Compare fitted models

Compares several fits of vlm_fit() by their (restricted) log-likelihoods and information criteria, and tests each model against the one before it by a likelihood ratio test: the comparison of covariance structures of Example 3.3 of the book (compound symmetry, first-order autoregressive, random coefficients).

Usage

vlm_compare(..., test = TRUE)

# Default S3 method
vlm_compare(..., test = TRUE)

Arguments

...: two or more results of vlm_fit(), optionally named.
test: logical; add likelihood ratio tests of each model against the previous one (default TRUE).

Value

A list of class c("vlm_compare", "vlm_result", "list") with models (one row per model: method, number of covariance parameters, -2 log-likelihood, AIC, AICC, BIC and the test against the previous model).

Details

With \(\ell\) the maximised (restricted) log-likelihood and \(d\) the number of covariance parameters (plus the rank of \(\mathbf{X}\) for ML), \(AIC = -2\ell + 2d\), \(AICC = -2\ell + 2dn^*/(n^* - d - 1)\) and \(BIC = -2\ell + d\log s\), where \(n^* = N - \mathrm{rank}(\mathbf{X})\) for REML (\(N\) for ML) and \(s\) is the number of subjects. The likelihood ratio statistic of two nested models is \(-2(\ell_0 - \ell_1)\), referred to \(\chi^2_{d_1 - d_0}\); when the larger model adds one variance component, the null value lies on the boundary and p_boundary gives the p-value of the 50:50 mixture of \(\chi^2_0\) and \(\chi^2_1\) (Self and Liang 1987). REML likelihoods are comparable only between models with the same fixed effects; the test is then left out.

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.

Self, S. G. and Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605-610.

Author

Muhammad Yaseen (myaseen208@gmail.com)

Examples

cs <-
  vlm_fit(
      fixed    = PCV ~ breed + breed:time
    , data     = ex33
    , random   = NULL
    , repeated = vlm_repeated(
          type    = c("vc", "cs", "ar1", "sp_pow", "un")[2]
        , subject = ~ animal_id:breed
        , order   = NULL
        )
    , method   = "REML"
    , ddfm     = "default"
    , type     = 3
    , class    = NULL
    , bound    = TRUE
    , control  = list()
    )
ar <-
  vlm_fit(
      fixed    = PCV ~ breed + breed:time
    , data     = ex33
    , random   = NULL
    , repeated = vlm_repeated(
          type    = c("vc", "cs", "ar1", "sp_pow", "un")[3]
        , subject = ~ animal_id:breed
        , order   = NULL
        )
    , method   = "REML"
    , ddfm     = "default"
    , type     = 3
    , class    = NULL
    , bound    = TRUE
    , control  = list()
    )
rc <-
  vlm_fit(
      fixed    = PCV ~ breed + breed:time
    , data     = ex33
    , random   = vlm_random(
          effects = ~ 1 + time
        , subject = ~ animal_id:breed
        , type    = c("vc", "un")[2]
        )
    , repeated = NULL
    , method   = "REML"
    , ddfm     = "default"
    , type     = 3
    , class    = NULL
    , bound    = TRUE
    , control  = list()
    )
vlm_compare(
    CS   = cs
  , AR1  = ar
  , RC   = rc
  , test = TRUE
  )
#> Comparison of 3 models; smallest AIC: RC
#> 
#> Models
#>  model method cov_parms   m2ll    AIC   AICC    BIC  chisq df      p p_boundary
#>     CS   REML         2 783.45 787.45 787.53 788.42         .      .          .
#>    AR1   REML         2 793.14 797.14 797.21 798.11         .      .          .
#>     RC   REML         4 767.56 775.56 775.81 777.50 25.583  2 <1e-04          .
#> 
#> Sources: Duchateau, Janssen and Rowlands (1998); Akaike (1974); Schwarz (1978).
#> Full references: vlm_cite(x).