LRTSim {RLRsim}R Documentation

Simulation of the (Restricted) Likelihood Ratio Statistic

Description

These functions simulate values from the (exact) finite sample distribution of the (restricted) likelihood ratio statistic for testing the presence of the variance component (and restrictions of the fixed effects) in a simple linear mixed model with known correlation structure of the random effect and i.i.d. errors. They are usually called by exactLRT or exactRLRT.

Usage

LRTSim(X, Z, q, sqrt.Sigma, seed = NA, nsim = 10000, log.grid.hi = 8,
       log.grid.lo=-10, gridlength=200)
RLRTSim(X, Z, sqrt.Sigma, lambda0 = NA, seed = NA, nsim = 10000, use.approx=0,
       log.grid.hi=8, log.grid.lo=-10, gridlength=200)

Arguments

X The fixed effects design matrix of the model under the alternative
Z The random effects design matrix of the model under the alternative
q The number of parameters restrictions on the fixed effects (see Details)
sqrt.Sigma The upper triangular cholesky factor of the correlation matrix of the random effect
lambda0 The value of the ratio of the variance of the random effect and the errors under the null
seed Specify a seed for set.seed
nsim Number of values to simulate
use.approx If 0, the exact distribution is simulated. If between 0 and 1, only the largest eigenvalues whose sum represents at least use.approx*(sum of all eigenvalues) are used.
log.grid.hi Lower value of the grid on the log scale. See Details
log.grid.lo Lower value of the grid on the log scale. See Details
gridlength Length of the grid for the grid search over lambda. See Details

Details

The model under the alternative must be a linear mixed model y=X*beta+Z*b+epsilon with a single random effect b with known correlation structure Sigma and i.i.d errors. The simulated distribution of the likelihood ratio statistic was derived by Crainiceanu & Ruppert (2004). The simulation algorithm uses a gridsearch over a log-regular grid of values of lambda=Var(b)/Var(epsilon) to maximize the likelihood under the alternative for nsim realizations of y drawn under the null hypothesis. log.grid.hi and log.grid.lo are the lower and upper limits of this grid on the log scale. gridlength is the number of points on the grid.\ These are just wrapper functions for the underlying C code.

Value

A vector with the simulated values of the (R)LRT under the null.

Author(s)

Fabian Scheipl

References

Crainiceanu, C. and Ruppert, D. (2004) Likelihood ratio tests in linear mixed models with one variance component, Journal of the Royal Statistical Society: Series B,66,165–185.

Scheipl, F. (2007) Testing for nonparametric terms and random effects in structured additive regression. Diploma thesis.\ http://www.statistik.lmu.de/~scheipl/downloads/DIPLOM.zip.

See Also

exactLRT, exactRLRT for tests

Examples

library(lme4)
g <- rep(1:10, e = 10)
x <- rnorm(100)
y <- 0.1 * x + rnorm(100)
m <- lmer(y ~ x + (1|g), method="ML")
m0 <- lm(y ~ 1)

(obs.LRT <- 2*(logLik(m)-logLik(m0)))
X <- m@X
Z <- t(as.matrix(m@Zt))
sim.LRT <- LRTSim(X, Z, 1, diag(10))
(pval <- mean(sim.LRT > obs.LRT))

[Package RLRsim version 2.0-2 Index]