%
  group_by(tumortype) %>%
  summarize(
            n       = length(genevalue)
          , Mean    = mean(genevalue)
          , Median  = median(genevalue)
          , Minimum = min(genevalue) 
          , Maximum = max(genevalue) 
          , Q1      = quantile(x = genevalue, probs = 0.25)
          , Q2      = quantile(x = genevalue, probs = 0.50)
          , Q3      = quantile(x = genevalue, probs = 0.75)
  )

# A tibble: 2 × 9
  tumortype     n  Mean Median Minimum Maximum    Q1    Q2    Q3
                   
1 ALL          27 1.89   1.93    0.458    2.77 1.80  1.93  2.18 
2 AML          11 0.636  0.738  -0.743    1.45 0.462 0.738 0.956

Measures of Dispersion

df7 %>%
  summarize(
            IQR      = IQR(Notebook)
          , Variance = var(Notebook)
          ,  SD      = sd(Notebook)
          )

# A tibble: 1 × 3
    IQR Variance    SD
       
1  2.25     2.25  1.50

df8 %>%
  summarize(
            IQR      = IQR(PlantHeight)
          , Variance = var(PlantHeight)
          ,  SD      = sd(PlantHeight)
          )

# A tibble: 1 × 3
    IQR Variance    SD
       
1  10.5     45.0  6.71

df10 %>%
  group_by(tumortype) %>%
  summarize(
            IQR      = IQR(genevalue)
          , Variance = var(genevalue)
          ,  SD      = sd(genevalue)
          )

# A tibble: 2 × 4
  tumortype   IQR Variance    SD
            
1 ALL       0.383    0.241 0.491
2 AML       0.494    0.338 0.582

Measures of Skewness

df7 %>% 
  summarize(
           SK = sum((Notebook - mean(Notebook))^3)/(n()*(sd(Notebook))^3)
          )

# A tibble: 1 × 1
     SK
  
1 0.217

df8 %>% 
  summarize(
           SK = sum((PlantHeight - mean(PlantHeight))^3)/(n()*(sd(PlantHeight))^3)
          )

# A tibble: 1 × 1
      SK
   
1 0.0574

df10 %>% 
  group_by(tumortype) %>%
  summarize(
           SK = sum((genevalue - mean(genevalue))^3)/(n()*(sd(genevalue))^3)
          )

# A tibble: 2 × 2
  tumortype     SK
        
1 ALL       -0.802
2 AML       -0.937

Measures of Skewness

df7 %>%
  summarize(
            K = sum((Notebook - mean(Notebook))^4)/(n()*(sd(Notebook))^4) - 3
          )

# A tibble: 1 × 1
      K
  
1 -1.19

df8 %>%
  summarize(
            K = sum((PlantHeight - mean(PlantHeight))^4)/(n()*(sd(PlantHeight))^4) - 3
          )

# A tibble: 1 × 1
       K
   
1 -0.960

df10 %>% 
  group_by(tumortype) %>%
  summarize(
            K = sum((genevalue - mean(genevalue))^4)/(n()*(sd(genevalue))^4) - 3
          )

# A tibble: 2 × 2
  tumortype     K
       
1 ALL       0.855
2 AML       0.345

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad and Yaseen, Muhammad},
  title = {Bioinformatics with {R}},
  date = {2018-09-20},
  url = {https://myaseen208.com/blogs/2018-09-20_BioInfoR/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad, and Muhammad Yaseen. 2018. “Bioinformatics with R.” September 20. https://myaseen208.com/blogs/2018-09-20_BioInfoR/.

An Introduction to Statistics using SPSS

Muhammad Yaseen — Thu, 16 Aug 2018 04:00:00 GMT

Introduction

Statistics

Statistics is the science of uncertainty & variability

Statistics turns data into information

Data -> Information -> Knowledge -> Wisdom

Statistics is the interpretation of Science

Statistics is the Art & Science of learning from data

Variable

Characteristic that may vary from individual to individual

Measurement

Process of assigning numbers or labels to objects or states in accordance with logically accepted rules

Measurement Scales

Nominal Scale: Obersvations may be classified into mutually exclusive & exhaustive categories

Ordinal Scale: Obersvations may be ranked

Interval Scale: Difference between obersvations is meaningful

Ratio Scale: Ratio between obersvations is meaningful & true zero point

Descriptive Statistics

No of observations

Measures of Central Tendency

Measures of Dispersion

Measures of Skewness

Measures of Kurtosis

Example

Following data presents a subset of Yarn strength and Elongation percentage from an experiment to test yarn strength and extension of blue jeans designed under Conducted in Sudan Textile Factory Method.

Single Strength	Elongation
419.7	3.22
442.7	2.78
457.4	1.60
435.2	1.88
404.1	1.88
479.1	2.34
450.8	2.44
432.7	2.12
378.1	2.88
413.3	2.37
365.1	1.40
492.0	2.02
389.5	2.02
467.8	1.92
442.3	2.44
444.3	1.74

Reference

Mousa, A.H.N. (1978). Analysis of the Effect of Size Formulation Variations on Yarn Tenacity and Elongation. Textile Research Journal, 48, 713-717.

Analyze > Descriptive Statistics > Descriptives …

Boxwhisker Diagram

Pictorial display of five number summary (Minimum, Q1, Q2, Q3 and Maximum)

Example

Effact of Two drying methods (Line Dry and Tumble Dry) are to be compared for some kind of shrinkage of some particular brand of fabric. Following data was observed from an experiment:

Drying Method	Length Shrinkage	Drying Method	Length Shrinkage
Line Dry	1.0	Tumble Dry	2.8
Line Dry	1.5	Tumble Dry	3.0
Line Dry	2.1	Tumble Dry	3.0
Line Dry	2.1	Tumble Dry	2.9
Line Dry	5.4	Tumble Dry	6.8
Line Dry	5.4	Tumble Dry	6.5
Line Dry	2.8	Tumble Dry	2.9
Line Dry	3.0	Tumble Dry	3.6
Line Dry	1.4	Tumble Dry	3.5
Line Dry	1.8	Tumble Dry	3.7
Line Dry	2.3	Tumble Dry	3.5
Line Dry	2.2	Tumble Dry	3.5
Line Dry	7.0	Tumble Dry	7.9
Line Dry	7.2	Tumble Dry	8.0
Line Dry	4.2	Tumble Dry	4.0
Line Dry	4.5	Tumble Dry	4.8
Line Dry	1.8	Tumble Dry	4.0
Line Dry	1.9	Tumble Dry	4.2
Line Dry	2.5	Tumble Dry	3.9
Line Dry	2.2	Tumble Dry	3.7
Line Dry	7.5	Tumble Dry	8.6
Line Dry	7.9	Tumble Dry	8.8
Line Dry	4.8	Tumble Dry	4.7
Line Dry	5.2	Tumble Dry	5.3

Reference

Higgins, L., Anand, S.C., Holmes, D.A., Hall, M.E., and Underly, K. (2003). Effects of Various Home Laundering on the Dimensional Stability, Wrinkling, and Other Properties of Plain Woven Cotton Fabrics: Part II: Effect of Rinse Cycle Softener and Drying Method and of Tumble Sheet Softener and Tumble Drying Time. Textile Research Journal, 73, 407-420.

Graphs > Legacy Dialogs > Boxplot …

Scatter Plot

Two dimensional graph to show the relationship among two variables

Example

An Experiment was conducted to test ballistic tests for some fabric. To test the effect, bullet volicity and depth of trauma was observed. Some obser vations as example are given in the following table. Each measurement is based on 6 replicates.

Bullet Velocity	Trauma Depth
372.83	3.87
370.67	5.20
370.50	4.85
369.33	7.12
371.67	2.73
370.50	5.75
369.67	5.32

Reference

Karahan, M. (2008). Comparison of Ballistic Performance and Energy Absorption Capabilities of Woven and Unidirectional Aramid Fabrics. Textile Research Journal, 78, 718-730.

Graphs > Legacy Dialogs > Scatter/Dot …

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

An Experiment was conducted to test ballistic tests for some fabric. To test the effect, bullet volicity and depth of trauma was observed. Some obser vations as example are given in the following table. Each measurement is based on 6 replicates.

Bullet Velocity	Trauma Depth
372.83	3.87
370.67	5.20
370.50	4.85
369.33	7.12
371.67	2.73
370.50	5.75
369.67	5.32

Reference

Karahan, M. (2008). Comparison of Ballistic Performance and Energy Absorption Capabilities of Woven and Unidirectional Aramid Fabrics. Textile Research Journal, 78, 718-730.

Analyze > Regression > Linear …

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Following data presents a Factorial experiment relating yarn strength and elongation to 3 factors: Corn Starch, Nordux, and Fats. Conducted in Sudan Textile Factory Method.

Corn Starch	Nordux	Fats	Single Strength	Enolgation
85	50	30.25	419.7	3.22
140	50	30.25	442.7	2.78
85	100	30.25	457.4	1.60
140	100	30.25	435.2	1.88
85	50	30.00	404.1	1.88
140	50	30.00	479.1	2.34
85	100	30.00	450.8	2.44
140	100	30.00	432.7	2.12
85	50	30.25	378.1	2.88
140	50	30.25	413.3	2.37
85	100	30.25	365.1	1.40
140	100	30.25	492.0	2.02
85	50	30.00	389.5	2.02
140	50	30.00	467.8	1.92
85	100	30.00	442.3	2.44
140	100	30.00	444.3	1.74

Reference

Mousa, A.H.N. (1978). Analysis of the Effect of Size Formulation Variations on Yarn Tenacity and Elongation. Textile Research Journal, 48, 713-717.

Analyze > Regression > Linear …

Polynomial Regression Analysis

Quantifying non-linear dependency of a normal response on quantitative explanatory variable(s)

Example

The Relationship Between Air Layers and Evaporative Resistance of Male Chinese Clothing is to be observed from Measurements of total air volume and total evaporative resistance of 39 Chinese clothing ensembles the following data.

Air Volume	Evaporative Resistance	Air Volume	Evaporative Resistance
22.99	13.7	32.29	21.2
27.98	16.8	33.60	21.1
24.41	16.5	34.83	21.0
33.21	17.3	36.37	22.9
42.07	18.5	38.52	21.2
44.30	19.0	30.02	19.3
46.00	18.0	50.18	23.4
93.06	23.2	32.70	20.5
26.19	18.8	33.57	20.2
30.76	17.5	64.02	24.4
28.76	19.4	60.16	23.2
30.30	21.2	35.85	22.3
49.43	20.5	39.89	22.7
28.06	20.1	47.83	24.3
33.61	21.2	30.60	18.1
28.93	19.5	35.19	23.0
20.07	18.5	45.72	23.6
84.67	21.1	74.20	26.6
27.95	20.2	65.07	30.2
31.07	22.4

Reference

Wang, F., Peng, H., and Shi, W. (2016). The Relationship Between Air Layers and Evaporative Resistance of Male Chinese Clothing. Applied Ergonomics, 56, 194-202.

Analyze > Regression > Curve Estimation …

Analysis of Variance (ANOVA)

Comparing means of Normal dependent variable for levels of different factor(s)

Example

Experiment comparing tensile strength and extension of blue jeans that were designed manually and with laser beams. Data from 20 pairs of jeans per design method is presented in the table.

Method	Jean ID	Sample ID	Strength	Extension
1	1	1	1266	68.06
1	2	1	1099	61.54
1	3	1	1068	40.48
1	4	1	1266	59.83
1	5	1	968	55.89
1	6	1	1039	54.12
1	7	1	1194	77.09
1	8	1	926	61.80
1	9	1	1009	58.31
1	10	1	1048	71.61
1	11	1	1203	74.40
1	12	1	884	63.38
1	13	1	1069	59.08
1	14	1	1034	73.21
1	15	1	1021	78.60
1	16	1	1022	57.44
1	17	1	1071	88.92
1	18	1	1089	76.77
1	19	1	1197	74.78
1	20	1	1035	71.11
2	21	1	1279	75.52
2	22	1	1197	71.44
2	23	1	1083	44.16
2	24	1	1418	70.56
2	25	1	1273	69.13
2	26	1	1232	62.52
2	27	1	1293	72.62
2	28	1	1121	68.96
2	29	1	1286	71.03
2	30	1	1416	69.34
2	31	1	1285	81.79
2	32	1	1268	69.29
2	33	1	1365	73.73
2	34	1	1309	75.01
2	35	1	1343	75.95
2	36	1	1292	72.64
2	37	1	1234	73.73
2	38	1	1226	75.97
2	39	1	1229	71.60
2	40	1	1254	70.48
1	1	2	1140	63.04
1	2	2	1104	65.20
1	3	2	1051	50.62
1	4	2	1128	56.21
1	5	2	1044	53.51
1	6	2	1303	59.53
1	7	2	1081	60.75
1	8	2	958	60.60
1	9	2	1058	69.80
1	10	2	1077	62.05
1	11	2	956	72.87
1	12	2	1010	59.41
1	13	2	1062	61.71
1	14	2	1073	71.51
1	15	2	1027	68.16
1	16	2	1048	60.76
1	17	2	1101	63.70
1	18	2	1076	80.69
1	19	2	1081	69.65
1	20	2	1084	68.45
2	21	2	1205	71.44
2	22	2	1214	69.00
2	23	2	1055	47.05
2	24	2	1402	74.50
2	25	2	1310	72.76
2	26	2	1324	73.01
2	27	2	1303	79.82
2	28	2	1083	65.43
2	29	2	1391	69.84
2	30	2	1225	74.16
2	31	2	1396	79.49
2	32	2	1273	75.30
2	33	2	1326	66.65
2	34	2	1393	79.21
2	35	2	1172	73.57
2	36	2	1241	71.80
2	37	2	1292	77.70
2	38	2	1354	73.83
2	39	2	1299	73.59
2	40	2	1202	68.45
1	1	3	1104	61.40
1	2	3	1113	63.44
1	3	3	1063	50.48
1	4	3	1200	63.08
1	5	3	1006	66.70
1	6	3	860	46.39
1	7	3	936	62.76
1	8	3	967	64.72
1	9	3	1018	61.55
1	10	3	1191	69.50
1	11	3	987	73.45
1	12	3	1155	79.59
1	13	3	982	57.91
1	14	3	1064	70.86
1	15	3	1112	68.17
1	16	3	1040	63.46
1	17	3	1075	61.44
1	18	3	1081	73.51
1	19	3	1045	71.19
1	20	3	1077	69.33
2	21	3	1283	78.62
2	22	3	1154	63.06
2	23	3	1117	42.43
2	24	3	1335	71.45
2	25	3	1277	73.41
2	26	3	1313	71.44
2	27	3	1361	74.27
2	28	3	1104	70.42
2	29	3	1450	72.40
2	30	3	1323	75.00
2	31	3	1315	74.82
2	32	3	1321	77.76
2	33	3	1315	66.90
2	34	3	1386	77.64
2	35	3	1165	75.60
2	36	3	1198	69.73
2	37	3	1223	69.42
2	38	3	1289	80.39
2	39	3	1339	71.40
2	40	3	1266	72.36

Reference

Ondogan, Z., Pamuk, O., Ondogan, E.N., and Ozguney, A. (2005). Improving the Appearance of All Textile Products from Clothing to Home Textile Using Laser Technology. Optics and Laser Technology, 37, 631-637.

Analyze > General Linear Model > Univariate …

Example

The analysis of variance of cotton yarn fineness is customer satisfaction with quality textile product. The purpose of this study is to minimize the faults of finished cotton fabric by maintaining actual yarn count during producing in ring frame. If the analysis shows the mean count difference due to shift and machine number then action should be taken to maintain required count of yarn. This action may be taken by setting of actual draft, uniform linear density of finisher draw frame sliver and maintaining proper atmospheric condition in ring and back section. The 210 data for yarn count of 30 KH (30 carded Hosiery) was summarized as follows:

Machine Number	Shift	YarnCount
1	A	29.52
2	A	29.58
3	A	28.62
4	A	30.50
5	A	29.03
6	A	29.75
7	A	31.28
1	B	30.91
2	B	29.33
3	B	29.32
4	B	30.04
5	B	30.33
6	B	30.91
7	B	29.95
1	C	30.14
2	C	30.00
3	C	29.98
4	C	28.99
5	C	29.82
6	C	29.85
7	C	30.05

Reference

Hossen, J., Haque, E., Khan, S.K., and Saha, S. (2012). Analysis of cotton yarn count variation by two way ANOVA. Conference: Annals of the University of Oradea, Fascicle of Textiles, Leatherwork. At Annals of the University of Oradea, XIII-02.

Analyze > General Linear Model > Univariate …

Analysis of Covariance (ANCOVA)

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Comparing means of Normal dependent variable for levels of different factor(s)

Example

Results of Experiment measuring top shrinkage under 3 treatments (untreated, ether-extracted, ether/alcohol extracted), measured at 3 levels of pH (2, 6, 10) is given in the following table.

Treatment	pH	Shrinkage
Untreated	2	42.8
Untreated	6	46.2
Untreated	10	45.6
Untreated	2	43.0
Untreated	6	47.0
Untreated	10	43.8
Untreated	2	43.5
Untreated	6	45.2
Untreated	10	45.7
Ether Extract	2	33.6
Ether Extract	6	40.4
Ether Extract	10	40.1
Ether Extract	2	35.0
Ether Extract	6	39.4
Ether Extract	10	39.5
Ether Extract	2	36.2
Ether Extract	6	37.9
Ether Extract	10	39.2
ether/alcohol extracted	2	35.6
ether/alcohol extracted	6	35.2
ether/alcohol extracted	10	36.6
ether/alcohol extracted	2	34.8
ether/alcohol extracted	6	33.0
ether/alcohol extracted	10	32.3
ether/alcohol extracted	2	34.8
ether/alcohol extracted	6	32.3
ether/alcohol extracted	10	35.6

Reference

Lindberg, J. (1953). Relationship Between Various Surface Properties of Wool Fibers: Part II: Frictional Properties. Textile Research Journal, 23, 225-237.

Same Slopes but different Intercepts

Analyze > General Linear Model > Univariate …

Different Intercepts and different Slopes

Analyze > General Linear Model > Univariate …

Factorial Experiments

Treatments are the combinations of the levels of different factors

Example

Following data presents the number of warp breaks per loom, where a loom corresponds to a fixed length of yarn for two factors, Wool Type with Two levels (A,B) and Tension with three levels.(Low, Medium, High). Analyze the data with interaction.

WoolType	Tension	Yarn Breaks
A	L	26
A	L	30
A	L	54
A	L	25
A	L	70
A	L	52
A	L	51
A	L	26
A	L	67
A	M	18
A	M	21
A	M	29
A	M	17
A	M	12
A	M	18
A	M	35
A	M	30
A	M	36
A	H	36
A	H	21
A	H	24
A	H	18
A	H	10
A	H	43
A	H	28
A	H	15
A	H	26
B	L	27
B	L	14
B	L	29
B	L	19
B	L	29
B	L	31
B	L	41
B	L	20
B	L	44
B	M	42
B	M	26
B	M	19
B	M	16
B	M	39
B	M	28
B	M	21
B	M	39
B	M	29
B	H	20
B	H	21
B	H	24
B	H	17
B	H	13
B	H	15
B	H	15
B	H	16
B	H	28

Reference

Tippett, L. H. C. (1950). Technological Applications of Statistics. John Wiley & Sons, New York, USA.

Analyze > General Linear Model > Univariate …

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {An {Introduction} to {Statistics} Using {SPSS}},
  date = {2018-08-16},
  url = {https://myaseen208.com/blogs/2018-08-16-PASTIC-NTU/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “An Introduction to Statistics Using SPSS.” August 16. https://myaseen208.com/blogs/2018-08-16-PASTIC-NTU/.

Emerging Technologies in Research: Google Apps and SPSS

Muhammad Yaseen — Tue, 15 May 2018 04:00:00 GMT

Introduction

Statistics

Statistics deals with uncertainty & variability

Statistics turns data into information

Data -> Information -> Knowledge -> Wisdom

Statistics is the interpretation of Science

Statistics is the Art & Science of learning from data

Variable

Characteristic that may vary from individual to individual

Measurement

Process of assigning numbers or labels to objects or states in accordance with logically accepted rules

Measurement Scales

Nominal Scale: Obersvations may be classified into mutually exclusive & exhaustive categories

Ordinal Scale: Obersvations may be ranked

Interval Scale: Difference between obersvations is meaningful

Ratio Scale: Ratio between obersvations is meaningful & true zero point

Descriptive Statistics

No of observations

Measures of Central Tendency

Measures of Dispersion

Measures of Skewness

Measures of Kurtosis

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

Analyze > Descriptive Statistics > Descriptives …

Boxwhisker Diagram

Pictorial display of five number summary (Minimum, Q1, Q2, Q3 and Maximum)

Example

Yield	Variety
5	V1
6	V1
7	V1
15	V2
16	V2
17	V2

Graphs > Legacy Dialogs > Scatter/Boxplot …

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

Graphs > Legacy Dialogs > Scatter/Dot …

Analyze > Regression > Linear …

Exercise

Fertilizer	Yield
0.3	10
0.6	15
0.9	30
1.2	35
1.5	25
1.8	30
2.1	50
2.4	45

Exercise

Weekly Income ($)	Weekly Expenditures ($)
80	70
100	65
120	90
140	95
160	110
180	115
200	120
220	140
240	155
260	150

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Fertilizer (Kg)	Rainfall (mm)	Yield (Kg)
100	10	40
200	20	50
300	10	50
400	30	70
500	20	65
600	20	65
700	30	80

Analyze > Regression > Linear …

Polynomial Regression Analysis

Quantifying non-linear dependency of a normal response on quantitative explanatory variable(s)

Example

An experiment was conducted to evaluate the effects of different levels of nitrogen. Three levels of nitrogen: 0, 10 and 20 grams per plot were used in the experiment. Each treatment was replicated twice and data is given below:

Nitrogen	Yield
0	5
0	7
10	15
10	17
20	9
20	11

Analysis of Variance (ANOVA)

Comparing means of Normal dependent variable for levels of different factor(s)

Example

Yield	Variety
5	V1
6	V1
7	V1
15	V2
16	V2
17	V2

General Linear Model > Univariate …

Exercise

Yield	Variety
5	V1
7	V1
15	V2
17	V2
17	V3
19	V3

Analysis of Covariance (ANCOVA)

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Comparing means of Normal dependent variable for levels of different factor(s)

Example

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Same intercepts but different slopes

Different intercepts and different slopes

Correlation Analysis

Linear Relationship between Quantitative Variables

Simple Correlation Analysis

Linear Relationship between Two Quantitative Variables

Example

Sparrow Wing length (cm)	Sparrow Tail length (cm)
10.4	7.4
10.8	7.6
11.1	7.9
10.2	7.2
10.3	7.4
10.2	7.1
10.7	7.4
10.5	7.2
10.8	7.8
11.2	7.7
10.6	7.8
11.4	8.3

Analyze > Correlate > Bivariate …

Partial Correlation Analysis

Linear Relationship between Quantitative Variables while holding/keeping all other constants

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Analyze > Correlate > Partial …

Multiple Correlation Analysis

Linear Relationship between a Quantitative Variable and set of other Quantitative Variables

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Completely Randomized Design (CRD)

Used when experimental material is homogeneous

Example

The following table shows some of the results of an experiment on the effects of applications of sulphur in reducing scab disease of potatoes. The object in applying sulphur is to increase the acidity of the soil, since scab does not thrive in very acid soil. In addition to untreated plots which serve as a control, 3 amounts of dressing were compared—300, 600, and 900 lb. per acre. Both a fall and a spring application of each amount was tested, so that in all there were seven distinct treatments. The sulphur was spread by hand on the surface of the soil, and then diced into a depth of about 4 inches. The quantity to be analyzed is the “scab index”. That is roughly speaking, the percentage of the surface area of the potato that is infected with scab. It is obtained by examining 100 potatoes at random from each plot, grading each potato on a scale from 0 to 100% infected, and taking the average.

Randomized Complete Block Design (RCBD)

Used when experimental material is heterogenous in one direction

Example

Yield : Yield of barley, SoilType : Soil Type, and Trt : 5 sources and a control

Latin Square Design

Used when experimental material is heterogenous in two perpendicular directions

Example

The following table shows the field layout and yield of a 5×5 Latin square experiment on the effect of spacing on yield of millet plants. Five levels of spacing were used. The data on yield (grams/plot) was recorded and is given below:

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {Emerging {Technologies} in {Research:} {Google} {Apps} and
    {SPSS}},
  date = {2018-05-15},
  url = {https://myaseen208.com/blogs/2018-05-15-PASTIC-FSD/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “Emerging Technologies in Research: Google Apps and SPSS.” May 15. https://myaseen208.com/blogs/2018-05-15-PASTIC-FSD/.

Spatial Analysis of Pakistan Population Census 2017

Muhammad Yaseen — Sun, 13 May 2018 04:00:00 GMT

The latest version of the ppcSpatial package for R is now on CRAN. It performs spatial analysis for exploration of Pakistan Population Census 2017 (http://www.pbscensus.gov.pk/). It uses data from R package PakPC2017.

The Spatial map is

and full view is here.

A video showing some functionality of the package is

ppcSpatial package video thumbnail

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {Spatial {Analysis} of {Pakistan} {Population} {Census} 2017},
  date = {2018-05-13},
  url = {https://myaseen208.com/blogs/2018-05-13_ppcSpatial/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “Spatial Analysis of Pakistan Population Census 2017.” May 13. https://myaseen208.com/blogs/2018-05-13_ppcSpatial/.

An introduction to R for Agricultural Research

Muhammad Yaseen — Wed, 09 May 2018 04:00:00 GMT

Introduction

R is a free, open-source programming language and software environment for statistical computing, bioinformatics, visualization and general computing. R provides a wide variety of statistical and graphical techniques, and is highly extensible. The latest version of R is 3.5.0 which can be obtained from https://cran.r-project.org/bin/windows/base/.

R R-3.5.0 for Windows (32/64 bit)

RStudio RStudio

Manual An Introduction to R

Ref Card R Reference Card

New York Times New York Times

Nature Article Nature Article

Regression Analysis

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Figure 1

Simple Linear Regression

Quantifying dependency of a normal response on a quantitative explanatory variable

Example

Fertilizer (Kg/acre)	Production (Bushels/acre)
100	70
200	70
400	80
500	100

# Fertilizer (Kg/acre)
# Production (Bushels/acre)
Fertilizer <- c(100, 200, 400, 500)
Production <- c( 70,  70,  80, 100)
df1        <- data.frame(Fertilizer, Production)
df1

  Fertilizer Production
1        100         70
2        200         70
3        400         80
4        500        100

library(ggplot2)
p1 <- 
  ggplot(data = df1, mapping = aes(x = Fertilizer, y = Production)) + 
  geom_point() +
  labs(x = "Fertilizer", y = "Production") +
  theme_bw()
print(p1)

fm1 <- lm(formula = Production ~ Fertilizer, data = df1)
summary(fm1)


Call:
lm(formula = Production ~ Fertilizer, data = df1)

Residuals:
 1  2  3  4 
 4 -3 -7  6 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 59.00000    7.95299   7.419   0.0177 *
Fertilizer   0.07000    0.02345   2.985   0.0963 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.416 on 2 degrees of freedom
Multiple R-squared:  0.8167,    Adjusted R-squared:  0.725 
F-statistic: 8.909 on 1 and 2 DF,  p-value: 0.0963

anova(fm1)

Analysis of Variance Table

Response: Production
           Df Sum Sq Mean Sq F value Pr(>F)  
Fertilizer  1    490     490  8.9091 0.0963 .
Residuals   2    110      55                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Fertilizer	Yield
0.3	10
0.6	15
0.9	30
1.2	35
1.5	25
1.8	30
2.1	50
2.4	45

Fert  <- c(0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.1, 2.4)
Yield <- c(10, 15, 30, 35, 25, 30, 50, 45)
df2   <- data.frame(Fert, Yield)
df2

  Fert Yield
1  0.3    10
2  0.6    15
3  0.9    30
4  1.2    35
5  1.5    25
6  1.8    30
7  2.1    50
8  2.4    45

library(ggplot2)
p2 <- 
  ggplot(data = df2, mapping = aes(x = Fert, y = Yield)) + 
  geom_point() +
  labs(x = "Fertilizer", y = "Yield") +
  theme_bw()
print(p2)

fm2 <- lm(formula = Yield ~ Fert, data = df2)
summary(fm2)


Call:
lm(formula = Yield ~ Fert, data = df2)

Residuals:
   Min     1Q Median     3Q    Max 
-7.441 -4.018 -2.441  7.351  7.798 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    8.036      5.504   1.460   0.1946   
Fert          16.270      3.633   4.478   0.0042 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.064 on 6 degrees of freedom
Multiple R-squared:  0.7697,    Adjusted R-squared:  0.7313 
F-statistic: 20.05 on 1 and 6 DF,  p-value: 0.004202

anova(fm2)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value   Pr(>F)   
Fert       1 1000.6  1000.6  20.052 0.004202 **
Residuals  6  299.4    49.9                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Weekly Income ($)	Weekly Expenditures ($)
80	70
100	65
120	90
140	95
160	110
180	115
200	120
220	140
240	155
260	150

# Weekly Income ($) of a Family
# Weekly Expenditures ($) of a Family
Income       <- seq(from = 80, to = 260, by = 20)
Expenditures <- c(70, 65, 90, 95, 110, 115, 120, 140, 155, 150)
df3          <- data.frame(Income, Expenditures)
df3

   Income Expenditures
1      80           70
2     100           65
3     120           90
4     140           95
5     160          110
6     180          115
7     200          120
8     220          140
9     240          155
10    260          150

library(ggplot2)
p3 <- 
  ggplot(data = df3, mapping = aes(x = Income, y = Expenditures)) + 
  geom_point() +
  scale_x_continuous(labels = scales::dollar) +
  scale_y_continuous(labels = scales::dollar) +
  labs(x = "Weekly income, $", y = "Weekly consumption expenditures, $") +
  theme_bw()
print(p3)

fm3 <- lm(formula = Expenditures ~ Income, data = df3)
summary(fm3)


Call:
lm(formula = Expenditures ~ Income, data = df3)

Residuals:
    Min      1Q  Median      3Q     Max 
-10.364  -4.977   1.409   4.364   8.364 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 24.45455    6.41382   3.813  0.00514 ** 
Income       0.50909    0.03574  14.243 5.75e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.493 on 8 degrees of freedom
Multiple R-squared:  0.9621,    Adjusted R-squared:  0.9573 
F-statistic: 202.9 on 1 and 8 DF,  p-value: 5.753e-07

anova(fm3)

Analysis of Variance Table

Response: Expenditures
          Df Sum Sq Mean Sq F value    Pr(>F)    
Income     1 8552.7  8552.7  202.87 5.753e-07 ***
Residuals  8  337.3    42.2                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Multiple Linear Regression

Quantifying dependency of a normal response on two or more quantitative explanatory variables

Example

Fertilizer (Kg)	Rainfall (mm)	Yield (Kg)
100	10	40
200	20	50
300	10	50
400	30	70
500	20	65
600	20	65
700	30	80

# Fertilizer (Kg)
# Rainfall (mm)
# Yield (Kg)
Fertilizer <- seq(from = 100, to = 700, by = 100)
Rainfall   <- c(10, 20, 10, 30, 20, 20, 30)
Yield      <- c(40, 50, 50, 70, 65, 65, 80)
df4        <- data.frame(Fertilizer, Rainfall, Yield)
df4

  Fertilizer Rainfall Yield
1        100       10    40
2        200       20    50
3        300       10    50
4        400       30    70
5        500       20    65
6        600       20    65
7        700       30    80

library(ggplot2)
ggplot(data = df4, mapping = aes(x = Fertilizer, y = Rainfall, color = Yield)) +
  geom_point(alpha = 0.8, size = 2) +
  scale_color_viridis_c() +
  theme_minimal()

fm4 <- lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)
summary(fm4)


Call:
lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)

Residuals:
      1       2       3       4       5       6       7 
-0.2381 -2.3810  2.1429  1.6667  1.1905 -2.6190  0.2381 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 28.095238   2.491482  11.277 0.000352 ***
Fertilizer   0.038095   0.005832   6.532 0.002838 ** 
Rainfall     0.833333   0.154303   5.401 0.005690 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.315 on 4 degrees of freedom
Multiple R-squared:  0.9814,    Adjusted R-squared:  0.972 
F-statistic: 105.3 on 2 and 4 DF,  p-value: 0.0003472

anova(fm4)

Analysis of Variance Table

Response: Yield
           Df Sum Sq Mean Sq F value    Pr(>F)    
Fertilizer  1 972.32  972.32 181.500 0.0001756 ***
Rainfall    1 156.25  156.25  29.167 0.0056899 ** 
Residuals   4  21.43    5.36                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(lm.beta)
lm.beta(fm4)


Call:
lm(formula = Yield ~ Fertilizer + Rainfall, data = df4)

Standardized Coefficients::
(Intercept)  Fertilizer    Rainfall 
         NA   0.5944301   0.4914732

Polynomial Regression Analysis

Quantifying non-linear dependency of a normal response on quantitative explanatory variable(s)

Example

An experiment was conducted to evaluate the effects of different levels of nitrogen. Three levels of nitrogen: 0, 10 and 20 grams per plot were used in the experiment. Each treatment was replicated twice and data is given below:

Nitrogen	Yield
0	5
0	7
10	15
10	17
20	9
20	11

Nitrogen <- c(0, 0, 10, 10, 20, 20)
Yield    <- c(5, 7, 15, 17,  9, 11)
df5      <- data.frame(Nitrogen, Yield)
df5

  Nitrogen Yield
1        0     5
2        0     7
3       10    15
4       10    17
5       20     9
6       20    11

library(ggplot2)
p5 <- 
  ggplot(data = df5, mapping = aes(x = Nitrogen, y = Yield)) + 
  geom_point() +
  labs(x = "Nitrogen", y = "Yield") +
  theme_bw()
print(p5)

fm5 <- lm(formula = Yield ~ Nitrogen + I(Nitrogen^2), data = df5)
# fm5 <- lm(formula = Yield ~ poly(x = Nitrogen, degree = 2, raw = TRUE), data = df5)
summary(fm5)


Call:
lm(formula = Yield ~ Nitrogen + I(Nitrogen^2), data = df5)

Residuals:
 1  2  3  4  5  6 
-1  1 -1  1 -1  1 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.00000    1.00000   6.000  0.00927 **
Nitrogen       1.80000    0.25495   7.060  0.00584 **
I(Nitrogen^2) -0.08000    0.01225  -6.532  0.00729 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.414 on 3 degrees of freedom
Multiple R-squared:  0.9441,    Adjusted R-squared:  0.9068 
F-statistic: 25.33 on 2 and 3 DF,  p-value: 0.01322

anova(fm5)

Analysis of Variance Table

Response: Yield
              Df Sum Sq Mean Sq F value   Pr(>F)   
Nitrogen       1 16.000  16.000   8.000 0.066276 . 
I(Nitrogen^2)  1 85.333  85.333  42.667 0.007292 **
Residuals      3  6.000   2.000                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Analysis of Variance (ANOVA)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 2

Example

Yield	Variety
5	V1
6	V1
7	V1
15	V2
16	V2
17	V2

Yield   <- c(5, 6, 7, 15, 16, 17)
Variety <- gl(n = 2, k = 3, length = 2*3, labels = c("V1", "V2"))
df6     <- data.frame(Yield, Variety)
df6

  Yield Variety
1     5      V1
2     6      V1
3     7      V1
4    15      V2
5    16      V2
6    17      V2

library(ggplot2)
p6 <- 
  ggplot(data = df6, mapping = aes(x = Variety, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Variety", y = "Yield") +
  theme_bw()
print(p6)

fm6 <- lm(formula = Yield ~ Variety, data = df6)
summary(fm6)


Call:
lm(formula = Yield ~ Variety, data = df6)

Residuals:
         1          2          3          4          5          6 
-1.000e+00 -4.774e-15  1.000e+00 -1.000e+00 -1.110e-16  1.000e+00 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   6.0000     0.5774   10.39 0.000484 ***
VarietyV2    10.0000     0.8165   12.25 0.000255 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1 on 4 degrees of freedom
Multiple R-squared:  0.974, Adjusted R-squared:  0.9675 
F-statistic:   150 on 1 and 4 DF,  p-value: 0.0002552

anova(fm6)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value    Pr(>F)    
Variety    1    150     150     150 0.0002552 ***
Residuals  4      4       1                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example

Yield	Variety
5	V1
7	V1
15	V2
17	V2
17	V3
19	V3

Yield   <- c(5, 7, 15, 17, 17, 19)
Variety <- gl(n = 3, k = 2, length = 3*2, labels = c("V1", "V2", "V3"))
df7     <- data.frame(Yield, Variety)
df7

  Yield Variety
1     5      V1
2     7      V1
3    15      V2
4    17      V2
5    17      V3
6    19      V3

library(ggplot2)
p7 <- 
  ggplot(data = df7, mapping = aes(x = Variety, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Variety", y = "Yield") +
  theme_bw()
print(p7)

fm7 <- lm(formula = Yield ~ Variety, data = df7)
summary(fm7)


Call:
lm(formula = Yield ~ Variety, data = df7)

Residuals:
 1  2  3  4  5  6 
-1  1 -1  1 -1  1 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.000      1.000   6.000  0.00927 **
VarietyV2     10.000      1.414   7.071  0.00582 **
VarietyV3     12.000      1.414   8.485  0.00344 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.414 on 3 degrees of freedom
Multiple R-squared:  0.965, Adjusted R-squared:  0.9416 
F-statistic: 41.33 on 2 and 3 DF,  p-value: 0.006553

anova(fm7)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value   Pr(>F)   
Variety    2 165.33  82.667  41.333 0.006553 **
Residuals  3   6.00   2.000                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Analysis of Covariance (ANCOVA)

Quantifying dependency of a normal response on quantitative explanatory variable(s)

Comparing means of Normal dependent variable for levels of different factor(s)

Figure 3

Example

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Yield    <- c(51, 52, 53, 54, 56, 57, 55, 56, 58, 59, 62, 63)
Fert     <- rep(x = c(80, 90, 100), each = 2)
Variety  <- gl(n = 2, k = 6, length = 2*6, labels = c("V1", "V2"), ordered = FALSE)
df8         <- data.frame(Yield, Fert, Variety)
df8

   Yield Fert Variety
1     51   80      V1
2     52   80      V1
3     53   90      V1
4     54   90      V1
5     56  100      V1
6     57  100      V1
7     55   80      V2
8     56   80      V2
9     58   90      V2
10    59   90      V2
11    62  100      V2
12    63  100      V2

Same intercepts but different slopes

fm8 <- lm(formula = Yield ~ Fert + Variety, data = df8)
summary(fm8)


Call:
lm(formula = Yield ~ Fert + Variety, data = df8)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.8333 -0.8333  0.1667  0.1667  1.1667 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 26.83333    2.54557   10.54 2.30e-06 ***
Fert         0.30000    0.02805   10.69 2.04e-06 ***
VarietyV2    5.00000    0.45812   10.91 1.72e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7935 on 9 degrees of freedom
Multiple R-squared:  0.9629,    Adjusted R-squared:  0.9546 
F-statistic: 116.7 on 2 and 9 DF,  p-value: 3.657e-07

anova(fm8)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value    Pr(>F)    
Fert       1 72.000   72.00  114.35 2.042e-06 ***
Variety    1 75.000   75.00  119.12 1.720e-06 ***
Residuals  9  5.667    0.63                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(ggplot2)
p8 <- 
  ggplot(data = df8, mapping = aes(x = Fert, y = Yield, color = Variety)) + 
  geom_point() +
  geom_line(mapping = aes(y = predict(fm8))) +
  labs(x = "Fert", y = "Yield") +
  theme_bw()
print(p8)

Different intercepts and different slopes

fm9 <- lm(formula = Yield ~ Fert * Variety, data = df8)
summary(fm9)


Call:
lm(formula = Yield ~ Fert * Variety, data = df8)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.83333 -0.33333 -0.08333  0.66667  0.66667 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)    31.33333    3.05903  10.243 7.09e-06 ***
Fert            0.25000    0.03385   7.385 7.73e-05 ***
VarietyV2      -4.00000    4.32612  -0.925   0.3822    
Fert:VarietyV2  0.10000    0.04787   2.089   0.0701 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.677 on 8 degrees of freedom
Multiple R-squared:  0.976, Adjusted R-squared:  0.967 
F-statistic: 108.4 on 3 and 8 DF,  p-value: 8.11e-07

anova(fm9)

Analysis of Variance Table

Response: Yield
             Df Sum Sq Mean Sq  F value    Pr(>F)    
Fert          1 72.000  72.000 157.0909 1.538e-06 ***
Variety       1 75.000  75.000 163.6364 1.315e-06 ***
Fert:Variety  1  2.000   2.000   4.3636   0.07013 .  
Residuals     8  3.667   0.458                       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(ggplot2)
p9 <- 
  ggplot(data = df8, mapping = aes(x = Fert, y = Yield, color = Variety)) + 
  geom_point() +
  geom_line(mapping = aes(y = predict(fm9))) +
  labs(x = "Fert", y = "Yield") +
  theme_bw()
print(p9)

Correlation Analysis

Linear Relationship between Quantitative Variables

Simple Correlation Analysis

Linear Relationship between Two Quantitative Variables

Example

Sparrow Wing length (cm)	Sparrow Tail length (cm)
10.4	7.4
10.8	7.6
11.1	7.9
10.2	7.2
10.3	7.4
10.2	7.1
10.7	7.4
10.5	7.2
10.8	7.8
11.2	7.7
10.6	7.8
11.4	8.3

WingLength  <- c(10.4, 10.8, 11.1, 10.2, 10.3, 10.2, 10.7, 10.5, 10.8, 11.2, 10.6, 11.4)
TailLength  <- c(7.4, 7.6, 7.9, 7.2, 7.4, 7.1, 7.4, 7.2, 7.8, 7.7, 7.8, 8.3)
df10        <- data.frame(WingLength, TailLength)
df10

   WingLength TailLength
1        10.4        7.4
2        10.8        7.6
3        11.1        7.9
4        10.2        7.2
5        10.3        7.4
6        10.2        7.1
7        10.7        7.4
8        10.5        7.2
9        10.8        7.8
10       11.2        7.7
11       10.6        7.8
12       11.4        8.3

library(ggplot2)
p10 <- 
  ggplot(data = df10, mapping = aes(x= WingLength, y = TailLength)) + 
  geom_point() +
  labs(x = "Sparrow Wing Length (cm)", y = "Sparrow Tail Length (cm)") +
  theme_bw()
print(p10)

cor(df10$WingLength, df10$TailLength)

[1] 0.8703546

cor.test(df10$WingLength, df10$TailLength)


    Pearson's product-moment correlation

data:  df10$WingLength and df10$TailLength
t = 5.5893, df = 10, p-value = 0.0002311
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.5923111 0.9631599
sample estimates:
      cor 
0.8703546

Partial Correlation Analysis

Linear Relationship between Quantitative Variables while holding/keeping all other constants

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

LeafArea    <- c(72, 174, 116, 78, 134, 95, 113, 98, 148, 42)
LeafMoist   <- c(75, 81, 83, 79, 81, 80, 81, 74, 78, 58)
TotShLength <- c(307, 529, 632, 527, 442, 525, 481, 710, 422, 345)
df11        <- data.frame(LeafArea, LeafMoist, TotShLength)
df11

   LeafArea LeafMoist TotShLength
1        72        75         307
2       174        81         529
3       116        83         632
4        78        79         527
5       134        81         442
6        95        80         525
7       113        81         481
8        98        74         710
9       148        78         422
10       42        58         345

library(ppcor)
pcor(df11)

$estimate
                LeafArea LeafMoist  TotShLength
LeafArea     1.000000000 0.6509096 -0.007012236
LeafMoist    0.650909618 1.0000000  0.324334659
TotShLength -0.007012236 0.3243347  1.000000000

$p.value
              LeafArea  LeafMoist TotShLength
LeafArea    0.00000000 0.05760462   0.9857154
LeafMoist   0.05760462 0.00000000   0.3944839
TotShLength 0.98571537 0.39448385   0.0000000

$statistic
               LeafArea LeafMoist TotShLength
LeafArea     0.00000000  2.268502 -0.01855309
LeafMoist    2.26850175  0.000000  0.90714704
TotShLength -0.01855309  0.907147  0.00000000

$n
[1] 10

$gp
[1] 1

$method
[1] "pearson"

Multiple Correlation Analysis

Linear Relationship between a Quantitative Variable and set of other Quantitative Variables

Example

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

fm12 <- lm(formula = TotShLength ~ LeafArea + LeafMoist, data = df11)
sqrt(summary(fm12)$r.squared)

[1] 0.421277

Completely Randomized Design (CRD)

Used when experimental material is homogeneous

Example

The following table shows some of the results of an experiment on the effects of applications of sulphur in reducing scab disease of potatoes. The object in applying sulphur is to increase the acidity of the soil, since scab does not thrive in very acid soil. In addition to untreated plots which serve as a control, 3 amounts of dressing were compared—300, 600, and 900 lb. per acre. Both a fall and a spring application of each amount was tested, so that in all there were seven distinct treatments. The sulphur was spread by hand on the surface of the soil, and then diced into a depth of about 4 inches. The quantity to be analyzed is the “scab index”. That is roughly speaking, the percentage of the surface area of the potato that is infected with scab. It is obtained by examining 100 potatoes at random from each plot, grading each potato on a scale from 0 to 100% infected, and taking the average.

ScabIndex <-
      c(
        12, 10,  24, 29, 
        30, 18,  32, 26, 
         9,  9,  16,  4,
        30,  7,  21,  9,
        16, 10,  18, 18,
        18, 24,  12, 19,
        10,  4,   4,  5,
        17,  7,  16, 17
      )
Trt  <- factor(
          x = rep(x = 1:7, times = c(8, 4, 4, 4, 4, 4, 4))
        , labels = c("O", "F3", "S3", "F6", "S6", "F9", "S9")
        )

df13 <- data.frame(Trt, ScabIndex)
df13

   Trt ScabIndex
1    O        12
2    O        10
3    O        24
4    O        29
5    O        30
6    O        18
7    O        32
8    O        26
9   F3         9
10  F3         9
11  F3        16
12  F3         4
13  S3        30
14  S3         7
15  S3        21
16  S3         9
17  F6        16
18  F6        10
19  F6        18
20  F6        18
21  S6        18
22  S6        24
23  S6        12
24  S6        19
25  F9        10
26  F9         4
27  F9         4
28  F9         5
29  S9        17
30  S9         7
31  S9        16
32  S9        17

library(ggplot2)
p13 <- 
  ggplot(data = df13, mapping = aes(x = Trt, y = ScabIndex)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Scab Index") +
  theme_bw()
print(p13)

fm13 <- lm(formula = ScabIndex ~ Trt, data = df13)
summary(fm13)


Call:
lm(formula = ScabIndex ~ Trt, data = df13)

Residuals:
    Min      1Q  Median      3Q     Max 
-12.625  -4.844   0.625   3.594  13.250 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   22.625      2.369   9.549 8.08e-10 ***
TrtF3        -13.125      4.104  -3.198 0.003734 ** 
TrtS3         -5.875      4.104  -1.432 0.164666    
TrtF6         -7.125      4.104  -1.736 0.094858 .  
TrtS6         -4.375      4.104  -1.066 0.296601    
TrtF9        -16.875      4.104  -4.112 0.000372 ***
TrtS9         -8.375      4.104  -2.041 0.051977 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6.702 on 25 degrees of freedom
Multiple R-squared:  0.4641,    Adjusted R-squared:  0.3355 
F-statistic: 3.608 on 6 and 25 DF,  p-value: 0.01026

anova(fm13)

Analysis of Variance Table

Response: ScabIndex
          Df  Sum Sq Mean Sq F value  Pr(>F)  
Trt        6  972.34 162.057  3.6081 0.01026 *
Residuals 25 1122.88  44.915                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Randomized Complete Block Design (RCBD)

Used when experimental material is heterogenous in one direction

Example

Yield : Yield of barley, SoilType : Soil Type, and Trt : 5 sources and a control

Yield <- c(
             32.1,  35.6,   41.9,   35.4
          ,  30.1,  31.5,   37.1,   30.8
          ,  25.4,  27.4,   33.8,   31.1
          ,  24.1,  33.0,   35.6,   31.4
          ,  26.1,  31.0,   33.8,   31.9
          ,  23.2,  24.8,   26.7,   26.7
          )
SoilType <- gl(n = 4, k = 6, length = 4*6
               , labels = c("I", "II", "III", "IV"), ordered = FALSE)
Trt      <- gl(n = 6, k = 1, length = 4*6
               , labels = c("(NH4)2SO4", "NH4NO3", "CO(NH2)2", "Ca(NO3)2", "NaNO3", "Control"), ordered = FALSE)

df14 <- data.frame(Yield, SoilType, Trt)
df14

   Yield SoilType       Trt
1   32.1        I (NH4)2SO4
2   35.6        I    NH4NO3
3   41.9        I  CO(NH2)2
4   35.4        I  Ca(NO3)2
5   30.1        I     NaNO3
6   31.5        I   Control
7   37.1       II (NH4)2SO4
8   30.8       II    NH4NO3
9   25.4       II  CO(NH2)2
10  27.4       II  Ca(NO3)2
11  33.8       II     NaNO3
12  31.1       II   Control
13  24.1      III (NH4)2SO4
14  33.0      III    NH4NO3
15  35.6      III  CO(NH2)2
16  31.4      III  Ca(NO3)2
17  26.1      III     NaNO3
18  31.0      III   Control
19  33.8       IV (NH4)2SO4
20  31.9       IV    NH4NO3
21  23.2       IV  CO(NH2)2
22  24.8       IV  Ca(NO3)2
23  26.7       IV     NaNO3
24  26.7       IV   Control

library(ggplot2)
p14 <- 
  ggplot(data = df14, mapping = aes(x = Trt, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Yield") +
  theme_bw()
print(p14)

fm14 <- lm(formula = Yield ~ SoilType + Trt, data = df14)
summary(fm14)


Call:
lm(formula = Yield ~ SoilType + Trt, data = df14)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.0208 -2.4229  0.0792  2.1354  6.7958 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   35.354      2.826  12.511 2.44e-09 ***
SoilTypeII    -3.500      2.664  -1.314   0.2087    
SoilTypeIII   -4.233      2.664  -1.589   0.1329    
SoilTypeIV    -6.583      2.664  -2.471   0.0259 *  
TrtNH4NO3      1.050      3.263   0.322   0.7521    
TrtCO(NH2)2   -0.250      3.263  -0.077   0.9399    
TrtCa(NO3)2   -2.025      3.263  -0.621   0.5442    
TrtNaNO3      -2.600      3.263  -0.797   0.4380    
TrtControl    -1.700      3.263  -0.521   0.6100    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.615 on 15 degrees of freedom
Multiple R-squared:  0.3512,    Adjusted R-squared:  0.005215 
F-statistic: 1.015 on 8 and 15 DF,  p-value: 0.4653

anova(fm14)

Analysis of Variance Table

Response: Yield
          Df Sum Sq Mean Sq F value Pr(>F)
SoilType   3 133.62  44.539  2.0915 0.1443
Trt        5  39.31   7.862  0.3692 0.8618
Residuals 15 319.43  21.295

Latin Square Design

Used when experimental material is heterogenous in two perpendicular directions

Example

The following table shows the field layout and yield of a 5×5 Latin square experiment on the effect of spacing on yield of millet plants. Five levels of spacing were used. The data on yield (grams/plot) was recorded and is given below:

Yield <- c(
           257, 230, 279, 287, 202,
           245, 283, 245, 280, 260,
           182, 252, 280, 246, 250,
           203, 204, 227, 193, 259,
           231, 271, 266, 334, 338
           )
Row <- factor(x = rep(x = 1:5, each = 5))
Col <- factor(x = rep(x = 1:5, times = 5))
Trt <- c(
          "B", "E", "A", "C", "D"
        , "D", "A", "E", "B", "C"
        , "E", "B", "C", "D", "A"
        , "A", "C", "D", "E", "B"
        , "C", "D", "B", "A", "E"
         )

df15 <- data.frame(Yield, Row, Col, Trt)
df15

   Yield Row Col Trt
1    257   1   1   B
2    230   1   2   E
3    279   1   3   A
4    287   1   4   C
5    202   1   5   D
6    245   2   1   D
7    283   2   2   A
8    245   2   3   E
9    280   2   4   B
10   260   2   5   C
11   182   3   1   E
12   252   3   2   B
13   280   3   3   C
14   246   3   4   D
15   250   3   5   A
16   203   4   1   A
17   204   4   2   C
18   227   4   3   D
19   193   4   4   E
20   259   4   5   B
21   231   5   1   C
22   271   5   2   D
23   266   5   3   B
24   334   5   4   A
25   338   5   5   E

library(ggplot2)
p15 <- 
  ggplot(data = df15, mapping = aes(x = Trt, y = Yield)) + 
  geom_point() +
  geom_boxplot() +
  labs(x = "Treatment", y = "Yield") +
  theme_bw()
print(p15)

fm15 <- lm(formula = Yield ~ Row + Col + Trt, data = df15)
summary(fm15)


Call:
lm(formula = Yield ~ Row + Col + Trt, data = df15)

Residuals:
   Min     1Q Median     3Q    Max 
-44.68 -12.48   1.12  16.52  54.92 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   240.08      23.43  10.247 2.75e-07 ***
Row2           11.60      20.55   0.565   0.5828    
Row3           -9.00      20.55  -0.438   0.6692    
Row4          -33.80      20.55  -1.645   0.1259    
Row5           37.00      20.55   1.801   0.0969 .  
Col2           24.40      20.55   1.187   0.2580    
Col3           35.80      20.55   1.742   0.1070    
Col4           44.40      20.55   2.161   0.0516 .  
Col5           38.20      20.55   1.859   0.0877 .  
TrtB           -7.00      20.55  -0.341   0.7393    
TrtC          -17.40      20.55  -0.847   0.4137    
TrtD          -31.60      20.55  -1.538   0.1500    
TrtE          -32.20      20.55  -1.567   0.1431    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 32.49 on 12 degrees of freedom
Multiple R-squared:  0.6536,    Adjusted R-squared:  0.3073 
F-statistic: 1.887 on 12 and 12 DF,  p-value: 0.1426

anova(fm15)

Analysis of Variance Table

Response: Yield
          Df  Sum Sq Mean Sq F value  Pr(>F)  
Row        4 13601.4  3400.3  3.2212 0.05164 .
Col        4  6146.2  1536.5  1.4556 0.27581  
Trt        4  4156.6  1039.1  0.9844 0.45229  
Residuals 12 12667.3  1055.6                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {An Introduction to {R} for {Agricultural} {Research}},
  date = {2018-05-09},
  url = {https://myaseen208.com/blogs/2018-05-09_ARRI/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “An Introduction to R for Agricultural Research.” May 9. https://myaseen208.com/blogs/2018-05-09_ARRI/.

ppcSpatial: Spatial Analysis of Pakistan Population Census

Muhammad Yaseen — Mon, 30 Apr 2018 04:00:00 GMT

A video showing some functionality of the package is

ppcSpatial package video thumbnail

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {ppcSpatial: {Spatial} {Analysis} of {Pakistan} {Population}
    {Census}},
  date = {2018-04-30},
  url = {https://myaseen208.com/blogs/2018-04-30_ppcSpatial/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “ppcSpatial: Spatial Analysis of Pakistan Population Census.” April 30. https://myaseen208.com/blogs/2018-04-30_ppcSpatial/.

VetResearchLMM v1.0.0 on CRAN

Muhammad Yaseen — Mon, 23 Apr 2018 04:00:00 GMT

The latest version of the VetResearchLMM package for R is now on CRAN. This package provides R Codes and Datasets for Duchateau, L. and Janssen, P. and Rowlands, G. J. (1998). Linear Mixed Models. An Introduction with applications in Veterinary Research. International Livestock Research Institute. The latest version of the package is prepare for new version of lmerTest.

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {VetResearchLMM V1.0.0 on {CRAN}},
  date = {2018-04-23},
  url = {https://myaseen208.com/blogs/2018-04-23_VetResearchLMM/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “VetResearchLMM V1.0.0 on CRAN.” April 23. https://myaseen208.com/blogs/2018-04-23_VetResearchLMM/.

Diallel Analysis with R

Muhammad Yaseen — Sat, 21 Apr 2018 04:00:00 GMT

The latest versin 0.1.1 of DiallelAnalysisR is aviable on CRAN. It performs Diallel Analysis with R using Griffing’s and Hayman’s approaches. Four different methods (1: Method-I (Parents + F1’s + reciprocals); 2: Method-II (Parents and one set of F1’s); 3: Method-III (One set of F1’s and reciprocals); 4: Method-IV (One set of F1’s only)) and two methods (1: Fixed Effects Model; 2: Random Effects Model) can be applied using Griffing’s approach.

DiallelAnalysisR has functions to perform Diallel Analysis with R using Griffing’s and Hayman’s approaches. Four different Methods:

Method-I (Parents + F1’s + reciprocals)
Method-II (Parents and one set of F1’s)
Method-III (One set of F1’s and reciprocals)
Method-IV (One set of F1’s only))

and two Models:

Fixed Effects Model
Random Effects Model

can be applied using Griffing’s approach.

Installation

Stable Version

Stable version of the package avialabe on CRAN can be installed using:

library(DiallelAnalysisR)

Development Version

Development version of the package avialabe on Github can be installed using:

Use the project package environment or install instructions from the package repository.

License

This package is free and open source software, licensed under GPL.

Citation

BibTeX citation:

@online{yaseen2018,
  author = {Yaseen, Muhammad},
  title = {Diallel {Analysis} with {R}},
  date = {2018-04-21},
  url = {https://myaseen208.com/blogs/2018-04-21_DiallelAnalysisR/},
  langid = {en}
}

For attribution, please cite this work as:

Yaseen, Muhammad. 2018. “Diallel Analysis with R.” April 21. https://myaseen208.com/blogs/2018-04-21_DiallelAnalysisR/.

Data Analysis with R: Introduction of R Language

Muhammad Arfan Dilber — Sun, 25 Mar 2018 04:00:00 GMT

The Workspace

The workspace is your current R working environment and includes any user-defined objects (vectors, matrices, data frames, lists, functions). At the end of an R session, the user can save an image of the current workspace that is automatically reloaded the next time R is started.

getwd()
# Use here::here() for project-relative paths.
save.image() # To save workspace image.
x<-c(1:4)
x
save(x, file = "mydat.RData")
rm(list = ls())
load("mydat.RData")
x

Data types

x<-2.56
class(x)

[1] "numeric"

[1] 2.56

y<-as.integer(x)
class(y)

[1] "integer"

[1] 2

z<-as.integer(2.56)
z

[1] 2

is.integer(z)

[1] TRUE

as.integer(is.integer(z))

[1] 1

as.integer(!is.integer(z))

[1] 0

as.integer(is.integer(x))

[1] 0

x<-round(x,digits = 1)
x

[1] 2.6

y<-"Male"
class(y)

[1] "character"

Vectors

A vector is a sequence of data elements of the same basic type. A combine function c() is used to form the vector. Here are examples of each type of vector:

x<-c(1,2,3,4)
x

[1] 1 2 3 4

y<-c("a","b","c","d")
y

[1] "a" "b" "c" "d"

c(x,y)

[1] "1" "2" "3" "4" "a" "b" "c" "d"

length(y)

[1] 4

y[c(2,4)]

[1] "b" "d"

y[c(2:4)]

[1] "b" "c" "d"

y[-3]

[1] "a" "b" "d"

names(x)<-c("first","second","third","fourth")
x

 first second  third fourth 
     1      2      3      4

x[c("second","fourth")]

second fourth 
     2      4

x[2]<-5
x

 first second  third fourth 
     1      5      3      4

x<-x*2
x

 first second  third fourth 
     2     10      6      8

seq(from = 1, to = 10, by = 2)

[1] 1 3 5 7 9

rep(5, time = 10)

 [1] 5 5 5 5 5 5 5 5 5 5

rep(c("Male","Female"),time = 10)

 [1] "Male"   "Female" "Male"   "Female" "Male"   "Female" "Male"   "Female"
 [9] "Male"   "Female" "Male"   "Female" "Male"   "Female" "Male"   "Female"
[17] "Male"   "Female" "Male"   "Female"

rep(c("Male","Female"), each = 10)

 [1] "Male"   "Male"   "Male"   "Male"   "Male"   "Male"   "Male"   "Male"  
 [9] "Male"   "Male"   "Female" "Female" "Female" "Female" "Female" "Female"
[17] "Female" "Female" "Female" "Female"

rep(seq(from = 2, to = 9, by = .25), time = 5)

  [1] 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50
 [16] 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 2.00
 [31] 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75
 [46] 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 2.00 2.25
 [61] 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00
 [76] 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 2.00 2.25 2.50
 [91] 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25
[106] 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 2.00 2.25 2.50 2.75
[121] 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50
[136] 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00

Matrices

A matrix is a collection of data elements arranged in a two dimensional rectangular layout. Matrices are created with the matrix function.

A <- matrix(
  c(1,2,3,4),
  nrow = 2,
  ncol = 2,
  byrow = TRUE)
B <- matrix(
  c(1,2,3,4,5,6,7,8,9),
  nrow = 3,
  ncol = 3,
  byrow = FALSE)
B

     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9

     [,1] [,2]
[1,]    1    2
[2,]    3    4

A[2,]

[1] 3 4

A[,2]

[1] 2 4

A[2,2]

[1] 4

t(A)

     [,1] [,2]
[1,]    1    3
[2,]    2    4

det(A)

[1] -2

diag(A)

[1] 1 4

B<-A + A
B

     [,1] [,2]
[1,]    2    4
[2,]    6    8

Asquared<- A %*% A
Asquared

     [,1] [,2]
[1,]    7   10
[2,]   15   22

Ainv<- solve(A)
Ainv

     [,1] [,2]
[1,] -2.0  1.0
[2,]  1.5 -0.5

Aev<- eigen(A)
Aev

eigen() decomposition
$values
[1]  5.3722813 -0.3722813

$vectors
           [,1]       [,2]
[1,] -0.4159736 -0.8245648
[2,] -0.9093767  0.5657675

Aev$values

[1]  5.3722813 -0.3722813

Aev$vectors

           [,1]       [,2]
[1,] -0.4159736 -0.8245648
[2,] -0.9093767  0.5657675

Lists

Lists are the most complex of the R data types. Basically, a list is an ordered collection of objects (components). A list allows you to gather a variety of (possibly unrelated) objects under one name. For example, a list may contain a combination of vectors, matrices, data frames, and even other lists. A list is created with the list() function.

g<-"My First List"
h<-c(25,26,18,39)
j<-matrix(1:10, nrow = 5)
k<-c("one", "two", "three")
mylist<-list(title = g, ages = h, j,k)
mylist

$title
[1] "My First List"

$ages
[1] 25 26 18 39

[[3]]
     [,1] [,2]
[1,]    1    6
[2,]    2    7
[3,]    3    8
[4,]    4    9
[5,]    5   10

[[4]]
[1] "one"   "two"   "three"

Data Frames

A Data Frame is used for storing data tables. It is a list of vectors of equal length. Different columns can contain different type of data (numeric,character, etc.). A data frame is created with the data.frame() function:

df<-data.frame(x,y)
df

        x y
first   2 a
second 10 b
third   6 c
fourth  8 d

nrow(df)

[1] 4

ncol(df)

[1] 2

df1<-df[df$y %in% c("a","d"),]
df1

       x y
first  2 a
fourth 8 d

df1<-df[df[,2] %in% c("b","c"),]
df1

        x y
second 10 b
third   6 c

library(PakPC2017)
data(PakPC2017Tehsil)
head(PakPC2017Tehsil)

# A tibble: 6 × 6
  Province Division   District   Tehsil            Pop1998 Pop2017
                                    
1 Punjab   Bahawalpur Bahawalpur Hasilpur           317513  456006
2 Punjab   Bahawalpur Bahawalpur Khairpur Tamewali  183903  262628
3 Punjab   Bahawalpur Bahawalpur Yazman             168950  262175
4 Punjab   Bahawalpur Bahawalpur Ahmadpur East      718297 1078683
5 Punjab   Bahawalpur Bahawalpur Bahawalpur City    419542  681696
6 Punjab   Bahawalpur Bahawalpur Bahawalpur Saddar  387038  574950

Reading/writing data from/to files (Import/Export Data)

library(readxl)
spi <- read_excel(here::here("blogs", "2018-03-25_Maldives_R", "data", "spi.xls"))
DT::datatable(spi,fillContainer = TRUE, editable = TRUE)
write.csv(PakPC2017Tehsil,"Population.csv")

Connecting to the database

library(dplyr)
con <- DBI::dbConnect(odbc::odbc(),
                      dsn = "mydata")
table13<-tbl(con, "RPI")
table13
#tehsil<-PakPC2017::PakPC2017Tehsil
#copy_to(con,tehsil,"tehsil",
#        temporary = FALSE
#      )
DBI::dbDisconnect(con)

Numerical or Graphical Summaries of data.

library(PakPC2017)
summary(PakPC2017Tehsil)

      Province        Division        District         Tehsil   
 Length   :543   Length   :543   Length   :543   Length   :543  
 N.unique :  5   N.unique : 29   N.unique :129   N.unique :538  
 N.blank  :  0   N.blank  :  0   N.blank  :  0   N.blank  :  0  
 Min.nchar:  4   Min.nchar:  4   Min.nchar:  4   Min.nchar:  3  
 Max.nchar: 18   Max.nchar: 19   Max.nchar: 19   Max.nchar: 28  
                                                                
                                                                
    Pop1998           Pop2017       
 Min.   :   3024   Min.   :   2665  
 1st Qu.:  57502   1st Qu.:  89801  
 Median : 157142   Median : 254647  
 Mean   : 240553   Mean   : 377084  
 3rd Qu.: 298074   3rd Qu.: 447854  
 Max.   :2219399   Max.   :4269079  
                   NAs    :1

pairs(PakPC2017Tehsil[,5:6])

cor(PakPC2017Tehsil[,5:6], use = "complete")

          Pop1998   Pop2017
Pop1998 1.0000000 0.9788354
Pop2017 0.9788354 1.0000000

popLm<-lm(formula = Pop2017 ~ Pop1998, data = PakPC2017Tehsil)
popLm


Call:
lm(formula = Pop2017 ~ Pop1998, data = PakPC2017Tehsil)

Coefficients:
(Intercept)      Pop1998  
  -3445.895        1.579

head(resid(popLm),10)

         1          2          3          4          5          6          7 
-42044.030 -24391.708  -1227.192 -52385.341  22496.260 -32911.295 -36767.021 
         8          9         10 
-92325.684 -73938.014  10353.258

head(fitted(popLm),10)

        1         2         3         4         5         6         7         8 
 498050.0  287019.7  263402.2 1131068.3  659199.7  607861.3  851910.0  783546.7 
        9        10 
 599536.0  143090.7

summary(popLm)


Call:
lm(formula = Pop2017 ~ Pop1998, data = PakPC2017Tehsil)

Residuals:
    Min      1Q  Median      3Q     Max 
-299426  -29128    -355   18117 1071215 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -3.446e+03  5.391e+03  -0.639    0.523    
Pop1998      1.579e+00  1.421e-02 111.147   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 96960 on 540 degrees of freedom
  (1 observation deleted due to missingness)
Multiple R-squared:  0.9581,    Adjusted R-squared:  0.958 
F-statistic: 1.235e+04 on 1 and 540 DF,  p-value: < 2.2e-16

par(mfrow = c(2,2))
plot(popLm)

Summarize Data

library(PakPC2017)
library(dplyr)

Bar charts and Dot charts

library(PakPC2017)
data("PakPC2017Tehsil")
head(PakPC2017Tehsil)

# A tibble: 6 × 6
  Province Division   District   Tehsil            Pop1998 Pop2017
                                    
1 Punjab   Bahawalpur Bahawalpur Hasilpur           317513  456006
2 Punjab   Bahawalpur Bahawalpur Khairpur Tamewali  183903  262628
3 Punjab   Bahawalpur Bahawalpur Yazman             168950  262175
4 Punjab   Bahawalpur Bahawalpur Ahmadpur East      718297 1078683
5 Punjab   Bahawalpur Bahawalpur Bahawalpur City    419542  681696
6 Punjab   Bahawalpur Bahawalpur Bahawalpur Saddar  387038  574950

province<-tapply(PakPC2017Tehsil$Pop2017,PakPC2017Tehsil$Province,sum,na.rm=TRUE)
par(mfrow = c(3,2))
dotchart(province)
barplot(province)
boxplot(Pop2017~Province,data = PakPC2017Tehsil)
pie(province, col = rainbow(length(province)))
plot(province, type = "o")
pop<-PakPC2017Tehsil$Pop2017
hist(pop)

hist(pop, breaks = seq(min(pop,na.rm = TRUE),max(pop,na.rm = TRUE), length.out = 31))

ggplot2 package

The mpg Data Frame

A data frame is a rectangular collection of variables (in the columns) and observations (in the rows). mpg contains observations collected by the US Environment Protection Agency on 38 models of cars; To learn more about mpg, open its help page by running ?mpg.

library(ggplot2)
ggplot(data = mpg)+
  geom_point(mapping = aes(x = displ, y = hwy))

ggplot(data = mpg)+
  geom_point(mapping = aes(x = displ, y = hwy, color = class))

ggplot(data = mpg)+
  geom_point(mapping = aes(x = displ, y = hwy, size = class))

ggplot(data = mpg)+
  geom_point(mapping = aes(x = displ, y = hwy, shape = class))

ggplot(data = mpg)+
  geom_point(mapping = aes(x = displ, y = hwy), color = "blue")

ggplot(data = mpg)+
  geom_smooth(mapping = aes(x = displ, y = hwy))

ggplot(data = mpg)+
  geom_smooth(mapping = aes(x = displ, y = hwy, linetype = drv))

ggplot(data = mpg)+
  geom_point(mapping = aes(x = displ, y = hwy, color = drv))+
  geom_smooth(mapping = aes(x = displ, y = hwy, linetype = drv))

ggplot(data = mpg, mapping = aes(x = displ, y = hwy, color = drv))+
  geom_point()+
  geom_smooth()

ggplot(data = mpg, mapping = aes(x = displ, y = hwy))+
  geom_point(mapping = aes(color = drv))+
  geom_smooth(mapping = aes(linetype = drv))

Data Transformation

To explore the basic data manipulation we’ll use PakPC2017::PakPC2017Tehsil. This data frame contains the popualtion of all 543 tehsils of Pakistan. The data comes from Pakistan Bureau of Statistics, Population Census 2017, and is documented in ?PakPc2017Tehsil.

library(PakPC2017)
library(dplyr)
tehsil<-as_tibble(PakPC2017Tehsil)
tehsil

# A tibble: 543 × 6
   Province Division   District     Tehsil            Pop1998 Pop2017
                                       
 1 Punjab   Bahawalpur Bahawalpur   Hasilpur           317513  456006
 2 Punjab   Bahawalpur Bahawalpur   Khairpur Tamewali  183903  262628
 3 Punjab   Bahawalpur Bahawalpur   Yazman             168950  262175
 4 Punjab   Bahawalpur Bahawalpur   Ahmadpur East      718297 1078683
 5 Punjab   Bahawalpur Bahawalpur   Bahawalpur City    419542  681696
 6 Punjab   Bahawalpur Bahawalpur   Bahawalpur Saddar  387038  574950
 7 Punjab   Bahawalpur Bahawalnagar Bahawalnagar       541553  815143
 8 Punjab   Bahawalpur Bahawalnagar Chishtian          498270  691221
 9 Punjab   Bahawalpur Bahawalnagar Haroonabad         381767  525598
10 Punjab   Bahawalpur Bahawalnagar Fort Abbas          92777  153444
# ℹ 533 more rows

Filter Rows with filter()

filter() allows you to subset observations based on their vaues. The first argument is the name of the data frame. The second and subsequent arguments are the expressions that filter the data frame. For example, we can select all tehsils of Division Faisalabad:

filter(tehsil, Division == "Faisalabad")

# A tibble: 17 × 6
   Province Division   District       Tehsil            Pop1998 Pop2017
                                         
 1 Punjab   Faisalabad Faisalabad     Faisalabad City   2140346 3237961
 2 Punjab   Faisalabad Faisalabad     Faisalabad Saddar  924110 1465411
 3 Punjab   Faisalabad Faisalabad     Chak Jhumra        253806  332461
 4 Punjab   Faisalabad Faisalabad     Sammundri          508637  643068
 5 Punjab   Faisalabad Faisalabad     Tandlianwala       540802  702733
 6 Punjab   Faisalabad Faisalabad     Jaranwala         1061846 1492276
 7 Punjab   Faisalabad Jhang          Jhang              999130 1465472
 8 Punjab   Faisalabad Jhang          18 Hazari          200036  295801
 9 Punjab   Faisalabad Jhang          Shorkot            373790  548626
10 Punjab   Faisalabad Jhang          Ahmadpur Sial      296465  433517
11 Punjab   Faisalabad Chiniot        Chiniot            382876  556147
12 Punjab   Faisalabad Chiniot        Lalian             309494  439323
13 Punjab   Faisalabad Chiniot        Bhawana            272754  374270
14 Punjab   Faisalabad Toba Tek Singh Toba Tek Singh     554231  739826
15 Punjab   Faisalabad Toba Tek Singh Gojra              495096  656007
16 Punjab   Faisalabad Toba Tek Singh Kamalia            283674  371851
17 Punjab   Faisalabad Toba Tek Singh Pir Mahal          288592  422331

newD<-filter(tehsil, grepl("F", tehsil$Division))
newd2<-filter(tehsil, grepl("abad", tehsil$Division))

If you want to save the result, you’ll need to use the assignment operator, <-: fsdDivision<-filter(tehsil, Division == “Faisalabad”)

Comparisons

To use filtering effectively, you have to know how to select the observations that you want using the comparison operators. R provides the standard suite: >, >=, <, <=, != (not equal), and == (equal).

filter(tehsil, Pop2017 >= 1000000)

# A tibble: 35 × 6
   Province Division   District       Tehsil            Pop1998 Pop2017
                                         
 1 Punjab   Bahawalpur Bahawalpur     Ahmadpur East      718297 1078683
 2 Punjab   Bahawalpur Rahim Yar Khan Rahim Yar Khan     986614 1530330
 3 Punjab   Bahawalpur Rahim Yar Khan Sadiqabad          771511 1264752
 4 Punjab   Faisalabad Faisalabad     Faisalabad City   2140346 3237961
 5 Punjab   Faisalabad Faisalabad     Faisalabad Saddar  924110 1465411
 6 Punjab   Faisalabad Faisalabad     Jaranwala         1061846 1492276
 7 Punjab   Faisalabad Jhang          Jhang              999130 1465472
 8 Punjab   Sahiwal    Sahiwal        Sahiwal           1056132 1491553
 9 Punjab   Sahiwal    Sahiwal        Chichawatni        787062 1026007
10 Punjab   Sahiwal    Okara          Okara              862364 1205655
# ℹ 25 more rows

Logical Operators

Multiple arguments to filter() are combined with “and”: every experessio must be true in order for a row to be included in the output. For other types of combinations, you’ll need to use Boolean operators yourself: & is “and”, | is “or”, and ! is “not”.

filter(tehsil, Pop2017 >= 2000000 | Pop1998 >= 1000000)

# A tibble: 15 × 6
   Province           Division   District    Tehsil           Pop1998 Pop2017
                                               
 1 Punjab             Faisalabad Faisalabad  Faisalabad City  2140346 3237961
 2 Punjab             Faisalabad Faisalabad  Jaranwala        1061846 1492276
 3 Punjab             Sahiwal    Sahiwal     Sahiwal          1056132 1491553
 4 Punjab             Sahiwal    Okara       Depalpur         1030836 1374912
 5 Punjab             Multan     Multan      Multan City      1381478 2258570
 6 Punjab             Sargodha   Sargodha    Sargodha         1081459 1537866
 7 Punjab             Lahore     Lahore      Lahore City      2219399 3655774
 8 Punjab             Lahore     Lahore      Model Town       1409228 2698235
 9 Punjab             Lahore     Lahore      Shalimar         1434823 2280308
10 Punjab             Lahore     Sheikhupura Sheikhupura      1049264 1555424
11 Punjab             Gujranwala Gujranwala  Gujranwal Saddar 1812137 2807054
12 Punjab             Gujranwala Gujrat      Gujrat           1093088 1497865
13 Punjab             Gujranwala Sialkot     Sialkot          1207744 1794658
14 Punjab             Rawalpindi Rawalpindi  Rawalpindi       1927612 3258547
15 Khyber Pakhtunkhwa Peshawar   Peshawar    Peshawar         2026851 4269079

filter(tehsil, Pop2017 >= 1000000 & Pop1998 >= 1000000)

# A tibble: 15 × 6
   Province           Division   District    Tehsil           Pop1998 Pop2017
                                               
 1 Punjab             Faisalabad Faisalabad  Faisalabad City  2140346 3237961
 2 Punjab             Faisalabad Faisalabad  Jaranwala        1061846 1492276
 3 Punjab             Sahiwal    Sahiwal     Sahiwal          1056132 1491553
 4 Punjab             Sahiwal    Okara       Depalpur         1030836 1374912
 5 Punjab             Multan     Multan      Multan City      1381478 2258570
 6 Punjab             Sargodha   Sargodha    Sargodha         1081459 1537866
 7 Punjab             Lahore     Lahore      Lahore City      2219399 3655774
 8 Punjab             Lahore     Lahore      Model Town       1409228 2698235
 9 Punjab             Lahore     Lahore      Shalimar         1434823 2280308
10 Punjab             Lahore     Sheikhupura Sheikhupura      1049264 1555424
11 Punjab             Gujranwala Gujranwala  Gujranwal Saddar 1812137 2807054
12 Punjab             Gujranwala Gujrat      Gujrat           1093088 1497865
13 Punjab             Gujranwala Sialkot     Sialkot          1207744 1794658
14 Punjab             Rawalpindi Rawalpindi  Rawalpindi       1927612 3258547
15 Khyber Pakhtunkhwa Peshawar   Peshawar    Peshawar         2026851 4269079

filter(tehsil, Pop2017 >= 2000000 & !Pop1998 >= 1000000)

# A tibble: 0 × 6
# ℹ 6 variables: Province , Division , District , Tehsil ,
#   Pop1998 , Pop2017

Missing Values

One important feature of R that can make comparison tricky is missing values, or NAs (“not available”). NA represents an unknown value so missing values are “contagious”; almost any operation involving an unknown value will also be unknown: is.na() is used to determine the missing value.

NA + 5

[1] NA

NA/2

[1] NA

NA*5

[1] NA

x<-c(NA,5,10)
is.na(x)

[1]  TRUE FALSE FALSE

Arrange Rows with arrange()

The vale of x is NA arrange() works similarly to filter() except that instead of selecting rows, it changes their order. It takes a data frame and a set of column names to order by.

arrange(tehsil, Pop2017)

# A tibble: 543 × 6
   Province    Division  District      Tehsil            Pop1998 Pop2017
                                          
 1 Balochistan Sibi      Sibi          Sangan               3024    2665
 2 Fata        Fata      Bajaur Agency Bar Chamer Kand      3247    2868
 3 Balochistan Zhob      Zhob          Kashatoo             5165    5180
 4 Balochistan Kalat     Kalat         Gazg                 3979    5726
 5 Sindh       Karachi   Karachi West  Manora Cantonment   10008    5874
 6 Balochistan Sibi      Sibi          Kotmandai            8057    7587
 7 Balochistan Nasirabad Jhal Magsi    Mirpur              11562    9444
 8 Balochistan Sibi      Harnai        Shahrig             17170    9454
 9 Sindh       Hyderabad Sujawal       Kharo Chan Taluka    9956   10235
10 Balochistan Sibi      Dera Bugti    Sangsillah          13700   10265
# ℹ 533 more rows

Use desc() to reorder by a column in descending order.

Select Columns with select()

It’s not uncommon to get datasets with hundreds or even thousands of variables. In this case, the first challenge is often narrowing in on the variables you’re actually interested in. select() allows you to rapidly zoom in on a useful subset using operations based on the names of the variables.

select(tehsil,Province, Tehsil, Pop2017)

# A tibble: 543 × 3
   Province Tehsil            Pop2017
                      
 1 Punjab   Hasilpur           456006
 2 Punjab   Khairpur Tamewali  262628
 3 Punjab   Yazman             262175
 4 Punjab   Ahmadpur East     1078683
 5 Punjab   Bahawalpur City    681696
 6 Punjab   Bahawalpur Saddar  574950
 7 Punjab   Bahawalnagar       815143
 8 Punjab   Chishtian          691221
 9 Punjab   Haroonabad         525598
10 Punjab   Fort Abbas         153444
# ℹ 533 more rows

select(tehsil,-Province, -Tehsil, -Pop2017)

# A tibble: 543 × 3
   Division   District     Pop1998
                   
 1 Bahawalpur Bahawalpur    317513
 2 Bahawalpur Bahawalpur    183903
 3 Bahawalpur Bahawalpur    168950
 4 Bahawalpur Bahawalpur    718297
 5 Bahawalpur Bahawalpur    419542
 6 Bahawalpur Bahawalpur    387038
 7 Bahawalpur Bahawalnagar  541553
 8 Bahawalpur Bahawalnagar  498270
 9 Bahawalpur Bahawalnagar  381767
10 Bahawalpur Bahawalnagar   92777
# ℹ 533 more rows

select(tehsil,Pop2017, everything())

# A tibble: 543 × 6
   Pop2017 Province Division   District     Tehsil            Pop1998
                                       
 1  456006 Punjab   Bahawalpur Bahawalpur   Hasilpur           317513
 2  262628 Punjab   Bahawalpur Bahawalpur   Khairpur Tamewali  183903
 3  262175 Punjab   Bahawalpur Bahawalpur   Yazman             168950
 4 1078683 Punjab   Bahawalpur Bahawalpur   Ahmadpur East      718297
 5  681696 Punjab   Bahawalpur Bahawalpur   Bahawalpur City    419542
 6  574950 Punjab   Bahawalpur Bahawalpur   Bahawalpur Saddar  387038
 7  815143 Punjab   Bahawalpur Bahawalnagar Bahawalnagar       541553
 8  691221 Punjab   Bahawalpur Bahawalnagar Chishtian          498270
 9  525598 Punjab   Bahawalpur Bahawalnagar Haroonabad         381767
10  153444 Punjab   Bahawalpur Bahawalnagar Fort Abbas          92777
# ℹ 533 more rows

Add New Variables with mutate()

Besides selecting sets of existing columns, it’s often useful to add new columns that are functions of existing columns. That’s the job of mutate(). mutate() always adds new columns at the end of your dataset. When you’re in RStudio use View() to see all columns.

dplyr::mutate(tehsil, diff = Pop2017-Pop1998)

# A tibble: 543 × 7
   Province Division   District     Tehsil            Pop1998 Pop2017   diff
                                         
 1 Punjab   Bahawalpur Bahawalpur   Hasilpur           317513  456006 138493
 2 Punjab   Bahawalpur Bahawalpur   Khairpur Tamewali  183903  262628  78725
 3 Punjab   Bahawalpur Bahawalpur   Yazman             168950  262175  93225
 4 Punjab   Bahawalpur Bahawalpur   Ahmadpur East      718297 1078683 360386
 5 Punjab   Bahawalpur Bahawalpur   Bahawalpur City    419542  681696 262154
 6 Punjab   Bahawalpur Bahawalpur   Bahawalpur Saddar  387038  574950 187912
 7 Punjab   Bahawalpur Bahawalnagar Bahawalnagar       541553  815143 273590
 8 Punjab   Bahawalpur Bahawalnagar Chishtian          498270  691221 192951
 9 Punjab   Bahawalpur Bahawalnagar Haroonabad         381767  525598 143831
10 Punjab   Bahawalpur Bahawalnagar Fort Abbas          92777  153444  60667
# ℹ 533 more rows

dplyr::mutate(tehsil, diff = Pop2017-Pop1998, add = Pop2017+Pop1998)

# A tibble: 543 × 8
   Province Division   District     Tehsil         Pop1998 Pop2017   diff    add
                                        
 1 Punjab   Bahawalpur Bahawalpur   Hasilpur        317513  456006 138493 7.74e5
 2 Punjab   Bahawalpur Bahawalpur   Khairpur Tame…  183903  262628  78725 4.47e5
 3 Punjab   Bahawalpur Bahawalpur   Yazman          168950  262175  93225 4.31e5
 4 Punjab   Bahawalpur Bahawalpur   Ahmadpur East   718297 1078683 360386 1.80e6
 5 Punjab   Bahawalpur Bahawalpur   Bahawalpur Ci…  419542  681696 262154 1.10e6
 6 Punjab   Bahawalpur Bahawalpur   Bahawalpur Sa…  387038  574950 187912 9.62e5
 7 Punjab   Bahawalpur Bahawalnagar Bahawalnagar    541553  815143 273590 1.36e6
 8 Punjab   Bahawalpur Bahawalnagar Chishtian       498270  691221 192951 1.19e6
 9 Punjab   Bahawalpur Bahawalnagar Haroonabad      381767  525598 143831 9.07e5
10 Punjab   Bahawalpur Bahawalnagar Fort Abbas       92777  153444  60667 2.46e5
# ℹ 533 more rows

newData<-dplyr::mutate(tehsil, new = ifelse(Province == "Punjab", 1,
                            ifelse(Province == "Sindh",2,3)))
newData

# A tibble: 543 × 7
   Province Division   District     Tehsil            Pop1998 Pop2017   new
                                        
 1 Punjab   Bahawalpur Bahawalpur   Hasilpur           317513  456006     1
 2 Punjab   Bahawalpur Bahawalpur   Khairpur Tamewali  183903  262628     1
 3 Punjab   Bahawalpur Bahawalpur   Yazman             168950  262175     1
 4 Punjab   Bahawalpur Bahawalpur   Ahmadpur East      718297 1078683     1
 5 Punjab   Bahawalpur Bahawalpur   Bahawalpur City    419542  681696     1
 6 Punjab   Bahawalpur Bahawalpur   Bahawalpur Saddar  387038  574950     1
 7 Punjab   Bahawalpur Bahawalnagar Bahawalnagar       541553  815143     1
 8 Punjab   Bahawalpur Bahawalnagar Chishtian          498270  691221     1
 9 Punjab   Bahawalpur Bahawalnagar Haroonabad         381767  525598     1
10 Punjab   Bahawalpur Bahawalnagar Fort Abbas          92777  153444     1
# ℹ 533 more rows

PakPC2017Tehsil Data Frame

plotUnit <-
  PakPC2017Tehsil %>%
  dplyr::group_by(Province, Division) %>%
  dplyr::summarize(
    Pop2017 = sum(Pop2017, na.rm = TRUE)
    , Pop1998 = sum(Pop1998, na.rm = TRUE)
  ) %>%
  dplyr::filter(Division %in% c("Faisalabad","Lahore")) %>%
  tidyr::gather(
    key   = "Census"
    , value = "Population"
    , c("Pop2017","Pop1998")
  )

head(plotUnit)

# A tibble: 4 × 4
# Groups:   Province [1]
  Province Division   Census  Population
                    
1 Punjab   Faisalabad Pop2017   14177081
2 Punjab   Lahore     Pop2017   19398081
3 Punjab   Faisalabad Pop1998    9885685
4 Punjab   Lahore     Pop1998   12015649

g<-ggplot(data = plotUnit, mapping = aes(x = Division, y = Population))
g

g<-ggplot(data = plotUnit, mapping = aes(x = Division, y = Population))+
  geom_col()
g

g<-ggplot(data = plotUnit, mapping = aes(x = Division, y = Population))+
  geom_col()+
  scale_y_continuous(labels= scales::comma, expand =  c(0, 0))
g

g<-ggplot(data = plotUnit, mapping = aes(x = Division, y = Population))+
  geom_bar(stat = "identity", position = position_dodge(width = .9))+
  scale_y_continuous(labels= scales::comma, expand =  c(0, 0))
g

g<-ggplot(data = plotUnit,
          mapping = aes(x = Division, y = Population, fill = Census))+
  geom_bar(stat = "identity", position = position_dodge(width = .9))+
  scale_y_continuous(labels= scales::comma, expand =  c(0, 0))
g

g<-ggplot(data = plotUnit,
          mapping = aes(x = Division, y = Population, fill = Census))+
  geom_bar(stat = "identity", position = position_dodge(width = .9))+
  scale_y_continuous(labels= scales::comma, expand =  c(0, 0))+
  geom_text(aes(label=scales::comma(Population)),
            position=position_dodge(width = 0.9), vjust = 0.8)
g

g<-ggplot(data = plotUnit,
          mapping = aes(x = Division, y = Population, fill = Census))+
  geom_bar(stat = "identity", position = position_dodge(width = .9))+
  scale_y_continuous(labels= scales::comma, expand =  c(0, 0))+
  geom_text(aes(label=scales::comma(Population)),
            position=position_dodge(width = 0.9), vjust = 0.8)+
  facet_wrap(~Province, scales = "free_x")
g

plotUnit <-
  PakPC2017Tehsil %>%
  dplyr::group_by(Province, Division) %>%
  dplyr::summarize(
    Pop2017 = sum(Pop2017, na.rm = TRUE)
    , Pop1998 = sum(Pop1998, na.rm = TRUE)
  ) %>%
  dplyr::filter(Division %in% c("Faisalabad","Lahore", "Karachi", "Quetta")) %>%
  tidyr::gather(
    key   = "Census"
    , value = "Population"
    , -Province, - Division
  )

g<-ggplot(data = plotUnit,
          mapping = aes(x = Division, y = Population, fill = Census))+
  geom_bar(stat = "identity", position = position_dodge(width = .9))+
  scale_y_continuous(labels= scales::comma, expand =  c(0, 0))+
  geom_text(aes(label=scales::comma(Population)),
            position=position_dodge(width = 0.9), vjust = 0.8)+
  facet_wrap(~Province, scales = "free_x")
g

g<-ggplot(data = PakPC2017Pakistan, mapping = aes(x = Province, y = Pop2017))
g+geom_point(aes(color = ResStatus))+theme(legend.position = "top")

Scatter Plot

ggplot(PakPC2017Tehsil, aes(Pop1998,Pop2017))+geom_point()+geom_smooth(method = lm)

dygraphs

The dygraphs package is an R interface to the dygraphs JavaScript charting library. It provides rich facilities for charting time-series data in R

library(dygraphs)
dygraph(nhtemp, main = "New Haven Temperatures", ylab = "Temp (F)")

lungDeaths <- cbind(mdeaths, fdeaths)
dygraph(lungDeaths)

dygraph(lungDeaths) %>% dyRangeSelector()

dygraph(lungDeaths) %>%
  dySeries("mdeaths", label = "Male") %>%
  dySeries("fdeaths", label = "Female") %>%
  dyOptions(stackedGraph = TRUE) %>%
  dyRangeSelector(height = 10)

Year<-seq(from = as.Date("1990/1/1"), to = as.Date("2018/1/1"), by = "years")
prod<-rnorm(29,0,1)
timeSeries<-xts::xts(prod,Year)
dygraph(timeSeries)

plotly

library(ggplot2)
library(plotly)
myPlot<-ggplot(data = mpg, mapping = aes(x = displ, y = hwy))+
  geom_point(mapping = aes(color = drv))+
  geom_smooth(mapping = aes(linetype = drv))
ggplotly(myPlot)

datatable

library(readxl)
spi <- read_excel(here::here("blogs", "2018-03-25_Maldives_R", "data", "spi.xls"))
DT::datatable(spi,fillContainer = TRUE, editable = TRUE)

Citation

BibTeX citation:

@online{arfan_dilber2018,
  author = {Arfan Dilber, Muhammad},
  title = {Data {Analysis} with {R:} {Introduction} of {R} {Language}},
  date = {2018-03-25},
  url = {https://myaseen208.com/blogs/2018-03-25_Maldives_R/},
  langid = {en}
}

For attribution, please cite this work as:

Arfan Dilber, Muhammad. 2018. “Data Analysis with R: Introduction of R Language.” March 25. https://myaseen208.com/blogs/2018-03-25_Maldives_R/.

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

WoolType	Tension	Yarn Breaks
A	L	26
A	L	30
A	L	54
A	L	25
A	L	70
A	L	52
A	L	51
A	L	26
A	L	67
A	M	18
A	M	21
A	M	29
A	M	17
A	M	12
A	M	18
A	M	35
A	M	30
A	M	36
A	H	36
A	H	21
A	H	24
A	H	18
A	H	10
A	H	43
A	H	28
A	H	15
A	H	26
B	L	27
B	L	14
B	L	29
B	L	19
B	L	29
B	L	31
B	L	41
B	L	20
B	L	44
B	M	42
B	M	26
B	M	19
B	M	16
B	M	39
B	M	28
B	M	21
B	M	39
B	M	29
B	H	20
B	H	21
B	H	24
B	H	17
B	H	13
B	H	15
B	H	15
B	H	16
B	H	28

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

WoolType	Tension	Yarn Breaks
A	L	26
A	L	30
A	L	54
A	L	25
A	L	70
A	L	52
A	L	51
A	L	26
A	L	67
A	M	18
A	M	21
A	M	29
A	M	17
A	M	12
A	M	18
A	M	35
A	M	30
A	M	36
A	H	36
A	H	21
A	H	24
A	H	18
A	H	10
A	H	43
A	H	28
A	H	15
A	H	26
B	L	27
B	L	14
B	L	29
B	L	19
B	L	29
B	L	31
B	L	41
B	L	20
B	L	44
B	M	42
B	M	26
B	M	19
B	M	16
B	M	39
B	M	28
B	M	21
B	M	39
B	M	29
B	H	20
B	H	21
B	H	24
B	H	17
B	H	13
B	H	15
B	H	15
B	H	16
B	H	28

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345

WoolType	Tension	Yarn Breaks
A	L	26
A	L	30
A	L	54
A	L	25
A	L	70
A	L	52
A	L	51
A	L	26
A	L	67
A	M	18
A	M	21
A	M	29
A	M	17
A	M	12
A	M	18
A	M	35
A	M	30
A	M	36
A	H	36
A	H	21
A	H	24
A	H	18
A	H	10
A	H	43
A	H	28
A	H	15
A	H	26
B	L	27
B	L	14
B	L	29
B	L	19
B	L	29
B	L	31
B	L	41
B	L	20
B	L	44
B	M	42
B	M	26
B	M	19
B	M	16
B	M	39
B	M	28
B	M	21
B	M	39
B	M	29
B	H	20
B	H	21
B	H	24
B	H	17
B	H	13
B	H	15
B	H	15
B	H	16
B	H	28

Yield	Fert	Variety
51	80	V1
52	80	V1
53	90	V1
54	90	V1
56	100	V1
57	100	V1
55	80	V2
56	80	V2
58	90	V2
59	90	V2
62	100	V2
63	100	V2

Leaf Area (cm^2)	Leaf Moisture (%)	Total Shoot Length (cm)
72	80	307
174	75	529
116	81	632
78	83	527
134	79	442
95	81	525
113	80	481
98	81	710
148	74	422
42	78	345