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```
#' Restricted Likelihood Ratio Tests for additive and linear mixed models
#'
#' This function provides an (exact) restricted likelihood ratio test based on
#' simulated values from the finite sample distribution for testing whether the
#' variance of a random effect is 0 in a linear mixed model with known
#' correlation structure of the tested random effect and i.i.d. errors.
#'
#' Testing in models with only a single variance component require only the
#' first argument \code{m}. For testing in models with multiple variance
#' components, the fitted model \code{m} must contain \bold{only} the random
#' effect set to zero under the null hypothesis, while \code{mA} and \code{m0}
#' are the models under the alternative and the null, respectively. For models
#' with a single variance component, the simulated distribution is exact if the
#' number of parameters (fixed and random) is smaller than the number of
#' observations. Extensive simulation studies (see second reference below)
#' confirm that the application of the test to models with multiple variance
#' components is safe and the simulated distribution is correct as long as the
#' number of parameters (fixed and random) is smaller than the number of
#' observations and the nuisance variance components are not superfluous or
#' very small. We use the finite sample distribution of the restricted
#' likelihood ratio test statistic as derived by Crainiceanu & Ruppert (2004).
#'
#' @param m The fitted model under the alternative or, for testing in models
#' with multiple variance components, the reduced model containing only the
#' random effect to be tested (see Details), an \code{lme}, \code{lmerMod} or
#' \code{spm} object
#' @param mA The full model under the alternative for testing in models with
#' multiple variance components
#' @param m0 The model under the null for testing in models with multiple
#' variance components
#' @param seed input for \code{set.seed}
#' @param nsim Number of values to simulate
#' @param log.grid.hi Lower value of the grid on the log scale. See
#' \code{\link{exactRLRT}}.
#' @param log.grid.lo Lower value of the grid on the log scale. See
#' \code{\link{exactRLRT}}.
#' @param gridlength Length of the grid. See \code{\link{exactLRT}}.
#' @param parallel The type of parallel operation to be used (if any). If
#' missing, the default is "no parallelization").
#' @param ncpus integer: number of processes to be used in parallel operation:
#' typically one would chose this to the number of available CPUs. Defaults to
#' 1, i.e., no parallelization.
#' @param cl An optional parallel or snow cluster for use if parallel = "snow".
#' If not supplied, a cluster on the local machine is created for the duration
#' of the call.
#' @return A list of class \code{htest} containing the following components:
#' @return A list of class \code{htest} containing the following components:
#' \itemize{
#' \item \code{statistic} the observed likelihood ratio
#' \item \code{p} p-value for the observed test statistic
#' \item \code{method} a character string indicating what type of test was
#' performed and how many values were simulated to determine the critical value
#' \item \code{sample} the samples from the null distribution returned by
#' \code{\link{RLRTSim}}
#' }
#' @author Fabian Scheipl, bug fixes by Andrzej Galecki, updates for
#' \pkg{lme4}-compatibility by Ben Bolker
#' @seealso \code{\link{RLRTSim}} for the underlying simulation algorithm;
#' \code{\link{exactLRT}} for likelihood based tests
#' @references Crainiceanu, C. and Ruppert, D. (2004) Likelihood ratio tests in
#' linear mixed models with one variance component, \emph{Journal of the Royal
#' Statistical Society: Series B},\bold{66},165--185.
#'
#' Greven, S., Crainiceanu, C., Kuechenhoff, H., and Peters, A. (2008)
#' Restricted Likelihood Ratio Testing for Zero Variance Components in Linear
#' Mixed Models, \emph{Journal of Computational and Graphical Statistics},
#' \bold{17} (4): 870--891.
#'
#' Scheipl, F., Greven, S. and Kuechenhoff, H. (2008) Size and power of tests
#' for a zero random effect variance or polynomial regression in additive and
#' linear mixed models. \emph{Computational Statistics & Data Analysis},
#' \bold{52}(7):3283--3299.
#' @keywords htest
#' @examples
#'
#' library(lme4)
#' data(sleepstudy)
#' mA <- lmer(Reaction ~ I(Days-4.5) + (1|Subject) + (0 + I(Days-4.5)|Subject),
#' data = sleepstudy)
#' m0 <- update(mA, . ~ . - (0 + I(Days-4.5)|Subject))
#' m.slope <- update(mA, . ~ . - (1|Subject))
#' #test for subject specific slopes:
#' exactRLRT(m.slope, mA, m0)
#'
#' library(mgcv)
#' data(trees)
#' #test quadratic trend vs. smooth alternative
#' m.q<-gamm(I(log(Volume)) ~ Height + s(Girth, m = 3), data = trees,
#' method = "REML")$lme
#' exactRLRT(m.q)
#' #test linear trend vs. smooth alternative
#' m.l<-gamm(I(log(Volume)) ~ Height + s(Girth, m = 2), data = trees,
#' method = "REML")$lme
#' exactRLRT(m.l)
#'
#' @export exactRLRT
#' @importFrom stats anova cov2cor logLik quantile
'exactRLRT' <- function(m, mA = NULL, m0 = NULL, seed = NA,
nsim = 10000, log.grid.hi = 8, log.grid.lo = -10, gridlength = 200,
parallel = c("no", "multicore", "snow"),
ncpus = 1L, cl = NULL) {
if (class(m) == "spm") {
m <- m$fit
class(m) <- "lme"
}
if (class(m) %in% c("amer", "mer"))
stop("Models fit with package <amer> or versions of <lme4> below 1.0 are no longer supported.")
if (!(c.m <- (class(m))) %in% c("lme", "lmerMod", "merModLmerTest"))
stop("Invalid <m> specified. \n")
if(c.m == "merModLmerTest") c.m <- "lmerMod"
if ("REML" != switch(c.m,
lme = m$method,
lmerMod = ifelse(lme4::isREML(m), "REML", "ML"))){
message("Using restricted likelihood evaluated at ML estimators.")
message("Refit with method=\"REML\" for exact results.")
}
d <- switch(c.m, lme = extract.lmeDesign(m),
lmerMod = extract.lmerModDesign(m))
X <- d$X
qrX <- qr(X)
Z <- d$Z
y <- d$y
Vr <- d$Vr
if(all(Vr == 0)){
# this only happens if the estimate of the tested variance component is 0.
# since we still want chol(cov2cor(Vr)) to work, this does the trick.
diag(Vr) <- 1
}
K <- ncol(Z)
n <- nrow(X)
p <- ncol(X)
if (is.null(mA) && is.null(m0)) {
if(length(d$lambda) != 1 || d$k != 1)
stop("multiple random effects in model -
exactRLRT needs <m> with only a single random effect.")
#2*restricted ProfileLogLik under H0: lambda=0
res <- qr.resid(qrX, y)
R <- qr.R(qrX)
detXtX <- det(t(R) %*% R)
reml.H0 <- -((n - p) * log(2 * pi) + (n - p) * log(sum(res^2)) +
log(detXtX) + (n - p) - (n - p) * log(n - p))
#observed value of the test-statistic
reml.obs <- 2 * logLik(m, REML = TRUE)[1]
rlrt.obs <- max(0, reml.obs - reml.H0)
lambda <- d$lambda
} else {
nonidentfixmsg <-
"Fixed effects structures of <mA> and <m0> not identical.
REML-based inference not appropriate."
if (c.m == "lme") {
if (any(mA$fixDF$terms != m0$fixDF$terms))
stop(nonidentfixmsg)
} else {
if (c.m == "mer") {
if (any(mA@X != m0@X))
stop(nonidentfixmsg)
} else {
if (c.m == "lmerMod") {
if (any(lme4::getME(mA,"X") != lme4::getME(m0,"X")))
stop(nonidentfixmsg)
}
}
}
## bug fix submitted by Andrzej Galecki 3/10/2009
DFx <- switch(c.m, lme = anova(mA,m0)$df,
lmerMod = anova(mA, m0, refit = FALSE)$Df)
if (abs(diff(DFx)) > 1) {
stop("Random effects not independent - covariance(s) set to 0 under H0.\n
exactRLRT can only test a single variance.\n")
}
rlrt.obs <- max(0, 2 * (logLik(mA, REML = TRUE)[1] -
logLik(m0, REML = TRUE)[1]))
}
p <- if (rlrt.obs != 0) {
sample <- RLRTSim(X, Z, qrX=qrX, sqrt.Sigma = chol(cov2cor(Vr)),
lambda0 = 0, seed = seed, nsim = nsim,
log.grid.hi = log.grid.hi,
log.grid.lo = log.grid.lo, gridlength = gridlength,
parallel = match.arg(parallel),
ncpus = ncpus, cl = cl)
if (quantile(sample, 0.9) == 0) {
warning("Null distribution has mass ", mean(sample ==
0), " at zero.\n")
}
mean(rlrt.obs < sample)
} else {
1
}
RVAL <- list(statistic = c(RLRT = rlrt.obs), p.value = p,
method = paste("simulated finite sample distribution of RLRT.\n
(p-value based on",
nsim, "simulated values)"), sample=sample)
class(RVAL) <- "htest"
return(RVAL)
}
```

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