Adjusted covariance matrix for linear mixed models according to Kenward and Roger

Kenward and Roger (1997) describe an improved small sample approximation to the covariance matrix estimate of the fixed parameters in a linear mixed model.

vcovAdj(object, details = 0)

# S3 method for class 'lmerMod'
vcovAdj(object, details = 0)

Arguments

object: An lmer model
details: If larger than 0 some timing details are printed.

Value

phiA: the estimated covariance matrix, this has attributed P, a list of matrices used in KR_adjust and the estimated matrix W of the variances of the covariance parameters of the random effects
SigmaG: list: Sigma: the covariance matrix of Y; G: the G matrices that sum up to Sigma; n.ggamma: the number (called M in the article) of G matrices)

Note

If $N$ is the number of observations, then the vcovAdj() function involves inversion of an $N x N$ matrix, so the computations can be relatively slow.

References

Ulrich Halekoh, Søren Højsgaard (2014)., A Kenward-Roger Approximation and Parametric Bootstrap Methods for Tests in Linear Mixed Models - The R Package pbkrtest., Journal of Statistical Software, 58(10), 1-30., https://www.jstatsoft.org/v59/i09/

Kenward, M. G. and Roger, J. H. (1997), Small Sample Inference for Fixed Effects from Restricted Maximum Likelihood, Biometrics 53: 983-997.

Author

Ulrich Halekoh uhalekoh@health.sdu.dk, Søren Højsgaard sorenh@math.aau.dk

Examples


fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy, REML=TRUE)
class(fm1)
#> [1] "lmerMod"
#> attr(,"package")
#> [1] "lme4"

set.seed(123)
sleepstudy2 <- sleepstudy[sample(nrow(sleepstudy), size=120), ]

fm2 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy2, REML=TRUE)

## Here the adjusted and unadjusted covariance matrices are identical,
## but that is not generally the case:

v1 <- vcov(fm1)
v1a <- vcovAdj(fm1, details=0)
v1a / v1
#> 2 x 2 Matrix of class "dgeMatrix"
#>             (Intercept) Days
#> (Intercept)           1    1
#> Days                  1    1

v2 <- vcov(fm2)
v2a <- vcovAdj(fm2, details=0)
v2a / v2
#> 2 x 2 Matrix of class "dgeMatrix"
#>             (Intercept)     Days
#> (Intercept)    1.008841 1.046870
#> Days           1.046870 1.009213

# For comparison, an alternative estimate of the
# variance-covariance matrix is based on parametric bootstrap (and
# this is easily parallelized):