Random-effects and residual covariance structures

vlm_random() declares random coefficients for a subject (the RANDOM ... / SUBJECT= TYPE= statement of SAS PROC MIXED), and vlm_repeated() declares the covariance structure of the residuals within a subject (the REPEATED / SUBJECT= TYPE= statement). Both are passed to vlm_fit().

Usage

vlm_random(effects = ~1, subject, type = c("vc", "un"))

vlm_repeated(
  type = c("vc", "cs", "ar1", "sp_pow", "un"),
  subject,
  order = NULL
)

Arguments

effects: one-sided formula of the random coefficients, such as ~ 1 + time (random intercept and slope) or ~ 1.
subject: one-sided formula of the subject, such as ~ animal_id:breed; its variables are treated as classification variables and their combination identifies a subject.
type: "vc" (variance components, independent coefficients; the default of vlm_random()) or "un" (unstructured) for vlm_random(); "vc", "cs", "ar1", "sp_pow" or "un" for vlm_repeated().
order: one-sided formula of the time (or order) variable within a subject, such as ~ time; required for "sp_pow" and "un"; for "ar1" the data order within a subject is used when it is NULL.

Value

An object of class vlm_random or vlm_repeated: a list with the arguments.

Details

A random coefficient specification adds, for subject $s$ with design rows $\mathbf{z}_{si}^{\top}$ (the values of effects for observation $i$), the term $\mathbf{z}_{si}^{\top}\mathbf{b}_s$ with $\mathbf{b}_s \sim N(\mathbf{0}, \mathbf{G}_0)$ independent between subjects. With type = "vc", $\mathbf{G}_0$ is diagonal (one variance per effect); with type = "un" it is unstructured and its parameters are reported as $UN(1,1)$, $UN(2,1)$, $UN(2,2)$, ..., as in SAS.

The residual structures for the observations of one subject, with $t_i$ the order (or time) of observation $i$, are $$\mathrm{vc}:\ \sigma^2\mathbf{I},\qquad \mathrm{cs}:\ \sigma_1\mathbf{J} + \sigma^2\mathbf{I},\qquad \mathrm{ar1}:\ \sigma^2\rho^{|r_i - r_j|},\qquad \mathrm{sp\_pow}:\ \sigma^2\rho^{|t_i - t_j|},\qquad \mathrm{un}:\ \sigma_{t_i t_j},$$ where $r_i$ is the rank of $t_i$ among the distinct times in the data, $\mathbf{J}$ a matrix of ones and $\sigma_1$ the compound-symmetry covariance. Observations of different subjects are independent.

References

Duchateau, L., Janssen, P. and Rowlands, G. J. (1998). Linear Mixed Models. An Introduction with Applications in Veterinary Research. International Livestock Research Institute, Nairobi. Section 3.4.

Littell, R. C., Milliken, G. A., Stroup, W. W. and Wolfinger, R. D. (1996). SAS System for Mixed Models. SAS Institute, Cary, NC.

Author

Muhammad Yaseen (myaseen208@gmail.com)

Examples

# random intercept and slope per animal, unstructured covariance (Example 3.3, Model 3)
ri <-
  vlm_random(
      effects = ~ 1 + time
    , subject = ~ animal_id:breed
    , type    = "un"
    )
ri
#> Random coefficients ~1 + time, subject ~animal_id:breed, type UN

# first-order autoregressive residuals within an animal (Example 3.3, Model 2)
ar <-
  vlm_repeated(
      type    = "ar1"
    , subject = ~ animal_id:breed
    , order   = ~ time
    )
ar
#> Residual structure AR1, subject ~animal_id:breed, order ~time