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inst/PSGLMM_MEE/R/psglmm.R
In TH.data: TH's Data Archive
## Function to fit a phylogenetic and spatial (generalised) linear mixed model
## (PSGLMM) in which the desgin matrices of the random effects are penalised
## using a known group correlation/covariance structure.
##
## Required by:
## Depends on: Packages lme4 and Matrix,
## Formulas adaptFormula(), randomTerms(), makeVARALL() and
## modelSelection() found in 'support.R'
## The function lformula() from lme4 needs to be modified in
## accordance with the file 'lFormula_tweak.R'
## Arguments:
## ----------
## formula: A linear formula as used by '(g)lmer' in package 'lme4'.
## VCV: A list of covariance/correlation matrices (preferably named) with
## wich the design matrices of the random effects are to be penalised.
## The list must contain one element per random effect and in the same
## order as the random effects in 'formula'. If a random effect
## shouldn't be penalised, 'NULL' is used in its corresponding place in
## 'VCV'. The dimensions of the penalty matrices must match the
## dimensions of the design matrices.
## data: A data frame containing the variables named in 'formula'. The 'data'
## argument is not optional.
## family: Error distribution and link function to be used in the model. This
## has to be a family function. Only 'gaussian()', 'binomial()' and
## 'poisson()' are supported.
## gf: An optional character vector containing the names of the grouping
## factors of each random effect.
## rd: To how many digits the variance estimates should be rounded. Takes
## the value 'NULL' if no rounding is to be done.
## msel: The proper scoring rule to be used for predictive cross-validation
## during model selection. For Gaussian responses the logarithmic score
## 'LS' and the Dawid-Sebastiani score 'DSS' are available, 'DSS' for
## Poisson responses and 'DSS' or the Brier score 'BS' for binary
## responses. For 'msel= FALSE' no model selection is performed.
## eps: The limit underneath which random effects variance estimates are to
## be considered as zero.
## msgs: If progress messages should be printed.
## REML: If parameter estimation for Gaussian responses should be REML (TRUE)
## or ML (FALSE).
## ...: Further arguments passed to '(g)lFormula'.
##
## Details:
## --------
## Model selection may only be performed if all random effects components are
## penalised.
## The predictive cross-validation is only performed if more than one random
## effects variance estimate is positive.
## The p value of the estimated variance parameter for the chosen covariance
## component is only calculated if model selection is selected.
## When model selection is performed, the covariance component with the highest
## score is marked with '*' in the column 'Selection' in '.$varall'. If only
## one variance parameter is estimated to be positive, no cross-validation is
## performed and the selected covariance component is marked with '**'.
##
## Value:
## ------
## A list containing
## - A fitted object of class 'merMod' or 'glmerMod'
## - A data frame containing all random effects variance estimates
## - A data frame containing the random effects variance estimate for the
## chosen covariance component if model selection is performed.
##
## See also:
## ---------
## 'lmer', 'glmer'
##
## Examples:
## ---------
##
##
psglmm <- function(formula, VCV= list(), data, family= gaussian(), gf, rd= NULL,
msel= FALSE, eps= 1e-5, msgs= TRUE, REML= FALSE, ...) {
## Required packages are loaded.
require("Matrix")
require("lme4")
## Version number for lme4 is checked.
stopifnot(compareVersion(packageDescription("lme4")$Version, "1.0-0") >= 0)
## Further checks are made.
if(length(VCV) == 0 || is.null(unlist(VCV)))
stop("Use '(g)lmer' if no penalising is wanted.")
if(!is.list(VCV)) stop("'VCV' must be a list.")
if(missing(formula)) stop("'formula' must be assigned.")
if(missing(data)) stop("'data' must be assigned.")
if(is.character(family)) try(family <- get(family, mode= "function"), TRUE)
if(is.function(family)) family <- family()
if(!family[[1]] %in% c("gaussian", "binomial", "poisson"))
stop("'family' must be 'gaussian', 'binomial' or 'poisson'.")
if(!is.null(rd) && rd < 0) stop("'rd' must be a positive integer.")
if(!(msel %in% c(TRUE, FALSE, "BS", "LS", "DSS"))) {
warning(paste("Selection criterium", msel,
"not recognised. No model selection performed."))
msel <- FALSE
}
## Call to psglmm is saved for later use.
if(msgs) cat("Setting up model\n")
call <- list(formula= formula, VCV= VCV, data= data, family= family)
## Formula and data are changed so that random effects with identical grouping
## factor are set up with unique grouping factors.
tmp <- adaptFormula(formula, data)
formula <- tmp$formula
data <- tmp$data
origGF <- tmp$origGF
f4selection <- tmp$f4selection
## Raw model is set up with lme4.
if(family == "gaussian" || family[[1]] == "gaussian") {
# tryCatch(source("lFormula_tweak.R"),
# error= function(e) print(cat(paste("Function lFormula() needs to be",
# "modified. Please load the file 'lformula_tweak.R' manually or place it",
# "in the current working directory.\n"))))
mtmp <- lFormula(formula= formula, data= data, REML= REML, ...)
}
if(family %in% c("binomial", "poisson") || family[[1]] %in% c("binomial", "poisson"))
mtmp <- glFormula(formula= formula, data= data, family= family, ...)
## The permutation of the order of random effects in 'formula' used by
## '(g)lFormula' is retrieved ('perm') together with the number of levels of
## the grouping factors ('dims') and an object ('rT') to replace
## 'mtmp$reTrms$cnms' later.
rT <- randomTerms(formula, data, mtmp, VCV)
nRE <- length(rT$rT) # number of random effects
VCVn <- names(VCV)[rT$perm] # names of penalties
if(is.null(VCVn)) {
VCVn <- as.character(rT$perm)
names(call$VCV) <- as.character(1:nRE)
}
## List of penalties is permuted to fit the order of the random effects used
## by (g)lFormula.
VCV <- lapply(rT$perm, function(i) VCV[[i]])
VCVNULL <- sapply(VCV, is.null)
## Diagonal matrices are added to 'VCV' for random effects which aren't to
## be penalised.
if(any(VCVNULL)) for(i in which(VCVNULL)) {
VCV[[i]] <- as(diag(rT$dims[i]), "dgCMatrix")
}
## A check is made, if the dimensions of the penalty matrices match the
## dimensions of the random effects.
VCVdims <- sapply(VCV, nrow)
if(!all(VCVdims == rT$dims)) {
tmp <- which(VCVdims != rT$dims)
tmp <- if(is.null(VCVn)) tmp else VCVn[tmp]
stop(paste("Dimension of elements",
paste("'",tmp,"'", sep= "", collapse= ", "), "don't match dimensions of",
"design matrices."))
}
## Cholesky factors of all penalty matrices are saved in a list.
CFl <- lapply(1:nRE, function(i) t(chol(VCV[[i]])))
## The Cholesky factors in 'CFl' are put together into a matrix containing all
## Cholesky factors in the right order on its "diagonals".
CF <- bdiag(CFl)
## A new 'Zt' matrix is calculated, which will replace 'mtmp$reTrms$Zt'.
## The old 'mtmp$reTrms$Zt' is penalised with 'CF'.
newZt <- t(crossprod(mtmp$reTrms$Zt, CF))
## New 'theta', 'lower' and 'Lind' vectors are calculated, for convenience
## this is done as lists.
tmp.cnms <- sapply(1:nRE, function(i) sum(1:length(mtmp$reTrms$cnms[[i]])))
tmp.rT <- sapply(1:nRE, function(i) sum(1:length(rT$rT[[i]])))
newtheta <- newlower <- vector("list", nRE)
## Calculations for first random effect outside of loop.
newtheta[[1]] <- mtmp$reTrms$theta[1:cumsum(tmp.cnms)[1]]
newlower[[1]] <- mtmp$reTrms$lower[1:cumsum(tmp.cnms)[1]]
## For all subsequent random effects calculations are made.
if(nRE > 1) for(i in 2:nRE) {
elem.cnms <- (cumsum(tmp.cnms)[(i - 1)] + 1):(cumsum(tmp.cnms)[i])
newtheta[[i]] <- mtmp$reTrms$theta[elem.cnms]
newlower[[i]] <- mtmp$reTrms$lower[elem.cnms]
}
## Elements of 'newtheta' and 'newlower' belonging to penalised random effects
## are set to 1 and 0 respectively, as only one variance parameter should be
## estimated for these.
for(i in which(!VCVNULL)) {
newtheta[[i]] <- 1
newlower[[i]] <- 0
}
newtheta <- unlist(newtheta)
newlower <- unlist(newlower)
## A new 'Lind' vector is calculated.
newLind <- vector("list", nRE)
for(i in 1:nRE) {
nSquare <- (-1 + sqrt(1 + 8 * tmp.rT[i])) / 2
nElem <- cumsum(tmp.rT)[i]
tmp <- sapply(1:nSquare,
function(i) {
(1:nElem)[(1+c(0, cumsum(nSquare:1))[i]):(c(0, cumsum(nSquare:1))[i] +
(nSquare - i + 1))]
})
tmp <- t(sapply(1:nSquare,
function(i) c(rep(0, nSquare - length(tmp[[i]])), tmp[[i]])))
tmp <- as.vector(tmp)
tmp <- tmp[tmp != 0]
tmp <- tmp + c(0, cumsum(tmp.rT))[i]
newLind[[i]] <- rep(tmp, rT$dims[i] / length(rT$rT[[i]]))
}
newLind <- unlist(newLind)
## If all random effects are penalised, 'mtmp$reTrms$Lambdat' is replaced by a
## diagonal matrix. This is also the case if the original matrix is already
## a diagonal matrix. If any of the random effects should not be penalised,
## and a non-diagonal structure of 'mtmp$reTrms$Lambdat' thus needs to be
## kept, the new matrix has to be constructed manually, as it will
## (probably) contain zeros as 'non-zero' elements.
newLambdat <- Matrix(0, nrow(newZt), nrow(newZt), sparse= TRUE)
newLambdat <- as(newLambdat, "dgCMatrix")
diag(newLambdat) <- rep(1, nrow(newZt))
if(!(all(!VCVNULL) ||
(length(mtmp$reTrms$Lambdat@x) == nrow(mtmp$reTrms$Lambdat) &&
all(mtmp$reTrms$Lambdat@x == 1)))) {
tmpi <- tmpp <- tmpx <- vector("list", nRE)
for(i in 1:nRE) {
levelsi <- length(rT$rT[[i]])
reps <- rT$dims[[i]] / levelsi
tmpp[[i]] <- cumsum(rep(1:levelsi, reps))
# number of indices in tmpp[[i-1] is added to the indices in tmpp[[i]]
if(i == 1) tmpp[[i]] <- c(0, tmpp[[i]]) else {
tmpp[[i]] <- tmpp[[i]] + tmpp[[i - 1]][length(tmpp[[i - 1]])]
}
tmp <- unlist(sapply(1:levelsi,
function(j) 1:((1:levelsi)[j])))
tmpi[[i]] <- rep(tmp, reps) +
rep(seq(0, rT$dims[[i]] - levelsi, by= levelsi), each= length(tmp)) - 1
tmpx[[i]] <- rep(unlist(sapply(1:levelsi, function(i) which(tmp==i))),
reps) +
rep(seq(0, (reps - 1) * length(tmp), by= length(tmp)),
each= length(tmp))
if(i > 1) {
tmpi[[i]] <- tmpi[[i]] + tmpi[[i - 1]][length(tmpi[[i - 1]])] + 1
tmpx[[i]] <- tmpx[[i]] + tmpx[[i - 1]][length(tmpx[[i - 1]])]
}
}
newLambdat@i <- as.integer(unlist(tmpi))
newLambdat@p <- as.integer(unlist(tmpp))
newLambdat@x[unlist(tmpx)] <- newtheta[newLind]
if(!.validateCsparse(newLambdat)) stop("Error in creating newLambdat.")
}
## Old vectors are replaced by new ones.
mtmp$reTrms$Zt <- newZt
mtmp$reTrms$theta <- newtheta
mtmp$reTrms$lower <- newlower
mtmp$reTrms$Lambdat <- newLambdat
mtmp$reTrms$Lind <- newLind
mtmp$reTrms$cnms <- rT$rT
# Manually changed models are fitted using lme4.
if(msgs) cat("Fitting model\n")
if(family[[1]] == "gaussian") {
devfun <- do.call(mkLmerDevfun, mtmp)
opt <- optimizeLmer(devfun)
m <- mkMerMod(environment(devfun), opt, mtmp$reTrms, fr= mtmp$fr)
}
if(family %in% c("binomial", "poisson") ||
family[[1]] %in% c("binomial", "poisson")) {
devfun <- do.call(mkGlmerDevfun, mtmp)
opt <- optimizeGlmer(devfun)
# Fehler: zwei Zeilen eingefuegt am 10.10.
devfun <- updateGlmerDevfun(devfun, mtmp$reTrms)
opt <- optimizeGlmer(devfun, stage= 2)
m <- mkMerMod(environment(devfun), opt, mtmp$reTrms, fr= mtmp$fr)
}
## Data frames with results for random effects variance estimates are set up.
varout <- list("varall"= makeVARALL(nRE, rT, origGF, VCVn, m, family, rd),
"varsel"= NULL)
## Model selection
if(msel != FALSE && all(!VCVNULL)) {
if(missing(gf)) {
gf <- varout$varall$Groups[!(varout$varall$Groups %in% c("", "Residual"))]
if(length(grep(":", gf)) > 0) {
tmp <- strsplit(gf[grep(":", gf)], ":")
gf[grep(":", gf)] <- sapply(1:length(tmp), function(j) tmp[[j]][[1]])
}
}
varout <- modelSelection(msel, varout, nRE, f4selection, call, family, gf,
eps, msgs, rd)
} else {
if(msel != FALSE && any(VCVNULL)) {
cat(paste("Model selection may only be done if all elements of 'VCV' are",
"penalised\n"))
}
}
return(list(modorig= m, varsel= varout$varsel, varall= varout$varall,
modsel= varout$modsel))
}
################################################################################
## SUPPORTING FUNCTIONS
################################################################################
## Changes the formula lFormula from lme4 to allow Gaussian models to be fitted
## with the number of random effects equal to the number of observations.
require("lme4")
assignInNamespace(x= "lFormula",
value=
lFormula <- function (formula, data = NULL, REML = TRUE, subset, weights,
na.action, offset, contrasts = NULL, control = lmerControl(),
...) {
control <- control$checkControl
mf <- mc <- match.call()
ignoreArgs <- c("start", "verbose", "devFunOnly", "control")
l... <- list(...)
l... <- l...[!names(l...) %in% ignoreArgs]
do.call(lme4:::checkArgs, c(list("lmer"), l...))
if (!is.null(list(...)[["family"]])) {
mc[[1]] <- quote(lme4::glFormula)
if (missing(control))
mc[["control"]] <- glmerControl()
return(eval(mc, parent.frame()))
}
denv <- lme4:::checkFormulaData(formula, data)
mc$formula <- formula <- as.formula(formula, env = denv)
m <- match(c("data", "subset", "weights", "na.action", "offset"),
names(mf), 0)
mf <- mf[c(1, m)]
mf$drop.unused.levels <- TRUE
mf[[1]] <- as.name("model.frame")
fr.form <- subbars(formula)
environment(fr.form) <- environment(formula)
mf$formula <- fr.form
fr <- eval(mf, parent.frame())
attr(fr, "formula") <- formula
attr(fr, "offset") <- mf$offset
n <- nrow(fr)
reTrms <- mkReTrms(findbars(formula[[3]]), fr)
# checkNlevels(reTrms$flist, n = n, control)
# checkZrank(reTrms$Zt, n = n, control, nonSmall = 1e+06)
fixedform <- formula
fixedform[[3]] <- if (is.null(nb <- nobars(fixedform[[3]])))
1
else nb
mf$formula <- fixedform
fixedfr <- eval(mf, parent.frame())
attr(attr(fr, "terms"), "predvars.fixed") <- attr(attr(fixedfr,
"terms"), "predvars")
X <- model.matrix(fixedform, fr, contrasts)
p <- ncol(X)
if ((rankX <- rankMatrix(X)) < p)
stop(gettextf("rank of X = %d < ncol(X) = %d", rankX,
p))
list(fr = fr, X = X, reTrms = reTrms, REML = REML, formula = formula)
}, ns= "lme4")
################################################################################
## Model set up
################################################################################
## (g)lFormula() may change the order of the random effects compared to how they
## are passed to the function. To be able to separate random effects with the
## same grouping factor, this function makes a new grouping factor for each
## of these random effects, simply adding a number in ascending order to the
## original name of the grouping factor. The data and formula are changed
## accordingly.
## Required by: psglmm()
## Depends on: nuniq()
##
## Arguments:
## ----------
## formula: A formula from which to extract the random terms, as passed to
## psglmm().
## data: data.frame as passed to psglmm().
##
## Returns: A list containing the new formula and data set using the new
## grouping factor names, the original grouping factor names and a
## processed form of the formula to be ud for model selection.
adaptFormula <- function(formula, data) {
terms <- unlist(strsplit(as.character(formula)[3], " + ", fixed= TRUE))
iRE <- grep("^\\(", terms)
f4selection <- list("y"= as.character(formula)[2],
"fe"= terms[setdiff(1:length(terms), iRE)], "re"= terms[iRE])
nRE <- length(iRE)
termsRE <- terms[iRE]
lastGF <- sapply(1:nRE, function(i) {
rev(unlist(strsplit(gsub("^.+ [|] ([a-zA-Z0-9_.]+)[:]*([a-zA-Z0-9_.]*)\\)$",
"\\1 \\2", termsRE[i]), " ")))[1]
})
if(length(lastGF) != nuniq(lastGF)) {
# how many times was the lastGF used (including its own use)
lastGFsuffix <- gsub("1", "",
sapply(1:nRE, function(i) sum(lastGF[1:i] == lastGF[i])))
lastGFnew <- vector("character", nRE)
for(i in 1:nRE) {
lastGFnew[i] <- paste(lastGF[i], lastGFsuffix[i], sep= "")
}
# new terms replace old
termsRE <- sapply(1:nRE,
function(i) gsub(lastGF[i], lastGFnew[i], termsRE[i]))
terms[iRE] <- termsRE
formula <- as.formula(paste(formula[[2]], formula[[1]],
paste(terms, collapse= "+")))
for(i in which(lastGFsuffix != "")) {
data[, lastGFnew[i]] <- data[, lastGF[i]]
}
}
return(list(formula= formula, data= data, origGF= lastGF,
f4selection= f4selection))
}
## Function for extracting the random effect variables and their grouping
## factors from a formula.
## Required by: psglmm()
## Depends on: nuniq()
##
## Arguments:
## ----------
## formula: A formula from which to extract the random terms, as passed to
## psglmm().
## data: data.frame as passed to psglmm().
## mtmp: Element of class 'merMod' or 'glmerMod'.
## VCV: List of Penalties for random effects as passed to psglmm().
##
## Returns: A list containing a list ('rT') which is similar to the
## '.$reTRms$cnms' element from a call to lFormula(), containing the levels of
## all random effects names according to their grouping factors. It also
## contains a vector 'perm' containing the permutation of the order of the
## random effects from 'formula' as they are used by (g)lmer() and a vector
## 'dims' containing the number of levels of the grouping factor of each
## random effect.
randomTerms <- function(formula, data, mtmp, VCV) {
allTerms <- strsplit(attr(terms(formula), "term.labels"), " ")
iRandom <- which(sapply(1:length(allTerms), function(i) "|" %in% allTerms[[i]]))
if(length(iRandom) == 0) stop("'formula' contains no random effects")
randomTerms <- lapply(iRandom, function(i) allTerms[[i]])
nRE <- length(randomTerms)
rT <- vector("list", nRE)
for(i in 1:nRE) {
names(rT)[i] <- randomTerms[[i]][length(randomTerms[[i]])]
RE <- setdiff(randomTerms[[i]], c("|", "+", "-", "0", "1", names(rT)[i]))
if(is.factor(data[, RE])) {
rT[[i]] <- paste(RE, levels(data[, RE]), sep= "")
} else {
if(!(all(c("0", "+") %in% randomTerms[[i]]) ||
all(c("-", "1") %in% randomTerms[[i]]))) {
rT[[i]] <- c("(Intercept)", RE)
} else rT[[i]] <- RE
}
if(length(RE) != 0 && !(all(c("-", "1") %in% randomTerms[[i]]) ||
all(c("0", "+") %in% randomTerms[[i]])) && is.factor(data[, RE])) {
rT[[i]][1] <- "(Intercept)"
}
if(length(RE) == 0) rT[[i]] <- "(Intercept)"
}
perm <- vector("numeric", nRE)
for(i in 1:nRE) {
perm[i] <- which(sapply(1:nRE,
function(j) {
names(rT)[j] == names(mtmp$reTrms$cnms)[i] &&
all(rT[[j]] == mtmp$reTrms$cnms[[i]])
}))
}
tmp <- strsplit(names(mtmp$reTrms$cnms), ":")
dims= vector("numeric", nRE)
for(i in 1:nRE) {
dims[i] <- prod(sapply(1:length(tmp[[i]]),
function(j) nuniq(data[, tmp[[i]][j]]))) * length(mtmp$reTrms$cnms[[i]])
}
if(length(VCV) != nRE) stop("'VCV' must contain one element per random effect.")
VCVNULL <- sapply(VCV, is.null)
for(i in which(!VCVNULL & sapply(1:nRE, function(i) length(rT[[i]])) > 1)) {
RE <- setdiff(randomTerms[[i]], c("|", "+", "-", "0", "1", names(rT)[i]))
if(!is.factor(data[, RE]) && rT[[i]][1] == "(Intercept)")
stop(paste("Penalising random effects for continous variables with ",
"intercept is not implemented. Remove intercept using '(", RE, " - 1 | ",
names(rT)[i], ")' or remove penalty for this random effect.", sep= ""))
rT[[i]] <- RE
}
rT <- lapply(perm, function(i) rT[[i]])
names(rT) <- names(mtmp$reTrms$cnms)
return(list(rT= rT, perm= perm, dims= dims))
}
## Function to prepare random effects variance estimates.
## Required by: psglmm()
## Depends on:
##
## Arguments:
## ----------
## nRE: The number of random effects in model.
## rT: A list 'rT' with data for the random terms as returned by the
## function randomTerms().
## origGF: Names of the grouping factors of each random effect as passed to
## psglmm().
## VCVn: The names of the random effects penalties as passed to psglmm() in
## the list VCV.
## m: A fitted 'merMod' or 'glmerMod' object.
## family: Family of responses.
## rd: Number of digits to round to.
##
## Returns: A list of two data.frames: one for the variance estimates of all
## random effects and one for the selected random effect if model
## selection is performed.
makeVARALL <- function(nRE, rT, origGF, VCVn, m, family, rd) {
RE <- as.data.frame(matrix(NA, sum(sapply(1:nRE,
function(i) length(rT$rT[[i]]))), 5,
dimnames= list(c(), c("Groups", "Name", "Penalty", "Variance", "Std.Dev"))))
RE$Groups <- unlist(sapply(1:nRE,
function(i) {
c(origGF[i], c(rep("", length(rT$rT[[rT$perm[i]]]) - 1)))
}))
RE$Name <- unlist(sapply(order(rT$perm), function(i) rT$rT[[i]]))
RE$Penalty <- unlist(sapply(rT$perm,
function(i) c(VCVn[i], c(rep("", length(rT$rT[[rT$perm[i]]]) - 1)))))
RE$Variance <- unlist(sapply(order(rT$perm),
function(i) {
sapply(1:length(rT$rT[[i]]), function(j) VarCorr(m)[[i]][j, j])
}))
RE$Std.Dev <- unlist(sapply(order(rT$perm),
function(i) {
sapply(1:length(rT$rT[[i]]),
function(j) attributes(VarCorr(m)[[i]])$stddev[j])
}))
if(family[[1]] == "gaussian") {
REresid <- data.frame("Residual", "", "", attr(VarCorr(m), "sc")^2,
attr(VarCorr(m), "sc"))
colnames(REresid) <- colnames(RE)
RE <- rbind(RE, REresid)
}
if(!is.null(rd)) {
RE$Variance <- round(RE$Variance, rd)
RE$Std.Dev <- round(RE$Std.Dev, rd)
}
return(RE)
}
################################################################################
## Model selection
################################################################################
## Function to perform model selection in psglmm(), i.e. to perform predictive
## cross-validataion and calculate the p-value of the variance estimate for
## the chosen correlation component.
## Required by: psglmm()
## Depends on: pred.xval()
##
## Arguments:
## ----------
## msel: A character vector with the name of the proper scoring rule to
## be used for the predictive cross-validation. For binary data the
## Brier score 'BS' and the logarithmic score 'LS are supported.
## For Gaussian responses 'LS' and the Dawid-Sebastian score 'DSS'
## are supported and for Poisson responses 'DSS' i supported.
## varout: The data frame for the results as returned by a call to
## makeVARALL().
## nRE: The number of random effects which are penalised.
## f4selection: The formula in a processed form as returned by adaptFormula().
## call: List of objects passed to psglmm().
## family: The family used.
## gf: The names of the grouping factors for each random effect.
## eps: The limit, above which variance estimates are consdiered to be
## "greater than zero". Variance estimates very close to zero seem
## to perform very good in the predictive cross-validation even if
## they are not significant.
## msgs: Logical value if messages conserning fitting progress should be
## returned.
## rd: Number of digits to round results to.
##
## Returns: A list of two data frame: one (varall) with the variance estimates
## for all random effects components (with cross-validation score) and
## one (varsel) containing information of the selected variance
## parameter and its p value.
modelSelection <- function(msel, varout, nRE, f4selection, call, family, gf,
eps, msgs, rd) {
if(msgs) cat("Model selection ")
RE <- varout$varall
VS <- varout$varsel
# number of decimal points is obained from 'eps'
ndp <- round(optimize(function(x) abs(10^-x - eps), c(1, 20))$minimum)
methods <- list()
methods$gaussian <- c("DSS", "LS")
methods$binomial <- c("BS", "LS")
methods$poisson <- c("DSS", "LS")
if(sum(RE[!(RE$Groups %in% c("", "Residual")), "Variance"] >= eps) > 1) {
if(!(msel %in% methods[[family[[1]]]]) || msel == TRUE) {
if(msel == TRUE) {
msel <- methods[[family[[1]]]][1]
} else {
warning(paste("Method '", msel, "' is not implemented for '",
family[[1]], "' data. 'msel' set to '", methods[[family[[1]]]][1],
"'.", sep= ""))
msel <- methods[[family[[1]]]][1]
}
}
if(msgs) cat(paste("using method '", msel, "'\n", sep=""))
VCV2test <- which(RE[!(RE$Groups %in% c("", "Residual")),
"Variance"] >= eps)
xval <- rep(NA, nRE)
subm <- vector("list", nRE)
subformula <- subformula.null <- vector("list", nRE)
var0 <- rep(FALSE, nRE)
# Predctive cross-validation is performed for all component with a variance
# estimate > eps
for(i in VCV2test) {
if(msgs) cat(paste(" for covariance component '", names(call$VCV)[i],
"':\n", sep= ""))
subformula[[i]] <- as.formula(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"), ifelse(is.null(f4selection$fe),
"", "+"),
f4selection$re[i]))
subformula.null[[i]] <- as.formula(paste(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"))))
subm[[i]] <- psglmm(subformula[[i]], data= call$data,
VCV= list("test"= call$VCV[[i]]), family= family,
msel= FALSE, eps= eps, msgs= FALSE)
# If a variance estimate is positive, but estimate in subm[[i]] is 0, this
# component should not be tested against the others, its pred.xval would
# be too good.
if(subm[[i]]$varall$Variance[1] > eps) {
xval[i] <- pred.xval(m= subm[[i]]$modorig,
y= as.character(subformula[[i]])[2], gf= gf[i], method= msel,
eps= eps, family= family, msgs= msgs)
if(is.na(xval[i])) break
} else {
var0[i] <- TRUE
if(msgs) cat(" crossvalidation skipped as variance estimated to 0\n")
}
}
if(any(var0)) VCV2test <- VCV2test[!(VCV2test %in% which(var0))]
RE[, msel] <- ""
xvalres <- round(xval, 5)
xvalres[setdiff(1:nRE, VCV2test)] <- ""
xvalres[is.na(xvalres)]<- "NA"
RE[!(RE$Groups %in% c("", "Residual")), msel] <- xvalres
RE[, "Selection"] <- ""
# if any of the results from the predictive cross-validataions returned NA,
# no model selection is performed and also no LR test.
if(!any(is.na(xval[VCV2test])) && length(VCV2test) > 0) {
VCVopt <- which.max(xval)
#? hier ?# component choice but no star or p value?
if(any(xval != 0, na.rm= TRUE) &&
(sum(!is.na(xval)) == length(VCV2test))) {
RE[!(RE$Groups %in% c("", "Residual")), "Selection"][VCVopt] <- "*"
}
ind <- c(VCVopt, which((RE$Groups %in% c("", "Residual"))))
VS <- RE[ind, 1:5]
VS[, c("Variance", "Std.Dev")] <- subm[[VCVopt]]$varall[, c("Variance",
"Std.Dev")]
if(!is.null(rd)) {
VS[, c("Variance", "Std.Dev")] <- round( VS[, c("Variance", "Std.Dev")],
rd)
}
## LR test if variance estimate for optimal covariance component is > 0
ll.alt <- logLik(subm[[VCVopt]]$modorig)[1]
ll.null <- logLik(glm(subformula.null[[VCVopt]], data= call$data,
family= family))[1]
C <- LR.test(ll.alt, ll.null)
VS[, "p (LR)"] <- ""
VS[1, "p (LR)"] <- C[, 2]
modsel <- subm[[VCVopt]]$modorig
} else modsel <- NULL
} else {
if(msgs) cat("\n")
RE[, "Selection"] <- ""
tmp <- round(RE[, "Std.Dev"], ndp)
tmp[RE$Groups %in% c("", "Residual")] <- NA
VCVopt <- which.max(tmp)
if(any(tmp != 0, na.rm= TRUE)) {
RE[VCVopt, "Selection"] <- "**"
ind <- c(VCVopt, which((RE$Groups %in% c("", "Residual"))))
VS <- RE[ind, 1:5]
## LR test if variance estimate for optimal covariance component is > 0
subformula <- as.formula(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"), ifelse(is.null(f4selection$fe),
"", "+"), f4selection$re[VCVopt]))
subformula.null <- as.formula(paste(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"))))
subm <- psglmm(subformula, data= call$data,
VCV= list("test"= call$VCV[[VCVopt]]), family= family, msel= FALSE,
eps= eps, msgs= FALSE)
VS[, c("Variance", "Std.Dev")] <- subm$varall[, c("Variance", "Std.Dev")]
if(!is.null(rd)) {
VS[, c("Variance", "Std.Dev")] <- round( VS[, c("Variance", "Std.Dev")],
rd)
}
ll.alt <- logLik(subm$modorig)[1]
ll.null <- logLik(glm(subformula.null, data= call$data, family= family))[1]
C <- LR.test(ll.alt, ll.null)
VS[, "p (LR)"] <- ""
VS[1, "p (LR)"] <- C[, 2]
modsel <- subm$modorig
} else modsel <- NULL
}
varout$varall <- RE
varout$varsel <- VS
varout$modsel <- modsel
return(varout)
}
## Function to perform LR test
## Required by: modelSelection()
## Depends on:
##
## Arguments:
## ----------
## ll.alt: The log-likelihood of the alternative model
## ll.null: The log-likelihood of the null model
##
## Returns: A data frame with the p value as numeric and text (if the p value
## is < 1e-5 the text string "< 1e-5" is returned.
LR.test <- function(ll.alt, ll.null) {
LR <- 2 * (ll.alt - ll.null)
C <- round(1 - 0.5 * pchisq(LR, 0) - 0.5 * pchisq(LR, 1), 5)
Ctxt <- if(C < 1e-5) paste("<", 1e-5) else as.character(C)
return(data.frame(C, Ctxt, stringsAsFactors= FALSE))
}
################################################################################
## Predictive cross-validation and proper scoring rules
################################################################################
## Function to perform predictive cross-validation as described by Braun et. al
## (2013).
## Required by: modelSelection()
## Depends on:
##
## Arguments:
## ----------
## m: A fitted model of class 'merMod' or 'glmerMod' with one random
# effect.
## y: A character string giving the name of the response variable used in
## m.
## gf: The name of the grouping factor of the rando meffect.
## method: The proper scoring rule to be used ('BS', 'LS' or 'DSS').
## eps: The limit, above which variance estimates are consdiered to be
## "greater than zero". Variance estimates very close to zero seem to
## perform very good in the predictive cross-validation even if they are
## not significant.
## family: The family of the responses.
## msgs: Logical value if messages conserning fitting progress should be
## returned.
##
## Returns: The score of the chosen proper scoring rule or NA id the algorithm
## fails to converge.
pred.xval <- function(m, y, gf, method, eps= 1e-5, family, msgs) {
data <- model.frame(m)
## Binary data saved as a factor is transformed to numerical.
if(family[[1]] == "binomial" & is.factor(data[, y])) {
data[, y] <- as.numeric(data[, y]) - 1
}
if(method %in% c("DSS", "LS") && is.factor(data[, y])) {
warning("Methods 'DSS' and 'LS' not compatible with factors.")
return(NA)
}
## The number of levels for the grouping factor is extracted as 'I'.
gf.levels <- unique(data[, gf])
I <- nlevels(gf.levels)
## If variance estimate is < eps (i.e. considered to be zero) NA is returned.
if(VarCorr(m)[[1]][1] < eps) {
warning("Variance estimate is 0, 'NA' returned.")
return(NA)
}
X <- model.matrix(m) # design matrix fixed effects
Z <- as(t(m@pp$Zt), "matrix") # design matrix random effects
b <- t(matrix(m@pp$b(1.), nrow(data) / I)) # estimated random effects
b <- as.vector(t(b))
scores <- vector("numeric", nrow(data))
error <- FALSE
## Cross-validation is performed.
## Clusters i are the levels of the grouping factor.
for(i in 1:I) {
if(error) break
if(msgs) { # progress bar
pb <- txtProgressBar(style= 3)
setTxtProgressBar(pb, i / I)
}
ind.i <- which(data[, gf] == gf.levels[i]) # observations belonging to level i of grouping factor.
J <- length(ind.i) # number of observations in cluster i
yi <- data[ind.i, y] # response subvector
Xi <- X[ind.i,] # fixed effects subdesign matrix for cluster i
Zi <- Z[ind.i, ind.i] # random effects subdesign matrix for cluster i
beta <- fixef(m) # fixed effects estimates
bi <- b[ind.i] # random effects estimates for cluster i
Q <- VarCorr(m)[[1]][1] * diag(J) # initial Q
## Observations j belonging to cluster i
for(j in 1:J) {
if(error) break
m.old <- bi
Wi <- matrix(0, J, J)
yi.tilde <- vector("numeric", J)
mj <- setdiff(1:J, j) # observations without the j-th ('mj')
sc <- 1
k <- 1
## For family Gaussian.
if(family[[1]] == "gaussian") {
while(sc >= eps) {
for(s in mj) {
yi.tilde[s] <- yi[s] - t(Xi[s,]) %*% beta
}
yimj.tilde <- yi.tilde[mj]
Zimj <- Zi[mj,] # Z_{i, -j}
C <- solve(solve(Q) + t(Zimj) %*% Zimj)
m.new <- C %*% t(Zimj) %*% yimj.tilde
sc <- max(abs(m.new - m.old) / abs(m.old + as.numeric(m.old == 0)))
m.old <- m.new
k <- k + 1
## If algorithm hasn't converged in 10000 iterations it's stopped
if(k > 10000) {
if(msgs) cat("\n")
warning(cat(paste("IWLS algorithm didn't converge in 10000 ",
"iterations for i= ", i, " and j = ", j, ".\n",
"No model selection possible.\n", sep= "")))
error <- TRUE
break
}
}
tau <- t(Xi[j,]) %*% beta + t(Zi[j,]) %*% m.new
sigma2 <- t(Zi[j,]) %*% C %*% Zi[j,]
## Score is calculated using the chosen proper scoring rule
scores[ind.i[j]] <- get(method, mode= "function")(data[ind.i[j], y],
tau, sigma2, family)
}
## For family binomial.
if(family[[1]] == "binomial") {
while(sc >= eps) {
for(s in mj) {
numerator <- exp(t(Xi[s,]) %*% beta + t(Zi[s,]) %*% m.old)
Wi[s, s] <- numerator / (1 + numerator)^2
if(Wi[s, s] == 0) {
tmp <- beta[which.max(abs(beta))]
warning(cat(paste("\nWi[s, s] is zero. Cross-validation stopped.",
" No model selection possible.\n", sep= "")))
error <- TRUE
}
if(error) break
yi.tilde[s] <- t(Zi[s,]) %*% m.old + yi[s] * (1 / Wi[s, s]) -
numerator - 1
}
if(error) break
Wimj <- Wi[mj, mj] # W_{i, -j}
yimj.tilde <- yi.tilde[mj]
Zimj <- Zi[mj,] # Z_{i, -j}
C <- solve(solve(Q) + t(Zimj) %*% Wimj %*% Zimj)
m.new <- C %*% t(Zimj) %*% Wimj %*% yimj.tilde
sc <- max(abs(m.new - m.old) / abs(m.old + as.numeric(m.old == 0)))
m.old <- m.new
k <- k + 1
if(k > 10000) {
if(msgs) cat("\n")
warning(cat(paste("IWLS algorithm didn't converge in 10000 ",
"iterations for i= ", i, " and j = ", j, ".\n",
"No model selection possible.\n", sep= "")))
error <- TRUE
break
}
}
tau <- t(Xi[j,]) %*% beta + t(Zi[j,]) %*% m.new
sigma2 <- t(Zi[j,]) %*% C %*% Zi[j,]
scores[ind.i[j]] <- get(method, mode= "function")(data[ind.i[j], y],
tau, sigma2, family)
}
## For family Poisson.
if(family[[1]] == "poisson") {
while(sc >= eps) {
for(s in mj) {
Wi[s, s] <- exp(t(Xi[s,]) %*% beta + t(Zi[s,]) %*% m.old)
if(Wi[s, s] == 0) {
tmp <- beta[which.max(abs(beta))]
warning(cat(paste("\nWi[s, s] is zero. Cross-validation stopped.",
" No model selection possible.\n", sep= "")))
error <- TRUE
}
if(error) break
yi.tilde[s] <- t(Zi[s,]) %*% m.old + yi[s] * (1 / Wi[s, s]) - 1
}
if(error) break
Wimj <- Wi[mj, mj] # W_{i, -j}
yimj.tilde <- yi.tilde[mj]
Zimj <- Zi[mj,] # Z_{i, -j}
C <- solve(solve(Q) + t(Zimj) %*% Wimj %*% Zimj)
m.new <- C %*% t(Zimj) %*% Wimj %*% yimj.tilde
if(!is.numeric(m.new)) {
cat("\n")
warning(cat("'m.new' not numeric. No model selection possible.\n"))
error <- TRUE
break
}
sc <- max(abs(m.new - m.old) / abs(m.old + as.numeric(m.old == 0)))
m.old <- m.new
k <- k + 1
if(k > 10000) {
if(msgs) cat("\n")
warning(cat(paste("IWLS algorithm didn't converge in 10000 ",
"iterations for i= ", i, " and j = ", j, ".\n",
"No model selection possible.\n", sep= "")))
error <- TRUE
break
}
}
tau <- t(Xi[j,]) %*% beta + t(Zi[j,]) %*% m.new
sigma2 <- t(Zi[j,]) %*% C %*% Zi[j,]
E.lambda <- exp(tau + 0.5 * sigma2)
V.lambda <- (exp(sigma2) - 1) * exp(2 * tau + sigma2)
E.yij <- E.lambda
V.yij <- E.lambda + V.lambda
if(method == "LS") {
# alpha <- E.yij^2 / V.yij
# if(V.yij <= 1) return(NA)
# phi <- E.yij
# scores[ind.i[j]] <- pnbinom(data[ind.i[j], y], size= phi, mu= alpha)
ydistr <- function(lambda, y, mu, var) {
((lambda^y) / factorial(y)) * exp(-lambda) *
(1 / (lambda * sqrt(2 * pi * var))) *
exp(-((log(lambda) - mu)^2) / 2 * var)
}
print("integration start")
scores[ind.i[j]] <- integrate(ydistr, y=data[ind.i[j], y],
mu= E.lambda, var= V.lambda, lower= 0, upper= Inf)
print("integration stop")
}
if(method == "DSS") {
scores[ind.i[j]] <- get(method, mode= "function")(data[ind.i[j], y],
E.yij, V.yij, family)
}
}
}
}
if(msgs) close(pb)
if(error) return(NA) # if algorithm didn't converge NA is returned
return(mean(scores))
}
## Proper scoring rule: Brier Score
## Required by: pred.xval()
## Depends on:
##
## Arguments:
## ----------
## y: A vector of actual responses.
## E: The expectation value obtained through cross-validation.
## V: The variance obtained through cross-validation.
## family: The family of the response. Not used.
##
## Returns: A numeric score. A higher score is better.
BS <- function(y, E, V, family){
brier.score <- -(pnorm(0, mean= E, sd= sqrt(V) + pi^2 / 3 * 225 / 256,
lower.tail= FALSE) - y)^2
return(brier.score)
}
## Proper scoring rule: Logarithmic Score
## Required by: pred.xval()
## Depends on:
##
## Arguments:
## ----------
## y: A vector of actual responses.
## E: The expectation value obtained through cross-validation.
## V: The variance obtained through cross-validation.
## family: The family of the response.
##
## Returns: A numeric score. A higher score is better.
LS <- function(y, E, V, family){
if(family[[1]] == "gaussian") {
ls <- log(dnorm(y, mean= E, sd= sqrt(V)))
}
if(family[[1]] == "binomial") {
ls <- log(dbinom(y, size= 1,
prob= pnorm(0, mean= E, sd= sqrt(V) + pi^2 / 3 * 225 / 256,
lower.tail= FALSE)))
}
if(family[[1]] == "poisson") {
# approximation using gamma distribution!
ls <- log(dpois(y, size= V, mu= E))
}
return(ls)
}
## Proper scoring rule: Dawid Sebastiani Score
## Required by: pred.xval()
## Depends on:
##
## Arguments:
## ----------
## y: A vector of actual responses.
## E: The expectation value obtained through cross-validation.
## V: The variance obtained through cross-validation.
## family: The family of the response. Not used.
##
## Returns: A numeric score. A higher score is better.
DSS <- function(y, E, V, family) {
-0.5 * (log(V) + ((y - E) / sqrt(V))^2)
}
# comment # for Poisson: if y==E and V<1 DSS is positive
################################################################################
## Misc.
################################################################################
## Function to retrieve the number of unique values to a vector 'x'. This as an
## alternative to 'nlevels', when only the number of actually used levels is
## needed.
## Required by: randomTerms(), adaptFormula()
## Depends on:
##
## Arguments:
## ----------
## x: A vector.
##
## Returns: The scalar number of unique values in 'x'.
nuniq <- function(x) {
if(is.factor(x)) x <- as.vector(x)
if(!(is.vector(x) || isMatrix(x))) stop("'x' must be a vector")
return(length(unique(as.vector(x))))
}
## Function to check if all elements of a vector are equal
## Required by:
## Depends on:
##
## Arguments:
## ----------
## x: A vector
##
## Returns: TRUE/FALSE
equal <- function(x) {
if(length(x) == 0 | all(is.na(x))) return(NA)
if(length(x) == 1) return(TRUE)
if(length(unique(x[!is.na(x)])) == 1) return(TRUE) else return(FALSE)
}
## Function returning 'yes' if 'test' is TRUE and 'no' if 'test' is FALSE.
## Contrary to ifelse() this function works for arbitrary objects to be used
## as 'yes' and 'no'.
## Required by:
## Depends on:
##
## Arguments:
## ----------
## test: A logical vector of length one.
## yes: An R object.
## no: An R object.
##
## Returns: The object 'yes' or 'no' depending on the value of 'test'.
IFELSE <- function(test, yes, no) {
if(test) return(yes)
if(!test) return(no)
}
## Function to create polygons of size 1x1 aligned in a rectangular grid.
## Required by:
## Depends on:
##
## Arguments:
## ----------
## rangex: range of plots' lower left horizontal starting coordinate.
## rangey: range of plots' lower left vertical starting coordinate.
##
## Returns: A list of 5x2 matrices containing the coordinates of each plot.
makePolys <- function(rangex, rangey) {
px <- length(rangex)
py <- length(rangey)
p <- px * py
polys <- vector("list", p)
names(polys) <- 1:p
for(i in 1:p) {
j <- ifelse(i %% px == 0, px, i %% px)
k <- (i %/% px) + ifelse(i %% px == 0, 0, 1)
x <- c(rangex[j], rep(rangex[j]+1, 2), rep(rangex[j], 2))
y <- c(rep(rangey[k], 2), rep(rangey[k]+1, 2), rangey[k])
polys[[i]] <- matrix(c(x, y), 5, 2)
}
return(polys)
}
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## Function to fit a phylogenetic and spatial (generalised) linear mixed model
## (PSGLMM) in which the desgin matrices of the random effects are penalised
## using a known group correlation/covariance structure.
##
## Required by:
## Depends on: Packages lme4 and Matrix,
## Formulas adaptFormula(), randomTerms(), makeVARALL() and
## modelSelection() found in 'support.R'
## The function lformula() from lme4 needs to be modified in
## accordance with the file 'lFormula_tweak.R'
## Arguments:
## ----------
## formula: A linear formula as used by '(g)lmer' in package 'lme4'.
## VCV: A list of covariance/correlation matrices (preferably named) with
## wich the design matrices of the random effects are to be penalised.
## The list must contain one element per random effect and in the same
## order as the random effects in 'formula'. If a random effect
## shouldn't be penalised, 'NULL' is used in its corresponding place in
## 'VCV'. The dimensions of the penalty matrices must match the
## dimensions of the design matrices.
## data: A data frame containing the variables named in 'formula'. The 'data'
## argument is not optional.
## family: Error distribution and link function to be used in the model. This
## has to be a family function. Only 'gaussian()', 'binomial()' and
## 'poisson()' are supported.
## gf: An optional character vector containing the names of the grouping
## factors of each random effect.
## rd: To how many digits the variance estimates should be rounded. Takes
## the value 'NULL' if no rounding is to be done.
## msel: The proper scoring rule to be used for predictive cross-validation
## during model selection. For Gaussian responses the logarithmic score
## 'LS' and the Dawid-Sebastiani score 'DSS' are available, 'DSS' for
## Poisson responses and 'DSS' or the Brier score 'BS' for binary
## responses. For 'msel= FALSE' no model selection is performed.
## eps: The limit underneath which random effects variance estimates are to
## be considered as zero.
## msgs: If progress messages should be printed.
## REML: If parameter estimation for Gaussian responses should be REML (TRUE)
## or ML (FALSE).
## ...: Further arguments passed to '(g)lFormula'.
##
## Details:
## --------
## Model selection may only be performed if all random effects components are
## penalised.
## The predictive cross-validation is only performed if more than one random
## effects variance estimate is positive.
## The p value of the estimated variance parameter for the chosen covariance
## component is only calculated if model selection is selected.
## When model selection is performed, the covariance component with the highest
## score is marked with '*' in the column 'Selection' in '.$varall'. If only
## one variance parameter is estimated to be positive, no cross-validation is
## performed and the selected covariance component is marked with '**'.
##
## Value:
## ------
## A list containing
## - A fitted object of class 'merMod' or 'glmerMod'
## - A data frame containing all random effects variance estimates
## - A data frame containing the random effects variance estimate for the
## chosen covariance component if model selection is performed.
##
## See also:
## ---------
## 'lmer', 'glmer'
##
## Examples:
## ---------
##
##
psglmm <- function(formula, VCV= list(), data, family= gaussian(), gf, rd= NULL,
msel= FALSE, eps= 1e-5, msgs= TRUE, REML= FALSE, ...) {
## Required packages are loaded.
require("Matrix")
require("lme4")
## Version number for lme4 is checked.
stopifnot(compareVersion(packageDescription("lme4")$Version, "1.0-0") >= 0)
## Further checks are made.
if(length(VCV) == 0 || is.null(unlist(VCV)))
stop("Use '(g)lmer' if no penalising is wanted.")
if(!is.list(VCV)) stop("'VCV' must be a list.")
if(missing(formula)) stop("'formula' must be assigned.")
if(missing(data)) stop("'data' must be assigned.")
if(is.character(family)) try(family <- get(family, mode= "function"), TRUE)
if(is.function(family)) family <- family()
if(!family[[1]] %in% c("gaussian", "binomial", "poisson"))
stop("'family' must be 'gaussian', 'binomial' or 'poisson'.")
if(!is.null(rd) && rd < 0) stop("'rd' must be a positive integer.")
if(!(msel %in% c(TRUE, FALSE, "BS", "LS", "DSS"))) {
warning(paste("Selection criterium", msel,
"not recognised. No model selection performed."))
msel <- FALSE
}
## Call to psglmm is saved for later use.
if(msgs) cat("Setting up model\n")
call <- list(formula= formula, VCV= VCV, data= data, family= family)
## Formula and data are changed so that random effects with identical grouping
## factor are set up with unique grouping factors.
tmp <- adaptFormula(formula, data)
formula <- tmp$formula
data <- tmp$data
origGF <- tmp$origGF
f4selection <- tmp$f4selection
## Raw model is set up with lme4.
if(family == "gaussian" || family[[1]] == "gaussian") {
# tryCatch(source("lFormula_tweak.R"),
# error= function(e) print(cat(paste("Function lFormula() needs to be",
# "modified. Please load the file 'lformula_tweak.R' manually or place it",
# "in the current working directory.\n"))))
mtmp <- lFormula(formula= formula, data= data, REML= REML, ...)
}
if(family %in% c("binomial", "poisson") || family[[1]] %in% c("binomial", "poisson"))
mtmp <- glFormula(formula= formula, data= data, family= family, ...)
## The permutation of the order of random effects in 'formula' used by
## '(g)lFormula' is retrieved ('perm') together with the number of levels of
## the grouping factors ('dims') and an object ('rT') to replace
## 'mtmp$reTrms$cnms' later.
rT <- randomTerms(formula, data, mtmp, VCV)
nRE <- length(rT$rT) # number of random effects
VCVn <- names(VCV)[rT$perm] # names of penalties
if(is.null(VCVn)) {
VCVn <- as.character(rT$perm)
names(call$VCV) <- as.character(1:nRE)
}
## List of penalties is permuted to fit the order of the random effects used
## by (g)lFormula.
VCV <- lapply(rT$perm, function(i) VCV[[i]])
VCVNULL <- sapply(VCV, is.null)
## Diagonal matrices are added to 'VCV' for random effects which aren't to
## be penalised.
if(any(VCVNULL)) for(i in which(VCVNULL)) {
VCV[[i]] <- as(diag(rT$dims[i]), "dgCMatrix")
}
## A check is made, if the dimensions of the penalty matrices match the
## dimensions of the random effects.
VCVdims <- sapply(VCV, nrow)
if(!all(VCVdims == rT$dims)) {
tmp <- which(VCVdims != rT$dims)
tmp <- if(is.null(VCVn)) tmp else VCVn[tmp]
stop(paste("Dimension of elements",
paste("'",tmp,"'", sep= "", collapse= ", "), "don't match dimensions of",
"design matrices."))
}
## Cholesky factors of all penalty matrices are saved in a list.
CFl <- lapply(1:nRE, function(i) t(chol(VCV[[i]])))
## The Cholesky factors in 'CFl' are put together into a matrix containing all
## Cholesky factors in the right order on its "diagonals".
CF <- bdiag(CFl)
## A new 'Zt' matrix is calculated, which will replace 'mtmp$reTrms$Zt'.
## The old 'mtmp$reTrms$Zt' is penalised with 'CF'.
newZt <- t(crossprod(mtmp$reTrms$Zt, CF))
## New 'theta', 'lower' and 'Lind' vectors are calculated, for convenience
## this is done as lists.
tmp.cnms <- sapply(1:nRE, function(i) sum(1:length(mtmp$reTrms$cnms[[i]])))
tmp.rT <- sapply(1:nRE, function(i) sum(1:length(rT$rT[[i]])))
newtheta <- newlower <- vector("list", nRE)
## Calculations for first random effect outside of loop.
newtheta[[1]] <- mtmp$reTrms$theta[1:cumsum(tmp.cnms)[1]]
newlower[[1]] <- mtmp$reTrms$lower[1:cumsum(tmp.cnms)[1]]
## For all subsequent random effects calculations are made.
if(nRE > 1) for(i in 2:nRE) {
elem.cnms <- (cumsum(tmp.cnms)[(i - 1)] + 1):(cumsum(tmp.cnms)[i])
newtheta[[i]] <- mtmp$reTrms$theta[elem.cnms]
newlower[[i]] <- mtmp$reTrms$lower[elem.cnms]
}
## Elements of 'newtheta' and 'newlower' belonging to penalised random effects
## are set to 1 and 0 respectively, as only one variance parameter should be
## estimated for these.
for(i in which(!VCVNULL)) {
newtheta[[i]] <- 1
newlower[[i]] <- 0
}
newtheta <- unlist(newtheta)
newlower <- unlist(newlower)
## A new 'Lind' vector is calculated.
newLind <- vector("list", nRE)
for(i in 1:nRE) {
nSquare <- (-1 + sqrt(1 + 8 * tmp.rT[i])) / 2
nElem <- cumsum(tmp.rT)[i]
tmp <- sapply(1:nSquare,
function(i) {
(1:nElem)[(1+c(0, cumsum(nSquare:1))[i]):(c(0, cumsum(nSquare:1))[i] +
(nSquare - i + 1))]
})
tmp <- t(sapply(1:nSquare,
function(i) c(rep(0, nSquare - length(tmp[[i]])), tmp[[i]])))
tmp <- as.vector(tmp)
tmp <- tmp[tmp != 0]
tmp <- tmp + c(0, cumsum(tmp.rT))[i]
newLind[[i]] <- rep(tmp, rT$dims[i] / length(rT$rT[[i]]))
}
newLind <- unlist(newLind)
## If all random effects are penalised, 'mtmp$reTrms$Lambdat' is replaced by a
## diagonal matrix. This is also the case if the original matrix is already
## a diagonal matrix. If any of the random effects should not be penalised,
## and a non-diagonal structure of 'mtmp$reTrms$Lambdat' thus needs to be
## kept, the new matrix has to be constructed manually, as it will
## (probably) contain zeros as 'non-zero' elements.
newLambdat <- Matrix(0, nrow(newZt), nrow(newZt), sparse= TRUE)
newLambdat <- as(newLambdat, "dgCMatrix")
diag(newLambdat) <- rep(1, nrow(newZt))
if(!(all(!VCVNULL) ||
(length(mtmp$reTrms$Lambdat@x) == nrow(mtmp$reTrms$Lambdat) &&
all(mtmp$reTrms$Lambdat@x == 1)))) {
tmpi <- tmpp <- tmpx <- vector("list", nRE)
for(i in 1:nRE) {
levelsi <- length(rT$rT[[i]])
reps <- rT$dims[[i]] / levelsi
tmpp[[i]] <- cumsum(rep(1:levelsi, reps))
# number of indices in tmpp[[i-1] is added to the indices in tmpp[[i]]
if(i == 1) tmpp[[i]] <- c(0, tmpp[[i]]) else {
tmpp[[i]] <- tmpp[[i]] + tmpp[[i - 1]][length(tmpp[[i - 1]])]
}
tmp <- unlist(sapply(1:levelsi,
function(j) 1:((1:levelsi)[j])))
tmpi[[i]] <- rep(tmp, reps) +
rep(seq(0, rT$dims[[i]] - levelsi, by= levelsi), each= length(tmp)) - 1
tmpx[[i]] <- rep(unlist(sapply(1:levelsi, function(i) which(tmp==i))),
reps) +
rep(seq(0, (reps - 1) * length(tmp), by= length(tmp)),
each= length(tmp))
if(i > 1) {
tmpi[[i]] <- tmpi[[i]] + tmpi[[i - 1]][length(tmpi[[i - 1]])] + 1
tmpx[[i]] <- tmpx[[i]] + tmpx[[i - 1]][length(tmpx[[i - 1]])]
}
}
newLambdat@i <- as.integer(unlist(tmpi))
newLambdat@p <- as.integer(unlist(tmpp))
newLambdat@x[unlist(tmpx)] <- newtheta[newLind]
if(!.validateCsparse(newLambdat)) stop("Error in creating newLambdat.")
}
## Old vectors are replaced by new ones.
mtmp$reTrms$Zt <- newZt
mtmp$reTrms$theta <- newtheta
mtmp$reTrms$lower <- newlower
mtmp$reTrms$Lambdat <- newLambdat
mtmp$reTrms$Lind <- newLind
mtmp$reTrms$cnms <- rT$rT
# Manually changed models are fitted using lme4.
if(msgs) cat("Fitting model\n")
if(family[[1]] == "gaussian") {
devfun <- do.call(mkLmerDevfun, mtmp)
opt <- optimizeLmer(devfun)
m <- mkMerMod(environment(devfun), opt, mtmp$reTrms, fr= mtmp$fr)
}
if(family %in% c("binomial", "poisson") ||
family[[1]] %in% c("binomial", "poisson")) {
devfun <- do.call(mkGlmerDevfun, mtmp)
opt <- optimizeGlmer(devfun)
# Fehler: zwei Zeilen eingefuegt am 10.10.
devfun <- updateGlmerDevfun(devfun, mtmp$reTrms)
opt <- optimizeGlmer(devfun, stage= 2)
m <- mkMerMod(environment(devfun), opt, mtmp$reTrms, fr= mtmp$fr)
}
## Data frames with results for random effects variance estimates are set up.
varout <- list("varall"= makeVARALL(nRE, rT, origGF, VCVn, m, family, rd),
"varsel"= NULL)
## Model selection
if(msel != FALSE && all(!VCVNULL)) {
if(missing(gf)) {
gf <- varout$varall$Groups[!(varout$varall$Groups %in% c("", "Residual"))]
if(length(grep(":", gf)) > 0) {
tmp <- strsplit(gf[grep(":", gf)], ":")
gf[grep(":", gf)] <- sapply(1:length(tmp), function(j) tmp[[j]][[1]])
}
}
varout <- modelSelection(msel, varout, nRE, f4selection, call, family, gf,
eps, msgs, rd)
} else {
if(msel != FALSE && any(VCVNULL)) {
cat(paste("Model selection may only be done if all elements of 'VCV' are",
"penalised\n"))
}
}
return(list(modorig= m, varsel= varout$varsel, varall= varout$varall,
modsel= varout$modsel))
}
################################################################################
## SUPPORTING FUNCTIONS
################################################################################
## Changes the formula lFormula from lme4 to allow Gaussian models to be fitted
## with the number of random effects equal to the number of observations.
require("lme4")
assignInNamespace(x= "lFormula",
value=
lFormula <- function (formula, data = NULL, REML = TRUE, subset, weights,
na.action, offset, contrasts = NULL, control = lmerControl(),
...) {
control <- control$checkControl
mf <- mc <- match.call()
ignoreArgs <- c("start", "verbose", "devFunOnly", "control")
l... <- list(...)
l... <- l...[!names(l...) %in% ignoreArgs]
do.call(lme4:::checkArgs, c(list("lmer"), l...))
if (!is.null(list(...)[["family"]])) {
mc[[1]] <- quote(lme4::glFormula)
if (missing(control))
mc[["control"]] <- glmerControl()
return(eval(mc, parent.frame()))
}
denv <- lme4:::checkFormulaData(formula, data)
mc$formula <- formula <- as.formula(formula, env = denv)
m <- match(c("data", "subset", "weights", "na.action", "offset"),
names(mf), 0)
mf <- mf[c(1, m)]
mf$drop.unused.levels <- TRUE
mf[[1]] <- as.name("model.frame")
fr.form <- subbars(formula)
environment(fr.form) <- environment(formula)
mf$formula <- fr.form
fr <- eval(mf, parent.frame())
attr(fr, "formula") <- formula
attr(fr, "offset") <- mf$offset
n <- nrow(fr)
reTrms <- mkReTrms(findbars(formula[[3]]), fr)
# checkNlevels(reTrms$flist, n = n, control)
# checkZrank(reTrms$Zt, n = n, control, nonSmall = 1e+06)
fixedform <- formula
fixedform[[3]] <- if (is.null(nb <- nobars(fixedform[[3]])))
1
else nb
mf$formula <- fixedform
fixedfr <- eval(mf, parent.frame())
attr(attr(fr, "terms"), "predvars.fixed") <- attr(attr(fixedfr,
"terms"), "predvars")
X <- model.matrix(fixedform, fr, contrasts)
p <- ncol(X)
if ((rankX <- rankMatrix(X)) < p)
stop(gettextf("rank of X = %d < ncol(X) = %d", rankX,
p))
list(fr = fr, X = X, reTrms = reTrms, REML = REML, formula = formula)
}, ns= "lme4")
################################################################################
## Model set up
################################################################################
## (g)lFormula() may change the order of the random effects compared to how they
## are passed to the function. To be able to separate random effects with the
## same grouping factor, this function makes a new grouping factor for each
## of these random effects, simply adding a number in ascending order to the
## original name of the grouping factor. The data and formula are changed
## accordingly.
## Required by: psglmm()
## Depends on: nuniq()
##
## Arguments:
## ----------
## formula: A formula from which to extract the random terms, as passed to
## psglmm().
## data: data.frame as passed to psglmm().
##
## Returns: A list containing the new formula and data set using the new
## grouping factor names, the original grouping factor names and a
## processed form of the formula to be ud for model selection.
adaptFormula <- function(formula, data) {
terms <- unlist(strsplit(as.character(formula)[3], " + ", fixed= TRUE))
iRE <- grep("^\\(", terms)
f4selection <- list("y"= as.character(formula)[2],
"fe"= terms[setdiff(1:length(terms), iRE)], "re"= terms[iRE])
nRE <- length(iRE)
termsRE <- terms[iRE]
lastGF <- sapply(1:nRE, function(i) {
rev(unlist(strsplit(gsub("^.+ [|] ([a-zA-Z0-9_.]+)[:]*([a-zA-Z0-9_.]*)\\)$",
"\\1 \\2", termsRE[i]), " ")))[1]
})
if(length(lastGF) != nuniq(lastGF)) {
# how many times was the lastGF used (including its own use)
lastGFsuffix <- gsub("1", "",
sapply(1:nRE, function(i) sum(lastGF[1:i] == lastGF[i])))
lastGFnew <- vector("character", nRE)
for(i in 1:nRE) {
lastGFnew[i] <- paste(lastGF[i], lastGFsuffix[i], sep= "")
}
# new terms replace old
termsRE <- sapply(1:nRE,
function(i) gsub(lastGF[i], lastGFnew[i], termsRE[i]))
terms[iRE] <- termsRE
formula <- as.formula(paste(formula[[2]], formula[[1]],
paste(terms, collapse= "+")))
for(i in which(lastGFsuffix != "")) {
data[, lastGFnew[i]] <- data[, lastGF[i]]
}
}
return(list(formula= formula, data= data, origGF= lastGF,
f4selection= f4selection))
}
## Function for extracting the random effect variables and their grouping
## factors from a formula.
## Required by: psglmm()
## Depends on: nuniq()
##
## Arguments:
## ----------
## formula: A formula from which to extract the random terms, as passed to
## psglmm().
## data: data.frame as passed to psglmm().
## mtmp: Element of class 'merMod' or 'glmerMod'.
## VCV: List of Penalties for random effects as passed to psglmm().
##
## Returns: A list containing a list ('rT') which is similar to the
## '.$reTRms$cnms' element from a call to lFormula(), containing the levels of
## all random effects names according to their grouping factors. It also
## contains a vector 'perm' containing the permutation of the order of the
## random effects from 'formula' as they are used by (g)lmer() and a vector
## 'dims' containing the number of levels of the grouping factor of each
## random effect.
randomTerms <- function(formula, data, mtmp, VCV) {
allTerms <- strsplit(attr(terms(formula), "term.labels"), " ")
iRandom <- which(sapply(1:length(allTerms), function(i) "|" %in% allTerms[[i]]))
if(length(iRandom) == 0) stop("'formula' contains no random effects")
randomTerms <- lapply(iRandom, function(i) allTerms[[i]])
nRE <- length(randomTerms)
rT <- vector("list", nRE)
for(i in 1:nRE) {
names(rT)[i] <- randomTerms[[i]][length(randomTerms[[i]])]
RE <- setdiff(randomTerms[[i]], c("|", "+", "-", "0", "1", names(rT)[i]))
if(is.factor(data[, RE])) {
rT[[i]] <- paste(RE, levels(data[, RE]), sep= "")
} else {
if(!(all(c("0", "+") %in% randomTerms[[i]]) ||
all(c("-", "1") %in% randomTerms[[i]]))) {
rT[[i]] <- c("(Intercept)", RE)
} else rT[[i]] <- RE
}
if(length(RE) != 0 && !(all(c("-", "1") %in% randomTerms[[i]]) ||
all(c("0", "+") %in% randomTerms[[i]])) && is.factor(data[, RE])) {
rT[[i]][1] <- "(Intercept)"
}
if(length(RE) == 0) rT[[i]] <- "(Intercept)"
}
perm <- vector("numeric", nRE)
for(i in 1:nRE) {
perm[i] <- which(sapply(1:nRE,
function(j) {
names(rT)[j] == names(mtmp$reTrms$cnms)[i] &&
all(rT[[j]] == mtmp$reTrms$cnms[[i]])
}))
}
tmp <- strsplit(names(mtmp$reTrms$cnms), ":")
dims= vector("numeric", nRE)
for(i in 1:nRE) {
dims[i] <- prod(sapply(1:length(tmp[[i]]),
function(j) nuniq(data[, tmp[[i]][j]]))) * length(mtmp$reTrms$cnms[[i]])
}
if(length(VCV) != nRE) stop("'VCV' must contain one element per random effect.")
VCVNULL <- sapply(VCV, is.null)
for(i in which(!VCVNULL & sapply(1:nRE, function(i) length(rT[[i]])) > 1)) {
RE <- setdiff(randomTerms[[i]], c("|", "+", "-", "0", "1", names(rT)[i]))
if(!is.factor(data[, RE]) && rT[[i]][1] == "(Intercept)")
stop(paste("Penalising random effects for continous variables with ",
"intercept is not implemented. Remove intercept using '(", RE, " - 1 | ",
names(rT)[i], ")' or remove penalty for this random effect.", sep= ""))
rT[[i]] <- RE
}
rT <- lapply(perm, function(i) rT[[i]])
names(rT) <- names(mtmp$reTrms$cnms)
return(list(rT= rT, perm= perm, dims= dims))
}
## Function to prepare random effects variance estimates.
## Required by: psglmm()
## Depends on:
##
## Arguments:
## ----------
## nRE: The number of random effects in model.
## rT: A list 'rT' with data for the random terms as returned by the
## function randomTerms().
## origGF: Names of the grouping factors of each random effect as passed to
## psglmm().
## VCVn: The names of the random effects penalties as passed to psglmm() in
## the list VCV.
## m: A fitted 'merMod' or 'glmerMod' object.
## family: Family of responses.
## rd: Number of digits to round to.
##
## Returns: A list of two data.frames: one for the variance estimates of all
## random effects and one for the selected random effect if model
## selection is performed.
makeVARALL <- function(nRE, rT, origGF, VCVn, m, family, rd) {
RE <- as.data.frame(matrix(NA, sum(sapply(1:nRE,
function(i) length(rT$rT[[i]]))), 5,
dimnames= list(c(), c("Groups", "Name", "Penalty", "Variance", "Std.Dev"))))
RE$Groups <- unlist(sapply(1:nRE,
function(i) {
c(origGF[i], c(rep("", length(rT$rT[[rT$perm[i]]]) - 1)))
}))
RE$Name <- unlist(sapply(order(rT$perm), function(i) rT$rT[[i]]))
RE$Penalty <- unlist(sapply(rT$perm,
function(i) c(VCVn[i], c(rep("", length(rT$rT[[rT$perm[i]]]) - 1)))))
RE$Variance <- unlist(sapply(order(rT$perm),
function(i) {
sapply(1:length(rT$rT[[i]]), function(j) VarCorr(m)[[i]][j, j])
}))
RE$Std.Dev <- unlist(sapply(order(rT$perm),
function(i) {
sapply(1:length(rT$rT[[i]]),
function(j) attributes(VarCorr(m)[[i]])$stddev[j])
}))
if(family[[1]] == "gaussian") {
REresid <- data.frame("Residual", "", "", attr(VarCorr(m), "sc")^2,
attr(VarCorr(m), "sc"))
colnames(REresid) <- colnames(RE)
RE <- rbind(RE, REresid)
}
if(!is.null(rd)) {
RE$Variance <- round(RE$Variance, rd)
RE$Std.Dev <- round(RE$Std.Dev, rd)
}
return(RE)
}
################################################################################
## Model selection
################################################################################
## Function to perform model selection in psglmm(), i.e. to perform predictive
## cross-validataion and calculate the p-value of the variance estimate for
## the chosen correlation component.
## Required by: psglmm()
## Depends on: pred.xval()
##
## Arguments:
## ----------
## msel: A character vector with the name of the proper scoring rule to
## be used for the predictive cross-validation. For binary data the
## Brier score 'BS' and the logarithmic score 'LS are supported.
## For Gaussian responses 'LS' and the Dawid-Sebastian score 'DSS'
## are supported and for Poisson responses 'DSS' i supported.
## varout: The data frame for the results as returned by a call to
## makeVARALL().
## nRE: The number of random effects which are penalised.
## f4selection: The formula in a processed form as returned by adaptFormula().
## call: List of objects passed to psglmm().
## family: The family used.
## gf: The names of the grouping factors for each random effect.
## eps: The limit, above which variance estimates are consdiered to be
## "greater than zero". Variance estimates very close to zero seem
## to perform very good in the predictive cross-validation even if
## they are not significant.
## msgs: Logical value if messages conserning fitting progress should be
## returned.
## rd: Number of digits to round results to.
##
## Returns: A list of two data frame: one (varall) with the variance estimates
## for all random effects components (with cross-validation score) and
## one (varsel) containing information of the selected variance
## parameter and its p value.
modelSelection <- function(msel, varout, nRE, f4selection, call, family, gf,
eps, msgs, rd) {
if(msgs) cat("Model selection ")
RE <- varout$varall
VS <- varout$varsel
# number of decimal points is obained from 'eps'
ndp <- round(optimize(function(x) abs(10^-x - eps), c(1, 20))$minimum)
methods <- list()
methods$gaussian <- c("DSS", "LS")
methods$binomial <- c("BS", "LS")
methods$poisson <- c("DSS", "LS")
if(sum(RE[!(RE$Groups %in% c("", "Residual")), "Variance"] >= eps) > 1) {
if(!(msel %in% methods[[family[[1]]]]) || msel == TRUE) {
if(msel == TRUE) {
msel <- methods[[family[[1]]]][1]
} else {
warning(paste("Method '", msel, "' is not implemented for '",
family[[1]], "' data. 'msel' set to '", methods[[family[[1]]]][1],
"'.", sep= ""))
msel <- methods[[family[[1]]]][1]
}
}
if(msgs) cat(paste("using method '", msel, "'\n", sep=""))
VCV2test <- which(RE[!(RE$Groups %in% c("", "Residual")),
"Variance"] >= eps)
xval <- rep(NA, nRE)
subm <- vector("list", nRE)
subformula <- subformula.null <- vector("list", nRE)
var0 <- rep(FALSE, nRE)
# Predctive cross-validation is performed for all component with a variance
# estimate > eps
for(i in VCV2test) {
if(msgs) cat(paste(" for covariance component '", names(call$VCV)[i],
"':\n", sep= ""))
subformula[[i]] <- as.formula(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"), ifelse(is.null(f4selection$fe),
"", "+"),
f4selection$re[i]))
subformula.null[[i]] <- as.formula(paste(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"))))
subm[[i]] <- psglmm(subformula[[i]], data= call$data,
VCV= list("test"= call$VCV[[i]]), family= family,
msel= FALSE, eps= eps, msgs= FALSE)
# If a variance estimate is positive, but estimate in subm[[i]] is 0, this
# component should not be tested against the others, its pred.xval would
# be too good.
if(subm[[i]]$varall$Variance[1] > eps) {
xval[i] <- pred.xval(m= subm[[i]]$modorig,
y= as.character(subformula[[i]])[2], gf= gf[i], method= msel,
eps= eps, family= family, msgs= msgs)
if(is.na(xval[i])) break
} else {
var0[i] <- TRUE
if(msgs) cat(" crossvalidation skipped as variance estimated to 0\n")
}
}
if(any(var0)) VCV2test <- VCV2test[!(VCV2test %in% which(var0))]
RE[, msel] <- ""
xvalres <- round(xval, 5)
xvalres[setdiff(1:nRE, VCV2test)] <- ""
xvalres[is.na(xvalres)]<- "NA"
RE[!(RE$Groups %in% c("", "Residual")), msel] <- xvalres
RE[, "Selection"] <- ""
# if any of the results from the predictive cross-validataions returned NA,
# no model selection is performed and also no LR test.
if(!any(is.na(xval[VCV2test])) && length(VCV2test) > 0) {
VCVopt <- which.max(xval)
#? hier ?# component choice but no star or p value?
if(any(xval != 0, na.rm= TRUE) &&
(sum(!is.na(xval)) == length(VCV2test))) {
RE[!(RE$Groups %in% c("", "Residual")), "Selection"][VCVopt] <- "*"
}
ind <- c(VCVopt, which((RE$Groups %in% c("", "Residual"))))
VS <- RE[ind, 1:5]
VS[, c("Variance", "Std.Dev")] <- subm[[VCVopt]]$varall[, c("Variance",
"Std.Dev")]
if(!is.null(rd)) {
VS[, c("Variance", "Std.Dev")] <- round( VS[, c("Variance", "Std.Dev")],
rd)
}
## LR test if variance estimate for optimal covariance component is > 0
ll.alt <- logLik(subm[[VCVopt]]$modorig)[1]
ll.null <- logLik(glm(subformula.null[[VCVopt]], data= call$data,
family= family))[1]
C <- LR.test(ll.alt, ll.null)
VS[, "p (LR)"] <- ""
VS[1, "p (LR)"] <- C[, 2]
modsel <- subm[[VCVopt]]$modorig
} else modsel <- NULL
} else {
if(msgs) cat("\n")
RE[, "Selection"] <- ""
tmp <- round(RE[, "Std.Dev"], ndp)
tmp[RE$Groups %in% c("", "Residual")] <- NA
VCVopt <- which.max(tmp)
if(any(tmp != 0, na.rm= TRUE)) {
RE[VCVopt, "Selection"] <- "**"
ind <- c(VCVopt, which((RE$Groups %in% c("", "Residual"))))
VS <- RE[ind, 1:5]
## LR test if variance estimate for optimal covariance component is > 0
subformula <- as.formula(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"), ifelse(is.null(f4selection$fe),
"", "+"), f4selection$re[VCVopt]))
subformula.null <- as.formula(paste(paste(f4selection$y, "~",
paste(f4selection$fe, collapse= "+"))))
subm <- psglmm(subformula, data= call$data,
VCV= list("test"= call$VCV[[VCVopt]]), family= family, msel= FALSE,
eps= eps, msgs= FALSE)
VS[, c("Variance", "Std.Dev")] <- subm$varall[, c("Variance", "Std.Dev")]
if(!is.null(rd)) {
VS[, c("Variance", "Std.Dev")] <- round( VS[, c("Variance", "Std.Dev")],
rd)
}
ll.alt <- logLik(subm$modorig)[1]
ll.null <- logLik(glm(subformula.null, data= call$data, family= family))[1]
C <- LR.test(ll.alt, ll.null)
VS[, "p (LR)"] <- ""
VS[1, "p (LR)"] <- C[, 2]
modsel <- subm$modorig
} else modsel <- NULL
}
varout$varall <- RE
varout$varsel <- VS
varout$modsel <- modsel
return(varout)
}
## Function to perform LR test
## Required by: modelSelection()
## Depends on:
##
## Arguments:
## ----------
## ll.alt: The log-likelihood of the alternative model
## ll.null: The log-likelihood of the null model
##
## Returns: A data frame with the p value as numeric and text (if the p value
## is < 1e-5 the text string "< 1e-5" is returned.
LR.test <- function(ll.alt, ll.null) {
LR <- 2 * (ll.alt - ll.null)
C <- round(1 - 0.5 * pchisq(LR, 0) - 0.5 * pchisq(LR, 1), 5)
Ctxt <- if(C < 1e-5) paste("<", 1e-5) else as.character(C)
return(data.frame(C, Ctxt, stringsAsFactors= FALSE))
}
################################################################################
## Predictive cross-validation and proper scoring rules
################################################################################
## Function to perform predictive cross-validation as described by Braun et. al
## (2013).
## Required by: modelSelection()
## Depends on:
##
## Arguments:
## ----------
## m: A fitted model of class 'merMod' or 'glmerMod' with one random
# effect.
## y: A character string giving the name of the response variable used in
## m.
## gf: The name of the grouping factor of the rando meffect.
## method: The proper scoring rule to be used ('BS', 'LS' or 'DSS').
## eps: The limit, above which variance estimates are consdiered to be
## "greater than zero". Variance estimates very close to zero seem to
## perform very good in the predictive cross-validation even if they are
## not significant.
## family: The family of the responses.
## msgs: Logical value if messages conserning fitting progress should be
## returned.
##
## Returns: The score of the chosen proper scoring rule or NA id the algorithm
## fails to converge.
pred.xval <- function(m, y, gf, method, eps= 1e-5, family, msgs) {
data <- model.frame(m)
## Binary data saved as a factor is transformed to numerical.
if(family[[1]] == "binomial" & is.factor(data[, y])) {
data[, y] <- as.numeric(data[, y]) - 1
}
if(method %in% c("DSS", "LS") && is.factor(data[, y])) {
warning("Methods 'DSS' and 'LS' not compatible with factors.")
return(NA)
}
## The number of levels for the grouping factor is extracted as 'I'.
gf.levels <- unique(data[, gf])
I <- nlevels(gf.levels)
## If variance estimate is < eps (i.e. considered to be zero) NA is returned.
if(VarCorr(m)[[1]][1] < eps) {
warning("Variance estimate is 0, 'NA' returned.")
return(NA)
}
X <- model.matrix(m) # design matrix fixed effects
Z <- as(t(m@pp$Zt), "matrix") # design matrix random effects
b <- t(matrix(m@pp$b(1.), nrow(data) / I)) # estimated random effects
b <- as.vector(t(b))
scores <- vector("numeric", nrow(data))
error <- FALSE
## Cross-validation is performed.
## Clusters i are the levels of the grouping factor.
for(i in 1:I) {
if(error) break
if(msgs) { # progress bar
pb <- txtProgressBar(style= 3)
setTxtProgressBar(pb, i / I)
}
ind.i <- which(data[, gf] == gf.levels[i]) # observations belonging to level i of grouping factor.
J <- length(ind.i) # number of observations in cluster i
yi <- data[ind.i, y] # response subvector
Xi <- X[ind.i,] # fixed effects subdesign matrix for cluster i
Zi <- Z[ind.i, ind.i] # random effects subdesign matrix for cluster i
beta <- fixef(m) # fixed effects estimates
bi <- b[ind.i] # random effects estimates for cluster i
Q <- VarCorr(m)[[1]][1] * diag(J) # initial Q
## Observations j belonging to cluster i
for(j in 1:J) {
if(error) break
m.old <- bi
Wi <- matrix(0, J, J)
yi.tilde <- vector("numeric", J)
mj <- setdiff(1:J, j) # observations without the j-th ('mj')
sc <- 1
k <- 1
## For family Gaussian.
if(family[[1]] == "gaussian") {
while(sc >= eps) {
for(s in mj) {
yi.tilde[s] <- yi[s] - t(Xi[s,]) %*% beta
}
yimj.tilde <- yi.tilde[mj]
Zimj <- Zi[mj,] # Z_{i, -j}
C <- solve(solve(Q) + t(Zimj) %*% Zimj)
m.new <- C %*% t(Zimj) %*% yimj.tilde
sc <- max(abs(m.new - m.old) / abs(m.old + as.numeric(m.old == 0)))
m.old <- m.new
k <- k + 1
## If algorithm hasn't converged in 10000 iterations it's stopped
if(k > 10000) {
if(msgs) cat("\n")
warning(cat(paste("IWLS algorithm didn't converge in 10000 ",
"iterations for i= ", i, " and j = ", j, ".\n",
"No model selection possible.\n", sep= "")))
error <- TRUE
break
}
}
tau <- t(Xi[j,]) %*% beta + t(Zi[j,]) %*% m.new
sigma2 <- t(Zi[j,]) %*% C %*% Zi[j,]
## Score is calculated using the chosen proper scoring rule
scores[ind.i[j]] <- get(method, mode= "function")(data[ind.i[j], y],
tau, sigma2, family)
}
## For family binomial.
if(family[[1]] == "binomial") {
while(sc >= eps) {
for(s in mj) {
numerator <- exp(t(Xi[s,]) %*% beta + t(Zi[s,]) %*% m.old)
Wi[s, s] <- numerator / (1 + numerator)^2
if(Wi[s, s] == 0) {
tmp <- beta[which.max(abs(beta))]
warning(cat(paste("\nWi[s, s] is zero. Cross-validation stopped.",
" No model selection possible.\n", sep= "")))
error <- TRUE
}
if(error) break
yi.tilde[s] <- t(Zi[s,]) %*% m.old + yi[s] * (1 / Wi[s, s]) -
numerator - 1
}
if(error) break
Wimj <- Wi[mj, mj] # W_{i, -j}
yimj.tilde <- yi.tilde[mj]
Zimj <- Zi[mj,] # Z_{i, -j}
C <- solve(solve(Q) + t(Zimj) %*% Wimj %*% Zimj)
m.new <- C %*% t(Zimj) %*% Wimj %*% yimj.tilde
sc <- max(abs(m.new - m.old) / abs(m.old + as.numeric(m.old == 0)))
m.old <- m.new
k <- k + 1
if(k > 10000) {
if(msgs) cat("\n")
warning(cat(paste("IWLS algorithm didn't converge in 10000 ",
"iterations for i= ", i, " and j = ", j, ".\n",
"No model selection possible.\n", sep= "")))
error <- TRUE
break
}
}
tau <- t(Xi[j,]) %*% beta + t(Zi[j,]) %*% m.new
sigma2 <- t(Zi[j,]) %*% C %*% Zi[j,]
scores[ind.i[j]] <- get(method, mode= "function")(data[ind.i[j], y],
tau, sigma2, family)
}
## For family Poisson.
if(family[[1]] == "poisson") {
while(sc >= eps) {
for(s in mj) {
Wi[s, s] <- exp(t(Xi[s,]) %*% beta + t(Zi[s,]) %*% m.old)
if(Wi[s, s] == 0) {
tmp <- beta[which.max(abs(beta))]
warning(cat(paste("\nWi[s, s] is zero. Cross-validation stopped.",
" No model selection possible.\n", sep= "")))
error <- TRUE
}
if(error) break
yi.tilde[s] <- t(Zi[s,]) %*% m.old + yi[s] * (1 / Wi[s, s]) - 1
}
if(error) break
Wimj <- Wi[mj, mj] # W_{i, -j}
yimj.tilde <- yi.tilde[mj]
Zimj <- Zi[mj,] # Z_{i, -j}
C <- solve(solve(Q) + t(Zimj) %*% Wimj %*% Zimj)
m.new <- C %*% t(Zimj) %*% Wimj %*% yimj.tilde
if(!is.numeric(m.new)) {
cat("\n")
warning(cat("'m.new' not numeric. No model selection possible.\n"))
error <- TRUE
break
}
sc <- max(abs(m.new - m.old) / abs(m.old + as.numeric(m.old == 0)))
m.old <- m.new
k <- k + 1
if(k > 10000) {
if(msgs) cat("\n")
warning(cat(paste("IWLS algorithm didn't converge in 10000 ",
"iterations for i= ", i, " and j = ", j, ".\n",
"No model selection possible.\n", sep= "")))
error <- TRUE
break
}
}
tau <- t(Xi[j,]) %*% beta + t(Zi[j,]) %*% m.new
sigma2 <- t(Zi[j,]) %*% C %*% Zi[j,]
E.lambda <- exp(tau + 0.5 * sigma2)
V.lambda <- (exp(sigma2) - 1) * exp(2 * tau + sigma2)
E.yij <- E.lambda
V.yij <- E.lambda + V.lambda
if(method == "LS") {
# alpha <- E.yij^2 / V.yij
# if(V.yij <= 1) return(NA)
# phi <- E.yij
# scores[ind.i[j]] <- pnbinom(data[ind.i[j], y], size= phi, mu= alpha)
ydistr <- function(lambda, y, mu, var) {
((lambda^y) / factorial(y)) * exp(-lambda) *
(1 / (lambda * sqrt(2 * pi * var))) *
exp(-((log(lambda) - mu)^2) / 2 * var)
}
print("integration start")
scores[ind.i[j]] <- integrate(ydistr, y=data[ind.i[j], y],
mu= E.lambda, var= V.lambda, lower= 0, upper= Inf)
print("integration stop")
}
if(method == "DSS") {
scores[ind.i[j]] <- get(method, mode= "function")(data[ind.i[j], y],
E.yij, V.yij, family)
}
}
}
}
if(msgs) close(pb)
if(error) return(NA) # if algorithm didn't converge NA is returned
return(mean(scores))
}
## Proper scoring rule: Brier Score
## Required by: pred.xval()
## Depends on:
##
## Arguments:
## ----------
## y: A vector of actual responses.
## E: The expectation value obtained through cross-validation.
## V: The variance obtained through cross-validation.
## family: The family of the response. Not used.
##
## Returns: A numeric score. A higher score is better.
BS <- function(y, E, V, family){
brier.score <- -(pnorm(0, mean= E, sd= sqrt(V) + pi^2 / 3 * 225 / 256,
lower.tail= FALSE) - y)^2
return(brier.score)
}
## Proper scoring rule: Logarithmic Score
## Required by: pred.xval()
## Depends on:
##
## Arguments:
## ----------
## y: A vector of actual responses.
## E: The expectation value obtained through cross-validation.
## V: The variance obtained through cross-validation.
## family: The family of the response.
##
## Returns: A numeric score. A higher score is better.
LS <- function(y, E, V, family){
if(family[[1]] == "gaussian") {
ls <- log(dnorm(y, mean= E, sd= sqrt(V)))
}
if(family[[1]] == "binomial") {
ls <- log(dbinom(y, size= 1,
prob= pnorm(0, mean= E, sd= sqrt(V) + pi^2 / 3 * 225 / 256,
lower.tail= FALSE)))
}
if(family[[1]] == "poisson") {
# approximation using gamma distribution!
ls <- log(dpois(y, size= V, mu= E))
}
return(ls)
}
## Proper scoring rule: Dawid Sebastiani Score
## Required by: pred.xval()
## Depends on:
##
## Arguments:
## ----------
## y: A vector of actual responses.
## E: The expectation value obtained through cross-validation.
## V: The variance obtained through cross-validation.
## family: The family of the response. Not used.
##
## Returns: A numeric score. A higher score is better.
DSS <- function(y, E, V, family) {
-0.5 * (log(V) + ((y - E) / sqrt(V))^2)
}
# comment # for Poisson: if y==E and V<1 DSS is positive
################################################################################
## Misc.
################################################################################
## Function to retrieve the number of unique values to a vector 'x'. This as an
## alternative to 'nlevels', when only the number of actually used levels is
## needed.
## Required by: randomTerms(), adaptFormula()
## Depends on:
##
## Arguments:
## ----------
## x: A vector.
##
## Returns: The scalar number of unique values in 'x'.
nuniq <- function(x) {
if(is.factor(x)) x <- as.vector(x)
if(!(is.vector(x) || isMatrix(x))) stop("'x' must be a vector")
return(length(unique(as.vector(x))))
}
## Function to check if all elements of a vector are equal
## Required by:
## Depends on:
##
## Arguments:
## ----------
## x: A vector
##
## Returns: TRUE/FALSE
equal <- function(x) {
if(length(x) == 0 | all(is.na(x))) return(NA)
if(length(x) == 1) return(TRUE)
if(length(unique(x[!is.na(x)])) == 1) return(TRUE) else return(FALSE)
}
## Function returning 'yes' if 'test' is TRUE and 'no' if 'test' is FALSE.
## Contrary to ifelse() this function works for arbitrary objects to be used
## as 'yes' and 'no'.
## Required by:
## Depends on:
##
## Arguments:
## ----------
## test: A logical vector of length one.
## yes: An R object.
## no: An R object.
##
## Returns: The object 'yes' or 'no' depending on the value of 'test'.
IFELSE <- function(test, yes, no) {
if(test) return(yes)
if(!test) return(no)
}
## Function to create polygons of size 1x1 aligned in a rectangular grid.
## Required by:
## Depends on:
##
## Arguments:
## ----------
## rangex: range of plots' lower left horizontal starting coordinate.
## rangey: range of plots' lower left vertical starting coordinate.
##
## Returns: A list of 5x2 matrices containing the coordinates of each plot.
makePolys <- function(rangex, rangey) {
px <- length(rangex)
py <- length(rangey)
p <- px * py
polys <- vector("list", p)
names(polys) <- 1:p
for(i in 1:p) {
j <- ifelse(i %% px == 0, px, i %% px)
k <- (i %/% px) + ifelse(i %% px == 0, 0, 1)
x <- c(rangex[j], rep(rangex[j]+1, 2), rep(rangex[j], 2))
y <- c(rep(rangey[k], 2), rep(rangey[k]+1, 2), rangey[k])
polys[[i]] <- matrix(c(x, y), 5, 2)
}
return(polys)
}
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