lme4/0000755000175100001440000000000012604662535011142 5ustar hornikuserslme4/inst/0000755000175100001440000000000012604503702012105 5ustar hornikuserslme4/inst/CITATION0000754000175100001440000000172712604503173013255 0ustar hornikusersbibentry(bibtype = "Article", title = "Fitting Linear Mixed-Effects Models Using {lme4}", author = c(person(given = "Douglas", family = "Bates"), person(given = "Martin", family = "M{\\\"a}chler"), person(given = "Ben", family = "Bolker"), person(given = "Steve", family = "Walker")), journal = "Journal of Statistical Software", year = "2015", volume = "67", number = "1", pages = "1--48", doi = "10.18637/jss.v067.i01", header = "To cite lme4 in publications use:", textVersion = paste("Douglas Bates, Martin Maechler, Ben Bolker, Steve Walker (2015).", "Fitting Linear Mixed-Effects Models Using lme4.", "Journal of Statistical Software, 67(1), 1-48.", "doi:10.18637/jss.v067.i01.") ) lme4/inst/tests/0000755000175100001440000000000012604534725013260 5ustar hornikuserslme4/inst/tests/test-resids.R0000754000175100001440000000555012406350752015654 0ustar hornikuserslibrary("testthat") library("lme4") context("residuals") test_that("lmer", { fm1 <- lmer(Reaction ~ Days + (Days|Subject),sleepstudy) ## fm2 <- lmer(Reaction ~ Days + (Days|Subject),sleepstudy, ## control=lmerControl(calc.derivs=FALSE)) ## all.equal(resid(fm1),resid(fm2)) expect_equal(range(resid(fm1)), c(-101.1789,132.5466), tolerance=1e-6) expect_equal(range(resid(fm1, scaled=TRUE)), c(-3.953567,5.179260), tolerance=1e-6) expect_equal(resid(fm1,"response"),resid(fm1)) expect_equal(resid(fm1,"response"),resid(fm1,type="working")) expect_equal(resid(fm1,"deviance"),resid(fm1,type="pearson")) expect_equal(resid(fm1),resid(fm1,type="pearson")) ## because no weights given expect_error(residuals(fm1,"partial"), "partial residuals are not implemented yet") sleepstudyNA <- sleepstudy na_ind <- c(10,50) sleepstudyNA[na_ind,"Days"] <- NA fm1NA <- update(fm1,data=sleepstudyNA) fm1NA_exclude <- update(fm1,data=sleepstudyNA,na.action="na.exclude") expect_equal(length(resid(fm1)),length(resid(fm1NA_exclude))) expect_true(all(is.na(resid(fm1NA_exclude)[na_ind]))) expect_true(!any(is.na(resid(fm1NA_exclude)[-na_ind]))) }) test_that("glmer", { gm1 <- glmer(incidence/size ~ period + (1|herd), cbpp, family=binomial, weights=size) gm2 <- update(gm1,control=glmerControl(calc.derivs=FALSE)) gm1.old <- update(gm1,control=glmerControl(calc.derivs=FALSE, use.last.params=TRUE)) expect_equal(resid(gm1),resid(gm2)) ## y, wtres, mu change ?? ## FIX ME:: why does turning on derivative calculation make these tests fail??? expect_equal(range(resid(gm1.old)), c(-3.197512,2.356677), tolerance=1e-6) expect_equal(range(resid(gm1)), c(-3.1975034,2.35668826), tolerance=1e-6) expect_equal(range(resid(gm1.old, "response")), c(-0.1946736,0.3184579), tolerance=1e-6) expect_equal(range(resid(gm1,"response")),c(-0.194674747774946, 0.318458889275477)) expect_equal(range(resid(gm1.old, "pearson")), c(-2.381643,2.879069),tolerance=1e-5) expect_equal(range(resid(gm1,"pearson")), c(-2.38163599828335, 2.87908806084918)) expect_equal(range(resid(gm1.old, "working")), c(-1.241733,5.410587),tolerance=1e-5) expect_equal(range(resid(gm1, "working")), c(-1.24173431447365, 5.41064465283686)) expect_equal(resid(gm1),resid(gm1,scaled=TRUE)) ## since sigma==1 expect_error(resid(gm1,"partial"), "partial residuals are not implemented yet") cbppNA <- cbpp na_ind <- c(10,50) cbppNA[na_ind,"period"] <- NA gm1NA <- update(gm1,data=cbppNA) gm1NA_exclude <- update(gm1,data=cbppNA,na.action="na.exclude") expect_equal(length(resid(gm1)),length(resid(gm1NA_exclude))) expect_true(all(is.na(resid(gm1NA_exclude)[na_ind]))) expect_true(!any(is.na(resid(gm1NA_exclude)[-na_ind]))) }) lme4/inst/tests/test-utils.R0000754000175100001440000000427512565075351015533 0ustar hornikuserslibrary("testthat") library("lme4") context("Utilities (including *non*-exported ones") test_that("namedList", { nList <- lme4:::namedList a <- b <- c <- 1 expect_identical(nList(a,b,c), list(a = 1, b = 1, c = 1)) expect_identical(nList(a,b,d=c),list(a = 1, b = 1, d = 1)) expect_identical(nList(a, d=pi, c), list(a = 1, d = pi, c = 1)) }) test_that("Var-Cov factor conversions", { ## from ../../R/vcconv.R mlist2vec <- lme4:::mlist2vec Cv_to_Vv <- lme4:::Cv_to_Vv Cv_to_Sv <- lme4:::Cv_to_Sv Sv_to_Cv <- lme4:::Sv_to_Cv Vv_to_Cv <- lme4:::Vv_to_Cv ## set.seed(1); cvec1 <- sample(10, 6) v1 <- Cv_to_Vv(cvec1) expect_equal(unname(v1), structure(c(9, 12, 15, 65, 34, 93), clen = 3)) expect_equal(2, as.vector(Vv_to_Cv(Cv_to_Vv(2)))) expect_equivalent(c(v1, 1), Cv_to_Vv(cvec1, s=3) / 3^2) expect_equal(as.vector(ss1 <- Sv_to_Cv(Cv_to_Sv(cvec1))), cvec1) expect_equal(as.vector(vv1 <- Vv_to_Cv(Cv_to_Vv(cvec1))), cvec1) ## for length-1 matrices, Cv_to_Sv should be equivalent ## to multiplying Cv by sigma and appending sigma .... clist2 <- list(matrix(1),matrix(2),matrix(3)) cvec2 <- mlist2vec(clist2) expect_equal(cvec2, structure(1:3, clen = rep(1,3)), tolerance=0) expect_true(all((cvec3 <- Cv_to_Sv(cvec2, s=2)) == c(cvec2*2,2))) n3 <- length(cvec3) expect_equivalent(Sv_to_Cv(cvec3, n=rep(1,3), s=2), cvec3[-n3]/cvec3[n3]) }) test_that("nobar", { rr <- lme4:::RHSForm expect_equal(nobars(y~1+(1|g)), y~1) expect_equal(nobars(y~1|g), y~1) expect_equal(nobars(y~1+(1||g)), y~1) expect_equal(nobars(y~1||g), y~1) expect_equal(nobars(y~1+(x:z|g)), y~1) expect_equal(nobars(y~1+(x*z|g/h)), y~1) expect_equal(nobars(y~(1|g)+x+(x|h)), y~x) expect_equal(nobars(y~(1|g)+x+(x+z|h)), y~x) expect_equal(nobars(~1+(1|g)), ~1) expect_equal(nobars(~(1|g)), ~1) expect_equal(nobars(rr(y~1+(1|g))), 1) expect_equal(nobars(rr(y~(1|g))), 1) }) lme4/inst/tests/test-doubleVertNotation.R0000754000175100001440000000430012406350752020202 0ustar hornikuserslibrary("lme4") library("testthat") context("testing '||' notation for independent ranefs") test_that("basic intercept + slope '||' works", { expect_equivalent( lFormula(Reaction ~ Days + (Days||Subject), sleepstudy)$reTrms, lFormula(Reaction ~ Days + (1|Subject) + (0 + Days|Subject), sleepstudy)$reTrms, ) expect_equivalent( fitted(lmer(Reaction ~ Days + (Days||Subject), sleepstudy)), fitted(lmer(Reaction ~ Days + (1|Subject) + (0 + Days|Subject), sleepstudy)), ) }) test_that("'||' works with nested, multiple, or interaction terms" , { #works with nested expect_equivalent(findbars(y ~ (x || id / id2)), findbars(y ~ (1 | id / id2) + (0 + x | id / id2))) #works with multiple expect_equivalent(findbars(y ~ (x1 + x2 || id / id2) + (x3 | id3) + (x4 || id4)), findbars(y ~ (1 | id / id2) + (0 + x1 | id / id2) + (0 + x2 | id / id2) + (x3 | id3) + (1 | id4) + (0 + x4| id4))) #interactions: expect_equivalent(findbars(y ~ (x1*x2 || id)), findbars(y ~ (1 | id) + (0+x1 | id) + (0 + x2 | id) + (0 + x1:x2 | id))) }) test_that("quoted terms work", { ## used to shit the bed in test-oldRZXFailure.R f <- quote(crab.speciesS + crab.sizeS + crab.speciesS:crab.sizeS + (snail.size | plot)) expect_equivalent(findbars(f)[[1]], (~(snail.size|plot))[[2]][[2]] ) }) test_that("leaves superfluous '||' alone", { expect_equivalent(findbars(y ~ z + (0 + x || id)), findbars(y ~ z + (0 + x | id))) }) test_that("plays nice with parens in fixed or random formulas", { expect_equivalent(findbars(y ~ (z + x)^2 + (x || id)), findbars(y ~ (z + x)^2 + (1 | id) + (0 + x | id))) expect_equivalent(findbars(y ~ ((x || id)) + (x2|id)), findbars(y ~ (1 | id) + (0 + x | id) + (x2|id))) }) test_that("update works as expected", { m <- lmer(Reaction ~ Days + (Days || Subject), sleepstudy) expect_equivalent(fitted(update(m, .~.-(0 + Days | Subject))), fitted(lmer(Reaction ~ Days + (1|Subject), sleepstudy))) }) lme4/inst/tests/napredict2.R0000754000175100001440000001216312402120407015423 0ustar hornikusers## refit with missing library(lme4) library(testthat) ## baseline model sleepstudy2 <- sleepstudy rownames(sleepstudy2) <- paste0("a",rownames(sleepstudy2)) fm1 <- lmer(Reaction~Days+(Days|Subject),sleepstudy2) sleepstudyNA <- sleepstudy2 sleepstudyNA$Reaction[1:3] = NA ## na.omit fm2 <- update(fm1,data=sleepstudyNA, control=lmerControl(check.conv.grad="ignore")) expect_equal(head(names(fitted(fm1))),paste0("a",1:6)) expect_equal(head(names(fitted(fm2))),paste0("a",4:9)) expect_equal(names(predict(fm2)),names(fitted(fm2))) expect_equal(length(p1 <- predict(fm2)),177) expect_equal(length(p2 <- predict(fm2,na.action=na.exclude)),180) expect_equal(length((s1 <- simulate(fm1,1))[[1]]),180) expect_equal(length((s2 <- simulate(fm2,1))[[1]]),177) expect_equal(head(rownames(s1)),paste0("a",1:6)) expect_equal(head(rownames(s2)),paste0("a",4:9)) ## na.pass (pretty messed up) fm3 <- update(fm1,data=sleepstudyNA, control=lmerControl(check.conv.grad="ignore"), na.action=na.pass) sleepstudyNA2 <- sleepstudy2 sleepstudyNA2$Days[1:3] = NA library(testthat) expect_error(fm4 <- update(fm1,data=sleepstudyNA2, control=lmerControl(check.conv.grad="ignore"), na.action=na.pass),"NA in Z") expect_is(suppressWarnings(confint(fm2,method="boot",nsim=3)),"matrix") ## fit.na.action <- attr(mfnew, "na.action") ## line 270 cake2 <- rbind(cake,tail(cake,1)) cake2[nrow(cake2),"angle"] <- NA fm0 <- lmer(angle ~ recipe * temperature + (1|recipe:replicate), cake) fm1 <- update(fm0,data=cake2) expect_that(update(fm1,na.action=na.fail), throws_error("missing values in object")) fm1_omit <- update(fm1,na.action=na.omit) ## check equal: expect_true(all.equal(fixef(fm0),fixef(fm1))) expect_true(all.equal(VarCorr(fm0),VarCorr(fm1))) expect_true(all.equal(ranef(fm0),ranef(fm1))) ## works, but doesn't make much sense fm1_pass <- update(fm1,na.action=na.pass) expect_true(all(is.na(fitted(fm1_pass)))) fm1_exclude <- update(fm1,na.action=na.exclude) expect_equal(length(fitted(fm1_omit)),270) expect_equal(length(fitted(fm1_exclude)),271) ## FIXME: fails on Windows (not on Linux!) expect_true(is.na(tail(predict(fm1_exclude),1))) ## test predict.lm d <- data.frame(x=1:10,y=c(rnorm(9),NA)) lm1 <- lm(y~x,data=d,na.action=na.exclude) predict(lm1) predict(lm1,newdata=data.frame(x=c(1:4,NA))) ## Triq examples ... m.lmer <- lmer (angle ~ temp + (1 | recipe) + (1 | replicate), data=cake) # Create new data frame with some NAs in fixed effect cake2 <- cake cake2$temp[1:5] <- NA p1_pass <- predict(m.lmer, newdata=cake2, re.form=NA, na.action=na.pass) expect_true(length(p1_pass)==nrow(cake2)) expect_true(all(is.na(p1_pass[1:5]))) p1_omit <- predict(m.lmer, newdata=cake2, re.form=NA, na.action=na.omit) p1_exclude <- predict(m.lmer, newdata=cake2, re.form=NA, na.action=na.exclude) expect_true(length(p1_omit)==nrow(na.omit(cake2))) expect_true(length(p1_exclude)==nrow(cake2)) expect_true(all.equal(c(na.omit(p1_exclude)),p1_omit)) expect_that(predict(m.lmer, newdata=cake2, re.form=NA, na.action=na.fail), throws_error("missing values in object")) ## now try it with re.form==NULL p2_pass <- predict(m.lmer, newdata=cake2, re.form=NULL, na.action=na.pass) expect_true(length(p2_pass)==nrow(cake2)) expect_true(all(is.na(p2_pass[1:5]))) p2_omit <- predict(m.lmer, newdata=cake2, re.form=NULL, na.action=na.omit) p2_exclude <- predict(m.lmer, newdata=cake2, re.form=NULL, na.action=na.exclude) expect_true(length(p2_omit)==nrow(na.omit(cake2))) expect_true(all.equal(c(na.omit(p2_exclude)),p2_omit)) expect_that(predict(m.lmer, newdata=cake2, re.form=NULL, na.action=na.fail), throws_error("missing values in object")) ## experiment with NA values in random effects -- should get ## treated cake3 <- cake cake3$replicate[1:5] <- NA expect_that(predict(m.lmer, newdata=cake3, re.form=NULL), throws_error("NAs are not allowed in prediction data")) p4 <- predict(m.lmer, newdata=cake3, re.form=NULL, allow.new.levels=TRUE) p4B <- predict(m.lmer, newdata=cake3, re.form=~1|recipe, allow.new.levels=TRUE) expect_true(all.equal(p4[1:5],p4B[1:5])) p4C <- predict(m.lmer, newdata=cake3, re.form=NA) library(lme4) library(testthat) d <- data.frame(x=runif(100),f=factor(rep(1:10,10))) set.seed(101) u <- rnorm(10) d <- transform(d,y=rnorm(100,1+2*x+u[f],0.2)) d0 <- d d[c(3,5,7),"x"] <- NA ## 'omit' and 'exclude' are the only choices under which ## we will see NA values in the results fm0 <- lmer(y~x+(1|f),data=d0) ## no 'na.action' attribute because no NAs in this data set stopifnot(is.null(attr(model.frame(fm0),"na.action"))) fm1 <- update(fm0,data=d) ## no NAs in predict or residuals because na.omit stopifnot(!any(is.na(predict(fm1)))) stopifnot(!any(is.na(residuals(fm1)))) fm2 <- update(fm1,na.action="na.exclude") ## no NAs in predict or residuals because na.omit nNA <- sum(is.na(d$x)) stopifnot(sum(is.na(predict(fm2)))==nNA) stopifnot(sum(is.na(residuals(fm2)))==nNA) expect_error(fm3 <- lmer(y~x+(1|f),data=d,na.action="na.pass"), "Error in qr.default") refit(fm0) refit(fm1) refit(fm2) refit(fm0,runif(100)) refit(fm1,runif(100)) refit(fm2,runif(100)) lme4/inst/tests/test-factors.R0000754000175100001440000000066712130605564016026 0ustar hornikuserslibrary("testthat") library("lme4") context("factor handling in grouping variables") test_that("factors", { set.seed(101) d <- data.frame(x=runif(1000),y=runif(1000),f1=rep(1:10,each=100),f2=rep(1:10,100)) d2 <- transform(d,f1=factor(f1),f2=factor(f2)) expect_that(lm1 <- lmer(y~x+(1|f1/f2),data=d), is_a("lmerMod")) expect_that(lm2 <- lmer(y~x+(1|f1/f2),data=d2),is_a("lmerMod")) expect_equivalent(lm1,lm2) }) lme4/inst/tests/test-oldRZXfailure.R0000754000175100001440000000076212374550357017124 0ustar hornikuserslibrary(lme4) library(testthat) load(system.file("testdata","crabs_randdata00.Rda",package="lme4")) test_that('RZX is being calculated properly', { # this is a test for an old problem, documented here: # http://stevencarlislewalker.github.io/notebook/RZX_problems.html fr <- cbind(final.snail.density, snails.lost) ~ crab.speciesS + crab.sizeS + crab.speciesS:crab.sizeS + (snail.size | plot) m <- glmer(fr, data = randdata00, family = binomial) expect_that(m, is_a("glmerMod")) }) lme4/inst/tests/test-glmer.R0000754000175100001440000003324712565075351015502 0ustar hornikuserslibrary("testthat") library("lme4") testLevel <- if (nzchar(s <- Sys.getenv("LME4_TEST_LEVEL"))) as.numeric(s) else 1 gives_error_or_warning <- function (regexp = NULL, all = FALSE, ...) { function(expr) { res <- try(evaluate_promise(expr),silent=TRUE) no_error <- !inherits(res, "try-error") if (no_error) { warnings <- res$warnings if (!is.null(regexp) && length(warnings) > 0) { return(matches(regexp, all = FALSE, ...)(warnings)) } else { return(expectation(length(warnings) > 0, "no warnings or errors given", paste0(length(warnings), " warnings created"))) } } if (!is.null(regexp)) { return(matches(regexp, ...)(res)) } else { expectation(TRUE, "no error thrown", "threw an error") } } } ## expect_that(stop("foo"),gives_error_or_warning("foo")) ## expect_that(warning("foo"),gives_error_or_warning("foo")) ## expect_that(TRUE,gives_error_or_warning("foo")) ## expect_that(stop("bar"),gives_error_or_warning("foo")) ## expect_that(warning("bar"),gives_error_or_warning("foo")) context("fitting glmer models") test_that("glmer", { set.seed(101) d <- data.frame(z=rbinom(200,size=1,prob=0.5), f=factor(sample(1:10,200,replace=TRUE))) expect_warning(glmer(z~ 1|f, d, family=binomial, method="abc"),"Use the nAGQ argument") expect_warning(glmer(z~ 1|f, d, family=binomial, method="Laplace"),"Use the nAGQ argument") expect_warning(glmer(z~ 1|f, d, sparseX=TRUE),"has no effect at present") expect_that(gm1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data = cbpp, family = binomial), is_a("glmerMod")) expect_that(gm1@resp, is_a("glmResp")) expect_that(gm1@pp, is_a("merPredD")) expect_equal(ge1 <- unname(fixef(gm1)), c(-1.39854982537216, -0.992335519118859, -1.12867532780426, -1.58030423764517), tolerance=5e-4) expect_equal(c(VarCorr(gm1)[[1]]), 0.41245527438386, tolerance=6e-4) ### expect_that(family(gm1), equals(binomial())) ### ?? binomial() has an 'initialize' component ... and the order is different expect_equal(deviance(gm1), 73.47428, tolerance=1e-5) ## was -2L = 184.05267459802 expect_equal(sigma(gm1), 1) expect_equal(extractAIC(gm1), c(5, 194.052674598026), tolerance=1e-5) expect_equal(theta <- unname(getME(gm1, "theta")), 0.642226809144453, tolerance=6e-4) expect_that(X <- getME(gm1, "X"), is_equivalent_to( model.matrix(model.frame(~ period, data=cbpp), cbpp))) expect_that(Zt <- getME(gm1, "Zt"), is_a("dgCMatrix")) expect_equal(dim(Zt), c(15L, 56L)) expect_equal(Zt@x, rep.int(1, 56L)) expect_that(Lambdat <- getME(gm1, "Lambdat"), is_a("dgCMatrix")) expect_equivalent(as(Lambdat, "matrix"), diag(theta, 15L, 15L)) expect_is(gm1_probit <- update(gm1,family=binomial(link="probit")),"merMod") expect_equal(family(gm1_probit)$link,"probit") ## FIXME: test user-specified/custom family? expect_error(glFormula(cbind(incidence, size - incidence) ~ period + (1 | herd), data = subset(cbpp, herd==levels(herd)[1]), family = binomial), "must have > 1") expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data = subset(cbpp, herd %in% levels(herd)[1:4]), family = binomial, control=glmerControl(check.nlev.gtreq.5="warning")), "< 5 sampled levels") expect_warning(fm1. <- glmer(Reaction ~ Days + (Days|Subject), sleepstudy), regexp="calling .* with family=gaussian .* as a shortcut") options(warn=2) options(glmerControl=list(junk=1,check.conv.grad="ignore")) expect_warning(glmer(z~ 1|f, d, family=binomial), "some options") options(glmerControl=NULL) cbppX <- transform(cbpp,prop=incidence/size) expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, start=NULL), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, verbose=0L), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, subset=TRUE), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, na.action="na.exclude"), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, offset=rep(0,nrow(cbppX))), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, contrasts=NULL), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, devFunOnly=FALSE), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, control=glmerControl(optimizer="Nelder_Mead")), "glmerMod") expect_is(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, control=glmerControl()), "glmerMod") options(warn=0) expect_warning(glmer(prop ~ period + (1 | herd), data = cbppX, family = binomial, weights=size, junkArg=TRUE), "extra argument.*disregarded") if(FALSE) { ## Hadley broke this expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data = cbpp, family = binomial, control=list()), "instead of passing a list of class") expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data = cbpp, family = binomial, control=lmerControl()), "instead of passing a list of class") } ## load(system.file("testdata","radinger_dat.RData",package="lme4")) mod <- glmer(presabs~predictor+(1|species),family=binomial, radinger_dat) expect_is(mod,"merMod") ## tolerance: 32-bit Windows (CRAN) reported ave.diff of 5.33e-8 expect_equal(unname(fixef(mod)), c(0.5425528,6.4289962), tolerance = 4e-7) set.seed(101) ## complete separation case d <- data.frame(y=rbinom(1000,size=1,p=0.5), x=runif(1000), f=factor(rep(1:20,each=50)), x2=rep(0:1,c(999,1))) mod2 <- glmer(y~x+x2+(1|f),data=d,family=binomial, control=glmerControl(check.conv.hess="ignore", check.conv.grad="ignore")) expect_equal(unname(fixef(mod2))[1:2], c(-0.10036244,0.03548523), tolerance=1e-4) expect_true(unname(fixef(mod2)[3] < -10)) mod3 <- update(mod2, family=binomial(link="probit")) # singular Hessian warning expect_equal(unname(fixef(mod3))[1:2], c(-0.062889, 0.022241), tolerance=1e-4) expect_true(fixef(mod3)[3] < -4) mod4 <- update(mod2, family=binomial(link="cauchit"), control=glmerControl(check.conv.hess="ignore", check.conv.grad="ignore"))#--> singular Hessian warning ## on-the-fly creation of index variables if (FALSE) { ## FIXME: fails in testthat context -- 'd' is not found ## in the parent environment of glmer() -- but works fine ## otherwise ... set.seed(101) d <- data.frame(y1=rpois(100,1), x=rnorm(100), ID=1:100) fit1 <- glmer(y1 ~ x+(1|ID),data=d,family=poisson) fit2 <- update(fit1, .~ x+(1|rownames(d))) expect_equal(unname(unlist(VarCorr(fit1))), unname(unlist(VarCorr(fit2)))) } ## if(testLevel > 1) { load(system.file("testdata","mastitis.rda",package="lme4")) t1 <- system.time(g1 <- glmer(NCM ~ birth + calvingYear + (1|sire) + (1|herd), mastitis, poisson, ## current (2014-04-24) default: control=glmerControl(optimizer=c("bobyqa","Nelder_Mead")))) t2 <- system.time(g2 <- update(g1, control=glmerControl(optimizer="bobyqa"))) ## 20 (then 13.0) seconds N-M vs 8 (then 4.8) seconds bobyqa ... ## problem is fairly ill-conditioned so parameters ## are relatively far apart even though likelihoods are OK expect_equal(logLik(g1),logLik(g2),tolerance=1e-7) } ## test bootstrap/refit with nAGQ>1 gm1AGQ <- update(gm1,nAGQ=2) s1 <- simulate(gm1AGQ) expect_equal(attr(bootMer(gm1AGQ,fixef),"bootFail"),0) ## do.call(new,...) bug new <- "foo" expect_that(gm1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data = cbpp, family = binomial), is_a("glmerMod")) rm("new") ## test issue #47, from Wolfgang Viechtbauer ## create some data n <- 100 ai <- rep(0:1, each = n/2) bi <- 1-ai ci <- c(rep(0,42), rep(1,8), rep(0,18), rep(1,32)) di <- 1-ci event <- c(rbind(ai,ci)) group <- rep(c(1,0), times=n) id <- rep(1:n, each=2) gm3 <- glmer(event ~ group + (1 | id), family=binomial, nAGQ=21) sd3 <- sqrt(diag(vcov(gm3))) expect_equal(sd3, c(0.4254254, 0.424922), tolerance=1e-5) expect_warning(vcov(gm3,use.hessian=FALSE), "finite-difference Hessian") expect_equal(suppressWarnings(sqrt(diag(vcov(gm3,use.hessian=FALSE)))), c(0.3840921, 0.3768747), tolerance=1e-7) expect_equal(sd3, unname(coef(summary(gm3))[,"Std. Error"])) ## test non-pos-def finite-difference Hessian ... if(getRversion() > "3.0.0") { ## saved fits are not safe with old R versions L <- load(system.file("testdata","polytomous_vcov_ex.RData", package="lme4", mustWork=TRUE)) expect_warning(vcov(polytomous_vcov_ex),"falling back to var-cov") } ## damage Hessian to make it singular ## (example thanks to J. Dushoff) gm1H <- gm1 gm1H@optinfo$derivs$Hessian[5,] <- 0 expect_warning(vcov(gm1H),"falling back to var-cov") ## test convergence warnings L <- load(system.file("testdata","gopherdat2.RData", package="lme4", mustWork=TRUE)) g0 <- glmer(shells~prev + (1|Site)+offset(log(Area)), family=poisson, data=Gdat) ## fit year as factor: OK gc <- glmerControl(check.conv.grad="stop") expect_is(update(g0,.~.+factor(year), control=gc), "glmerMod") ## error/warning with year as numeric: ## don't have full knowledge of which platforms lead to which ## results, and can't detect whether we're running on valgrind, ## which changes the result on 32-bit linux ... ## SEGFAULT on MacOS? why? if (FALSE) { expect_that(update(g0,.~.+year), gives_error_or_warning("(failed to converge|pwrssUpdate did not converge)")) } ## ("(failed to converge|pwrssUpdate did not converge in)")) ## if (sessionInfo()$platform=="i686-pc-linux-gnu (32-bit)") { ## expect_warning(update(g0, .~. +year), "failed to converge") ## } else { ## ## MacOS x86_64-apple-darwin10.8.0 (64-bit) ## ## MM's platform ## ## "pwrssUpdate did not converge in (maxit) iterations" ## expect_error(update(g0, .~. +year), "pwrssUpdate did not converge in") ## } ## OK if we scale & center it expect_is(update(g0,.~. + scale(year), control=gc), "glmerMod") ## not OK if we scale and don't center expect_warning(update(g0,.~. + scale(year,center=FALSE)), "failed to converge with max|grad|") ## OK if center and don't scale expect_is(update(g0,.~. + scale(year,center=TRUE,scale=FALSE), control=gc), "glmerMod") ## try higher-order AGQ expect_is(update(gm1,nAGQ=90),"glmerMod") expect_error(update(gm1,nAGQ=101),"ord < 101L") ## non-numeric response variables ss <- transform(sleepstudy, Reaction = as.character(Reaction)) expect_error(glmer(Reaction~(1|Days), family="poisson", data=ss), "response must be numeric") expect_error(glmer(Reaction~(1|Days), family="binomial", data=ss), "response must be numeric or factor") ss2 <- transform(ss,rr=rep(c(TRUE,FALSE),length.out=nrow(ss))) ## should work OK with logical too expect_is(glmer(rr~(1|Days),family="binomial",data=ss2),"merMod") ## starting values with log(.) link -- thanks to Eric Weese @ Yale: grp <- rep(letters[1:5], 20); set.seed(1); x <- rnorm(100) expect_error(glmer(x ~ 1 + (1|grp), family=gaussian(link="log")), "valid starting values") ## related to GH 231 ## fails on some platforms, skip for now if (FALSE) { rr <- gm1@resp$copy() ff <- setdiff(ls(gm1@resp),c("copy","initialize","initialize#lmResp","ptr", "updateMu","updateWts","resDev","setOffset","wrss")) for (i in ff) { expect_equal(gm1@resp[[i]],rr[[i]]) } } }) lme4/inst/tests/test-summary.R0000754000175100001440000000075012542427271016057 0ustar hornikuserslibrary("testthat") library("lme4") context("summarizing/printing models") test_that("lmer", { set.seed(0) J <- 8 n <- 10 N <- J * n beta <- c(5, 2, 4) u <- matrix(rnorm(J * 3), J, 3) x.1 <- rnorm(N) x.2 <- rnorm(N) g <- rep(1:J, rep(n, J)) y <- 1 * (beta[1] + u[g,1]) + x.1 * (beta[2] + u[g,2]) + x.2 * (beta[3] + u[g,3]) + rnorm(N) summary(lmer(y ~ x.1 + x.2 + (1 + x.1 | g))) summary(lmer(y ~ x.1 + x.2 + (1 + x.1 + x.2 | g))) }) lme4/inst/tests/tmp.html0000754000175100001440000005763612324542612014762 0ustar hornikusers R: lme4 News
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lme4/inst/tests/test-rank.R0000754000175100001440000000660412565075351015324 0ustar hornikuserslibrary("testthat") library("lme4") context("testing fixed-effect design matrices for full rank") test_that("lmerRank", { set.seed(101) n <- 20 x <- y <- rnorm(n) d <- data.frame(x,y, z = rnorm(n), r = sample(1:5, size=n, replace=TRUE), y2 = y + c(0.001, rep(0,n-1))) expect_message(fm <- lmer( z ~ x + y + (1|r), data=d), "fixed-effect model matrix is .*rank deficient") ## test reconstitution of full parameter vector (with NAs) expect_equal(names(fixef(fm,add.dropped=TRUE)), c("(Intercept)","x","y")) expect_equal(fixef(fm,add.dropped=TRUE)[1:2], fixef(fm)) expect_equal(nrow(anova(fm)), 1L) expect_error(lmer( z ~ x + y + (1|r), data=d, control=lmerControl(check.rankX="stop")), "rank deficient") expect_error(lmer( z ~ x + y + (1|r), data=d, control=lmerControl(check.rankX="ignore")), "not positive definite") ## should work: expect_is(lmer( z ~ x + y2 + (1|r), data=d), "lmerMod") d2 <- expand.grid(a=factor(1:4),b=factor(1:4),rep=1:10) n <- nrow(d2) d2 <- transform(d2,r=sample(1:5, size=n, replace=TRUE), z=rnorm(n)) d2 <- subset(d2,!(a=="4" & b=="4")) expect_error(lmer( z ~ a*b + (1|r), data=d2, control=lmerControl(check.rankX="stop")), "rank deficient") expect_message(fm <- lmer( z ~ a*b + (1|r), data=d2), "fixed-effect model matrix is rank deficient") d2 <- transform(d2, ab=droplevels(interaction(a,b))) ## should work: expect_is(fm2 <- lmer( z ~ ab + (1|r), data=d2), "lmerMod") expect_equal(logLik(fm), logLik(fm2)) expect_equal(sum(anova(fm)[, "Df"]), anova(fm2)[, "Df"]) expect_equal(sum(anova(fm)[, "Sum Sq"]), anova(fm2)[, "Sum Sq"]) }) test_that("glmerRank", { set.seed(111) n <- 100 x <- y <- rnorm(n) d <- data.frame(x, y, z = rbinom(n,size=1,prob=0.5), r = sample(1:5, size=n, replace=TRUE), y2 = ## y + c(0.001,rep(0,n-1)), ## too small: get convergence failures ## FIXME: figure out how small a difference will still fail? rnorm(n)) expect_message(fm <- glmer( z ~ x + y + (1|r), data=d, family=binomial), "fixed-effect model matrix is rank deficient") expect_error(glmer( z ~ x + y + (1|r), data=d, family=binomial, control=glmerControl(check.rankX="stop")), "rank deficient.*rank.X.") expect_is(glmer( z ~ x + y2 + (1|r), data=d, family=binomial), "glmerMod") }) test_that("nlmerRank", { set.seed(101) n <- 1000 nblock <- 15 x <- abs(rnorm(n)) y <- rnorm(n) z <- rnorm(n,mean=x^y) r <- sample(1:nblock, size=n, replace=TRUE) d <- data.frame(x,y,z,r) ## save("d","nlmerRank.RData") ## see what's going on with difference in contexts fModel <- function(a,b) (exp(a)*x)^(b*y) fModf <- deriv(body(fModel), namevec = c("a","b"), func = fModel) fModel2 <- function(a,b,c) (exp(a+c)*x)^(b*y) fModf2 <- deriv(body(fModel2), namevec = c("a","b","c"), func = fModel2) ## should be OK: fails in test mode? nlmer(y ~ fModf(a,b) ~ a|r, d, start = c(a=1,b=1)) ## FIXME: this doesn't get caught where I expected expect_error(nlmer(y ~ fModf2(a,b,c) ~ a|r, d, start = c(a=1,b=1,c=1)),"Downdated VtV") }) lme4/inst/tests/test-start.R0000754000175100001440000000543312565075351015525 0ustar hornikuserslibrary("testthat") library("lme4") context("specifying starting values") ## is "Nelder_Mead" default optimizer? isNM <- formals(lmerControl)$optimizer == "Nelder_Mead" test_that("lmer", { frm <- as.formula("Reaction ~ Days + (Days|Subject)") ctrl <- lmerControl(optCtrl=list(maxfun= if(isNM) 50 else 100)) x <- suppressWarnings(lmer(frm, data=sleepstudy, control=ctrl, REML=FALSE)) x2 <- suppressWarnings(update(x,start=c(1,0,1))) x3 <- suppressWarnings(update(x,start=list(theta=c(1,0,1)))) ff <- update(x,devFunOnly=TRUE) x2@call <- x3@call <- x@call ## hack call component expect_equal(x,x2) expect_equal(x,x3) expect_error(update(x,start=c("a")),"start must be a list or a numeric vector") ## misspelled expect_error(update(x,start=list(Theta=c(1,0,1))),"incorrect components") th0 <- getME(x,"theta") y <- suppressWarnings(update(x,start=th0)) if(isNM) { expect_equal(AIC(x), 1768.025, tolerance=1e-6) expect_equal(AIC(y), 1763.949, tolerance=1e-6) } else { ## only "bobyqa" tested: expect_equal(AIC(x), 1763.939344, tolerance=1e-6) expect_equal(AIC(x), AIC(y)) } if(isNM) expect_equal(suppressWarnings(optimizeLmer(ff,control=list(maxfun=50),start=c(1,0,1))$fval), unname(deviance(x))) expect_equal(suppressWarnings(optimizeLmer(ff,control=list(maxfun=50),start=th0)$fval), unname(deviance(y))) }) test_that("glmer", { ctrl <- glmerControl(optCtrl=list(maxfun=50)) x <- suppressWarnings(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data = cbpp, family = binomial, control=ctrl)) ## theta only x2 <- suppressWarnings(update(x,start=c(1))) x3 <- suppressWarnings(update(x,start=list(theta=c(1)))) ff <- update(x,devFunOnly=TRUE) x2@call <- x3@call <- x@call ## hack call component expect_equal(x,x2) expect_equal(x,x3) expect_error(update(x,start=c("a")),"start must be a list or a numeric vector") expect_error(update(x,start=list(Theta=c(1))),"bad name\\(s\\)") th0 <- getME(x,"theta") y <- suppressWarnings(update(x,start=th0)) ## theta and beta x0 <- update(x,nAGQ=0) x4 <- suppressWarnings(update(x,start=list(theta=1,fixef=fixef(x0)))) x4@call <- x@call expect_equal(x,x4) x5 <- suppressWarnings(update(x,start=list(theta=1,fixef=rep(0,4)))) expect_equal(AIC(x5),221.5823,tolerance=1e-6) x6 <- expect_error(update(x,start=list(theta=1,fixef=rep(0,5))), "incorrect number of fixef components") ## beta only x7 <- suppressWarnings(update(x,start=list(fixef=fixef(x0)))) x7@call <- x@call expect_equal(x,x7) x8 <- suppressWarnings(update(x,start=list(fixef=rep(0,4)))) x8@call <- x5@call expect_equal(x5,x8) }) lme4/inst/tests/wenseleers.rda0000754000175100001440000000160712210167752016125 0ustar hornikusers‹íØwxeÀñËèJKw«‚€¢ ¢Ô6Ý ˆZE¡E°Ž3M¥š&%­Tœ lœ Š¢à@q‚{/\l™. 2Lý^“»ò¼—KKë?öžçsùݻ߻'¿Ü“Ò’2»­Ì&I’Y²˜8[­fN&É*ÅñS.ûä§O’,iJÚQh¢t£"X…h¥ b¡tW†ŽGB Ÿ”ˆ$$#©PÆMG2qÇh8Ñ qŽFƒcÑÝpºãxôÀ 8=‘…“Ø‘‹<䣅(B1NÆ)è…ÞèƒSѧátœœ‰³Ðg£?à Ä Æœ‹óPŠ¡8Ã0  #p!Fb.Ÿ—BÆep NTàrŒF%®À•p¡ nxP1ð¢<_é*Ô¢W£×`,ÆáZ\‡ë17àFLÀDÜ„›q nÅm¸“0wàNLÁTLÃtÌÀLÌÂlÜ…»qîÅ}¸s0ó0àA<„‡ñàQ<†…xO`žÄSXŒ%xÏàY<‡¥x/`^ÄKx¯àU¼†åX×ñÞÄ[xïà]¼‡÷ñ>ÄGøŸàS|†Ï±_àK¬Âj|…¯ñ ¾ÅwXƒµX‡õ؀؄ï±[°Û°;ð~ÄOø¿`'~Åoø`vãOü…¿±{±ûñüh|õM¦@°˜Úü™Û´ y}ˆv9k.å&¹±6PjÊQ{Hû(‡K®Q›kƒTÈŸÇ+5¦Ÿ°é¦%·biÖEµh1Í+×D¶ñ a´ke‰‡(©™’›!%©Ò ¤ÿ2tdêSÒ‡¹!x2 &äªAžä«AªA‘sN¶åh‘]‹rµ(O‹òµ¨@‹ µ¨H‹´9ìÚvm{0CZÔ- “¢šÕãÀX ©7 ®õÚè¢ùÂõ×6t\£C´vQy¸=]‹Ö$ÚÞzý/‰Úˆæ­[olQ_½5‰æ6Ú¸{ésÔ[¿¨,’¶¡ë3º'FûjÊšôúëíUÔ>ܼzc|?~§rËUNõÊ,Œ­‘«ª]N·WM'Ëã®^ÅUË>o¥«ÒíTÇðyåJ_ÈÀq^O]–:¸ò“iÏ©¡Áï×ÉY6åϤ¬ /]¤@Þòÿ ’¡õ‚lme4/inst/tests/test-formulaEval.R0000754000175100001440000001416512565075351016647 0ustar hornikuserslibrary("testthat") library("lme4") context("data= argument and formula evaluation") test_that("glmerFormX", { set.seed(101) n <- 50 x <- rbinom(n, 1, 1/2) y <- rnorm(n) z <- rnorm(n) r <- sample(1:5, size=n, replace=TRUE) d <- data.frame(x,y,z,r) F <- "z" rF <- "(1|r)" modStr <- (paste("x ~", "y +", F, "+", rF)) modForm <- as.formula(modStr) ## WARNING: these drop/environment tests are extremely sensitive to environment ## they may fail/not fail, or fail differently, within a "testthat" environment vs. ## when run interactively ## AICvec <- c(77.0516381151634, 75.0819116367084, 75.1915023640827) expect_that(m_data.3 <- glmer( modStr , data=d, family="binomial"), is_a("glmerMod")) expect_that(m_data.4 <- glmer( "x ~ y + z + (1|r)" , data=d, family="binomial"), is_a("glmerMod")) ## interactively: ## expect_equal(drop1(m_data.3)$AIC,AICvec) ## expect_equal(drop1(m_data.4)$AIC,AICvec) ## in test environment: expect_error(drop1(m_data.3),"'data' not found") expect_error(drop1(m_data.4),"'data' not found") }) test_that("glmerForm", { set.seed(101) n <- 50 x <- rbinom(n, 1, 1/2) y <- rnorm(n) z <- rnorm(n) r <- sample(1:5, size=n, replace=TRUE) d <- data.frame(x,y,z,r) F <- "z" rF <- "(1|r)" modStr <- (paste("x ~", "y +", F, "+", rF)) modForm <- as.formula(modStr) ## formulas have environments associated, but character vectors don't ## data argument not specified: ## should work, but documentation warns against it expect_that(m_nodata.0 <- glmer( x ~ y + z + (1|r) , family="binomial"), is_a("glmerMod")) expect_that(m_nodata.1 <- glmer( as.formula(modStr) , family="binomial"), is_a("glmerMod")) expect_that(m_nodata.2 <- glmer( modForm , family="binomial"), is_a("glmerMod")) expect_that(m_nodata.3 <- glmer( modStr , family="binomial"), is_a("glmerMod")) expect_that(m_nodata.4 <- glmer( "x ~ y + z + (1|r)" , family="binomial"), is_a("glmerMod")) ## apply drop1 to all of these ... m_nodata_List <- list(m_nodata.0, m_nodata.1,m_nodata.2,m_nodata.3,m_nodata.4) d_nodata_List <- lapply(m_nodata_List,drop1) rm(list=c("x","y","z","r")) ## data argument specified expect_that(m_data.0 <- glmer( x ~ y + z + (1|r) , data=d, family="binomial"), is_a("glmerMod")) expect_that(m_data.1 <- glmer( as.formula(modStr) , data=d, family="binomial"), is_a("glmerMod")) expect_that(m_data.2 <- glmer( modForm , data=d, family="binomial"), is_a("glmerMod")) expect_that(m_data.3 <- glmer( modStr , data=d, family="binomial"), is_a("glmerMod")) expect_that(m_data.4 <- glmer( "x ~ y + z + (1|r)" , data=d, family="binomial"), is_a("glmerMod")) ff <- function() { set.seed(101) n <- 50 x <- rbinom(n, 1, 1/2) y <- rnorm(n) z <- rnorm(n) r <- sample(1:5, size=n, replace=TRUE) d2 <- data.frame(x,y,z,r) glmer( x ~ y + z + (1|r), data=d2, family="binomial") } m_data.5 <- ff() ff2 <- function() { set.seed(101) n <- 50 x <- rbinom(n, 1, 1/2) y <- rnorm(n) z <- rnorm(n) r <- sample(1:5, size=n, replace=TRUE) glmer( x ~ y + z + (1|r), family="binomial") } m_data.6 <- ff2() m_data_List <- list(m_data.0,m_data.1,m_data.2,m_data.3,m_data.4,m_data.5,m_data.6) badNums <- 4:5 d_data_List <- lapply(m_data_List[-badNums],drop1) ## these do NOT fail if there is a variable 'd' living in the global environment -- ## they DO fail in the testthat context expect_error(drop1(m_data.3),"'data' not found") expect_error(drop1(m_data.4),"'data' not found") ## expect_error(lapply(m_data_List[4],drop1)) ## expect_error(lapply(m_data_List[5],drop1)) ## d_data_List <- lapply(m_data_List,drop1,evalhack="parent") ## fails on element 1 ## d_data_List <- lapply(m_data_List,drop1,evalhack="formulaenv") ## fails on element 4 ## d_data_List <- lapply(m_data_List,drop1,evalhack="nulldata") ## succeeds ## drop1(m_data.5,evalhack="parent") ## 'd2' not found ## drop1(m_data.5,evalhack="nulldata") ## 'x' not found (d2 is in environment ...) ## should we try to make update smarter ... ?? ## test equivalence of (i vs i+1) for all models, all drop1() results for (i in 1:(length(m_nodata_List)-1)) { expect_equivalent(m_nodata_List[[i]],m_nodata_List[[i+1]]) expect_equivalent(d_nodata_List[[i]],d_nodata_List[[i+1]]) } expect_equivalent(m_nodata_List[[1]],m_data_List[[1]]) expect_equivalent(d_nodata_List[[1]],d_data_List[[1]]) for (i in 1:(length(m_data_List)-1)) { expect_equivalent(m_data_List[[i]],m_data_List[[i+1]]) } ## allow for dropped 'bad' vals for (i in 1:(length(d_data_List)-1)) { expect_equivalent(d_data_List[[i]],d_data_List[[i+1]]) } }) test_that("lmerForm", { set.seed(101) x <- rnorm(10) y <- rnorm(10) z <- rnorm(10) r <- sample(1:3, size=10, replace=TRUE) d <- data.frame(x,y,z,r) ## example from Joehanes Roeby m2 <- suppressWarnings(lmer(x ~ y + z + (1|r), data=d)) ff <- function() { m1 <- suppressWarnings(lmer(x ~ y + z + (1|r), data=d)) return(anova(m1)) } ff1 <- Reaction ~ Days + (Days|Subject) fm1 <- lmer(ff1, sleepstudy) fun <- function () { ff1 <- Reaction ~ Days + (Days|Subject) fm1 <- suppressWarnings(lmer(ff1, sleepstudy)) return (anova(fm1)) } anova(m2) ff() expect_equal(anova(m2),ff()) anova(fm1) fun() expect_equal(anova(fm1),fun()) ## test deparsing of long RE terms varChr <- paste0("varname_",outer(letters,letters,paste0)[1:100]) rvars <- varChr[1:9] form <- as.formula(paste("y ~",paste(varChr,collapse="+"), "+", paste0("(",paste(rvars,collapse="+"),"|f)"))) ff <- lme4:::reOnly(form) environment(ff) <- .GlobalEnv expect_equal(ff, ~(varname_aa + varname_ba + varname_ca + varname_da + varname_ea + varname_fa + varname_ga + varname_ha + varname_ia | f)) }) lme4/inst/tests/test-catch.R0000754000175100001440000000105712406350752015443 0ustar hornikuserslibrary("testthat") library("lme4") context("storing warnings, convergence status, etc.") test_that("storewarning", { gCtrl <- glmerControl(optimizer = "Nelder_Mead", optCtrl = list(maxfun=3)) expect_warning(gm1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), data=cbpp, family=binomial, control=gCtrl), "failure to converge in 3") expect_equal(gm1@optinfo$warnings[[1]],"failure to converge in 3 evaluations") ## FIXME: why is conv==0 here? }) lme4/inst/tests/test-glmmFail.R0000754000175100001440000000321512565075351016114 0ustar hornikuserslibrary("testthat") library("lme4") source(system.file("testdata/lme-tst-funs.R", package="lme4", mustWork=TRUE)) ##-> gSim(), a general simulation function ... set.seed(101) dBc <- gSim(family=binomial(link="cloglog"), nbinom = 1) # {0,1} Binomial ## m1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), ## family = binomial, data = cbpp) context("Errors and warnings from glmer") test_that("glmer", { expect_error(glmer(y ~ 1 + (1|block), data=dBc, family=binomial(link="cloglog")), "Response is constant") expect_warning(lmer(cbind(incidence, size - incidence) ~ period + (1 | herd), family = binomial, data = cbpp), "calling lmer with .*family.* is deprecated.*") ## expect_equal(m1,m2) expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), family = binomial, data = cbpp, REML=TRUE), "extra argument.*REML") expect_warning(glmer(Reaction ~ Days + (Days|Subject), sleepstudy), "calling glmer.*family=gaussian.*deprecated") expect_warning(glmer(Reaction ~ Days + (Days|Subject), sleepstudy, family=gaussian), "calling glmer.*family=gaussian.*deprecated") m3 <- suppressWarnings(glmer(Reaction ~ Days + (Days|Subject), sleepstudy)) m4 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy) m5 <- suppressWarnings(glmer(Reaction ~ Days + (Days|Subject), sleepstudy, family=gaussian)) expect_equal(fixef(m3),fixef(m5)) m3@call[[1]] <- m5@call[[1]] <- quote(lmer) ## hack call expect_equal(m3,m4) expect_equal(m3,m5) }) lme4/inst/tests/test-glmFamily.R0000754000175100001440000001177412406350752016311 0ustar hornikuserslibrary("testthat") library("lme4") eps <- .Machine$double.eps oneMeps <- 1 - eps set.seed(1) ## sample linear predictor values for the unconstrained families etas <- list(seq.int(-8, 8, by=1), # equal spacing to asymptotic area runif(17, -8, 8), # random sample from wide uniform dist rnorm(17, 0, 8), # random sample from wide normal dist c(-10^30, rnorm(15, 0, 4), 10^30)) ## sample linear predictor values for the families in which eta must be positive etapos <- list(seq.int(1, 20, by=1), rexp(20), rgamma(20, 3), pmax(.Machine$double.eps, rnorm(20, 2, 1))) ## values of mu in the (0,1) interval mubinom <- list(runif(100, 0, 1), rbeta(100, 1, 3), pmin(pmax(eps, rbeta(100, 0.1, 3)), oneMeps), pmin(pmax(eps, rbeta(100, 3, 0.1)), oneMeps)) context("glmFamily linkInv and muEta") test_that("inverse link and muEta functions", { tst.lnki <- function(fam, frm) { ff <- glmFamily$new(family=fam) sapply(frm, function(x) expect_that(fam$linkinv(x), equals(ff$linkInv(x)))) } tst.muEta <- function(fam, frm) { ff <- glmFamily$new(family=fam) sapply(frm, function(x) expect_that(fam$mu.eta(x), equals(ff$muEta(x)))) } tst.lnki(binomial(), etas) # binomial with logit link tst.muEta(binomial(), etas) tst.lnki(binomial("probit"), etas) # binomial with probit link tst.muEta(binomial("probit"), etas) tst.lnki(binomial("cloglog"), etas) # binomial with cloglog link tst.muEta(binomial("cloglog"), etas) tst.lnki(binomial("cauchit"), etas) # binomial with cauchit link tst.muEta(binomial("cauchit"), etas) tst.lnki(poisson(), etas) # Poisson with log link tst.muEta(poisson(), etas) tst.lnki(gaussian(), etas) # Gaussian with identity link tst.muEta(gaussian(), etas) tst.lnki(Gamma(), etapos) # gamma family tst.muEta(Gamma(), etapos) tst.lnki(inverse.gaussian(), etapos) # inverse Gaussian tst.muEta(inverse.gaussian(), etapos) }) context("glmFamily linkFun and variance") test_that("link and variance functions", { tst.link <- function(fam, frm) { ff <- glmFamily$new(family=fam) sapply(frm, function(x) expect_that(fam$linkfun(x), equals(ff$link(x)))) } tst.variance <- function(fam, frm) { ff <- glmFamily$new(family=fam) sapply(frm, function(x) expect_that(fam$variance(x), equals(ff$variance(x)))) } tst.link( binomial(), mubinom) tst.variance(binomial(), mubinom) tst.link( binomial("probit"), mubinom) tst.link( binomial("cauchit"), mubinom) tst.link( gaussian(), etas) tst.variance(gaussian(), etas) tst.link( Gamma(), etapos) tst.variance(Gamma(), etapos) tst.link( inverse.gaussian(), etapos) tst.variance(inverse.gaussian(), etapos) tst.variance(MASS::negative.binomial(1), etapos) tst.variance(MASS::negative.binomial(0.5), etapos) tst.link( poisson(), etapos) tst.variance(poisson(), etapos) }) context("glmFamily devResid and aic") test_that("devResid and aic", { tst.devres <- function(fam, frm) { ff <- glmFamily$new(family=fam) sapply(frm, function(x) { nn <- length(x) wt <- rep.int(1, nn) n <- wt y <- switch(fam$family, binomial = rbinom(nn, 1L, x), gaussian = rnorm(nn, x), poisson = rpois(nn, x), error("Unknown family")) dev <- ff$devResid(y, x, wt) expect_that(fam$dev.resids(y, x, wt), equals(dev)) dd <- sum(dev) expect_that(fam$aic(y, n, x, wt, dd), equals(ff$aic(y, n, x, wt, dd))) }) } tst.devres(binomial(), mubinom) tst.devres(gaussian(), etas) tst.devres(poisson(), etapos) }) context("negative binomial") test_that("variance", { tst.variance <- function(fam, frm) { ff <- glmFamily$new(family=fam) sapply(frm, function(x) expect_that(fam$variance(x), equals(ff$variance(x)))) } tst.variance(MASS::negative.binomial(1.), etapos) nb1 <- MASS::negative.binomial(1.) cppnb1 <- glmFamily$new(family=nb1) expect_that(cppnb1$theta(), equals(1)) nb2 <- MASS::negative.binomial(2.) cppnb1$setTheta(2) sapply(etapos, function(x) expect_that(cppnb1$variance(x), equals(nb2$variance(x)))) bfam <- glmFamily$new(family=binomial()) if (FALSE) { ## segfaults on MacOS mavericks 3.1.0 ... ?? expect_error(bfam$theta())#, "theta accessor applies only to negative binomial") expect_error(bfam$setTheta(2))#, "setTheta applies only to negative binomial") } }) lme4/inst/tests/test-stepHalving.R0000754000175100001440000000061012374550357016646 0ustar hornikuserslibrary(lme4) library(testthat) load(system.file("testdata","survdat_reduced.Rda",package="lme4")) test_that('Step-halving works properly', { # this example is known to require step-halving (or at least has in the past # required step-halving) form <- survprop~(1|nobs) m <- glmer(form,weights=eggs,data=survdat_reduced,family=binomial,nAGQ=1L) expect_that(m, is_a("glmerMod")) }) lme4/inst/tests/test-lmer.R0000754000175100001440000003222512566374200015322 0ustar hornikusersstopifnot(require("testthat"), require("lme4")) context("fitting lmer models") ## is "Nelder_Mead" default optimizer? -- no longer (isNM <- formals(lmerControl)$optimizer == "Nelder_Mead") test_that("lmer", { set.seed(101) d <- data.frame(z=rnorm(200), f=factor(sample(1:10,200,replace=TRUE))) expect_warning(lmer(z~ 1|f, d, method="abc"),"Use the REML argument") expect_warning(lmer(z~ 1|f, d, method="Laplace"),"Use the REML argument") expect_warning(lmer(z~ 1|f, d, sparseX=TRUE),"has no effect at present") expect_is(fm1 <- lmer(Yield ~ 1|Batch, Dyestuff), "lmerMod") expect_is(fm1_noCD <- update(fm1,control=lmerControl(calc.derivs=FALSE)), "lmerMod") expect_equal(VarCorr(fm1),VarCorr(fm1_noCD)) ## backward compatibility version {for optimizer="Nelder-Mead" only}: if(isNM) expect_is(fm1.old <- update(fm1,control=lmerControl(use.last.params=TRUE)), "lmerMod") expect_is(fm1@resp, "lmerResp") expect_is(fm1@pp, "merPredD") expect_that(fe1 <- fixef(fm1), is_equivalent_to(1527.5)) expect_that(VarCorr(fm1)[[1]][1,1], ## "bobyqa" : 1764.050060 equals(1764.0375195, tolerance = 1e-5)) ## back-compatibility ... if(isNM) expect_that(VarCorr(fm1.old)[[1]][1,1], equals(1764.0726543)) expect_that(isREML(fm1), equals(TRUE)) expect_is(REMLfun <- as.function(fm1), "function") expect_that(REMLfun(1), equals(319.792389042002)) expect_that(REMLfun(0), equals(326.023232155879)) expect_that(family(fm1), equals(gaussian())) expect_that(isREML(fm1ML <- refitML(fm1)), equals(FALSE)) expect_that(REMLcrit(fm1), equals(319.654276842342)) expect_that(deviance(fm1ML), equals(327.327059881135)) ## "bobyqa": 49.51009984775 expect_that(sigma(fm1), equals(49.5101272946856, tolerance=1e-6)) if(isNM) expect_that(sigma(fm1.old), equals(49.5100503990048)) expect_that(sigma(fm1ML), equals(49.5100999308089)) expect_that(extractAIC(fm1), equals(c(3, 333.327059881135))) expect_that(extractAIC(fm1ML), equals(c(3, 333.327059881135))) ## "bobyqa": 375.71667627943 expect_that(vcov(fm1) [1,1], equals(375.714676744, tolerance=1e-5)) if(isNM) expect_that(vcov(fm1.old)[1,1], equals(375.72027872986)) expect_that(vcov(fm1ML) [1,1], equals(313.09721874266, tolerance=1e-7)) # was 313.0972246957 expect_is(fm2 <- refit(fm1, Dyestuff2$Yield), "lmerMod") expect_that(fixef(fm2), is_equivalent_to(5.6656)) expect_that(VarCorr(fm2)[[1]][1,1], is_equivalent_to(0)) expect_that(getME(fm2, "theta"), is_equivalent_to(0)) expect_that(X <- getME(fm1, "X"), is_equivalent_to(array(1, c(1, 30)))) expect_is(Zt <- getME(fm1, "Zt"), "dgCMatrix") expect_that(dim(Zt), equals(c(6L, 30L))) expect_that(Zt@x, equals(rep.int(1, 30L))) expect_equal(dimnames(Zt), list(levels(Dyestuff$Batch), rownames(Dyestuff))) ## "bobyqa": 0.8483237982 expect_that(theta <- getME(fm1, "theta"), equals(0.84832031, tolerance=6e-6, check.attributes=FALSE)) if(isNM) expect_that(getME(fm1.old, "theta"), is_equivalent_to(0.848330078)) expect_is(Lambdat <- getME(fm1, "Lambdat"), "dgCMatrix") expect_that(as(Lambdat, "matrix"), is_equivalent_to(diag(theta, 6L, 6L))) expect_is(fm3 <- lmer(Reaction ~ Days + (1|Subject) + (0+Days|Subject), sleepstudy), "lmerMod") expect_that(getME(fm3,"n_rtrms"), equals(2L)) expect_that(getME(fm3,"n_rfacs"), equals(1L)) expect_error(fm4 <- lmer(Reaction ~ Days + (1|Subject), subset(sleepstudy,Subject==levels(Subject)[1])), "must have > 1") expect_warning(fm4 <- lFormula(Reaction ~ Days + (1|Subject), subset(sleepstudy,Subject==levels(Subject)[1]), control=lmerControl(check.nlev.gtr.1="warning")), "must have > 1") expect_warning(fm4 <- lmer(Reaction ~ Days + (1|Subject), subset(sleepstudy,Subject %in% levels(Subject)[1:4]), control=lmerControl(check.nlev.gtreq.5="warning")), "< 5 sampled levels") sstudy9 <- subset(sleepstudy, Days == 1 | Days == 9) expect_error(lmer(Reaction ~ 1 + Days + (1 + Days | Subject), data = sleepstudy, subset = (Days == 1 | Days == 9)), "number of observations \\(=36\\) <= number of random effects \\(=36\\)") expect_error(lFormula(Reaction ~ 1 + Days + (1 + Days | Subject), data = sleepstudy, subset = (Days == 1 | Days == 9)), "number of observations \\(=36\\) <= number of random effects \\(=36\\)") ## with most recent Matrix (1.1-1), should *not* flag this ## for insufficient rank dat <- readRDS(system.file("testdata", "rankMatrix.rds", package="lme4")) expect_is(lFormula(y ~ (1|sample) + (1|day) + (1|day:sample) + (1|operator) + (1|day:operator) + (1|sample:operator) + (1|day:sample:operator), data = dat, control = lmerControl(check.nobs.vs.rankZ = "stop")), "list") ## check scale ss <- within(sleepstudy, Days <- Days*1e6) expect_warning(lmer(Reaction ~ Days + (1|Subject), data=ss), "predictor variables are on very different scales") ## Promote warning to error so that warnings or errors will stop the test: options(warn=2) expect_is(lmer(Yield ~ 1|Batch, Dyestuff, REML=TRUE), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, start=NULL), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, verbose=0L), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, subset=TRUE), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, weights=rep(1,nrow(Dyestuff))), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, na.action="na.exclude"), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, offset=rep(0,nrow(Dyestuff))), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, contrasts=NULL), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, devFunOnly=FALSE), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, control=lmerControl(optimizer="Nelder_Mead")), "lmerMod") expect_is(lmer(Yield ~ 1|Batch, Dyestuff, control=lmerControl()), "lmerMod") expect_error(lmer(Yield ~ 1|Batch, Dyestuff, control=lmerControl(optimizer="optimx")),"must be loaded") expect_error(lmer(Yield ~ 1|Batch, Dyestuff, control=lmerControl(optimizer="junk")), "couldn't find optimizer function") ## disable test ... should be no warning expect_is(lmer(Reaction ~ 1 + Days + (1 + Days | Subject), data = sleepstudy, subset = (Days == 1 | Days == 9), control=lmerControl(check.nobs.vs.rankZ="ignore", check.nobs.vs.nRE="ignore", check.conv.hess="ignore", ## need to ignore relative gradient check too; ## surface is flat so *relative* gradient gets large check.conv.grad="ignore")), "merMod") expect_is(lmer(Reaction ~ 1 + Days + (1|obs), data = transform(sleepstudy,obs=seq(nrow(sleepstudy))), control=lmerControl(check.nobs.vs.nlev="ignore", check.nobs.vs.nRE="ignore", check.nobs.vs.rankZ="ignore")), "merMod") expect_error(lmer(Reaction ~ 1 + Days + (1|obs), data = transform(sleepstudy,obs=seq(nrow(sleepstudy))), "number of levels of each grouping factor")) ## check for errors with illegal input checking options flags <- lme4:::.get.checkingOpts(names(formals(lmerControl))) .t <- lapply(flags, function(OPT) { ## set each to invalid string: ## cat(OPT,"\n") expect_error(lFormula(Reaction~1+Days+(1|Subject), data = sleepstudy, control = do.call(lmerControl, ## Deliberate: fake typo ## vvv setNames(list("warnign"), OPT))), "invalid control level") }) ## disable warning via options options(lmerControl=list(check.nobs.vs.rankZ="ignore",check.nobs.vs.nRE="ignore")) expect_is(fm4 <- lmer(Reaction ~ Days + (1|Subject), subset(sleepstudy,Subject %in% levels(Subject)[1:4])), "merMod") expect_is(lmer(Reaction ~ 1 + Days + (1 + Days | Subject), data = sleepstudy, subset = (Days == 1 | Days == 9), control=lmerControl(check.conv.hess="ignore", check.conv.grad="ignore")), "merMod") options(lmerControl=NULL) ## check for when ignored options are set options(lmerControl=list(junk=1,check.conv.grad="ignore")) expect_warning(lmer(Reaction ~ Days + (1|Subject),sleepstudy), "some options") options(lmerControl=NULL) options(warn=0) expect_warning(lmer(Yield ~ 1|Batch, Dyestuff, junkArg=TRUE),"extra argument.*disregarded") expect_warning(lmer(Yield ~ 1|Batch, Dyestuff, control=list()), "passing control as list is deprecated") if(FALSE) ## Hadley broke this expect_warning(lmer(Yield ~ 1|Batch, Dyestuff, control=glmerControl()), "passing control as list is deprecated") ss <- transform(sleepstudy,obs=factor(seq(nrow(sleepstudy)))) expect_warning(lmer(Reaction ~ 1 + (1|obs), data=ss, control=lmerControl(check.nobs.vs.nlev="warning", check.nobs.vs.nRE="ignore")), "number of levels of each grouping factor") ## test deparsing of very long terms inside mkReTrms set.seed(101) longNames <- sapply(letters[1:25], function(x) paste(rep(x,8),collapse="")) tstdat <- data.frame(Y=rnorm(10), F=factor(1:10), matrix(runif(250),ncol=25, dimnames=list(NULL, longNames))) expect_is(lFormula(Y~1+(aaaaaaaa+bbbbbbbb+cccccccc+dddddddd+ eeeeeeee+ffffffff+gggggggg+hhhhhhhh+ iiiiiiii+jjjjjjjj+kkkkkkkk+llllllll|F), data=tstdat, control=lmerControl(check.nobs.vs.nlev="ignore", check.nobs.vs.nRE="ignore", check.nobs.vs.rankZ="ignore")),"list") ## do.call(new,...) bug new <- "foo" expect_is(refit(fm1),"merMod") rm("new") ## test subset-with-( printing from summary fm1 <- lmer(z~1|f,d,subset=(z<1e9)) expect_equal(sum(grepl("Subset: \\(",capture.output(summary(fm1)))),1) ## test messed-up Hessian fm1 <- lmer(z~ as.numeric(f) + 1|f, d) fm1@optinfo$derivs$Hessian[2,2] <- NA expect_warning(lme4:::checkConv(fm1@optinfo$derivs, coefs=c(1,1), ctrl=lmerControl()$checkConv,lbound=0), "Problem with Hessian check") ## test ordering of Ztlist names ## this is a silly model, just using it for a case ## where nlevs(RE term 1) < nlevs(RE term 2)x data(cbpp) cbpp <- transform(cbpp,obs=factor(1:nrow(cbpp))) fm0 <- lmer(incidence~1+(1|herd)+(1|obs),cbpp, control=lmerControl(check.nobs.vs.nlev="ignore", check.nobs.vs.rankZ="ignore", check.nobs.vs.nRE="ignore", check.conv.grad="ignore", check.conv.singular="ignore", check.conv.hess="ignore")) fm0B <- update(fm0, .~1+(1|obs)+(1|herd)) expect_equal(names(getME(fm0,"Ztlist")), c("obs.(Intercept)", "herd.(Intercept)")) ## stable regardless of order in formula expect_equal(getME(fm0,"Ztlist"),getME(fm0B,"Ztlist")) }) ## test_that(..) test_that("coef_lmer", { ## test coefficient extraction in the case where RE contain ## terms that are missing from the FE ... set.seed(101) d <- data.frame(resp=runif(100), var1=factor(sample(1:5,size=100,replace=TRUE)), var2=runif(100), var3=factor(sample(1:5,size=100,replace=TRUE))) library(lme4) mix1 <- lmer(resp ~ 0 + var1 + var1:var2 + (1|var3), data=d) c1 <- coef(mix1) expect_is(c1, "coef.mer") cd1 <- c1$var3 expect_is (cd1, "data.frame") n1 <- paste0("var1", 1:5) nn <- c(n1, paste(n1, "var2", sep=":")) expect_identical(names(cd1), c("(Intercept)", nn)) expect_equal(fixef(mix1), setNames(c(0.27039541, 0.38329083, 0.45127874, 0.65288384, 0.61098249, 0.49497978, 0.12227105, 0.087020934,-0.28564318,-0.015968354), nn), tolerance= 7e-7)# 64-bit: 6.73e-9 }) lme4/inst/tests/test-lmerResp.R0000754000175100001440000000435712406350752016160 0ustar hornikuserslibrary("lme4") library("testthat") data(Dyestuff, package="lme4") n <- nrow(Dyestuff) ones <- rep.int(1, n) zeros <- rep.int(0, n) YY <- Dyestuff$Yield mYY <- mean(YY) context("lmerResp objects") test_that("lmerResp", { mres <- YY - mYY rr <- lmerResp$new(y=YY) expect_that(rr$weights, equals(ones)) expect_that(rr$sqrtrwt, equals(ones)) expect_that(rr$sqrtXwt, equals(ones)) expect_that(rr$offset, equals(zeros)) expect_that(rr$mu, equals(zeros)) expect_that(rr$wtres, equals(YY)) expect_that(rr$wrss(), equals(sum(YY^2))) expect_that(rr$updateMu(rep.int(mYY, n)), equals(sum(mres^2))) expect_that(rr$REML, equals(0L)) rr$REML <- 1L expect_that(rr$REML, equals(1L)) }) mlYY <- mean(log(YY)) gmeanYY <- exp(mlYY) # geometric mean context("glmResp objects") test_that("glmResp", { mres <- YY - gmeanYY gmean <- rep.int(gmeanYY, n) rr <- glmResp$new(family=poisson(), y=YY) expect_that(rr$weights, equals(ones)) expect_that(rr$sqrtrwt, equals(ones)) expect_that(rr$sqrtXwt, equals(ones)) expect_that(rr$offset, equals(zeros)) expect_that(rr$mu, equals(zeros)) expect_that(rr$wtres, equals(YY)) expect_that(rr$n, equals(ones)) ## wrss() causes an update of mu which becomes ones, wtres also changes expect_that(rr$wrss(), equals(sum((YY-1)^2))) expect_that(rr$mu, equals(ones)) expect_that(rr$wtres, equals(YY-ones)) expect_that(rr$updateMu(rep.int(mlYY, n)), equals(sum(mres^2))) expect_that(rr$mu, equals(gmean)) expect_that(rr$muEta(), equals(gmean)) expect_that(rr$variance(), equals(gmean)) rr$updateWts() expect_that(1/sqrt(rr$variance()), equals(rr$sqrtrwt)) expect_that(as.vector(rr$sqrtXwt), equals(rr$sqrtrwt * rr$muEta())) }) lme4/inst/tests/test-methods.R0000754000175100001440000005137712604534664016044 0ustar hornikuserslibrary("testthat") library("lme4") L <- load(system.file("testdata", "lme-tst-fits.rda", package="lme4", mustWork=TRUE)) ## FIXME: should test for old R versions, skip reloading test data in that ## case? fm0 <- fit_sleepstudy_0 fm1 <- fit_sleepstudy_1 fm2 <- fit_sleepstudy_2 gm1 <- fit_cbpp_1 gm2 <- fit_cbpp_2 ## More objects to use in all contexts : set.seed(101) dNA <- data.frame( xfac= sample(letters[1:10], 100, replace=TRUE), xcov= runif(100), resp= rnorm(100)) dNA[sample(1:100, 10), "xcov"] <- NA CI.boot <- function(fm, nsim=10, seed=101, ...) suppressWarnings(confint(fm, method="boot", nsim=nsim, quiet=TRUE, seed=seed, ...)) ## rSimple <- function(rep = 2, m.u = 2, kinds = c('fun', 'boring', 'meh')) { stopifnot(is.numeric(rep), rep >= 1, is.numeric(m.u), m.u >= 1, is.character(kinds), (nk <- length(kinds)) >= 1) nobs <- rep * m.u * nk data.frame(kind= rep(kinds, each=rep*m.u), unit = gl(m.u, 1, nobs), y = round(50*rnorm(nobs))) } d12 <- rSimple() data("Pixel", package="nlme") nPix <- nrow(Pixel) fmPix <- lmer(pixel ~ day + I(day^2) + (day | Dog) + (1 | Side/Dog), data = Pixel) context("summary") test_that("summary", { ## test for multiple-correlation-warning bug and other 'correlation = *' variants ## Have 2 summary() versions, each with 3 print(.) ==> 6 x capture.output(.) sf.aa <- summary(fit_agridat_archbold) msg1 <- "Correlation.* not shown by default" ## message => *not* part of capture.*(.) expect_message(c1 <- capture.output(sf.aa), msg1) # correlation = NULL - default cF <- capture.output(print(sf.aa, correlation=FALSE)) ## TODO? ensure the above gives *no* message/warning/error expect_identical(c1, cF) expect_message( cT <- capture.output(print(sf.aa, correlation=TRUE)) , "Correlation.* could have been required in summary()") expect_identical(cF, cT[seq_along(cF)]) sfT.aa <- summary(fit_agridat_archbold, correlation=TRUE) expect_message(cT2 <- capture.output(sfT.aa), msg1) expect_identical(cF, cT2) cT3 <- capture.output(print(sfT.aa, correlation=TRUE)) expect_identical(cT, cT3) cF2 <- capture.output(print(sfT.aa, correlation=FALSE)) expect_identical(cF, cF2) }) context("anova") test_that("lmer", { expect_that(suppressMessages(anova(fm0,fm1)), is_a("anova")) expect_warning(do.call(anova,list(fm0,fm1)), "assigning generic names") ## dat <- data.frame(y = 1:5, u = c(rep("A",2), rep("B",3)), t = c(rep("A",3), rep("B",2))) datfun <- function(x) dat aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa <- dat expect_is(stats::anova(lmer(y ~ u + (1 | t), dat = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa, REML=FALSE), lmer(y ~ 1 + (1 | t), dat = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa, REML=FALSE)), "anova") expect_equal(rownames(stats::anova(lmer(y ~ u + (1 | t), dat = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa, REML=FALSE), lmer(y ~ 1 + (1 | t), dat = aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa, REML=FALSE), model.names=c("a","b"))), c("b","a")) expect_error(rownames(stats::anova(lmer(y ~ u + (1 | t), dat = dat, REML=FALSE), lmer(y ~ 1 + (1 | t), dat = dat, REML=FALSE), model.names=c("a","b","c"))), "different lengths") z <- 1 stats::anova(lmer(y ~ u + (1 | t), data = datfun(z), REML=FALSE), lmer(y ~ 1 + (1 | t), data = datfun(z), REML=FALSE)) ## ## from Roger Mundry via Roman Lustrik full <- lmer(resp ~ xcov + (1|xfac), data=dNA) null <- lmer(resp ~ 1 + (1|xfac), data=dNA) expect_error(anova(null,full), "models were not all fitted to the same size of dataset") ## Github issue #256 from Jonas Lindeløv -- issue is *not* specific for this dataset ## Two models with subset() within lmer() full3 <- lmer(y ~ kind + (1|unit), subset(d12, kind != 'boring'), REML=FALSE) null3 <- update(full3, .~. - kind) op <- options(warn = 2) # no warnings! ano3 <- anova(full3, null3)## issue #256: had warning in data != data[[1]] : ... o3 <- capture.output(ano3) # now prints with only one 'Data:' expect_equal(1, grep("^Data:", o3)) ## ## no problem with subset()ting outside lmer() call: d12sub <- subset(d12, kind != 'boring') expect_is(full3s <- lmer(y ~ kind + (1|unit), d12sub, REML=FALSE), "lmerMod") expect_is(null3s <- update(full3s, .~. - kind), "lmerMod") expect_is(ano3s <- anova(full3s, null3s), "anova") expect_equal(ano3, ano3s, check.attributes=FALSE) options(op) ## anova() of glmer+glm models: gm1 <- glmer(y~(1|u),data=dat[1:4,],family=poisson) gm0 <- glm(y~1,data=dat[1:4,],family=poisson) gm2 <- glmer(y~(1|u),data=dat[1:4,],family=poisson,nAGQ=2) aa <- anova(gm1,gm0) expect_equal(aa[2,"Chisq"],0) expect_error(anova(gm2,gm0),"incommensurate") ## anova() of lmer+glm models: dat2 <- dat set.seed(101) dat2$y <- rnorm(5) fm1 <- lmer(y~(1|u),data=dat2,REML=FALSE) fm0 <- lm(y~1,data=dat2) aa2 <- anova(fm1,fm0) expect_equal(aa2[2,"Chisq"],0) }) context("bootMer confint()") set.seed(47) test_that("bootMer", { ## testing bug-fix for ordering of sd/cor components in sd/cor matrix with >2 rows m1 <- lmer(strength~1+(cask|batch),Pastes) ci <- CI.boot(m1) corvals <- ci[grep("^cor_",rownames(ci)),] expect_true(all(abs(corvals) <= 1)) ## test bootMer with GLMM, multiple RE ci1 <- CI.boot(gm2, nsim=5) ci2 <- CI.boot(gm2, nsim=5, parm=3:6) ci2B <- CI.boot(gm2, nsim=5, parm="beta_") ## previously tested with nsim=5 vs nsim=3 expect_true(nrow(ci2) == 4) expect_equal(ci2,ci2B) expect_equal(ci1[3:6,], ci2) ## , tolerance = 0.4)# 0.361 ## bootMer with NA values PastesNA <- Pastes PastesNA$cask[1:3] <- NA ## previously set 'Sample' (sic) -- no effect! m2 <- update(m1, data=PastesNA) ci3 <- CI.boot(m2) expect_equal(ci, ci3, tol=0.2) sleepstudyNA <- sleepstudy sleepstudyNA$Days[1:3] <- NA m4 <- update(fm2, data = sleepstudyNA) expect_true(nrow(ci4 <- CI.boot(m4)) == 6) # could check more ## ## semipar bootstrapping m5 <- lmer(Yield ~ 1|Batch, Dyestuff) set.seed(1) suppressPackageStartupMessages(require(boot)) boo01.sp <- bootMer(m5, fixef, nsim = 100, use.u = TRUE, type = "semiparametric") expect_equal(sd(boo01.sp$t), 8.215586, tol = 1e-4) ## passing FUN to confint: Torbjørn HÃ¥kan Ergon testFun <- function(fit) fixef(fit)[1] expect_equal(c(unclass( suppressWarnings(confint(fm1, method="boot", FUN=testFun, nsim=10, quiet=TRUE)))), c(243.7551,256.9104),tol=1e-3) ## passing re.form to bootMer FUN <- function(.){ predict(., type="response") } fm2 <- lmer(strength ~ (1|batch/cask), Pastes) expect_is(bootMer(fm2, predict, nsim=3),"boot") expect_is(bootMer(fm2, predict, re.form=NULL, nsim=3),"boot") expect_is(bootMer(fm2, predict, re.form=~(1|batch)+(1|cask:batch), nsim=3), "boot") expect_is(b3 <- bootMer(fm2, predict, re.form=~(1|batch), nsim=3), "boot") }) context("confint_other") test_that("confint", { load(system.file("testdata", "gotway_hessianfly.rda", package = "lme4")) ## generated via: ## gotway_hessianfly_fit <- glmer(cbind(y, n-y) ~ gen + (1|block), ## data=gotway.hessianfly, family=binomial, ## control=glmerControl(check.nlev.gtreq.5="ignore")) ## gotway_hessianfly_prof <- profile(gotway_hessianfly_fit,which=1) ## save(list=ls(pattern="gotway"),file="gotway_hessianfly.rda") expect_equal(confint(gotway_hessianfly_prof)[1,1],0) ## FIXME: should add tests for {-1,1} bounds on correlations as well expect_equal(c(confint(fm1,method="Wald",parm="beta_")), c(232.301892,8.891041,270.508318,12.043531), tol=1e-5) ## Wald gives NA for theta values expect_true(all(is.na(confint(fm1,method="Wald",parm="theta_")))) ## check names ci1.p <- suppressWarnings(confint(fm1,quiet=TRUE)) ci1.w <- confint(fm1,method="Wald") ci1.b <- CI.boot(fm1, nsim=2) expect_equal(dimnames(ci1.p), list(c(".sig01", ".sigma", "(Intercept)", "Days"), c("2.5 %", "97.5 %"))) expect_equal(dimnames(ci1.p),dimnames(ci1.w)) expect_equal(dimnames(ci1.p),dimnames(ci1.b)) ci1.p.n <- suppressWarnings(confint(fm1,quiet=TRUE,oldNames=FALSE)) ci1.w.n <- confint(fm1,method="Wald", oldNames=FALSE) ci1.b.n <- CI.boot(fm1, nsim=2, oldNames=FALSE) expect_equal(dimnames(ci1.p.n), list(c("sd_(Intercept)|Subject", "sigma", "(Intercept)", "Days"), c("2.5 %", "97.5 %"))) expect_equal(dimnames(ci1.p.n),dimnames(ci1.w.n)) expect_equal(dimnames(ci1.p.n),dimnames(ci1.b.n)) ## test case of slightly wonky (spline fit fails) but monotonic profiles: ## simfun <- function(J,n_j,g00,g10,g01,g11,sig2_0,sig01,sig2_1){ N <- sum(rep(n_j,J)) x <- rnorm(N) z <- rnorm(J) mu <- c(0,0) sig <- matrix(c(sig2_0,sig01,sig01,sig2_1),ncol=2) u <- MASS::mvrnorm(J,mu=mu,Sigma=sig) b_0j <- g00 + g01*z + u[,1] b_1j <- g10 + g11*z + u[,2] y <- rep(b_0j,each=n_j)+rep(b_1j,each=n_j)*x + rnorm(N,0,sqrt(0.5)) sim_data <- data.frame(Y=y,X=x,Z=rep(z,each=n_j), group=rep(1:J,each=n_j)) } set.seed(102) dat <- simfun(10,5,1,.3,.3,.3,(1/18),0,(1/18)) fit <- lmer(Y~X+Z+X:Z+(X||group),data=dat) if (Sys.info()["sysname"] != "SunOS") { ## doesn't produce warnings on Solaris ... expect_warning(pp <- profile(fit,"theta_",quiet=TRUE), "non-monotonic profile") expect_warning(cc <- confint(pp),"falling back to linear interpolation") ## very small/unstable problem, needs large tolerance expect_equal(unname(cc[2,]),c(0,0.5427609),tolerance=1e-2) } badprof <- readRDS(system.file("testdata","badprof.rds", package="lme4")) expect_warning(cc <- confint(badprof), "falling back to linear") expect_equal(cc, structure(c(0, -1, 2.50856219044636, 48.8305727797906, NA, NA, 33.1204478717389, 1, 7.33374326592662, 68.7254711217912, -6.90462047196017, NA), .Dim = c(6L, 2L), .Dimnames = list(c(".sig01", ".sig02", ".sig03", ".sigma", "(Intercept)", "cYear"), c("2.5 %", "97.5 %"))), tol=1e-3) }) context("refit") test_that("refit", { s1 <- simulate(fm1) expect_is(refit(fm1,s1), "merMod") s2 <- simulate(fm1,2) expect_error(refit(fm1,s2), "refit not implemented .* lists") data(Orthodont,package = "nlme") fmOrth <- fm <- lmer(distance ~ I(age - 11) + (I(age - 11) | Subject), data = Orthodont) expect_equal(s1 <- simulate(fm,newdata = Orthodont,seed = 101), s2 <- simulate(fm,seed = 101)) }) context("predict") test_that("predict", { d1 <- with(cbpp, expand.grid(period = unique(period), herd = unique(herd))) d2 <- data.frame(period = "1", herd = unique(cbpp$herd)) d3 <- expand.grid(period = as.character(1:3), herd = unique(cbpp$herd)) p0 <- predict(gm1) p1 <- predict(gm1,d1) p2 <- predict(gm1,d2) p3 <- predict(gm1,d3) expect_equal(p0[1], p1[1]) expect_equal(p0[1], p2[1]) expect_equal(p0[1], p3[1]) expect_warning(predict(gm1, ReForm=NA), "is deprecated") ## matrix-valued predictors: Github #201 from Fabian S. sleepstudy$X <- cbind(1, sleepstudy$Days) m <- lmer(Reaction ~ -1 + X + (Days | Subject), sleepstudy) pm <- predict(m, newdata=sleepstudy) expect_is(pm, "numeric") expect_equal(quantile(pm, names = FALSE), c(211.006525, 260.948978, 296.87331, 328.638297, 458.155583)) if (require("MEMSS",quietly=TRUE)) { ## test spurious warning with factor as response variable data("Orthodont", package = "MEMSS") # (differently "coded" from the 'default' "nlme" one) silly <- glmer(Sex ~ distance + (1|Subject), data = Orthodont, family = binomial) } sillypred <- data.frame(distance = c(20, 25)) op <- options(warn = 2) # no warnings! ps <- predict(silly, sillypred, re.form=NA, type = "response") expect_is(ps, "numeric") expect_equal(unname(ps), c(0.999989632, 0.999997201)) ## a case with interactions (failed in one temporary version): expect_warning(fmPixS <<- update(fmPix, .~. + Side), "nearly unidentifiable|unable to evaluate scaled gradient|failed to converge") ## (1|2|3); 2 and 3 seen (as Error??) on CRAN's Windows 32bit options(op) set.seed(1); ii <- sample(nrow(Pixel), 16) expect_equal(predict(fmPix, newdata = Pixel[ii,]), fitted(fmPix )[ii]) expect_equal(predict(fmPixS, newdata = Pixel[ii,]), fitted(fmPixS)[ii]) set.seed(7); n <- 100; y <- rnorm(n) dd <- data.frame(id = factor(sample(10, n, replace = TRUE)), x1 = 1, y = y, x2 = rnorm(n, mean = sign(y))) expect_message(m <- lmer(y ~ x1 + x2 + (1 | id), data = dd), "fixed-effect model matrix is rank deficient") expect_is(summary(m),"summary.merMod") ii <- sample(n, 16) expect_equal(predict(m, newdata = dd[ii,]), fitted(m)[ii]) ## predict(*, new..) gave Error in X %*% fixef(object) - now also drops col. ## predict(*, new..) with NA in data {and non-simple model}, issue #246: m1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy) sleepst.NA <- sleepstudy ; sleepst.NA$Days[2] <- NA m2 <- update(fm1, data = sleepst.NA) ## maybe tricky for evaluation; fm1 was defined elsewhere, so data expect_equal(length(predict(m2, sleepst.NA[1:4,])),4) ## Wrong 'b' constructed in mkNewReTrms() -- issue #257 data(Orthodont,package="nlme") Orthodont <- within(Orthodont, nsex <- as.numeric(Sex == "Male")) m3 <- lmer(distance ~ age + (age|Subject) + (0 + Sex |Subject), data=Orthodont, control=lmerControl(check.conv.hess="ignore")) m4 <- lmer(distance ~ age + (age|Subject) + (0 + nsex|Subject), data=Orthodont) expect_equal(p3 <- predict(m3, Orthodont), fitted(m3), tol=1e-14) expect_equal(p4 <- predict(m4, Orthodont), fitted(m4), tol=1e-14) ## related to GH #275 (*passes*), ss <- sleepstudy set.seed(1) ss$noiseChar <- ifelse(runif(nrow(sleepstudy)) > 0.8, "Yes", "No") ss$noiseFactor <- factor(ss$noiseChar) fm2 <- lmer(Reaction ~ Days + noiseChar + (Days | Subject), ss) expect_equal(predict(fm2, newdata = model.frame(fm2)[2:3, ])[2], predict(fm2, newdata = model.frame(fm2)[3, ])) fm3 <- lmer(Reaction ~ Days + noiseFactor + (Days | Subject), ss) expect_equal(predict(fm3, newdata = model.frame(fm3)[2:3, ])[2], predict(fm3, newdata = model.frame(fm3)[3, ])) ## complex-basis functions in RANDOM effect: (currently) fm5 <- lmer(Reaction~Days+(poly(Days,2)|Subject),sleepstudy) expect_equal(predict(fm5,sleepstudy[1,]),fitted(fm5)[1]) ## complex-basis functions in FIXED effect are fine fm6 <- lmer(Reaction~poly(Days,2)+(1|Subject),sleepstudy) expect_equal(predict(fm6,sleepstudy[1,]),fitted(fm6)[1]) }) context("simulate") test_that("simulate", { expect_is(simulate(gm2), "data.frame") expect_warning(simulate(gm2, ReForm = NA), "is deprecated") expect_warning(simulate(gm2, REForm = NA), "is deprecated") p1 <- simulate(gm2, re.form = NULL, seed = 101) p2 <- simulate(gm2, re.form = ~0, seed = 101) p3 <- simulate(gm2, re.form = NA, seed = 101) p4 <- simulate(gm2, re.form = NULL, seed = 101) expect_warning(p5 <- simulate(gm2, ReForm = ~0, seed = 101), "is deprecated") p6 <- simulate(gm2, re.form = NA, seed = 101) expect_warning(p7 <- simulate(gm2, REForm = NULL, seed = 101), "is deprecated") p8 <- simulate(gm2, re.form = ~0, seed = 101) p9 <- simulate(gm2, re.form = NA, seed = 101) p10 <- simulate(gm2,use.u = FALSE, seed = 101) p11 <- simulate(gm2,use.u = TRUE, seed = 101) ## minimal check of content: expect_identical(colSums(p1[,1]), c(incidence = 95, 747)) expect_identical(colSums(p2[,1]), c(incidence = 109, 733)) ## equivalences: ## group ~0: expect_equal(p2,p3) expect_equal(p2,p5) expect_equal(p2,p6) expect_equal(p2,p8) expect_equal(p2,p9) expect_equal(p2,p10) ## group 1: expect_equal(p1,p4) expect_equal(p1,p7) expect_equal(p1,p11) expect_error(simulate(gm2,use.u = TRUE, re.form = NA), "should specify only one") ## ## hack: test with three REs p1 <- lmer(diameter ~ (1|plate) + (1|plate) + (1|sample), Penicillin, control = lmerControl(check.conv.hess = "ignore", check.conv.grad = "ignore")) expect_is(sp1 <- simulate(p1, seed=123), "data.frame") expect_identical(dim(sp1), c(nrow(Penicillin), 1L)) expect_equal(fivenum(sp1[,1]), c(20.9412, 22.5805, 23.5575, 24.6095, 27.6997), tol=1e-5) ## Pixel example expect_identical(dim(simulate(fmPixS)), c(nPix, 1L)) expect_identical(dim(simulate(fmPix )), c(nPix, 1L)) ## simulation with newdata smaller/larger different from original fm <- lmer(diameter ~ 1 + (1|plate) + (1|sample), Penicillin) expect_is(simulate(fm,newdata=Penicillin[1:10,],allow.new.levels=TRUE),"data.frame") expect_is(simulate(fm,newdata=do.call(rbind,replicate(4,Penicillin,simplify=FALSE))),"data.frame") ## negative binomial sims set.seed(101) dd <- data.frame(f=factor(rep(1:10,each=20)), x=runif(200), y=rnbinom(200,size=2,mu=2)) g1 <- glmer.nb(y ~ x + (1|f), data=dd) th.g1 <- getME(g1, "glmer.nb.theta") ts1 <- table(s1 <- simulate(g1)[,1]) expect_equal(fixef(g1), c("(Intercept)" = 0.630067, x = -0.0167248), tol = 1e-5) expect_equal(th.g1, 2.013, tol = 1e-4) expect_equal(th.g1, g1@call$family[["theta"]])# <- important for pkg{effects} eval() expect_identical(sum(s1), 403) expect_identical(as.vector(ts1[as.character(0:5)]), c(51L, 54L, 36L, 21L, 14L, 9L)) ## Simulate with newdata with *new* RE levels: d <- sleepstudy[-1] # droping the response ("Reaction") ## d$Subject <- factor(rep(1:18, each=10)) ## Add 18 new subjects: d <- rbind(d, d) d$Subject <- factor(rep(1:36, each=10)) d$simulated <- simulate(fm1, seed=1, newdata = d, re.form=NULL, allow.new.levels = TRUE)[,1] expect_equal(mean(d$simulated), 299.9384608) }) context("misc") test_that("misc", { expect_equal(df.residual(fm1),176) if (require(ggplot2)) { expect_is(fortify(fm1), "data.frame") expect_is(fortify(gm1), "data.frame") } expect_is(as.data.frame(VarCorr(fm1)), "data.frame") }) context("plot") test_that("plot", { ## test getData() within plot function: reported by Dieter Menne doFit <- function(){ data(Orthodont,package = "nlme") data1 <- Orthodont lmer(distance ~ age + (age|Subject), data = data1) } data(Orthodont, package = "nlme") fm0 <- lmer(distance ~ age + (age|Subject), data = Orthodont) expect_is(plot(fm0), "trellis") suppressWarnings(rm("Orthodont")) fm <- doFit() pp <- plot(fm, resid(., scaled = TRUE) ~ fitted(.) | Sex, abline = 0) expect_is(pp, "trellis") ## test qqmath/getIDLabels() expect_is(q1 <- lattice::qqmath(fm,id=0.05),"trellis") cake2 <- transform(cake,replicate=as.numeric(replicate), recipe=as.numeric(recipe)) fm2 <- lmer(angle ~ recipe + temp + (1|recipe:replicate), cake2, REML= FALSE) expect_is(lattice::qqmath(fm2,id=0.05), "trellis") expect_is(lattice::qqmath(fm2,id=0.05, idLabels=~recipe), "trellis") }) context("misc") test_that("summary", { ## test that family() works when $family element is weird gnb <- glmer(TICKS~1+(1|BROOD), family=MASS::negative.binomial(theta=2), data=grouseticks) expect_is(family(gnb),"family") }) lme4/inst/tests/test-glmer.Rout0000754000175100001440000002173712227505347016231 0ustar hornikusers R Under development (unstable) (2013-09-09 r63889) -- "Unsuffered Consequences" Copyright (C) 2013 The R Foundation for Statistical Computing Platform: i686-pc-linux-gnu (32-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. 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Type 'q()' to quit R. > library("testthat") > library("lme4") Loading required package: lattice Loading required package: Matrix > > testLevel <- if (nzchar(s <- Sys.getenv("LME4_TEST_LEVEL"))) + as.numeric(s) else 1 > > context("fitting glmer models") > test_that("glmer", { + expect_warning(glmer(z~ 1|f, family=binomial, method="abc"),"Use the nAGQ argument") + expect_warning(glmer(z~ 1|f, family=binomial, method="Laplace"),"Use the nAGQ argument") + expect_warning(glmer(z~ 1|f, sparseX=TRUE),"has no effect at present") + expect_that(gm1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), + data = cbpp, family = binomial), is_a("glmerMod")) + expect_that(gm1@resp, is_a("glmResp")) + expect_that(gm1@pp, is_a("merPredD")) + expect_equal(ge1 <- unname(fixef(gm1)), c(-1.39854982537216, -0.992335519118859, + -1.12867532780426, -1.58030423764517), + tol=5e-4) + expect_equal(c(VarCorr(gm1)[[1]]), 0.41245527438386, tol=6e-4) + ### expect_that(family(gm1), equals(binomial())) + ### ?? binomial() has an 'initialize' component ... and the order is different + expect_equal(deviance(gm1), 184.052674598026, tol=1e-5) + expect_equal(sigma(gm1), 1) + expect_equal(extractAIC(gm1), c(5, 194.052674598026), tol=1e-5) + + expect_equal(theta <- unname(getME(gm1, "theta")), 0.642226809144453, tol=6e-4) + ###expect_that(X <- getME(gm1, "X"), is_equivalent_to(array(1, c(1, 30)))) + expect_that(Zt <- getME(gm1, "Zt"), is_a("dgCMatrix")) + expect_equal(dim(Zt), c(15L, 56L)) + expect_equal(length(Zt@x), 56L) + expect_equal(Zt@x, rep.int(1, 56L)) + expect_that(Lambdat <- getME(gm1, "Lambdat"), is_a("dgCMatrix")) + expect_equivalent(as(Lambdat, "matrix"), diag(theta, 15L, 15L)) + expect_error(glFormula(cbind(incidence, size - incidence) ~ period + (1 | herd), + data = subset(cbpp, herd==levels(herd)[1]), family = binomial), + "must have > 1") + expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), + data = subset(cbpp, herd %in% levels(herd)[1:4]), + family = binomial, + control=glmerControl(check.nlev.gtreq.5="warning")), + "< 5 sampled levels") + expect_warning(fm1. <- glmer(Reaction ~ Days + (Days|Subject), sleepstudy), + regexp="calling .* with family=gaussian .* as a shortcut") + options(warn=2) + cbppX <- transform(cbpp,prop=incidence/size) + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size), "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, start=NULL), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, verbose=0L), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, subset=TRUE), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, na.action="na.exclude"), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, offset=rep(0,nrow(cbppX))), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, contrasts=NULL), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, devFunOnly=FALSE), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, + control=glmerControl(optimizer="Nelder_Mead")), + "glmerMod") + expect_is(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, control=glmerControl()), + "glmerMod") + options(warn=0) + expect_warning(glmer(prop ~ period + (1 | herd), + data = cbppX, family = binomial, weights=size, junkArg=TRUE), + "extra argument.*disregarded") + expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), + data = cbpp, family = binomial, + control=list()), + "instead of passing a list of class") + expect_warning(glmer(cbind(incidence, size - incidence) ~ period + (1 | herd), + data = cbpp, family = binomial, + control=lmerControl()), + "instead of passing a list of class") + + ## + load(system.file("testdata","radinger_dat.RData",package="lme4")) + mod <- glmer(presabs~predictor+(1|species),family=binomial, + radinger_dat) + expect_is(mod,"merMod") + ## TODO: is this reliable across platforms or do we have to loosen? + expect_equal(unname(fixef(mod)),c(0.5425528,6.4289962)) + set.seed(101) + d <- data.frame(y=rbinom(1000,size=1,p=0.5), + x=runif(1000), + f=factor(rep(1:20,each=50)), + x2=rep(0:1,c(999,1))) + mod2 <- glmer(y~x+x2+(1|f),data=d,family=binomial) + expect_equal(unname(fixef(mod2))[1:2], + c(-0.10036244,0.03548523),tol=1e-4) + expect_true(unname(fixef(mod2)[3]<(-10))) + mod3 <- update(mod2,family=binomial(link="probit")) + expect_equal(unname(fixef(mod3))[1:2], + c(-0.06288878,0.02224270),tol=1e-4) + expect_true(unname(fixef(mod3)[3]<(-4))) + mod4 <- update(mod2,family=binomial(link="cauchit")) + + ## on-the-fly creation of index variables + set.seed(101) + d <- data.frame(y1=rpois(100,1), x=rnorm(100), ID=1:100) + fit1 <- glmer(y1 ~ x+(1|ID),data=d,family=poisson) + ## fit2 <- update(fit1, .~ x+(1|rownames(d))) + fit2 <- glmer(y1 ~ x+(1|rownames(.GlobalEnv$d)),data=d,family=poisson) + expect_equal(unname(unlist(VarCorr(fit1))), + unname(unlist(VarCorr(fit2)))) + + ## + if (testLevel>1) { + load(system.file("testdata","mastitis.rda",package="lme4")) + t1 <- system.time(g1 <- + glmer(NCM ~ birth + calvingYear + (1|sire) + + (1|herd),mastitis,poisson)) + t2 <- system.time(g2 <- update(g1, + control=glmerControl(optimizer="bobyqa"))) + ## 20 seconds N-M vs 8 seconds bobyqa ... + ## problem is fairly ill-conditioned so parameters + ## are relatively far apart even though likelihoods are OK + expect_equal(logLik(g1),logLik(g2),tol=1e-7) + } + + }) Error: Test failed: 'glmer' invalid type (NULL) for variable 'rownames(.GlobalEnv$d)' 1: glmer(y1 ~ x + (1 | rownames(.GlobalEnv$d)), data = d, family = poisson) 2: eval(mc, parent.frame(1L)) 3: eval(expr, envir, enclos) 4: lme4::glFormula(formula = y1 ~ x + (1 | rownames(.GlobalEnv$d)), data = d, family = poisson) 5: eval(mf, parent.frame()) 6: eval(expr, envir, enclos) 7: model.frame(data = d, drop.unused.levels = TRUE, formula = y1 ~ x + (1 + rownames(.GlobalEnv$d))) 8: model.frame.default(data = d, drop.unused.levels = TRUE, formula = y1 ~ x + (1 + rownames(.GlobalEnv$d))) 9: .handleSimpleError(function (e) { e$calls <- head(sys.calls()[-seq_len(frame + 7)], -2) signalCondition(e) }, "invalid type (NULL) for variable 'rownames(.GlobalEnv$d)'", quote(model.frame.default(data = d, drop.unused.levels = TRUE, formula = y1 ~ x + (1 + rownames(.GlobalEnv$d))))) Execution halted lme4/inst/tests/test-NAhandling.R0000754000175100001440000001625412565075351016376 0ustar hornikusersstopifnot(require("testthat"), require("lme4")) context("NA (and Inf) handling") ## Modified sleepstudy data : sleepst.a <- sleepstudy rownames(sleepst.a) <- paste0("a", rownames(sleepstudy)) sleepstudyNA <- within(sleepst.a, Reaction[1:3] <- NA) sleepstudyNA2 <- within(sleepst.a, Days[1:3] <- NA) sleepInf <- within(sleepstudy, Reaction[Reaction > 400] <- Inf) ## Modified cake data : cakeNA <- rbind(cake, tail(cake,1)) cakeNA[nrow(cakeNA), "angle"] <- NA ## Create new data frame with some NAs in fixed effect cakeNA.X <- within(cake, temp[1:5] <- NA) ## NA values in random effects -- should get treated cakeNA.Z <- within(cake, replicate[1:5] <- NA) test_that("naming", { ## baseline model fm1 <- lmer(Reaction~Days+(Days|Subject), sleepst.a) ## default: na.omit fm2 <- update(fm1, data=sleepstudyNA, control=lmerControl(check.conv.grad="ignore")) expect_equal(head(names(fitted(fm1))), paste0("a",1:6)) expect_equal(head(names(fitted(fm2))), paste0("a",4:9)) expect_equal(names(predict(fm2)), names(fitted(fm2))) expect_equal(length(p1 <- predict(fm2)), 177) ## predict with na.exclude -> has 3 NA's, but otherwise identical: expect_equal(length(p2 <- predict(fm2, na.action=na.exclude)), 180) expect_identical(p1, p2[!is.na(p2)]) expect_equal(length((s1 <- simulate(fm1,1))[[1]]),180) expect_equal(length((s2 <- simulate(fm2,1))[[1]]),177) expect_equal(head(rownames(s1)),paste0("a",1:6)) expect_equal(head(rownames(s2)),paste0("a",4:9)) ## test simulation expect_is(attr(simulate(fm2),"na.action"),"omit") expect_is(refit(fm2,simulate(fm2)),"merMod") expect_equal(fixef(fm2), fixef(refit(fm2, sleepstudyNA$Reaction)), tolerance = 1e-5) fm2ex <- update(fm2, na.action=na.exclude) expect_equal(nrow(ss2 <- simulate(fm2ex)),180) expect_is(refit(fm2,ss2[[1]]),"merMod") ## issue #197, 18 new subjects; some with NA in y d2 <- sleepstudyNA[c(1:180, 1:180),] d2[,"Subject"] <- factor(rep(1:36, each=10)) d2[d2$Subject == 19, "Reaction"] <- NA expect_equal(dim( simulate(fm1, newdata=d2, allow.new.levels=TRUE) ), c(360,1)) ## na.pass (pretty messed up) expect_error(update(fm1,data=sleepstudyNA, control=lmerControl(check.conv.grad="ignore"), na.action=na.pass), "NA/NaN/Inf in 'y'") sleepstudyNA2 <- within(sleepst.a, Days[1:3] <- NA) expect_error(fm4 <- update(fm1, data = sleepstudyNA2, control=lmerControl(check.conv.grad="ignore"), na.action=na.pass),"NA in Z") expect_is(suppressWarnings(confint(fm2,method="boot",nsim=3, quiet=TRUE)),"matrix") expect_error(update(fm1, data = sleepstudyNA2, control = lmerControl(check.conv.grad="ignore"), na.action = na.pass), "NA in Z") expect_is(suppressWarnings( ci2 <- confint(fm2, method="boot", nsim=3, quiet=TRUE)), "matrix") }) test_that("other_NA", { expect_error(lmer(Reaction ~ Days + (Days | Subject), sleepInf), "\\") fm0 <- lmer(angle ~ recipe * temperature + (1|recipe:replicate), cake) ## NA's in response : fm1 <- update(fm0, data = cakeNA) expect_true(all.equal( fixef(fm0), fixef(fm1))) expect_true(all.equal(VarCorr(fm0),VarCorr(fm1))) expect_true(all.equal( ranef(fm0), ranef(fm1))) fm1_omit <- update(fm1, na.action = na.omit) fm1_excl <- update(fm1, na.action = na.exclude) expect_error(update(fm1, na.action = na.pass), "NA/NaN") expect_error(update(fm1, na.action = na.fail), "missing values in object") fm1_omit@call <- fm1@call ## <- just for comparing: expect_equal(fm1, fm1_omit) expect_equal(length(fitted(fm1_omit)), 270) expect_equal(length(fitted(fm1_excl)), 271) expect_true(is.na(tail(predict(fm1_excl),1))) ## test predict.lm d <- data.frame(x = 1:10, y = c(rnorm(9),NA)) lm1 <- lm(y~x, data=d, na.action=na.exclude) expect_is(predict(lm1), "numeric") expect_equal(1, sum(is.na(predict(lm1, newdata = data.frame(x=c(1:4,NA)))))) ## Triq examples ... m.lmer <- lmer (angle ~ temp + (1 | recipe) + (1 | replicate), data=cake) ## NAs in fixed effect p1_pass <- predict(m.lmer, newdata=cakeNA.X, re.form=NA, na.action=na.pass) expect_true(length(p1_pass)==nrow(cakeNA.X)) expect_true(all(is.na(p1_pass[1:5]))) p1_omit <- predict(m.lmer, newdata=cakeNA.X, re.form=NA, na.action=na.omit) p1_exclude <- predict(m.lmer, newdata=cakeNA.X, re.form=NA, na.action=na.exclude) expect_true(length(p1_omit)==nrow(na.omit(cakeNA.X))) expect_true(length(p1_exclude)==nrow(cakeNA.X)) expect_true(all.equal(c(na.omit(p1_exclude)),p1_omit)) expect_error(predict(m.lmer, newdata=cakeNA.X, re.form=NA, na.action=na.fail), "missing values in object") ## now try it with re.form==NULL p2_pass <- predict(m.lmer, newdata=cakeNA.X, re.form=NULL, na.action=na.pass) expect_true(length(p2_pass)==nrow(cakeNA.X)) expect_true(all(is.na(p2_pass[1:5]))) p2_omit <- predict(m.lmer, newdata=cakeNA.X, re.form=NULL, na.action=na.omit) p2_exclude <- predict(m.lmer, newdata=cakeNA.X, re.form=NULL, na.action=na.exclude) expect_true(length(p2_omit)==nrow(na.omit(cakeNA.X))) expect_true(all.equal(c(na.omit(p2_exclude)),p2_omit)) expect_error(predict(m.lmer, newdata=cakeNA.X, re.form=NULL, na.action=na.fail), "missing values in object") ## experiment with NA values in random effects -- should get treated expect_error(predict(m.lmer, newdata=cakeNA.Z, re.form=NULL), "NAs are not allowed in prediction data") p4 <- predict(m.lmer, newdata=cakeNA.Z, re.form=NULL, allow.new.levels=TRUE) p4B <- predict(m.lmer, newdata=cakeNA.Z, re.form=~1|recipe, allow.new.levels=TRUE) expect_true(all.equal(p4[1:5],p4B[1:5])) p4C <- predict(m.lmer, newdata=cakeNA.Z, re.form=NA) d <- data.frame(x=runif(100),f=factor(rep(1:10,10))) set.seed(101) u <- rnorm(10) d <- transform(d,y=rnorm(100,1+2*x+u[f],0.2)) d0 <- d d[c(3,5,7),"x"] <- NA ## 'omit' and 'exclude' are the only choices under which ## we will see NA values in the results fm0 <- lmer(y~x+(1|f), data=d0) ## no 'na.action' attribute because no NAs in this data set expect_equal(attr(model.frame(fm0),"na.action"),NULL) fm1 <- update(fm0, data=d) ## no NAs in predict or residuals because na.omit expect_false(any(is.na(predict(fm1)))) expect_false(any(is.na(residuals(fm1)))) fm2 <- update(fm1,na.action="na.exclude") ## no NAs in predict or residuals because na.omit nNA <- sum(is.na(d$x)) expect_equal(sum(is.na(predict(fm2))),nNA) expect_equal(sum(is.na(residuals(fm2))),nNA) expect_error(fm3 <- lmer(y~x+(1|f), data=d, na.action="na.pass"), "Error in qr.default") expect_is(refit(fm0),"merMod") expect_is(refit(fm1),"merMod") expect_is(refit(fm2),"merMod") }) lme4/inst/tests/test-glmernb.R0000754000175100001440000000125212565075351016011 0ustar hornikuserslibrary("testthat") library("lme4") context("glmer.nb") test_that("basic", { set.seed(101) dd <- expand.grid(f1 = factor(1:3), f2 = LETTERS[1:2], g=1:9, rep=1:15, KEEP.OUT.ATTRS=FALSE) mu <- 5*(-4 + with(dd, as.integer(f1) + 4*as.numeric(f2))) dd$y <- rnbinom(nrow(dd), mu = mu, size = 0.5) require("MASS") m.glm <- glm.nb(y ~ f1*f2, data=dd) m.nb <- glmer.nb(y ~ f1*f2 + (1|g), data=dd, verbose=TRUE) expect_null(m.nb@call$verbose) ## check: GH #321 if (FALSE) { ## FIXME: still trying to work this one out (GH #319) expect_equal(fixef(m.nb), coef (m.glm), tol=1e-5) } } ) lme4/inst/NEWS.Rd0000754000175100001440000007605212604503342013164 0ustar hornikusers\newcommand{\PR}{\Sexpr[results=rd]{tools:::Rd_expr_PR(#1)}} \name{NEWS} \title{lme4 News} \encoding{UTF-8} \section{CHANGES IN VERSION 1.1-10 (2015-10-05)}{ This release is primarily a version bump for the release of the paper in J. Stat. Software. \subsection{USER-VISIBLE CHANGES}{ \itemize{ \item updated CITATION file. } } \subsection{NEW FEATURES}{ \itemize{ \item We export set of about a dozen printing utility functions which are used in our \code{print} methods. \item \code{bootMer} now allows the use of \code{re.form}. } } \subsection{BUG FIXES}{ \itemize{ \item fixed reordering bug in names of \code{getME(.,"Ztlist")} (terms are reordered in decreasing order of the number of levels of the grouping variable, but names were not being reordered) \item fixed issue with simulation when complex forms (such as nested random effects terms) are included in the model (Github #335) } } } \section{CHANGES IN VERSION 1.1-9 (2015-08-20)}{ \subsection{USER-VISIBLE CHANGES}{ \itemize{ \item explicit \code{maxit} arguments for various functions (\code{refit}, \code{mkGlmerDevfun}, ...) } } \subsection{NEW FEATURES}{ \itemize{ \item \code{terms} and \code{formula} methods now have \code{random.only} options \item \code{getME} gains a \code{glmer.nb.theta} option. It is now (an S3) generic with an \code{"merMod"} method in \pkg{lme4} and potentially other methods in dependent packages. \item \code{simulate} now works for \code{glmer.nb} models (Github #284: idea from @aosmith16) } } \subsection{BUG FIXES}{ \itemize{ \item prediction and simulation now work when random-effects terms have data-dependent bases (e.g., \code{poly(.)} or \code{ns(.)} terms) (Github #313, Edgar Gonzalez) \item \code{logLik} for \code{glmer.nb} models now includes the overdispersion parameter in the parameter count (\code{df} attribute) \item \code{lmList} handles offsets and weights better \item lots of fixes to \code{glmer.nb} (Github #176, #266, #287, #318). \strong{Please note that glmer.nb is still somewhat unstable/under construction.} } } \subsection{CRAN-COMPATIBILITY UPDATES}{ \itemize{ \item import functions from base packages to pass CRAN checks \item tweak to failing tests on Windows } } } \section{CHANGES IN VERSION 1.1-8 (2015-06-22)}{ \subsection{NEW FEATURES}{ \itemize{ \item \code{getME} gains a \code{"Tlist"} option (returns a vector of template matrices from which the blocks of \code{Lambda} are generated) \item \code{hatvalues} method returns the diagonal of the hat matrix of LMMs \item \code{nlminbwrap} convenience function allows use of \code{nlminb} without going through the \code{optimx} package \item \code{as.data.frame.VarCorr.merMod} gains an \code{order} option that allows the results to be sorted with variances first and covariances last (default) or in lower-triangle order \item allow more flexibility in \code{scales} for \code{xyplot.thpr} method (John Maindonald) \item models with only random effects of the form \code{1|f} have better starting values for \code{lmer} optimization (Gabor Grothendieck) \item \code{glmer} now allows a logical vector as the response for binomial models \item \code{anova} will now do (sequential) likelihood ratio tests for two or more models including both \code{merMod} and \code{glm} or \code{lm} models (at present, only for GLMMs fitted with the Laplace approximation) } } \subsection{USER-VISIBLE CHANGES}{ \itemize{ \item \code{deviance()} now returns the deviance, rather than half the negative log-likelihood, for GLMMs fitted with Laplace (the behaviour for LMMs and GLMMs fitted with \code{nAGQ>1} has not changed) \item convergence warning and diagnostic test issues are now reported in \code{print} and \code{summary} methods \item \code{update} now (attempts to) re-evaluate the original fit in the environment of its formula (as is done with \code{drop1}) \item \code{refit} of a nonlinear mixed model fit now throws an error, but this will hopefully change in future releases (related to bug fixes for Github #231) \item \code{lmList} now returns objects of class \code{lmList4}, to avoid overwriting \code{lmList} methods from the recommended \code{nlme} package \item names of random effects parameters in \code{confint} changed (modified for consistency across methods); \code{oldNames=TRUE} (default) gives \code{".sig01"}-style names, \code{oldNames=FALSE} gives \code{"sd_(Intercept)|Subject"}-style names \item \code{confint(.,method="Wald")} result now contains rows for random effects parameters (values set to \code{NA}) as well as for fixed-effect parameters } } \subsection{BUG FIXES}{ \itemize{ \item \code{simulate} and \code{predict} now work more consistently with different-length data, differing factor levels, and \code{NA} values (Github #153, #197, #246, #275) \item \code{refit} now works correctly for \code{glmer} fits (Github #231) \item fixed bug in \code{family.merMod}; non-default links were not retrieved correctly (Alessandro Moscatelli) \item fixed \code{bootMer} bug for \code{type=="parametric"}, \code{use.u=TRUE} (Mark Lai) \item gradient scaling for convergence checks now uses the Cholesky factor of the Hessian; while it is more correct, this will lead to some additional (probably false-positive) convergence warnings \item As with \code{lm()}, users now get an error for non-finite (\code{Inf}, \code{NA}, or \code{NaN}) values in the response unless \code{na.action} is set to exclude or omit them (Github #310) } } } \section{CHANGES IN VERSION 1.1-7 (2014-07-19)}{ \subsection{NEW FEATURES}{ \itemize{ \item the \code{nloptr} package is now imported; a wrapper function (\code{nloptwrap}) is provided so that \code{lmerControl(optimizer="nloptwrap")} is all that's necessary to use \code{nloptr} optimizers in the nonlinear optimization stage (the default algorithm is NLopt's implementation of BOBYQA: see \code{?nloptwrap} for examples) \item preliminary implementation of checks for scaling of model matrix columns (see \code{check.scaleX} in \code{?lmerControl}) \item \code{beta} is now allowed as a synonym for \code{fixef} when specifying starting parameters (Github #194) } } \subsection{USER-VISIBLE CHANGES}{ \itemize{ \item the use of \code{deviance} to return the REML criterion is now deprecated; users should use \code{REMLcrit()} instead (Github #211) \item changed the default value of \code{check.nobs.vs.rankZ} to \code{"ignore"} (Github #214) } } \subsection{BUG FIXES}{ \itemize{ \item change gradient testing from absolute to relative \item fix \code{confint(.,method="boot")} to allow/work properly with \code{boot.type} values other than \code{"perc"} (reported by Alan Zaslavsky) \item allow \code{plot()} to work when data are specified in a different environment (reported by Dieter Menne) \item \code{predict} and \code{simulate} work for matrix-valued predictors (Github #201) \item other \code{simulate} bugs (Github #212) \item \code{predict} no longer warns spuriously when original response was a factor (Github #205) \item fix memory access issues (Github #200) } } } \section{CHANGES IN VERSION 1.1-6 (2014-04-13)}{ This version incorporates no changes in functionality, just modifications to testing and dependencies for CRAN/backward compatibility. \subsection{BUG FIXES}{ \itemize{ \item change \code{drop1} example to prevent use of old/incompatible \code{pbkrtest} versions, for 2.15.3 compatibility \item explicitly \code{require(mlmRev)} for tests to prevent cyclic dependency \item bump \code{RcppEigen} Imports: requirement from >0.3.1.2.3 to >=0.3.2.0; \code{Rcpp} dependency to >= 0.10.5 } } } \section{CHANGES IN VERSION 1.1-5 (2014-03-14)}{ \subsection{BUG FIXES}{ \itemize{ \item improved NA handling in \code{simulate} and \code{refit} \item made internal handling of \code{weights}/\code{offset} arguments slightly more robust (Github #191) \item handle non-positive-definite estimated fixed effect variance-covariance matrices slightly more generally/robustly (fall back on RX approximation, with a warning, if finite-difference Hessian is non-PD; return \code{NA} matrix if RX approximation is also bad) } } \subsection{MINOR USER-VISIBLE CHANGES}{ \itemize{ \item Added output specifying when Gauss-Hermite quadrature was used to fit the model, and specifying number of GHQ points (Github #190) } } } \section{CHANGES IN VERSION 1.1-4}{ \subsection{BUG FIXES}{ \itemize{ \item Models with prior weights returned an incorrect sigma and deviance (Github issue #155). The deviance bug was only a practical issue in model comparisons, not with inferences given a particular model. Both bugs are now fixed. \item Profiling failed in some cases for models with vector random effects (Github issue #172) \item Standard errors of fixed effects are now computed from the approximate Hessian by default (see the \code{use.hessian} argument in \code{vcov.merMod}); this gives better (correct) answers when the estimates of the random- and fixed-effect parameters are correlated (Github #47) } } \subsection{MAJOR USER-VISIBLE CHANGES}{ \itemize{ \item The default optimizer for \code{lmer} fits has been switched from "Nelder_Mead" to "bobyqa" because we have generally found the latter to be more reliable. To switch back to the old behaviour, use \code{control=lmerControl(optimizer="Nelder_Mead")}. \item Better handling of rank-deficient/overparameterized fixed-effect model matrices; see \code{check.rankX} option to \code{[g]lmerControl}. The default value is "message+drop.cols", which automatically drops redundant columns and issues a message (not a warning). (Github #144) } } \subsection{MINOR USER-VISIBLE CHANGES}{ \itemize{ \item slight changes in convergence checking; tolerances can be specified where appropriate, and some default tolerances have changed (e.g., \code{check.conv.grad}) \item improved warning messages about rank-deficiency in X and Z etc. (warnings now try to indicate whether the unidentifiability is in the fixed- or random-effects part of the model) \item \code{predict} and \code{simulate} now prefer \code{re.form} as the argument to specify which random effects to condition on, but allow \code{ReForm}, \code{REForm}, or \code{REform}, giving a message (not a warning) that they are deprecated (addresses Github #170) \item small fixes for printing consistency in models with no fixed effects \item we previously exported a \code{fortify} function identical to the one found in \code{ggplot2} in order to be able to define a \code{fortify.merMod} S3 method without inducing a dependency on \code{ggplot2}. This has now been unexported to avoid masking \code{ggplot2}'s own \code{fortify} methods; if you want to add diagnostic information to the results of a model, use \code{fortify.merMod} explicitly. \item \code{simulate.formula} now checks for names associated with the \code{theta} and \code{beta} parameter vectors. If missing, it prints a message (not a warning); otherwise, it re-orders the parameter vectors to match the internal representation. \item preliminary implementation of a \code{check.scaleX} argument in \code{[g]lmerControl} that warns about scaling if some columns of the fixed-effect model matrix have large standard deviations (relative to 1, or to each other) } } } \section{CHANGES IN VERSION 1.1-3}{ \subsection{NEW FEATURES}{ \itemize{ \item The gradient and Hessian are now computed via finite differencing after the nonlinear fit is done, and the results are used for additional convergence tests. Control of the behaviour is available through the \code{check.conv.*} options in \code{[g]lmerControl}. Singular fits (fits with estimated variances of zero or correlations of +/- 1) can also be tested for, although the current default value of the \code{check.conv.singular} option is \code{"ignore"}; this may be changed to \code{"warning"} in the future. The results are stored in \code{@optinfo$derivs}. (Github issue #120; based on code by Rune Christensen.) \item The \code{simulate} method will now work to generate simulations "from scratch" by providing a model formula, a data frame holding the predictor variables, and a list containing the values of the model parameters: see \code{?simulate.merMod}. (Github issue #115) \item \code{VarCorr.merMod} objects now have an \code{as.data.frame} method, converting the list of matrices to a more convenient form for reporting and post-processing. (Github issue #129) } } \subsection{MINOR USER-VISIBLE CHANGES}{ \itemize{ \item results of \code{fitted()}, \code{predict()}, and \code{residuals()} now have names in all cases (previously results were unnamed, or named only when predicting from new data) \item the \code{anova} method now has a \code{refit} argument that controls whether objects of class \code{lmerMod} should be refitted with ML before producing the \code{anova} table. (Github issues #141, #165; contributed by Henrik Singmann.) \item the \code{print} method for \code{VarCorr} objects now has a \code{formatter} argument for finer control of standard deviation and variance formats \item the \code{optinfo} slot now stores slightly more information, including the number of function evaluations (\code{$feval}). \item \code{dotplot.ranef.mer} now adds titles to sub-plots by default, like \code{qqmath.ranef.mer} } } \subsection{BUG FIXES}{ \itemize{ \item \code{fitted} now respects \code{na.action} settings (Github issue #149) \item \code{confint(.,method="boot")} now works when there are \code{NA} values in the original data set (Github issue #158) \item previously, the code stored the results (parameter values, residuals, etc.) based on the \emph{last} set of parameters evaluated, rather than the optimal parameters. These were not always the same, but were almost always very close, but some previous results will change slightly (Github issue #166) } } } \section{CHANGES IN VERSION 1.1-0}{ \subsection{MINOR USER-VISIBLE CHANGES}{ \itemize{ \item when using the default \code{method="profile"}, \code{confint} now returns appropriate upper/lower bounds (-1/1 for correlations, 0/Inf for standard deviations) rather than \code{NA} when appropriate } } \subsection{BUG FIXES}{ \itemize{ \item in a previous development version, \code{ranef} returned incorrect conditional variances (github issue #148). this is now fixed } } } \section{CHANGES IN VERSION 1.0-6 (2014-02-02)}{ \subsection{BUG FIXES}{ \itemize{ \item prediction now works when new data have fewer factor levels than are present in the original data (Github issue #143, reported by Rune Haubo) \item the existence of a variable "new" in the global environment would mess \code{lme4} up: reported at \url{http://stackoverflow.com/questions/19801070/error-message-glmer-using-r-what-must-be-a-character-string-or-a-function} } } } \section{CHANGES IN VERSION 1.0-5 (2013-10-24)}{ \subsection{USER-VISIBLE CHANGES}{ \itemize{ \item \code{confint.merMod} and \code{vcov.merMod} are now exported, for downstream package-author convenience \item the package now depends on Matrix >=1.1-0 and RcppEigen >=0.3.1.2.3 \item new \code{rename.response} option for \code{refit} (see BUG FIXES section) } } \subsection{BUG FIXES}{ \itemize{ \item eliminated redundant messages about suppressed fixed-effect correlation matrices when p>20 \item most inverse-link functions are now bounded where appropriate by \code{.Machine$double.eps}, allowing fitting of GLMMs with extreme parameter values \item \code{merMod} objects created with \code{refit} did not work with \code{update}: optional \code{rename.response} option added to \code{refit.merMod}, to allow this (but the default is still \code{FALSE}, for back-compatibility) (reported by A. Kuznetsova) \item fixed buglet preventing on-the-fly creation of index variables, e.g. \code{y~1+(1|rownames(data))} (reported by J. Dushoff) \item \code{predict} now works properly for \code{glmer} models with basis-creating terms (e.g. \code{poly}, \code{ns}) \item step sizes determined from fixed effect coefficient standard errors after first state of \code{glmer} fitting are now bounded, allowing some additional models to be fitted } } } \section{CHANGES IN VERSION 1.0-4 (2013-09-08)}{ \subsection{BUG FIXES}{ \itemize{ \item \code{refit()} now works, again, with lists of length 1, so that e.g. \code{refit(.,simulate(.))} works. (Reported by Gustaf Granath) \item \code{getME(.,"ST")} was returning a list containing the Cholesky factorizations that get repeated in Lambda. But this was inconsistent with what \code{ST} represents in \code{lme4.0}. This inconsistency has now been fixed and \code{getME(.,"ST")} is now consistent with the definition of the \code{ST} matrix in \code{lme4.0}. See \code{https://github.com/lme4/lme4/issues/111} for more detail. Thanks to Vince Dorie. \item Corrected order of unpacking of standard deviation/correlation components, which affected results from \code{confint(.,method="boot")}. (Reported by Reinhold Kliegl) \item fixed a copying bug that made \code{refitML()} modify the original model } } } \section{CHANGES IN VERSION 1.0-1 (2013-08-17)}{ \subsection{MINOR USER-VISIBLE CHANGES}{ \itemize{ \item \code{check.numobs.*} and \code{check.numlev.*} in \code{(g)lmerControl} have been changed (from recent development versions) to \code{check.nobs.*} and \code{check.nlev.*} respectively, and the default values of \code{check.nlev.gtreq.5} and \code{check.nobs.vs.rankZ} have been changed to \code{"ignore"} and \code{"warningSmall"} respectively \item in \code{(g)lmerControl}, arguments to the optimizer should be passed as a list called \code{optCtrl}, rather than specified as additional (ungrouped) arguments \item the \code{postVar} argument to \code{ranef} has been changed to the (more sensible) \code{condVar} ("posterior variance" was a misnomer, "conditional variance" -- short for "variance of the conditional mode" -- is preferred) \item the \code{REform} argument to \code{predict} has been changed to \code{ReForm} for consistency \item the \code{tnames} function, briefly exported, has been unexported \item \code{getME(.,"cnms")} added \item \code{print} method for \code{merMod} objects is now more terse, and different from \code{summary.merMod} \item the \code{objective} method for the \code{respMod} reference class now takes an optional \code{sigma.sq} parameter (defaulting to \code{NULL}) to allow calculation of the objective function with a residual variance different from the profiled value (Vince Dorie) } } } \section{CHANGES IN VERSION 1.0-0 (2013-08-01)}{ \subsection{MAJOR USER-VISIBLE CHANGES}{ \itemize{ \item Because the internal computational machinery has changed, results from the newest version of \code{lme4} will not be numerically identical to those from previous versions. For reasonably well- defined fits, they will be extremely close (within numerical tolerances of 1e-4 or so), but for unstable or poorly-defined fits the results may change, and very unstable fits may fail when they (apparently) succeeded with previous versions. Similarly, some fits may be slower with the new version, although on average the new version should be faster and more stable. More numerical tuning options are now available (see below); non-default settings may restore the speed and/or ability to fit a particular model without an error. If you notice significant or disturbing changes when fitting a model with the new version of \code{lme4}, \emph{please notify the maintainers}. \item \code{VarCorr} returns its results in the same format as before (as a list of variance-covariance matrices with \code{correlation} and \code{stddev} attributes, plus a \code{sc} attribute giving the residual standard deviation/scale parameter when appropriate), but prints them in a different (nicer) way. \item By default \code{residuals} gives deviance (rather than Pearson) residuals when applied to \code{glmer} fits (a side effect of matching \code{glm} behaviour more closely). \item As another side effect of matching \code{\link{glm}} behaviour, reported log-likelihoods from \code{\link{glmer}} models are no longer consistent with those from pre-1.0 \code{lme4}, but \emph{are} consistent with \code{glm}; see \code{\link{glmer}} examples. } } \subsection{MINOR USER-VISIBLE CHANGES}{ \itemize{ \item More use is made of S3 rather than S4 classes and methods: one side effect is that the \code{nlme} and \code{lme4} packages are now much more compatible; methods such as \code{fixef} no longer conflict. \item The internal optimizer has changed. \code{[gn]lmer} now has an \code{optimizer} argument; \code{"Nelder_Mead"} is the default for \code{[n]lmer}, while a combination of \code{"bobyqa"} (an alternative derivative-free method) and \code{"Nelder_Mead"} is the default for \code{glmer}. To use the \code{nlminb} optimizer as in the old version of \code{lme4}, you can use \code{optimizer="optimx"} with \code{control=list(method="nlminb")} (you will need the \code{optimx} package to be installed and loaded). See \code{\link{lmerControl}} for details. \item Families in GLMMs are no longer restricted to built-in/hard- coded families; any family described in \code{\link{family}}, or following that design, is usable (although there are some hard-coded families, which will be faster). \item \code{[gn]lmer} now produces objects of class \code{merMod} rather than class \code{mer} as before. \item the structure of the \code{Zt} (transposed random effect design matrix) as returned by \code{getME(.,"Zt")}, and the corresponding order of the random effects vector (\code{getME(.,"u")}) have changed. To retrieve \code{Zt} in the old format, use \code{do.call(Matrix::rBind,getME(.,"Ztlist"))}. \item the package checks input more thoroughly for non-identifiable or otherwise problematic cases: see \code{\link{lmerControl}} for fine control of the test behaviour. } } \subsection{NEW FEATURES}{ \itemize{ \item A general-purpose \code{\link{getME}} accessor method allows extraction of a wide variety of components of a mixed-model fit. \code{getME} also allows a vector of objects to be returned as a list of mixed-model components. This has been backported to be compatible with older versions of \code{lme4} that still produce \code{mer} objects rather than \code{merMod} objects. However, backporting is incomplete; some objects are only extractable in newer versions of \code{lme4}. \item Optimization information (convergence codes, warnings, etc.) is now stored in an \code{@optinfo} slot. \item \code{\link{bootMer}} provides a framework for obtaining parameter confidence intervals by parametric bootstrapping. \item \code{\link{plot.merMod}} provides diagnostic plotting methods similar to those from the \code{nlme} package (although missing \code{augPred}). \item A \code{\link{predict.merMod}} method gives predictions; it allows an effect-specific choice of conditional prediction or prediction at the population level (i.e., with random effects set to zero). \item Likelihood profiling for \code{lmer} and \code{glmer} results (see \code{link{profile-methods}}). \item Confidence intervals by likelihood profiling (default), parametric bootstrap, or Wald approximation (fixed effects only): see \code{\link{confint.merMod}} \item \code{nAGQ=0}, an option to do fast (but inaccurate) fitting of GLMMs. \item Using \code{devFunOnly=TRUE} allows the user to extract a deviance function for the model, allowing further diagnostics/customization of model results. \item The internal structure of [gn]lmer is now more modular, allowing finer control of the different steps of argument checking; construction of design matrices and data structures; parameter estimation; and construction of the final \code{merMod} object (see \code{?modular}). \item the \code{formula}, \code{model.frame}, and \code{terms} methods return full versions (including random effect terms and input variables) by default, but a \code{fixed.only} argument allows access to the fixed effect submodel. } } \subsection{EXPERIMENTAL FEATURES}{ \itemize{ \item \code{\link{glmer.nb}} provides an embryonic negative binomial fitting capability. } } \subsection{STILL NON-EXISTENT FEATURES}{ \itemize{ \item Adaptive Gaussian quadrature (AGQ) is not available for multiple and/or non-scalar random effects. \item Posterior variances of conditional models for non-scalar random effects. \item Standard errors for \code{\link{predict.merMod}} results. \item Automatic MCMC sampling based on the fit turns out to be very difficult to implement in a way that is really broadly reliable and robust; \code{mcmcsamp} will not be implemented in the near future. See \code{\link{pvalues}} for alternatives. \item "R-side" structures (within-block correlation and heteroscedasticity) are not on the current timetable. } } \subsection{BUG FIXES}{ \itemize{ \item In a development version, prior weights were not being used properly in the calculation of the residual standard deviation, but this has been fixed. Thanks to Simon Wood for pointing this out. \item In a development version, the step-halving component of the penalized iteratively reweighted least squares algorithm was not working, but this is now fixed. \item In a development version, square \code{RZX} matrices would lead to a \code{pwrssUpdate did not converge in 30 iterations} error. This has been fixed by adding an extra column of zeros to \code{RZX}. } } \subsection{DEPRECATED AND DEFUNCT}{ \itemize{ \item Previous versions of \code{lme4} provided the \code{mcmcsamp} function, which efficiently generated a Markov chain Monte Carlo sample from the posterior distribution of the parameters, assuming flat (scaled likelihood) priors. Due to difficulty in constructing a version of \code{mcmcsamp} that was reliable even in cases where the estimated random effect variances were near zero (e.g. \url{https://stat.ethz.ch/pipermail/r-sig-mixed-models/2009q4/003115.html}), \code{mcmcsamp} has been withdrawn (or more precisely, not updated to work with \code{lme4} versions >=1.0). \item Calling \code{glmer} with the default \code{gaussian} family redirects to \code{lmer}, but this is deprecated (in the future \code{glmer(...,family="gaussian")} may fit a LMM using the penalized iteratively reweighted least squares algorithm). Please call \code{lmer} directly. \item Calling \code{lmer} with a \code{family} argument redirects to \code{glmer}; this is deprecated. Please call \code{glmer} directly. } } } \section{CHANGES IN VERSION 0.999375-16 (2008-06-23)}{ \subsection{MAJOR USER-VISIBLE CHANGES}{ \itemize{ \item The underlying algorithms and representations for all the mixed-effects models fit by this package have changed - for the better, we hope. The class "mer" is a common mixed-effects model representation for linear, generalized linear, nonlinear and generalized nonlinear mixed-effects models. \item ECME iterations are no longer used at all, nor are analytic gradients. Components named 'niterEM', 'EMverbose', or 'gradient' can be included in the 'control' argument to lmer(), glmer() or nlmer() but have no effect. \item PQL iterations are no longer used in glmer() and nlmer(). Only the Laplace approximation is currently available. AGQ, for certain classes of GLMMs or NLMMs, is being added. \item The 'method' argument to lmer(), glmer() or nlmer() is deprecated. Use the 'REML = FALSE' in lmer() to obtain ML estimates. Selection of AGQ in glmer() and nlmer() will be controlled by the argument 'nAGQ', when completed. } } \subsection{NEW FEATURES}{ \itemize{ \item The representation of mixed-effects models has been dramatically changed to allow for smooth evaluation of the objective as the variance-covariance matrices for the random effects approach singularity. Beta testers found this representation to be more robust and usually faster than previous versions of lme4. \item The mcmcsamp function uses a new sampling method for the variance-covariance parameters that allows recovery from singularity. The update is not based on a sample from the Wishart distribution. It uses a redundant parameter representation and a linear least squares update. \item CAUTION: Currently the results from mcmcsamp look peculiar and are probably incorrect. I hope it is just a matter of my omitting a scaling factor but I have seen patterns such as the parameter estimate for some variance-covariance parameters being the maximum value in the chain, which is highly unlikely. \item The 'verbose' argument to lmer(), glmer() and nlmer() can be used instead of 'control = list(msVerbose = TRUE)'. } } } lme4/inst/vignettedata/0000755000175100001440000000000012225565275014600 5ustar hornikuserslme4/inst/vignettedata/calcium.txt0000754000175100001440000000233212213177042016744 0ustar hornikusersPlant Leaf Det Calcium 1 1 1 3.28 1 1 2 3.09 1 2 1 3.52 1 2 2 3.48 1 3 1 2.88 1 3 2 2.80 2 1 1 2.46 2 1 2 2.44 2 2 1 1.87 2 2 2 1.92 2 3 1 2.19 2 3 2 2.19 3 1 1 2.77 3 1 2 2.66 3 2 1 3.74 3 2 2 3.44 3 3 1 2.55 3 3 2 2.55 4 1 1 3.78 4 1 2 3.87 4 2 1 4.07 4 2 2 4.12 4 3 1 3.31 4 3 2 3.31 lme4/inst/vignettedata/mcmcsampdat.RData0000754000175100001440000107527412213623321020010 0ustar 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{formatcom={\vspace{-1ex}},fontfamily=courier,fontseries=b,% fontsize=\footnotesize} %%\VignetteIndexEntry{Computational Methods} %%\VignetteDepends{lme4} \title{Computational methods for mixed models} \author{Douglas Bates\\Department of Statistics\\% University of Wisconsin -- Madison} \begin{document} \SweaveOpts{engine=R,eps=FALSE,pdf=TRUE,strip.white=true,keep.source=TRUE} \SweaveOpts{include=FALSE} \setkeys{Gin}{width=\textwidth} \newcommand{\code}[1]{\texttt{\small{#1}}} \newcommand{\package}[1]{\textsf{\small{#1}}} \newcommand{\trans}{\ensuremath{^\prime}} \renewcommand{\vec}{\operatorname{vec}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\bc}[1]{\ensuremath{\bm{\mathcal{#1}}}} <>= options(width=65,digits=5) #library(lme4) @ \maketitle \begin{abstract} The \package{lme4} package provides R functions to fit and analyze several different types of mixed-effects models, including linear mixed models, generalized linear mixed models and nonlinear mixed models. In this vignette we describe the formulation of these models and the computational approach used to evaluate or approximate the log-likelihood of a model/data/parameter value combination. \end{abstract} \section{Introduction} \label{sec:intro} The \package{lme4} package provides \code{R} functions to fit and analyze linear mixed models, generalized linear mixed models and nonlinear mixed models. These models are called \emph{mixed-effects models} or, more simply, \emph{mixed models} because they incorporate both \emph{fixed-effects} parameters, which apply to an entire population or to certain well-defined and repeatable subsets of a population, and \emph{random effects}, which apply to the particular experimental units or observational units in the study. Such models are also called \emph{multilevel} models because the random effects represent levels of variation in addition to the per-observation noise term that is incorporated in common statistical models such as linear regression models, generalized linear models and nonlinear regression models. We begin by describing common properties of these mixed models and the general computational approach used in the \package{lme4} package. The estimates of the parameters in a mixed model are determined as the values that optimize an objective function --- either the likelihood of the parameters given the observed data, for maximum likelihood (ML) estimates, or a related objective function called the REML criterion. Because this objective function must be evaluated at many different values of the model parameters during the optimization process, we focus on the evaluation of the objective function and a critical computation in this evalution --- determining the solution to a penalized, weighted least squares (PWLS) problem. The dimension of the solution of the PWLS problem can be very large, perhaps in the millions. Furthermore, such problems must be solved repeatedly during the optimization process to determine parameter estimates. The whole approach would be infeasible were it not for the fact that the matrices determining the PWLS problem are sparse and we can use sparse matrix storage formats and sparse matrix computations \citep{davis06:csparse_book}. In particular, the whole computational approach hinges on the extraordinarily efficient methods for determining the Cholesky decomposition of sparse, symmetric, positive-definite matrices embodied in the CHOLMOD library of C functions \citep{Cholmod}. % The three types of mixed models -- linear, generalized linear and % nonlinear -- share common characteristics in that the model is % specified in whole or in part by a \emph{mixed model formula} that % describes a \emph{linear predictor} and a variance-covariance % structure for the random effects. In the next section we describe % the mixed model formula and the forms of these matrices. The % following section presents a general formulation of the Laplace % approximation to the log-likelihood of a mixed model. % In subsequent sections we describe computational methods for specific % kinds of mixed models. In particular, we should how a profiled % log-likelihood for linear mixed models, and for some nonlinear mixed % models, can be evaluated exactly. In the next section we describe the general form of the mixed models that can be represented in the \package{lme4} package and the computational approach embodied in the package. In the following section we describe a particular form of mixed model, called a linear mixed model, and the computational details for those models. In the fourth section we describe computational methods for generalized linear mixed models, nonlinear mixed models and generalized nonlinear mixed models. \section{Formulation of mixed models} \label{sec:form-mixed-models} A mixed-effects model incorporates two vector-valued random variables: the $n$-dimensional response vector, $\bc Y$, and the $q$-dimensional random effects vector, $\bc B$. We observe the value, $\bm y$, of $\bc Y$. We do not observe the value of $\bc B$. The random variable $\bc Y$ may be continuous or discrete. That is, the observed data, $\bm y$, may be on a continuous scale or they may be on a discrete scale, such as binary responses or responses representing a count. In our formulation, the random variable $\bc B$ is always continous. We specify a mixed model by describing the unconditional distribution of $\bc B$ and the conditional distribution $(\bc Y|\bc B=\bm b)$. \subsection{The unconditional distribution of $\bc B$} \label{sec:uncond-distr-B} In our formulation, the unconditional distribution of $\bc B$ is always a $q$-dimensional multivariate Gaussian (or ``normal'') distribution with mean $\bm 0$ and with a parameterized covariance matrix, \begin{equation} \label{eq:2} \bc B\sim\mathcal{N}\left(\bm 0,\sigma^2\bm\Lambda(\bm\theta) \bm\Lambda\trans(\bm\theta)\right) . \end{equation} The scalar, $\sigma$, in (\ref{eq:2}), is called the \emph{common scale parameter}. As we will see later, not all types of mixed models incorporate this parameter. We will include $\sigma^2$ in the general form of the unconditional distribution of $\bc B$ with the understanding that, in some models, $\sigma\equiv 1$. The $q\times q$ matrix $\bm\Lambda(\bm\theta)$, which is a left factor of the covariance matrix (when $\sigma=1$) or the relative covariance matrix (when $\sigma\ne 1$), depends on an $m$-dimensional parameter $\bm\theta$. Typically $m\ll q$; in the examples we show below it is always the case that $m<5$, even when $q$ is in the thousands. The fact that $m$ is very small is important because, as we shall see, determining the parameter estimates in a mixed model can be expressed as an optimization problem with respect to $\bm\theta$ only. The parameter $\bm\theta$ may be, and typically is, subject to constraints. For ease of computation, we require that the constraints be expressed as ``box'' constraints of the form $\theta_{iL}\le\theta_i\le\theta_{iU},i=1,\dots,m$ for constants $\theta_{iL}$ and $\theta_{iU}, i=1,\dots,m$. We shall write the set of such constraints as $\bm\theta_L\le\bm\theta\le\bm\theta_R$. The matrix $\bm\Lambda(\bm\theta)$ is required to be non-singular (i.e.{} invertible) when $\bm\theta$ is not on the boundary. \subsection{The conditional distribution, $(\bc Y|\bc B=\bm b)$} \label{sec:cond-distr-YB} The conditional distribution, $(\bc Y|\bc B=\bm b)$, must satisfy: \begin{enumerate} \item The conditional mean, $\bm\mu_{\bc Y|\bc B}(\bm b) = \mathrm{E}[\bc Y|\bc B=\bm b]$, depends on $\bm b$ only through the value of the \emph{linear predictor}, $\bm Z\bm b+\bm X\bm\beta$, where $\bm\beta$ is the $p$-dimensional \emph{fixed-effects} parameter vector and the \emph{model matrices}, $\bm Z$ and $\bm X$, are fixed matrices of the appropriate dimension. That is, the two model matrices must have the same number of rows and must have $q$ and $p$ columns, respectively. The number of rows in $\bm Z$ and $\bm X$ is a multiple of $n$, the dimension of $\bm y$. \item The scalar distributions, $(\mathcal{Y}_i|\bc B=\bm b),i=1,\dots,n$, all have the same form and are completely determined by the conditional mean, $\bm\mu_{\bc Y|\bc B}(\bm b)$ and, at most, one additional parameter, $\sigma$, which is the common scale parameter. \item The scalar distributions, $(\mathcal{Y}_i|\bc B=\bm b),i=1,\dots,n$, are independent. That is, the components of $\bc Y$ are \emph{conditionally independent} given $\bc B$. \end{enumerate} An important special case of the conditional distribution is the multivariate Gaussian distribution of the form \begin{equation} \label{eq:1} (\bc Y|\bc B=\bm b)\sim\mathcal{N}(\bm Z\bm b+\bm X\bm\beta,\sigma^2\bm I_n) \end{equation} where $\bm I_n$ denotes the identity matrix of size $n$. In this case the conditional mean, $\bm\mu_{\bc Y|\bc B}(\bm b)$, is exactly the linear predictor, $\bm Z\bm b+\bm X\bm\beta$, a situation we will later describe as being an ``identity link'' between the conditional mean and the linear predictor. Models with conditional distribution (\ref{eq:1}) are called \emph{linear mixed models}. \subsection{A change of variable to ``spherical'' random effects} \label{sec:change-vari-spher} Because the conditional distribution $(\bc Y|\bc B=\bm b)$ depends on $\bm b$ only through the linear predictor, it is easy to express the model in terms of a linear transformation of $\bc B$. We define the linear transformation from a $q$-dimensional ``spherical'' Gaussian random variable, $\bc U$, to $\bc B$ as \begin{equation} \label{eq:3} \bc B=\bm\Lambda(\bm\theta)\bc U,\quad \bc U\sim\mathcal{N}(\bm 0,\sigma^2\bm I_q). \end{equation} (The term ``spherical'' refers to the fact that contours of constant probability density for $\bc U$ are spheres centered at the mean --- in this case, $\bm0$.) When $\bm\theta$ is not on the boundary this is an invertible transformation. When $\bm\theta$ is on the boundary the transformation can fail to be invertible. However, we will only need to be able to express $\bc B$ in terms of $\bc U$ and that transformation is well-defined, even when $\bm\theta$ is on the boundary. The linear predictor, as a function of $\bm u$, is \begin{equation} \label{eq:4} \bm\gamma(\bm u)=\bm Z\bm\Lambda(\bm\theta)\bm u + \bm X\bm\beta. \end{equation} When we wish to emphasize the role of the model parameters, $\bm\theta$ and $\bm\beta$, in the formulation of $\bm\gamma$, we will write the linear predictor as $\bm\gamma(\bm u,\bm\theta,\bm\beta)$. \subsection{The conditional density $(\bc U|\bc Y=\bm y)$} \label{sec:cond-dens-bc} Because we observe $\bm y$ and do not observe $\bm b$ or $\bm u$, the conditional distribution of interest, for the purposes of statistical inference, is $(\bc U|\bc Y=\bm y)$ (or, equivalently, $(\bc B|\bc Y=\bm y)$). This conditional distribution is always a continuous distribution with conditional probability density $f_{\bc U|\bc Y}(\bm u|\bm y)$. We can evaluate $f_{\bc U|\bc Y}(\bm u|\bm y)$ , up to a constant, as the product of the unconditional density, $f_{\bc U}(\bm u)$, and the conditional density (or the probability mass function, whichever is appropriate), $f_{\bc Y|\bc U}(\bm y|\bm u)$. We write this unnormalized conditional density as \begin{equation} \label{eq:5} h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma) = f_{\bc Y|\bc U}(\bm y|\bm u,\bm\theta,\bm\beta,\sigma) f_{\bc U}(\bm u|\sigma) . \end{equation} We say that $h$ is the ``unnormalized'' conditional density because all we know is that the conditional density is proportional to $h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)$. To obtain the conditional density we must normalize $h$ by dividing by the value of the integral \begin{equation} \label{eq:6} L(\bm\theta,\bm\beta,\sigma|\bm y) = \int_{\mathbb{R}^q}h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)\,d\bm u . \end{equation} We write the value of the integral (\ref{eq:6}) as $L(\bm\theta,\bm\beta,\sigma|\bm y)$ because it is exactly the \emph{likelihood} of the parameters $\bm\theta$, $\bm\beta$ and $\sigma$, given the observed data $\bm y$. The \emph{maximum likelihood (ML) estimates} of these parameters are the values that maximize $L$. \subsection{Determining the ML estimates} \label{sec:DeterminingML} The general problem of maximizing $L(\bm\theta,\bm\beta,\sigma|\bm y)$ with respect to $\bm\theta$, $\bm\beta$ and $\sigma$ can be formidable because each evaluation of this function involves a potentially high-dimensional integral and because the dimension of $\bm\beta$ can be large. However, this general optimization problem can be split into manageable subproblems. Given a value of $\bm\theta$ we can determine the \emph{conditional mode}, $\tilde{\bm u}(\bm\theta)$, of $\bm u$ and the \emph{conditional estimate}, $\tilde{\bm\beta}(\bm\theta)$ simultaneously using \emph{penalized, iteratively re-weighted least squares} (PIRLS). The conditional mode and the conditional estimate are defined as \begin{equation} \label{eq:condmode} \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\ \tilde{\bm\beta}(\bm\theta) \end{bmatrix}=\arg\max_{\bm u,\bm\beta}h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma) . \end{equation} (It may look as if we have missed the dependence on $\sigma$ on the left-hand side but it turns out that the scale parameter does not affect the location of the optimal values of quantities in the linear predictor.) As is common in such optimization problems, we re-express the conditional density on the \emph{deviance scale}, which is negative twice the logarithm of the density, where the optimization becomes \begin{equation} \label{eq:condmode2} \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\ \tilde{\bm\beta}(\bm\theta) \end{bmatrix}=\arg\min_{\bm u,\bm\beta}-2\log\left(h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)\right) . \end{equation} It is this optimization problem that can be solved quite efficiently using PIRLS. In fact, for linear mixed models, which are described in the next section, $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$ can be directly evaluated. The second-order Taylor series expansion of $-2\log h$ at $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$ provides the Laplace approximation to the profiled deviance. Optimizing this function with respect to $\bm\theta$ provides the ML estimates of $\bm\theta$, from which the ML estimates of $\bm\beta$ and $\sigma$ (if used) are derived. \section{Methods for linear mixed models} \label{sec:pwls-problem} As indicated in the introduction, a critical step in our methods for determining the maximum likelihood estimates of the parameters in a mixed model is solving a penalized, weighted least squares (PWLS) problem. We will motivate the general form of the PWLS problem by first considering computational methods for linear mixed models that result in a penalized least squares (PLS) problem. Recall from \S\ref{sec:cond-distr-YB} that, in a linear mixed model, both the conditional distribution, $(\bc Y|\bc U=\bm u)$, and the unconditional distribution, $\bc U$, are spherical Gaussian distributions and that the conditional mean, $\bm\mu_{\bc Y|\bc U}(\bm u)$, is the linear predictor, $\bm\gamma(\bm u)$. Because all the distributions determining the model are continuous distributions, we consider their densities. On the deviance scale these are \begin{equation} \label{eq:7} \begin{aligned} -2\log(f_{\bc U}(\bm u))&=q\log(2\pi\sigma^2)+\frac{\|\bm u\|^2}{\sigma^2}\\ -2\log(f_{\bc Y|\bc U}(\bm y|\bm u))&=n\log(2\pi\sigma^2)+ \frac{\|\bm y-\bm Z\bm\Lambda(\bm\theta)\bm u-\bm X\bm\beta\|^2}{\sigma^2}\\ -2\log(h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)) &= (n+q)\log(2\pi\sigma^2)+ \frac{\|\bm y-\bm\gamma(\bm u,\bm\theta,\bm\beta)\|^2+\|\bm u\|^2}{\sigma^2}\\ &= (n+q)\log(2\pi\sigma^2)+ \frac{d(\bm u|\bm y,\bm\theta,\bm\beta)}{\sigma^2} \end{aligned} \end{equation} In (\ref{eq:7}) the \emph{discrepancy} function, \begin{equation} \label{eq:9} d(\bm u|\bm y,\bm\theta,\bm\beta) = \|\bm y-\bm\gamma(\bm u,\bm\theta,\bm\beta)\|^2+\|\bm u\|^2 \end{equation} has the form of a penalized residual sum of squares in that the first term, $\|\bm y-\bm\gamma(\bm u,\bm\theta,\bm\beta)\|^2$ is the residual sum of squares for $\bm y$, $\bm u$, $\bm\theta$ and $\bm\beta$ and the second term, $\|\bm u\|^2$, is a penalty on the size of $\bm u$. Notice that the discrepancy does not depend on the common scale parameter, $\sigma$. \subsection{The canonical form of the discrepancy} \label{sec:conditional-mode-bm} Using a so-called ``pseudo data'' representation, we can write the discrepancy as a residual sum of squares for a regression model that is linear in both $\bm u$ and $\bm\beta$ \begin{equation} \label{eq:10} d(\bm u|\bm y,\bm\theta,\bm\beta) =\left\| \begin{bmatrix} \bm y\\\bm 0 \end{bmatrix} - \begin{bmatrix} \bm Z\bm\Lambda(\bm\theta) & \bm X \\ \bm I_q & \bm0 \end{bmatrix} \begin{bmatrix}\bm u\\\bm\beta\end{bmatrix} \right\|^2 . \end{equation} The term ``pseudo data'' reflects the fact that we have added $q$ ``pseudo observations'' to the observed response, $\bm y$, and to the linear predictor, $\bm\gamma(\bm u,\bm\theta,\bm\beta)=\bm Z\bm\Lambda(\bm\theta)\bm u+\bm X\bm\beta$, in such a way that their contribution to the overall residual sum of squares is exactly the penalty term in the discrepancy. In the form (\ref{eq:10}) we can see that the discrepancy is a quadratic form in both $\bm u$ and $\bm\beta$. Furthermore, because we require that $\bm X$ has full column rank, the discrepancy is a positive-definite quadratic form in $\bm u$ and $\bm\beta$ that is minimized at $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$ satisfying \begin{equation} \label{eq:13} \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta) +\bm I_q&\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm X\\ \bm X\trans\bm Z\bm\Lambda(\theta) &\bm X\trans\bm X \end{bmatrix} \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\\tilde{\bm\beta}(\bm\theta) \end{bmatrix} = \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm y\\ \bm X\trans\bm y \end{bmatrix} \end{equation} An effective way of determining the solution to a sparse, symmetric, positive definite system of equations such as (\ref{eq:13}) is the sparse Cholesky decomposition \citep{davis06:csparse_book}. If $\bm A$ is a sparse, symmetric positive definite matrix then the sparse Cholesky factor with fill-reducing permutation $\bm P$ is the lower-triangular matrix $\bm L$ such that \begin{equation} \label{eq:14} \bm L\bm L\trans=\bm P\bm A\bm P\trans . \end{equation} (Technically, the factor $\bm L$ is only determined up to changes in the sign of the diagonal elements. By convention we require the diagonal elements to be positive.) The fill-reducing permutation represented by the permutation matrix $\bm P$, which is determined from the pattern of nonzeros in $\bm A$ but does not depend on particular values of those nonzeros, can have a profound impact on the number of nonzeros in $\bm L$ and hence on the speed with which $\bm L$ can be calculated from $\bm A$. In most applications of linear mixed models the matrix $\bm Z\bm\Lambda(\bm\theta)$ is sparse while $\bm X$ is dense or close to it so the permutation matrix $\bm P$ can be restricted to the form \begin{equation} \label{eq:15} \bm P=\begin{bmatrix}\bm P_{\bm Z}&\bm0\\ \bm0&\bm P_{\bm X}\end{bmatrix} \end{equation} without loss of efficiency. In fact, in most cases we can set $\bm P_{\bm X}=\bm I_p$ without loss of efficiency. Let us assume that the permutation matrix is required to be of the form (\ref{eq:15}) so that we can write the Cholesky factorization for the positive definite system (\ref{eq:13}) as \begin{multline} \label{eq:16} \begin{bmatrix} \bm L_{\bm Z}&\bm0\\\bm L_{\bm{XZ}}&\bm L_{\bm X} \end{bmatrix} \begin{bmatrix} \bm L_{\bm Z}&\bm0\\\bm L_{\bm{XZ}}&\bm L_{\bm X} \end{bmatrix}\trans =\\ \begin{bmatrix}\bm P_{\bm Z}&\bm0\\ \bm0&\bm P_{\bm X}\end{bmatrix} \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta) +\bm I_q&\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm X\\ \bm X\trans\bm Z\bm\Lambda(\theta) &\bm X\trans\bm X \end{bmatrix} \begin{bmatrix}\bm P_{\bm Z}&\bm0\\ \bm0&\bm P_{\bm X}\end{bmatrix}\trans . \end{multline} The discrepancy can now be written in the canonical form \begin{equation} \label{eq:17} d(\bm u|\bm y,\bm\theta,\bm\beta) =\tilde{d}(\bm y,\bm\theta) + \left\| \begin{bmatrix} \bm L_{\bm Z}\trans&\bm L_{\bm{XZ}}\trans\\ \bm 0&\bm L_{\bm X}\trans \end{bmatrix} \begin{bmatrix} \bm P_{\bm Z}(\bm u-\tilde{\bm u})\\ \bm P_{\bm X}(\bm\beta-\tilde{\bm\beta}) \end{bmatrix} \right\|^2 \end{equation} where \begin{equation} \label{eq:18} \tilde{d}(\bm y,\bm\theta)= d(\tilde{\bm u}(\bm\theta)|\bm y,\bm\theta,\tilde{\bm\beta}(\bm\theta)) \end{equation} is the minimum discrepancy, given $\bm\theta$. \subsection{The profiled likelihood for linear mixed models} \label{sec:prof-log-likel} Substituting (\ref{eq:17}) into (\ref{eq:7}) provides the unnormalized conditional density $h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)$ on the deviance scale as \begin{multline} \label{eq:32} -2\log(h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma))\\= (n+q)\log(2\pi\sigma^2)+\frac{\tilde{d}(\bm y,\bm\theta) + \left\| \begin{bmatrix} \bm L_{\bm Z}\trans&\bm L_{\bm{XZ}}\trans\\ \bm 0&\bm L_{\bm X}\trans \end{bmatrix} \begin{bmatrix} \bm P_{\bm Z}(\bm u-\tilde{\bm u})\\ \bm P_{\bm X}(\bm\beta-\tilde{\bm\beta}) \end{bmatrix} \right\|^2}{\sigma^2} . \end{multline} As shown in Appendix \ref{sec:integr-quadr-devi}, the integral of a quadratic form on the deviance scale, such as (\ref{eq:32}), is easily evaluated, providing the log-likelihood, $\ell(\bm\theta,\bm\beta,\sigma|\bm y)$, as \begin{multline} \label{eq:lmmdev} -2\ell(\bm\theta,\bm\beta,\sigma|\bm y)\\ =-2\log\left(L(\bm\theta,\bm\beta,\sigma|\bm y)\right)\\ =n\log(2\pi\sigma^2)+\log(|\bm L_{\bm Z}|^2)+\frac{\tilde{d}(\bm y,\bm\theta) + \left\|\bm L_{\bm X}\trans\bm P_{\bm X}(\bm\beta-\tilde{\bm\beta})\right\|^2}{\sigma^2}, \end{multline} from which we can see that the conditional estimate of $\bm\beta$, given $\bm\theta$, is $\tilde{\bm\beta}(\bm\theta)$ and the conditional estimate of $\sigma$, given $\bm\theta$, is \begin{equation} \label{eq:condsigma} \tilde{\sigma^2}(\bm\theta)= \frac{\tilde{d}(\bm\theta|\bm y)}{n} . \end{equation} Substituting these conditional estimates into (\ref{eq:lmmdev}) produces the \emph{profiled likelihood}, $\tilde{L}(\bm\theta|\bm y)$, as \begin{equation} \label{eq:19} -2\tilde{\ell}(\bm\theta|\bm y))= \log(|\bm L_{\bm Z}(\bm\theta)|^2)+ n\left(1+\log\left(\frac{2\pi\tilde{d}(\bm y,\bm\theta)}{n}\right)\right) . \end{equation} The maximum likelihood estimate of $\bm\theta$ can then be expressed as \begin{equation} \label{eq:29} \widehat{\bm\theta}_L=\arg\min_{\bm\theta} \left(-2\tilde{\ell}(\bm\theta|\bm y)\right) . \end{equation} from which the ML estimates of $\sigma^2$ and $\bm\beta$ are evaluated as \begin{align} \label{eq:30} \widehat{\sigma^2_L}&= \frac{\tilde{d}(\widehat{\bm\theta}_L,\bm y)}{n}\\ \widehat{\bm\beta}_L&=\tilde{\bm\beta}(\widehat{\bm\theta}_L) . \end{align} The important thing to note about optimizing the profiled likelihood, (\ref{eq:19}), is that it is a $m$-dimensional optimization problem and typically $m$ is very small. \subsection{The REML criterion} \label{sec:reml-criterion} In practice the so-called REML estimates of variance components are often preferred to the maximum likelihood estimates. (``REML'' can be considered to be an acronym for ``restricted'' or ``residual'' maximum likelihood, although neither term is completely accurate because these estimates do not maximize a likelihood.) We can motivate the use of the REML criterion by considering a linear regression model, \begin{equation} \label{eq:20} \bc Y\sim\mathcal{N}(\bm X\bm\beta,\sigma^2\bm I_n), \end{equation} in which we typically estimate $\sigma^2$ by \begin{equation} \label{eq:21} \widehat{\sigma^2_R}=\frac{\|\bm y-\bm X\widehat{\bm\beta}\|^2}{n-p} \end{equation} even though the maximum likelihood estimate of $\sigma^2$ is \begin{equation} \label{eq:22} \widehat{\sigma^2_{L}}=\frac{\|\bm y-\bm X\widehat{\bm\beta}\|^2}{n} . \end{equation} The argument for preferring $\widehat{\sigma^2_R}$ to $\widehat{\sigma^2_{L}}$ as an estimate of $\sigma^2$ is that the numerator in both estimates is the sum of squared residuals at $\widehat{\bm\beta}$ and, although the residual vector $\bm y-\bm X\bm\beta$ is an $n$-dimensional vector, the residual at $\widehat{\bm\theta}$ satisfies $p$ linearly independent constraints, $\bm X\trans(\bm y-\bm X\widehat{\bm\beta})=\bm 0$. That is, the residual at $\widehat{\bm\theta}$ is the projection of the observed response vector, $\bm y$, into an $(n-p)$-dimensional linear subspace of the $n$-dimensional response space. The estimate $\widehat{\sigma^2_R}$ takes into account the fact that $\sigma^2$ is estimated from residuals that have only $n-p$ \emph{degrees of freedom}. The REML criterion for determining parameter estimates $\widehat{\bm\theta}_R$ and $\widehat{\sigma_R^2}$ in a linear mixed model has the property that these estimates would specialize to $\widehat{\sigma^2_R}$ from (\ref{eq:21}) for a linear regression model. Although not usually derived in this way, the REML criterion can be expressed as \begin{equation} \label{eq:23} c_R(\bm\theta,\bm\sigma|\bm y)=-2\log \int_{\mathbb{R}^p}L(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)\,d\bm\beta \end{equation} on the deviance scale. The REML estimates $\widehat{\bm\theta}_R$ and $\widehat{\sigma_R^2}$ minimize $c_R(\bm\theta,\bm\sigma|\bm y)$. The profiled REML criterion, a function of $\bm\theta$ only, is \begin{equation} \label{eq:24} \tilde{c}_R(\bm\theta|\bm y)= \log(|\bm L_{\bm Z}(\bm\theta)|^2|\bm L_{\bm X}(\bm\theta)|^2)+(n-p) \left(1+\log\left(\frac{2\pi\tilde{d}(\bm\theta|\bm y)}{n-p}\right)\right) \end{equation} and the REML estimate of $\bm\theta$ is \begin{equation} \label{eq:31} \widehat{\bm\theta}_R = \arg\min_{\bm\theta}\tilde{c}_R(\bm\theta,\bm y) . \end{equation} The REML estimate of $\sigma^2$ is $\widehat{\sigma^2_R}=\tilde{d}(\widehat{\bm\theta}_R|\bm y)/(n-p)$. It is not entirely clear how one would define a ``REML estimate'' of $\bm\beta$ because the REML criterion, $c_R(\bm\theta,\bm\sigma|\bm y)$, defined in (\ref{eq:23}), does not depend on $\bm\beta$. However, it is customary (and not unreasonable) to use $\widehat{\bm\beta}_R=\tilde{\bm\beta}(\widehat{\bm\theta}_R)$ as the REML estimate of $\bm\beta$. Note that the profiled REML criterion can be evaluated from a sparse Cholesky decomposition like that in (\ref{eq:16}) but without the requirement that the permutation can be applied to the columns of $\bm Z\bm\Lambda(\bm\theta)$ separately from the columnns of $\bm X$. That is, we can use a general fill-reducing permutation rather than the specific form (\ref{eq:15}) with separate permutations represented by $\bm P_{\bm Z}$ and $\bm P_{\bm X}$. This can be useful in cases where both $\bm Z$ and $\bm X$ are large and sparse. \subsection{Summary for linear mixed models} \label{sec:lmmsummary} A linear mixed model is characterized by the conditional distribution \begin{equation} \label{eq:lmmcond} (\bc Y|\bc U=\bm u)\sim\mathcal{N}(\bm\gamma(\bm u,\bm\theta,\bm\beta),\sigma^2\bm I_n)\text{ where } \bm\gamma(\bm u,\bm\theta,\bm\beta)=\bm Z\bm\Lambda(\bm\theta)\bm u+\bm X\bm\beta \end{equation} and the unconditional distribution $\bc U\sim\mathcal{N}(\bm 0,\sigma^2\bm I_q)$. The discrepancy function, \begin{displaymath} d(\bm u|\bm y,\bm\theta,\bm\beta)= \left\|\bm y-\bm\gamma(\bm u,\bm\theta,\bm\beta)\right\|^2+\|\bm u\|^2, \end{displaymath} is minimized at the conditional mode, $\tilde{\bm u}(\bm\theta)$, and the conditional estimate, $\tilde{\bm\beta}(\bm\theta)$, which are the solutions to the sparse, positive-definite linear system \begin{displaymath} \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta) +\bm I_q&\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm X\\ \bm X\trans\bm Z\bm\Lambda(\theta) &\bm X\trans\bm X \end{bmatrix} \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\\tilde{\bm\beta}(\bm\theta) \end{bmatrix} = \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm y\\ \bm X\trans\bm y \end{bmatrix} . \end{displaymath} In the process of solving this system we create the sparse left Cholesky factor, $L_{\bm Z}(\bm\theta)$, which is a lower triangular sparse matrix satisfying \begin{displaymath} \bm L_{\bm Z}(\bm\theta)\bm L_{\bm Z}(\bm\theta)\trans=\bm P_{\bm Z}\left(\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta)+\bm I_q\right)\bm P_{\bm Z}\trans \end{displaymath} where $\bm P_{\bm Z}$ is a permutation matrix representing a fill-reducing permutation formed from the pattern of nonzeros in $\bm Z\bm\Lambda(\bm\theta)$ for any $\bm\theta$ not on the boundary of the parameter region. (The values of the nonzeros depend on $\bm\theta$ but the pattern doesn't.) The profiled log-likelihood, $\tilde{\ell}(\bm\theta|\bm y)$, is \begin{displaymath} -2\tilde{\ell}(\bm\theta|\bm y)= \log(|\bm L_{\bm Z}(\bm\theta)|^2)+ n\left(1+\log\left(\frac{2\pi\tilde{d}(\bm y,\bm\theta)}{n}\right)\right) \end{displaymath} where $\tilde{d}(\bm y,\bm\theta)=d(\tilde{\bm u}(\bm\theta)|\bm y,\tilde{\bm\beta}(\bm\theta),\bm\theta)$. \section{Generalizing the discrepancy function} \label{sec:generalizations} Because one of the factors influencing the choice of implementation for linear mixed models is the extent to which the methods can also be applied to other mixed models, we describe several other classes of mixed models before discussing the implementation details for linear mixed models. At the core of our methods for determining the maximum likelihood estimates (MLEs) of the parameters in the mixed model are methods for minimizing the discrepancy function with respect to the coefficients $\bm u$ and $\bm\beta$ in the linear predictor $\bm\gamma(\bm u,\bm\theta,\bm\beta)$. In this section we describe the general form of the discrepancy function that we will use and a penalized iteratively reweighted least squares (PIRLS) algorithm for determining the conditional modes $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$. We then describe several types of mixed models and the form of the discrepancy function for each. \subsection{A weighted residual sum of squares} \label{sec:weighted} As shown in \S\ref{sec:conditional-mode-bm}, the discrepancy function for a linear mixed model has the form of a penalized residual sum of squares from a linear model (\ref{eq:10}). In this section we generalize that definition to \begin{equation} \label{eq:11} d(\bm u|\bm y,\bm\theta,\bm\beta) =\left\|\bm W^{1/2}(\bm\mu) \left[\bm y-\bm\mu_{\bc Y|\bc U}(\bm u,\bm\theta,\bm\beta)\right]\right\|^2+ \|\bm 0-\bm u\|^2 . \end{equation} where $\bm W$ is an $n\times n$ diagonal matrix, called the \emph{weights matrix}, with positive diagonal elements and $\bm W^{1/2}$ is the diagonal matrix with the square roots of the weights on the diagonal. The $i$th weight is inversely proportional to the conditional variances of $(\mathcal{Y}|\bc U=\bm u)$ and may depend on the conditional mean, $\bm\mu_{\bc Y|\bc U}$. We allow the conditional mean to be a nonlinear function of the linear predictor, but with certain restrictions. We require that the mapping from $\bm u$ to $\bm\mu_{\bc Y|\bc U=\bm u}$ be expressed as \begin{equation} \label{eq:uGammaEtaMu} \bm u\;\rightarrow\;\bm\gamma\;\rightarrow\;\bm\eta\;\rightarrow\;\bm\mu \end{equation} where $\bm\gamma=\bm Z\bm\Lambda(\bm\theta)\bm u+\bm X\bm\theta$ is an $ns$-dimensional vector ($s > 0$) while $\bm\eta$ and $\bm\mu$ are $n$-dimensional vectors. The map $\bm\eta\rightarrow\bm\mu$ has the property that $\mu_i$ depends only on $\eta_i$, $i=1,\dots,n$. The map $\bm\gamma\rightarrow\bm\eta$ has a similar property in that, if we write $\bm\gamma$ as an $n\times s$ matrix $\bm\Gamma$ such that \begin{equation} \label{eq:vecGamma} \bm\gamma=\vec{\bm\Gamma} \end{equation} (i.e.{} concatenating the columns of $\bm\Gamma$ produces $\bm\gamma$) then $\eta_i$ depends only on the $i$th row of $\bm\Gamma$, $i=1,\dots,n$. Thus the Jacobian matrix $\frac{d\bm\mu}{d\bm\eta\trans}$ is an $n\times n$ diagonal matrix and the Jacobian matrix $\frac{d\bm\eta}{d\bm\gamma\trans}$ is the horizontal concatenation of $s$ diagonal $n\times n$ matrices. For historical reasons, the function that maps $\eta_i$ to $\mu_i$ is called the \emph{inverse link} function and is written $\mu=g^{-1}(\eta)$. The \emph{link function}, naturally, is $\eta=g(\mu)$. When applied component-wise to vectors $\bm\mu$ or $\bm\eta$ we write these as $\bm\eta=\bm g(\bm\mu)$ and $\bm\mu=\bm g^{-1}(\bm\eta)$. Recall that the conditional distribution, $(\mathcal{Y}_i|\bc U=\bm u)$, is required to be independent of $(\mathcal{Y}_j|\bc U=\bm u)$ for $i,j=1,\dots,n,\,i\ne j$ and that all the component conditional distributions must be of the same form and differ only according to the value of the conditional mean. Depending on the family of the conditional distributions, the allowable values of the $\mu_i$ may be in a restricted range. For example, if the conditional distributions are Bernoulli then $0\le\mu_i\le1,i=1,\dots,n$. If the conditional distributions are Poisson then $0\le\mu_i,i=1,\dots,n$. A characteristic of the link function, $g$, is that it must map the restricted range to an unrestricted range. That is, a link function for the Bernoulli distribution must map $[0,1]$ to $[-\infty,\infty]$ and must be invertible within the range. The mapping from $\bm\gamma$ to $\bm\eta$ is defined by a function $m:\mathbb{R}^s\rightarrow\mathbb{R}$, called the \emph{nonlinear model} function, such that $\eta_i=m(\bm\gamma_i),i=1,\dots,n$ where $\bm\gamma_i$ is the $i$th row of $\bm\Gamma$. The vector-valued function is $\bm\eta=\bm m(\bm\gamma)$. Determining the conditional modes, $\tilde{\bm u}(\bm y|\bm\theta)$, and $\tilde{\bm\beta}(\bm y|\bm\theta)$, that jointly minimize the discrepancy, \begin{equation} \label{eq:12} \begin{bmatrix} \tilde{\bm u}(\bm y|\bm\theta)\\ \tilde{\bm\beta}(\bm y|\bm\theta) \end{bmatrix} =\arg\min_{\bm u,\bm\beta}\left[(\bm y-\bm\mu)\trans\bm W(\bm y-\bm\mu)+\|\bm u\|^2\right] \end{equation} becomes a weighted, nonlinear least squares problem except that the weights, $\bm W$, can depend on $\bm\mu$ and, hence, on $\bm u$ and $\bm\beta$. In describing an algorithm for linear mixed models we called $\tilde{\bm\beta}(\bm\theta)$ the \emph{conditional estimate}. That name reflects that fact that this is the maximum likelihood estimate of $\bm\beta$ for that particular value of $\bm\theta$. Once we have determined the MLE, $\widehat(\bm\theta)_L$ of $\bm\theta$, we have a ``plug-in'' estimator, $\widehat{\bm\beta}_L=\tilde{\bm\beta}(\bm\theta)$ for $\bm\beta$. This property does not carry over exactly to other forms of mixed models. The values $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$ are conditional modes in the sense that they are the coefficients in $\bm\gamma$ that jointly maximize the unscaled conditional density $h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)$. Here we are using the adjective ``conditional'' more in the sense of conditioning on $\bc Y=\bm y$ than in the sense of conditioning on $\bm\theta$, although these values are determined for a fixed value of $\bm\theta$. \subsection{The PIRLS algorithm for $\tilde{\bm u}$ and $\tilde{\bm\beta}$} \label{sec:pirls-algor-tild} The penalized, iteratively reweighted, least squares (PIRLS) algorithm to determine $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$ is a form of the Fisher scoring algorithm. We fix the weights matrix, $\bm W$, and use penalized, weighted, nonlinear least squares to minimize the penalized, weighted residual sum of squares conditional on these weights. Then we update the weights to those determined by the current value of $\bm\mu$ and iterate. To describe this algorithm in more detail we will use parenthesized superscripts to denote the iteration number. Thus $\bm u^{(0)}$ and $\bm\beta^{(0)}$ are the initial values of these parameters, while $\bm u^{(i)}$ and $\bm\beta^{(i)}$ are the values at the $i$th iteration. Similarly $\bm\gamma^{(i)}=\bm Z\bm\Lambda(\bm\theta)\bm u^{(i)}+\bm X\bm\beta^{(i)}$, $\bm\eta^{(i)}=\bm m(\bm\gamma^{(i)})$ and $\bm\mu^{(i)}=\bm g^{-1}(\bm\eta^{(i)})$. We use a penalized version of the Gauss-Newton algorithm \citep[ch.~2]{bateswatts88:_nonlin} for which we define the weighted Jacobian matrices \begin{align} \label{eq:Jacobian} \bm U^{(i)}&=\bm W^{1/2}\left.\frac{d\bm\mu}{d\bm u\trans}\right|_{\bm u=\bm u^{(i)},\bm\beta=\bm\beta^{(i)}}=\bm W^{1/2} \left.\frac{d\bm\mu}{d\bm\eta\trans}\right|_{\bm\eta^{(i)}} \left.\frac{d\bm\eta}{d\bm\gamma\trans}\right|_{\bm\gamma^{(i)}} \bm Z\bm\Lambda(\bm\theta)\\ \bm V^{(i)}&=\bm W^{1/2}\left.\frac{d\bm\mu}{d\bm\beta\trans}\right|_{\bm u=\bm u^{(i)},\bm\beta=\bm\beta^{(i)}}=\bm W^{1/2} \left.\frac{d\bm\mu}{d\bm\eta\trans}\right|_{\bm\eta^{(i)}} \left.\frac{d\bm\eta}{d\bm\gamma\trans}\right|_{\bm\gamma^{(i)}} \bm X \end{align} of dimension $n\times q$ and $n\times p$, respectively. The increments at the $i$th iteration, $\bm\delta_{\bm u}^{(i)}$ and $\bm\delta_{\bm\beta}^{(i)}$, are the solutions to \begin{equation} \label{eq:PNLSinc} \begin{bmatrix} {\bm U^{(i)}}\trans\bm U^{(i)}+\bm I_q&{\bm U^{(i)}}\trans\bm V^{(i)}\\ {\bm V^{(i)}}\trans\bm U^{(i)}&{\bm V^{(i)}}\trans\bm V^{(i)} \end{bmatrix} \begin{bmatrix} \bm\delta_{\bm u}^{(i)}\\ \bm\delta_{\bm\beta}^{(i)} \end{bmatrix} = \begin{bmatrix} {\bm U^{(i)}}\trans\bm W^{1/2}(\bm y-\bm\mu^{(i)})-\bm u^{(i)}\\ {\bm U^{(i)}}\trans\bm W^{1/2}(\bm y-\bm\mu^{(i)}) \end{bmatrix} \end{equation} providing the updated parameter values \begin{equation} \label{eq:33} \begin{bmatrix}\bm u^{(i+1)}\\\bm\beta^{(i+1)}\end{bmatrix}= \begin{bmatrix}\bm u^{(i)}\\\bm\beta^{(i)}\end{bmatrix}+\lambda \begin{bmatrix}\bm\delta_{\bm u}^{(i)}\\\bm\delta_{\bm\beta}^{(i)} \end{bmatrix} \end{equation} where $\lambda>0$ is a step factor chosen to ensure that \begin{equation} \label{eq:34} (\bm y-\bm\mu^{(i+1)})\trans\bm W(\bm y-\bm\mu^{(i+1)})+\|\bm u^{(i+1)}\|^2 < (\bm y-\bm\mu^{(i)})\trans\bm W(\bm y-\bm\mu^{(i)})+\|\bm u^{(i)}\|^2 . \end{equation} In the process of solving for the increments we form the sparse, lower triangular, Cholesky factor, $\bm L^{(i)}$, satisfying \begin{equation} \label{eq:35} \bm L^{(i)} {\bm L^{(i)}}\trans = \bm P_{\bm Z}\left({\bm U^{(i)}}\trans\bm U^{(i)}+ \bm I_n\right)\bm P_{\bm Z}\trans . \end{equation} After each successful iteration, determining new values of the coefficients, $\bm u^{(i+1)}$ and $\bm\beta^{(i+1)}$, that reduce the penalized, weighted residual sum of squqres, we update the weights matrix to $\bm W(\bm\mu^{(i+1)})$ and the weighted Jacobians, $\bm U^{(i+1)}$ and $\bm V^{(i+1)}$, then iterate. Convergence is determined according to the orthogonality convergence criterion~\citep[ch.~2]{bateswatts88:_nonlin}, suitably adjusted for the weights matrix and the penalty. \subsection{Weighted linear mixed models} \label{sec:weightedLMM} One of the simplest generalizations of linear mixed models is a weighted linear mixed model where $s=1$, the link function, $g$, and the nonlinear model function, $m$, are both the identity, the weights matrix, $\bm W$, is constant and the conditional distribution family is Gaussian. That is, the conditional distribution can be written \begin{equation} \label{eq:weightedLMM} (\bc Y|\bc U=\bm u)\sim\mathcal{N}(\bm\gamma(\bm u,\bm\theta,\bm\beta),\sigma^2\bm W^{-1}) \end{equation} with discrepancy function \begin{equation} \label{eq:wtddisc} d(\bm u|\bm y,\bm\theta,\bm\beta)=\left\|\bm W^{1/2}(\bm y-\bm Z\bm\Lambda(\bm\theta)\bm u-\bm X\bm\theta)\right\|^2+\|\bm u\|^2 . \end{equation} The conditional mode, $\tilde{\bm u}(\bm\theta)$, and the conditional estimate, $\tilde{\bm\beta}(\bm\theta)$, are the solutions to \begin{equation} \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm W\bm Z\bm\Lambda(\theta) +\bm I_q&\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm W\bm X\\ \bm X\trans\bm W\bm Z\bm\Lambda(\theta) &\bm X\trans\bm W\bm X \end{bmatrix} \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\\tilde{\bm\beta}(\bm\theta) \end{bmatrix} = \begin{bmatrix} \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm W\bm y\\ \bm X\trans\bm W\bm y \end{bmatrix} , \end{equation} which can be solved directly, and the Cholesky factor, $\bm L_{\bm Z}(\bm\theta)$, satisfies \begin{equation} \bm L_{\bm Z}(\bm\theta)\bm L_{\bm Z}(\bm\theta)\trans=\bm P_{\bm Z}\left(\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm W\bm Z\bm\Lambda(\theta)+\bm I_q\right)\bm P_{\bm Z}\trans . \end{equation} The profiled log-likelihood, $\tilde{\ell}(\bm\theta|\bm y)$, is \begin{equation} \label{eq:wtdprofilelik} -2\tilde{\ell}(\bm\theta|\bm y)= \log\left(\frac{|\bm L_{\bm Z}(\bm\theta)|^2}{|\bm W|}\right)+ n\left(1+\log\left(\frac{2\pi\tilde{d}(\bm y,\bm\theta)}{n}\right)\right) . \end{equation} If the matrix $\bm W$ is fixed then we can ignore the term $|\bm W|$ in (\ref{eq:wtdprofilelik}) when determining the MLE, $\widehat{\bm\theta}_L$. However, in some models, we use a parameterized weight matrix, $\bm W(\bm\phi)$, and wish to determine the MLEs, $\widehat{\bm\phi}_L$ and $\widehat{\bm\theta}_L$ simultaneously. In these cases we must include the term involving $|\bm W(\bm\phi)|$ when evaluating the profiled log-likelihood. Note that we must define the parameterization of $\bm W(\bm\phi)$ such that $\sigma^2$ and $\bm\phi$ are not a redundant parameterization of $\sigma^2\bm W(\bm\phi)$. For example, we could require that the first diagonal element of $\bm W$ be unity. \subsection{Nonlinear mixed models} \label{sec:NLMMs} In an unweighted, nonlinear mixed model the conditional distribution is Gaussian, the link, $g$, is the identity and the weights matrix, $\bm W=\bm I_n$. That is, \begin{equation} \label{eq:conddistNLMM} (\bc Y|\bc U=\bm u)\sim\mathcal{N}(\bm m(\bm\gamma),\sigma^2\bm I_n) \end{equation} with discrepancy function \begin{equation} \label{eq:discNLMM} d(\bm u|\bm y,\bm\theta,\bm\beta)= \|\bm y-\bm\mu\|^2 + \|\bm u\|^2 . \end{equation} For a given value of $\bm\theta$ we determine the conditional modes, $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$, as the solution to the penalized nonlinear least squares problem \begin{equation} \label{eq:NLMMpnls} \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\ \tilde{\bm\beta}(\bm\theta) \end{bmatrix} = \arg\min_{\bm u,\bm\theta}d(\bm u|\bm y,\bm\theta,\bm\beta) \end{equation} and we write the minimum discrepancy, given $\bm y$ and $\bm\theta$, as \begin{equation} \label{eq:25} \tilde{d}(\bm y,\bm\theta)=d(\tilde{\bm u}(\bm\theta)|\bm y,\bm\theta,\tilde{\bm\beta}(\bm\theta)). \end{equation} Let $\tilde{\bm L}_Z(\bm\theta)$ and $\tilde{\bm L}_X(\bm\theta)$ be the Cholesky factors at $\bm\theta$, $\tilde{\bm\beta}(\bm\theta)$ and $\tilde{\bm u}(\bm\theta)$. Then the \emph{Laplace approximation} to the log-likelihood is \begin{equation} \label{eq:36} -2\ell_P(\bm\theta,\bm\beta,\sigma|\bm y)\approx n\log(2\pi\sigma^2)+\log(|\tilde{\bm L}_{\bm Z}|^2)+ \frac{\tilde{d}(\bm y,\bm\theta) + \left\|\tilde{\bm L}_{\bm X}\trans(\bm\beta-\tilde{\bm\beta})\right\|^2}{\sigma^2}, \end{equation} producing the approximate profiled log-likelihood, $\tilde{\ell}_P(\bm\theta|\bm y)$, \begin{equation} \label{eq:37} -2\tilde{\ell}_P(\bm\theta|\bm y)\approx \log(|\tilde{\bm L}_{\bm Z}|^2)+n\left(1+\log(2\pi \tilde{d}(\bm y,\bm\theta)/n) \right). \end{equation} \subsubsection{Nonlinear mixed model summary} \label{sec:nonl-mixed-model} In a nonlinear mixed model we determine the parameter estimate, $\widehat{\bm\theta}_P$, from the Laplace approximation to the log-likelihood as \begin{equation} \label{eq:38} \widehat{\bm\theta}_P = \arg\max_{\bm\theta}\tilde{\ell}_P(\bm\theta|\bm y) =\arg\min_{\bm\theta} \log(|\tilde{\bm L}_{\bm Z}|^2)+ n\left(1+\log(2\pi \tilde{d}(\bm y,\bm\theta)/n) \right). \end{equation} Each evaluation of $\tilde{\ell}_P(\bm\theta|\bm y)$ requires a solving the penalized nonlinear least squares problem (\ref{eq:NLMMpnls}) simultaneously with respect to both sets of coefficients, $\bm u$ and $\bm\beta$, in the linear predictor, $\bm\gamma$. For a weighted nonlinear mixed model with fixed weights, $\bm W$, we replace the unweighted discrepancy function $d(\bm u|\bm y,\bm\theta,\bm\beta)$ with the weighted discrepancy function, %% Finish this off \section{Details of the implementation} \label{sec:details} \subsection{Implementation details for linear mixed models} \label{sec:impl-line-mixed} The crucial step in implementing algorithms for determining ML or REML estimates of the parameters in a linear mixed model is evaluating the factorization (\ref{eq:16}) for any $\bm\theta$ satisfying $\bm\theta_L\le\bm\theta\le\bm\theta_U$. We will assume that $\bm Z$ is sparse as is $\bm Z\bm\Lambda(\bm\theta)$. When $\bm X$ is not sparse we will use the factorization (\ref{eq:16}) setting $\bm P_{\bm X}=\bm I_p$ and storing $\bm L_{\bm X\bm Z}$ and $\bm L_{\bm X}$ as dense matrices. The permutation matrix $\bm P_{\bm Z}$ is determined from the pattern of non-zeros in $\bm Z\bm\Lambda(\bm\theta)$ which is does not depend on $\bm\theta$, as long as $\bm\theta$ is not on the boundary. In fact, in most cases the pattern of non-zeros in $\bm Z\bm\Lambda(\bm\theta)$ is the same as the pattern of non-zeros in $\bm Z$. For many models, in particular models with scalar random effects (described later), the matrix $\bm\Lambda(\bm\theta)$ is diagonal. Given a value of $\bm\theta$ we determine the Cholesky factor $\bm L_{\bm Z}$ satisfying \begin{equation} \label{eq:LZ} \bm L_{\bm Z}\bm L_{\bm Z}\trans=\bm P_{\bm Z}( \bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta) +\bm I_q)\bm P_{\bm Z}\trans . \end{equation} The CHOLMOD package allows for $\bm L_{\bm Z}$ to be calculated directly from $\bm\Lambda\trans(\bm\theta)\bm Z\trans$ or from $\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta)$. The choice in implementation is whether to store $\bm Z\trans$ and update it to $\bm\Lambda\trans(\bm\theta)\bm Z$ or to store $\bm Z\trans\bm Z$ and use it to form $\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm Z\bm\Lambda(\theta)$ at each evaluation. In the \package{lme4} package we store $\bm Z\trans$ and use it to form $\bm\Lambda\trans(\bm\theta)\bm Z\trans$ from which $\bm L_{\bm Z}$ is evaluated. There are two reasons for this choice. First, the calculations for the more general forms of mixed models cannot be reduced to calculations involving $\bm Z\trans\bm Z$ and by expressing these calculations in terms of $\bm\Lambda(\bm\theta)\bm Z\trans$ for linear mixed models we can reuse the code for the more general models. Second, the calculation of $\bm\Lambda(\bm\theta)\trans\left(\bm Z\trans\bm Z\right)\bm\Lambda(\bm\theta)$ from $\bm Z\trans\bm Z$ is complicated compared to the calculation of $\bm\Lambda(\bm\theta)\trans\bm Z\trans$ from $\bm Z\trans$. This choice is disadvantageous when $n\gg q$ because $\bm Z\trans$ is much larger than $\bm Z\trans\bm Z$, even when they are stored as sparse matrices. Evaluation of $\bm L_{\bm Z}$ directly from $\bm Z\trans$ requires more storage and more calculation that evaluating $\bm L_{\bm Z}$ from $\bm Z\trans\bm Z$. Next we evaluate $\bm L_{\bm X\bm Z}\trans$ as the solution to \begin{equation} \label{eq:LXZ} \bm L_{\bm Z}\bm L_{\bm X\bm Z}\trans=\bm P_{\bm Z}\bm\Lambda\trans(\bm\theta)\bm Z\trans\bm X . \end{equation} Again we have the choice of calculating and storing $\bm Z\trans\bm X$ or storing $\bm X$ and using it to reevaluate $\bm Z\trans\bm X$. In the \package{lme4} package we store $\bm X$, because the calculations for the more general models cannot be expressed in terms of $\bm Z\trans\bm X$. Finally $\bm L_{\bm X}$ is evaluated as the (dense) solution to \begin{equation} \label{eq:LX} \bm L_{\bm X}\bm L_{\bm X}\trans= \bm X\trans\bm X-\bm L_{\bm X\bm Z}\bm L_{\bm X\bm Z} . \end{equation} from which $\tilde{\bm\beta}$ can be determined as the solution to dense system \begin{equation} \label{eq:tildebeta} \bm L_{\bm X}\bm L_{\bm X}\tilde{\bm\beta}=\bm X\trans\bm y \end{equation} and $\tilde{\bm u}$ as the solution to the sparse system \begin{equation} \label{eq:tildeu} \bm L_{\bm Z}\bm L_{\bm Z}\tilde{u}=\bm\Lambda\trans\bm Z\trans\bm y \end{equation} For many models, in particular models with scalar random effects, which are described later, the matrix $\bm\Lambda(\bm\theta)$ is diagonal. For such a model, if both $\bm Z$ and $\bm X$ are sparse and we plan to use the REML criterion then we create and store \begin{equation} \label{eq:8} \bm A= \begin{bmatrix} \bm Z\trans\bm Z & \bm Z\trans\bm X\\ \bm X\trans\bm Z & \bm X\trans\bm X \end{bmatrix}\quad\text{and}\quad \bm c =\begin{bmatrix}\bm Z\trans\bm y\\\bm X\trans\bm y\end{bmatrix} \end{equation} and determine a fill-reducing permutation, $\bm P$, for $\bm A$. Given a value of $\bm\theta$ we create the factorization \begin{equation} \label{eq:26} \bm L(\bm\theta)\bm L(\bm\theta)\trans=\bm P\left( \begin{bmatrix} \bm\Lambda(\bm\theta) & \bm0\\\bm0&\bm I_p \end{bmatrix} \bm A \begin{bmatrix} \bm\Lambda(\bm\theta) & \bm0\\\bm0&\bm I_p \end{bmatrix}+ \begin{bmatrix}\bm I_q&\bm0\\\bm0&\bm0\end{bmatrix}\right) \bm P\trans \end{equation} solve for $\tilde{\bm u}(\bm\theta)$ and $\tilde{\bm\beta}(\bm\theta)$ in \begin{equation} \label{eq:28} \bm L\bm L\trans\bm P \begin{bmatrix} \tilde{\bm u}(\bm\theta)\\\tilde{\bm\beta}(\bm\theta) \end{bmatrix}= \bm P \begin{bmatrix}\bm\Lambda(\bm\theta) & \bm0\\\bm0&\bm I_p \end{bmatrix} \bm c \end{equation} then evaluate $\tilde{d}(\bm y|\bm\theta)$ and the profiled REML criterion as \begin{equation} \label{eq:27} \tilde{d}_R(\bm\theta|\bm y)=\log(|\bm L(\bm\theta)|^2)+ (n-p)\left(1+\log\left(\frac{2\pi\tilde{d}(\bm y|\bm\theta)} {n-p}\right)\right) . \end{equation} \bibliography{lme4} \appendix{} \section{Notation} \label{sec:notation} \subsection{Random variables in the model} \label{sec:rand-vari-model} \begin{description} \item[$\bc B$] Random-effects vector of dimension $q$, $\bc{B}\sim\mathcal{N}(\bm 0,\sigma^2\bm V(\bm\theta)\bm V(\bm\theta)\trans)$. \item[$\bm U$] ``Spherical'' random-effects vector of dimension $q$, $\bc U\sim\mathcal{N}(\bm 0,\sigma^2\bm I_q)$, $\bc B=\bm V(\bm\theta)\bc U$. \item[$\bc Y$] Response vector of dimension $n$. \end{description} \subsection{Parameters of the model} \label{sec:parameters-model} \begin{description} \item[$\bm\beta$] Fixed-effects parameters (dimension $p$). \item[$\bm\theta$] Parameters determining the left factor, $\bm\Lambda(\bm\theta)$ of the relative covariance matrix of $\bc B$ (dimension $m$). \item[$\sigma$] the common scale parameter - not used in some generalized linear mixed models and generalized nonlinear mixed models. \end{description} \subsection{Dimensions} \label{sec:dimensions} \begin{description} \item[$m$] dimension of the parameter $\bm\theta$. \item[$n$] dimension of the response vector, $\bm y$, and the random variable, $\bm{\mathcal{Y}}$. \item[$p$] dimension of the fixed-effects parameter, $\bm\beta$. \item[$q$] dimension of the random effects, $\bc B$ or $\bc U$. \item[$s$] dimension of the parameter vector, $\bm\phi$, in the nonlinear model function. \end{description} \subsection{Matrices} \label{sec:matrices} \begin{description} \item[$\bm L$] Left Cholesky factor of a positive-definite symmetric matrix. $\bm L_{\bm Z}$ is $q\times q$; $\bm L_{\bm X}$ is $p\times p$. \item[$\bm P$] Fill-reducing permutation for the random effects model matrix. (Size $q\times q$.) \item[$\bm V$] Left factor of the relative covariance matrix of the random effects. (Size $q\times q$.) \item[$\bm X$] Model matrix for the fixed-effects parameters, $\bm\beta$. (Size $(ns)\times p$.) \item[$\bm Z$] Model matrix for the random effects. (Size $(ns)\times q$.) \end{description} \section{Integrating a quadratic deviance expression} \label{sec:integr-quadr-devi} In (\ref{eq:6}) we defined the likelihood of the parameters given the observed data as \begin{displaymath} L(\bm\theta,\bm\beta,\sigma|\bm y) = \int_{\mathbb{R}^q}h(\bm u|\bm y,\bm\theta,\bm\beta,\sigma)\,d\bm u . \end{displaymath} which is often alarmingly described as ``an intractable integral''. In point of fact, this integral can be evaluated exactly in the case of a linear mixed model and can be approximated quite accurately for other forms of mixed models. \end{document} lme4/inst/doc/Theory.R0000754000175100001440000000035112512030564014247 0ustar hornikusers### R code from vignette source 'Theory.Rnw' ################################################### ### code chunk number 1: preliminaries ################################################### options(width=65,digits=5) #library(lme4) lme4/inst/doc/PLSvGLS.R0000754000175100001440000000035212512030564014170 0ustar hornikusers### R code from vignette source 'PLSvGLS.Rnw' ################################################### ### code chunk number 1: preliminaries ################################################### options(width=65,digits=5) #library(lme4) lme4/inst/doc/lmer.Rnw0000754000175100001440000042554712512030564014323 0ustar hornikusers%\VignetteEngine{knitr::knitr} %\VignetteDepends{ggplot2} %\VignetteDepends{splines} %\VignetteIndexEntry{Fitting Linear Mixed-Effects Models using lme4} \documentclass[nojss]{jss} \usepackage[T1]{fontenc}% for correct hyphenation and T1 encoding \usepackage[utf8]{inputenc}% \usepackage{lmodern}% latin modern font \usepackage[american]{babel} %% for texi2dvi ~ bug \usepackage{bm,amsmath,thumbpdf,amsfonts}%,minted} \usepackage{blkarray} \usepackage{array} \newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}} \newcommand{\matindex}[1]{\mbox{\scriptsize#1}}% Matrix index \newcommand{\github}{Github} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\VEC}{vec} \newcommand{\bmb}[1]{{\color{red} \emph{#1}}} \newcommand{\scw}[1]{{\color{blue} \emph{#1}}} \newcommand{\dmb}[1]{{\color{magenta} \emph{#1}}} \shortcites{bolker_strategies_2013,sleepstudy,gelman2013bayesian} \author{Douglas Bates\\U. of Wisconsin - Madison\And Martin M\"achler\\ETH Zurich\And Benjamin M. Bolker\\McMaster University\And Steven C. Walker\\McMaster University } \Plainauthor{Douglas Bates, Martin M\"achler, Ben Bolker, Steve Walker} \title{Fitting Linear Mixed-Effects Models using \pkg{lme4}} \Plaintitle{Fitting Linear Mixed-Effects Models using lme4} \Shorttitle{Linear Mixed Models with lme4} \Abstract{% Maximum likelihood or restricted maximum likelihood (REML) estimates of the parameters in linear mixed-effects models can be determined using the \code{lmer} function in the \pkg{lme4} package for \proglang{R}. As for most model-fitting functions in \proglang{R}, the model is described in an \code{lmer} call by a formula, in this case including both fixed- and random-effects terms. The formula and data together determine a numerical representation of the model from which the profiled deviance or the profiled REML criterion can be evaluated as a function of some of the model parameters. The appropriate criterion is optimized, using one of the constrained optimization functions in \proglang{R}, to provide the parameter estimates. We describe the structure of the model, the steps in evaluating the profiled deviance or REML criterion, and the structure of classes or types that represents such a model. Sufficient detail is included to allow specialization of these structures by users who wish to write functions to fit specialized linear mixed models, such as models incorporating pedigrees or smoothing splines, that are not easily expressible in the formula language used by \code{lmer}.} \Keywords{% sparse matrix methods, linear mixed models, penalized least squares, Cholesky decomposition} \Address{ Douglas Bates\\ Department of Statistics, University of Wisconsin\\ 1300 University Ave.\\ Madison, WI 53706, U.S.A.\\ E-mail: \email{bates@stat.wisc.edu} \par\bigskip Martin M\"achler\\ Seminar f\"ur Statistik, HG G~16\\ ETH Zurich\\ 8092 Zurich, Switzerland\\ E-mail: \email{maechler@stat.math.ethz.ch}\\ % URL: \url{http://stat.ethz.ch/people/maechler} \par\bigskip Benjamin M. Bolker\\ Departments of Mathematics \& Statistics and Biology \\ McMaster University \\ 1280 Main Street W \\ Hamilton, ON L8S 4K1, Canada \\ E-mail: \email{bolker@mcmaster.ca} \par\bigskip Steven C. Walker\\ Department of Mathematics \& Statistics \\ McMaster University \\ 1280 Main Street W \\ Hamilton, ON L8S 4K1, Canada \\ E-mail: \email{scwalker@math.mcmaster.ca } } \newcommand{\thetavec}{{\bm\theta}} \newcommand{\betavec}{{\bm\beta}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\abs}{\operatorname{abs}} \newcommand{\bLt}{\ensuremath{\bm\Lambda_{\bm\theta}}} \newcommand{\mc}[1]{\ensuremath{\mathcal{#1}}} \newcommand{\trans}{\ensuremath{^\top}} % JSS wants \top \newcommand{\yobs}{\ensuremath{\bm y_{\mathrm{obs}}}} \newcommand*{\eq}[1]{eqn.~\ref{#1}}% or just {(\ref{#1})} <>= options(width=70, show.signif.stars=FALSE, str=strOptions(strict.width="cut"), prompt = "R> ", continue = "+ ", useFancyQuotes = FALSE) library("knitr") library("lme4") library("ggplot2") theme_set(theme_bw()) library("grid") zmargin <- theme(panel.margin=unit(0,"lines")) library("lattice") opts_chunk$set(engine='R',dev='pdf',fig.width=9,prompt=TRUE, fig.height=5.5,cache=TRUE,tidy=FALSE,comment=NA) @ \setkeys{Gin}{width=\textwidth} \setkeys{Gin}{height=3.5in} \begin{document} Submitted to \emph{Journal of Statistical Software} \section{Introduction} \label{sec:intro} The \pkg{lme4} package for \proglang{R} provides functions to fit and analyze linear mixed models, generalized linear mixed models and nonlinear mixed models. In each of these names, the term ``mixed'' or, more fully, ``mixed effects'', denotes a model that incorporates both fixed- and random-effects terms in a linear predictor expression from which the conditional mean of the response can be evaluated. In this paper we describe the formulation and representation of linear mixed models. The techniques used for generalized linear and nonlinear mixed models will be described separately, in a future paper. The development of general software for fitting mixed models remains an active area of research with many open problems. Consequently, the \pkg{lme4} package has evolved since it was first released, and continues to improve as we learn more about mixed models. However, we recognize the need to maintain stability and backward compatibility of \pkg{lme4} so that it continues to be broadly useful. In order to maintain stability while continuing to advance mixed-model computation, we have developed several additional frameworks that draw on the basic ideas of \code{lme4} but modify its structure or implementation in various ways. These descendants include the \mbox{\pkg{MixedModels}} package in \proglang{Julia}, the \pkg{lme4pureR} package in \proglang{R}, and the \code{flexLambda} development branch of \pkg{lme4}. The current article is largely restricted to describing the current stable version of the \pkg{lme4} package (1.1-7), with Appendix~\ref{sec:modularExamples} describing hooks into the computational machinery that are designed for extension development. The \pkg{gamm4} \citep{gamm4} and \pkg{blme} \citep{blme} packages currently make use of these hooks. Another goal of this article is to contrast the approach used by \pkg{lme4} with previous formulations of mixed models. The expressions for the profiled log-likelihood and profiled REML (restricted maximum likelihood) criteria derived in Section~\ref{sec:profdev} are similar to those presented in \citet{bates04:_linear} and, indeed, are closely related to ``Henderson's mixed-model equations''~\citep{henderson_1982}. Nonetheless there are subtle but important changes in the formulation of the model and in the structure of the resulting penalized least squares (PLS) problem to be solved (Section~\ref{sec:PLSpureR}). We derive the current version of the PLS problem (Section~\ref{sec:plsMath}) and contrast this result with earlier formulations (Section~\ref{sec:previous_lmm_form}). This article is organized into four main sections (\ref{sec:lFormula}; \ref{sec:mkLmerDevfun}; \ref{sec:optimizeLmer}; \ref{sec:mkMerMod}), each of which corresponds to one of the four largely separate modules that comprise \pkg{lme4}. Before describing the details of each module, we describe the general form of the linear mixed model underlying \pkg{lme4} (Section~\ref{sec:LMMs}); introduce the \code{sleepstudy} data that will be used as an example throughout (Section~\ref{sec:sleepstudy}); and broadly outline \pkg{lme4}'s modular structure (Section~\ref{sec:modular}). \subsection{Linear mixed models} \label{sec:LMMs} Just as a linear model is described by the distribution of a vector-valued random response variable, $\mc{Y}$, whose observed value is $\yobs$, a linear mixed model is described by the distribution of two vector-valued random variables: $\mc{Y}$, the response, and $\mc{B}$, the vector of random effects. In a linear model the distribution of $\mc Y$ is multivariate normal,%\begin{linenomath} \begin{equation} \label{eq:linearmodel} \mc Y\sim\mc{N}(\bm X\bm\beta+\bm o,\sigma^2\bm W^{-1}), \end{equation} where $n$ is the dimension of the response vector, $\bm W$ is a diagonal matrix of known prior weights, $\bm\beta$ is a $p$-dimensional coefficient vector, $\bm X$ is an $n\times p$ model matrix, and $\bm o$ is a vector of known prior offset terms. The parameters of the model are the coefficients $\bm\beta$ and the scale parameter $\sigma$. In a linear mixed model it is the \emph{conditional} distribution of $\mc Y$ given $\mc B=\bm b$ that has such a form, \begin{equation} \label{eq:LMMcondY} ( \mc Y|\mc B=\bm b)\sim\mc{N}(\bm X\bm\beta+\bm Z\bm b+\bm o,\sigma^2\bm W^{-1}), % | <- for ESS \end{equation} where $\bm Z$ is the $n\times q$ model matrix for the $q$-dimensional vector-valued random effects variable, $\mc B$, whose value we are fixing at $\bm b$. The unconditional distribution of $\mc B$ is also multivariate normal with mean zero and a parameterized $q\times q$ variance-covariance matrix, $\bm\Sigma$, \begin{equation} \label{eq:LMMuncondB} \mc B\sim\mc N(\bm0,\bm\Sigma) . \end{equation} As a variance-covariance matrix, $\bm\Sigma$ must be positive semidefinite. It is convenient to express the model in terms of a \emph{relative covariance factor}, $\bLt$, which is a $q\times q$ matrix, depending on the \emph{variance-component parameter}, $\bm\theta$, and generating the symmetric $q\times q$ variance-covariance matrix, $\bm\Sigma$, according to%\begin{linenomath} \begin{equation} \label{eq:relcovfac} \bm\Sigma_{\bm\theta}=\sigma^2\bLt\bLt\trans , \end{equation}%\end{linenomath} where $\sigma$ is the same scale factor as in the conditional distribution (\ref{eq:LMMcondY}). Although equations \ref{eq:LMMcondY}, \ref{eq:LMMuncondB}, and \ref{eq:relcovfac} fully describe the class of linear mixed models that \pkg{lme4} can fit, this terse description hides many important details. Before moving on to these details, we make a few observations: \begin{itemize} \item This formulation of linear mixed models allows for a relatively compact expression for the profiled log-likelihood of $\bm\theta$ (Section~\ref{sec:profdev}, Equation~\ref{eq:profiledDeviance}). \item The matrices associated with random effects, $\bm Z$ and $\bLt$, typically have a sparse structure with a sparsity pattern that encodes various model assumptions. Sections~\ref{sec:LMMmatrix} and \ref{sec:CSCmats} provide details on these structures, and how to represent them efficiently. \item The interface provided by \pkg{lme4}'s \code{lmer} function is slightly less general than the model described by equations \ref{eq:LMMcondY}, \ref{eq:LMMuncondB}, and \ref{eq:relcovfac}. To take advantage of the entire range of possibilities, one may use the modular functions (Sections~\ref{sec:modular} and Appendix~\ref{sec:modularExamples}) or explore the experimental \code{flexLambda} branch of \pkg{lme4} on \github. \end{itemize} \subsection{Example} \label{sec:sleepstudy} Throughout our discussion of \pkg{lme4}, we will work with a data set on the average reaction time per day for subjects in a sleep deprivation study \citep{sleepstudy}. On day 0 the subjects had their normal amount of sleep. Starting that night they were restricted to 3 hours of sleep per night. The response variable, \code{Reaction}, represents average reaction times in milliseconds (ms) on a series of tests given each \code{Day} to each \code{Subject} (Figure~\ref{fig:sleepPlot}), <>= str(sleepstudy) @ <>= ## BMB: seemed more pleasing to arrange by increasing slope rather than ## intercept ... xyplot(Reaction ~ Days | Subject, sleepstudy, aspect = "xy", layout = c(9, 2), type = c("g", "p", "r"), index.cond = function(x, y) coef(lm(y ~ x))[2], xlab = "Days of sleep deprivation", ylab = "Average reaction time (ms)", as.table = TRUE) @ % | Each subject's reaction time increases approximately linearly with the number of sleep-deprived days. However, subjects also appear to vary in the slopes and intercepts of these relationships, which suggests a model with random slopes and intercepts. As we shall see, such a model may be fitted by minimizing the REML criterion (Equation~\ref{eq:REMLdeviance}) using <>= fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy) @ % | The estimates of the standard deviations of the random effects for the intercept and the slope are \Sexpr{round(sqrt(VarCorr(fm1)$Subject[1,1]), 2)} ms % $ and \Sexpr{round(sqrt(VarCorr(fm1)$Subject[2,2]), 2)} ms/day. % $ The fixed-effects coefficients, $\betavec$, are \Sexpr{round(fixef(fm1)[1], 1)} ms and \Sexpr{round(fixef(fm1)[2], 2)} ms/day for the intercept and slope. In this model, one interpretation of these fixed effects is that they are the estimated population mean values of the random intercept and slope (Section~\ref{sec:intuitiveFormulas}). \subsection{High level modular structure} \label{sec:modular} The \code{lmer} function is composed of four largely independent modules. In the first module, a mixed-model formula is parsed and converted into the inputs required to specify a linear mixed model (Section~\ref{sec:lFormula}). The second module uses these inputs to construct an \proglang{R} function which takes the covariance parameters, $\bm\theta$, as arguments and returns negative twice the log profiled likelihood or the REML criterion (Section~\ref{sec:mkLmerDevfun}). The third module optimizes this objective function to produce maximum likelihood (ML) or REML estimates of $\bm\theta$ (Section~\ref{sec:optimizeLmer}). Finally, the fourth module provides utilities for interpreting the optimized model (Section~\ref{sec:mkMerMod}). \begin{table}[tb] \centering \begin{tabular}{lllp{2.1in}} \hline Module & & \proglang{R} function & Description \\ \hline Formula module & (Section~\ref{sec:lFormula}) & \code{lFormula} & Accepts a mixed-model formula, data, and other user inputs, and returns a list of objects required to fit a linear mixed model. \\ Objective function module & (Section~\ref{sec:mkLmerDevfun}) & \code{mkLmerDevfun} & Accepts the results of \code{lFormula} and returns a function to calculate the deviance (or restricted deviance) as a function of the covariance parameters, $\bm\theta$.\\ Optimization module & (Section~\ref{sec:optimizeLmer}) & \code{optimizeLmer} & Accepts a deviance function returned by \code{mkLmerDevfun} and returns the results of the optimization of that deviance function. \\ Output module & (Section~\ref{sec:mkMerMod}) & \code{mkMerMod} & Accepts an optimized deviance function and packages the results into a useful object. \\ \hline \end{tabular} \caption{The high-level modular structure of \code{lmer}.} \label{tab:modular} \end{table} To illustrate this modularity, we recreate the \code{fm1} object by a series of four modular steps; the formula module, <>= parsedFormula <- lFormula(formula = Reaction ~ Days + (Days | Subject), data = sleepstudy) @ the objective function module, <>= devianceFunction <- do.call(mkLmerDevfun, parsedFormula) @ the optimization module, <>= optimizerOutput <- optimizeLmer(devianceFunction) @ and the output module, <>= mkMerMod( rho = environment(devianceFunction), opt = optimizerOutput, reTrms = parsedFormula$reTrms, fr = parsedFormula$fr) @ % | \section{Formula module} \label{sec:lFormula} \subsection{Mixed-model formulas} \label{sec:formulas} Like most model-fitting functions in \proglang{R}, \code{lmer} takes as its first two arguments a \emph{formula} specifying the model and the \emph{data} with which to evaluate the formula. This second argument, \code{data}, is optional but recommended and is usually the name of an \proglang{R} data frame. In the \proglang{R} \code{lm} function for fitting linear models, formulas take the form \verb+resp ~ expr+, where \code{resp} determines the response variable and \code{expr} is an expression that specifies the columns of the model matrix. Formulas for the \code{lmer} function contain special random-effects terms, <>= resp ~ FEexpr + (REexpr1 | factor1) + (REexpr2 | factor2) + ... @ where \code{FEexpr} is an expression determining the columns of the fixed-effects model matrix, $\bm X$, and the random-effects terms, \code{(REexpr1 | factor1)} and \code{(REexpr2 | factor2)}, determine both the random-effects model matrix, $\bm Z$ (Section~\ref{sec:mkZ}), and the structure of the relative covariance factor, $\bLt$ (Section~\ref{sec:mkLambdat}). In principle, a mixed-model formula may contain arbitrarily many random-effects terms, but in practice the number of such terms is typically low. \subsection{Understanding mixed-model formulas} \label{sec:intuitiveFormulas} Before describing the details of how \pkg{lme4} parses mixed-model formulas (Section~\ref{sec:LMMmatrix}), we provide an informal explanation and then some examples. Our discussion assumes familiarity with the standard \proglang{R} modelling paradigm \citep{Chambers:1993}. Each random-effects term is of the form \code{(expr | factor)}. The expression \code{expr} is evaluated as a linear model formula, producing a model matrix following the same rules used in standard \proglang{R} modelling functions (e.g., \code{lm} or \code{glm}). The expression \code{factor} is evaluated as an \proglang{R} factor. One way to think about the vertical bar operator is as a special kind of interaction between the model matrix and the grouping factor. This interaction ensures that the columns of the model matrix have different effects for each level of the grouping factor. What makes this a special kind of interaction is that these effects are modelled as unobserved random variables, rather than unknown fixed parameters. Much has been written about important practical and philosophical differences between these two types of interactions (e.g., \citet{henderson_1982}; \citet{gelman2005analysis}). For example, the random-effects implementation of such interactions can be used to obtain shrinkage estimates of regression coefficients (e.g., \citet{1977EfronAndMorris}), or account for lack of independence in the residuals due to block structure or repeated measurements (e.g., \citet{laird_ware_1982}). Table~\ref{tab:formulas} provides several examples of the right-hand-sides of mixed-model formulas. The first example, \code{(1 | g)}, % | is the simplest possible mixed-model formula, where each level of the grouping factor, \code{g}, has its own random intercept. The mean and standard deviation of these intercepts are parameters to be estimated. Our description of this model incorporates any non-zero mean of the random effects as fixed-effects parameters. If one wishes to specify that a random intercept has \emph{a priori} known means, one may use the \code{offset} function as in the second model in Table~\ref{tab:formulas}. This model contains no fixed effects, or more accurately the fixed-effects model matrix, $\bm X$, has zero columns and $\bm\beta$ has length zero. \begin{table}[tb] \centering \begin{tabular}{llP{1.5in}} %% see new column type for ragged right \hline Formula & Alternative & Meaning \\ \hline%------------------------------------------------ \code{(1 | g)} & \code{1 + (1 | g)} & Random intercept with fixed mean \\ \code{0 + offset(o) + (1 | g)} & \code{-1 + offset(o) + (1 | g)} & Random intercept with \emph{a priori} means \\ \code{(1 | g1/g2)} & \code{(1 | g1)+(1 | g1:g2)} % | & Intercept varying among \code{g1} and \code{g2} within \code{g1} \\ \code{(1 | g1)+(1 | g2)} & \code{1 + (1 | g1) + (1 | g2)} & Intercept varying among \code{g1} and \code{g2} \\ \code{x + (x | g)} & \code{1 + x + (1 + x | g)} & Correlated random intercept and slope \\ \code{x + (x || g)} & \code{1 + x + (1 | g) + (0 + x | g)} & Uncorrelated random intercept and slope \\ \hline \end{tabular} \caption{Examples of the right-hand sides of mixed-effects model formulas. The names of grouping factors are denoted \code{g}, \code{g1}, and \code{g2}, and covariates and \emph{a priori} known offsets as \code{x} and \code{o}.} \label{tab:formulas} \end{table} We may also construct models with multiple grouping factors. For example, if the observations are grouped by \code{g2}, which is nested within \code{g1}, then the third formula in Table \ref{tab:formulas} can be used to model variation in the intercept. A common objective in mixed modeling is to account for such nested (or hierarchical) structure. However, one of the most useful aspects of \pkg{lme4} is that it can be used to fit random-effects associated with non-nested grouping factors. For example, suppose the data are grouped by fully crossing two factors, \code{g1} and \code{g2}, then the fourth formula in Table \ref{tab:formulas} may be used. Such models are common in item response theory, where \code{subject} and \code{item} factors are fully crossed \citep{doran2007estimating}. In addition to varying intercepts, we may also have varying slopes (e.g., the \code{sleepstudy} data, Section~\ref{sec:sleepstudy}). The fifth example in Table~\ref{tab:formulas} gives a model where both the intercept and slope vary among the levels of the grouping factor. \subsubsection{Specifying uncorrelated random effects} \label{sec:uncor} By default, \pkg{lme4} assumes that all coefficients associated with the same random-effects term are correlated. To specify an uncorrelated slope and intercept (for example), one may either use double-bar notation, \code{(x || g)}, or equivalently use multiple random effects terms, \code{x + (1 | g) + (0 + x | g)}, as in the final example of Table~\ref{tab:formulas}. For example, from the printout of model \code{fm1} of the \code{sleepstudy} data (Section~\ref{sec:sleepstudy}), we see that the estimated correlation between the slope for \code{Days} and the intercept is fairly low (\Sexpr{round(attr(VarCorr(fm1)$Subject, "correlation")[2],3)}) % $ (See Section~\ref{sec:summary} below for more on how to extract the random effects covariance matrix.) We may use double-bar notation to fit a model that excludes a correlation parameter: <>= fm2 <- lmer(Reaction ~ Days + (Days || Subject), sleepstudy) @ Although mixed models where the random slopes and intercepts are assumed independent are commonly used to reduce the complexity of random-slopes models, they do have one subtle drawback. Models in which the slopes and intercepts are allowed to have a non-zero correlation (e.g., \code{fm1}) are invariant to additive shifts of the continuous predictor (\code{Days} in this case). This invariance breaks down when the correlation is constrained to zero; any shift in the predictor will necessarily lead to a change in the estimated correlation, and in the likelihood and predictions of the model. For example, we can eliminate the correlation in \code{fm1} simply by shifting \code{Days} by an amount equal to the ratio of the estimated among-subject standard deviations multiplied by the estimated correlation (i.e., $\sigma_{\text{\small slope}}/\sigma_{\text{\small intercept}} \cdot \rho_{\text{\small slope:intercept}}$). The use of such models should ideally be restricted to cases where the predictor is measured on a ratio scale (i.e., the zero point on the scale is meaningful, not just a location defined by convenience or convention). %% <>= %% sleepstudyShift <- within(sleepstudy, { %% Days <- Days + (24.74*0.07)/5.92 }) %% lmer(Reaction ~ Days + (Days | Subject), sleepstudyShift) %% @ \subsection{Algebraic and computational account of mixed-model formulas} \label{sec:LMMmatrix} The fixed-effects terms of a mixed-model formula are parsed to produce the fixed-effects model matrix, $\bm X$, in the same way that the \proglang{R} \code{lm} function generates model matrices. However, a mixed-model formula incorporates $k\ge1$ random-effects terms of the form \code{(r | f)} as well. % | These $k$ terms are used to produce the random effects model matrix, $\bm Z$ (Equation~\ref{eq:LMMcondY}; Section~\ref{sec:mkZ}), and the structure of the relative covariance factor, $\bLt$ (Equation~\ref{eq:relcovfac}; Section~\ref{sec:mkLambdat}), which are matrices that typically have a sparse structure. We now describe how one might construct these matrices from the random-effects terms, considering first a single term, \code{(r | f)}, % | and then generalizing to multiple terms. Tables~\ref{tab:dim} and \ref{tab:algebraic} summarize the matrices and vectors that determine the structure of $\bm Z$ and $\bLt$. \begin{table}[tb] \centering \begin{tabular}{lll} \hline Symbol & Size \\ \hline $n$ & Length of the response vector, $\mc{Y}$ \\ $p$ & Number of columns of fixed effects model matrix, $\bm X$ \\ $q = \sum_i^k q_i$ & Number of columns of random effects model matrix, $\bm Z$ \\ $p_i$ & Number of columns of the raw model matrix, $\bm X_i$ \\ $\ell_i$ & Number of levels of the grouping factor indices, $\bm i_i$ \\ $q_i = p_i\ell_i$ & Number of columns of the term-wise model matrix, $\bm Z_i$ \\ $k$ & Number of random effects terms \\ $m_i = \binom{p_i+1}{2}$ & Number of covariance parameters for term $i$ \\ $m = \sum_i^k m_i$ & Total number of covariance parameters \\ \hline \end{tabular} \caption{Dimensions of linear mixed models. The subscript $i = 1, \dots, k$ denotes a specific random effects term.} \label{tab:dim} \end{table} \begin{table}[tb] \centering \begin{tabular}{lll} \hline Symbol & Size & Description \\ \hline $\bm X_i$ & $n\times p_i$ & Raw random effects model matrix \\ $\bm J_i$ & $n\times \ell_i$ & Indicator matrix of grouping factor indices\\ $\bm X_{ij}$ & $p_i\times 1$ & Column vector containing $j$th row of $\bm X_i$ \\ $\bm J_{ij}$ & $\ell_i\times 1$ & Column vector containing $j$th row of $\bm J_i$ \\ $\bm i_i$ & $n$ & Vector of grouping factor indices \\ $\bm Z_i$ & $n\times q_i$ & Term-wise random effects model matrix \\ $\bm\theta$ & $m$ & Covariance parameters \\ $\bm T_i$ & $p_i\times p_i$ & Lower triangular template matrix \\ $\bm\Lambda_i$ & $q_i\times q_i$ & Term-wise relative covariance factor \\ \hline \end{tabular} \caption{Symbols used to describe the structure of the random effects model matrix and the relative covariance factor. The subscript $i = 1, \dots, k$ denotes a specific random effects term.} \label{tab:algebraic} \end{table} The expression, \code{r}, is a linear model formula that evaluates to an \proglang{R} model matrix, $\bm X_i$, of size $n\times p_i$, called the \emph{raw random effects model matrix} for term $i$. A term is said to be a \emph{scalar} random-effects term when $p_i=1$, otherwise it is \emph{vector-valued}. For a \emph{simple, scalar} random-effects term of the form \code{(1 | f)}, $\bm X_i$ is the % | $n\times 1$ matrix of ones, which implies a random intercept model. The expression \code{f} evaluates to an \proglang{R} factor, called the \emph{grouping factor}, for the term. For the $i$th term, we represent this factor mathematically with a vector $\bm i_i$ of \emph{factor indices}, which is an $n$-vector of values from $1,\dots,\ell_i$.\footnote{In practice, fixed-effects model matrices and random-effects terms are evaluated with respect to a \emph{model frame}, ensuring that any expressions for grouping factors have been coerced to factors and any unused levels of these factors have been dropped. That is, $\ell_i$, the number of levels in the grouping factor for the $i$th random-effects term, is well-defined.} Let $\bm J_i$ be the $n\times \ell_i$ matrix of indicator columns for $\bm i_i$. Using the \pkg{Matrix} package \citep{Matrix_pkg} in \proglang{R}, we may construct the transpose of $\bm J_i$ from a factor vector, \code{f}, by coercing \code{f} to a \code{sparseMatrix} object. For example, <>= set.seed(2) @ <>= (f <- gl(3, 2)) (Ji <- t(as(f, Class = "sparseMatrix"))) @ When $k>1$ we order the random-effects terms so that $\ell_1\ge\ell_2\ge\dots\ge\ell_k$; in general, this ordering reduces ``fill-in'' (i.e., the proportion of elements that are zero in the lower triangle of $\bLt\trans\bm Z\trans\bm W\bm Z\bLt+\bm I$ but not in the lower triangle of its left Cholesky factor, $\bm L_{\bm\theta}$, described below in Equation~\ref{eq:blockCholeskyDecomp}). This reduction in fill-in provides more efficient matrix operations within the penalized least squares algorithm (Section~\ref{sec:plsMath}). \subsubsection{Constructing the random effects model matrix} \label{sec:mkZ} The $i$th random effects term contributes $q_i=\ell_ip_i$ columns to the model matrix $\bm Z$. We group these columns into a matrix, $\bm Z_i$, which we refer to as the \emph{term-wise model matrix} for the $i$th term. Thus $q$, the number of columns in $\bm Z$ and the dimension of the random variable, $\mc{B}$, is \begin{equation} \label{eq:qcalc} q=\sum_{i=1}^k q_i = \sum_{i=1}^k \ell_i\,p_i . \end{equation} Creating the matrix $\bm Z_i$ from $\bm X_i$ and $\bm J_i$ is a straightforward concept that is, nonetheless, somewhat awkward to describe. Consider $\bm Z_i$ as being further decomposed into $\ell_i$ blocks of $p_i$ columns. The rows in the first block are the rows of $\bm X_i$ multiplied by the 0/1 values in the first column of $\bm J_i$. Similarly for the subsequent blocks. With these definitions we may define the term-wise random-effects model matrix, $\bm Z_i$, for the $i$th term as a transposed Khatri-Rao product, \begin{equation} \label{eq:Zi} \bm Z_i = (\bm J_i\trans * \bm X_i\trans)\trans = \begin{bmatrix} \bm J_{i1}\trans \otimes \bm X_{i1}\trans \\ \bm J_{i2}\trans \otimes \bm X_{i2}\trans \\ \vdots \\ \bm J_{in}\trans \otimes \bm X_{in}\trans \\ \end{bmatrix} \end{equation} where $*$ and $\otimes$ are the Khatri-Rao\footnote{Note that the original definition of the Khatri-Rao product is more general than the definition used in \pkg{Matrix} package, which is the definition we use here.} \citep{khatri1968solutions} and Kronecker products, and $\bm J_{ij}\trans$ and $\bm X_{ij}\trans$ are row vectors of the $j$th rows of $\bm J_i$ and $\bm X_i$. These rows correspond to the $j$th sample in the response vector, $\mc Y$, and thus $j$ runs from $1 \dots n$. The \pkg{Matrix} package for \proglang{R} contains a \code{KhatriRao} function, which can be used to form $\bm Z_i$. For example, if we begin with a raw model matrix, <>= (Xi <- cbind(1, rep.int(c(-1, 1), 3L))) @ then the term-wise random effects model matrix is, <>= (Zi <- t(KhatriRao(t(Ji), t(Xi)))) @ <>= ## alternative formulation of Zi (eq:Zi) rBind(Ji[1,] %x% Xi[1,], Ji[2,] %x% Xi[2,], Ji[3,] %x% Xi[3,], Ji[4,] %x% Xi[4,], Ji[5,] %x% Xi[5,], Ji[6,] %x% Xi[6,]) @ In particular, for a simple, scalar term, $\bm Z_i$ is exactly $\bm J_i$, the matrix of indicator columns. For other scalar terms, $\bm Z_i$ is formed by element-wise multiplication of the single column of $\bm X_i$ by each of the columns of $\bm J_i$. Because each $\bm Z_i$ is generated from indicator columns, its cross-product, $\bm Z_i\trans\bm Z_i$ is block-diagonal consisting of $\ell_i$ diagonal blocks each of size $p_i$.\footnote{To see this, note that by the properties of Kronecker products we may write the cross-product matrix $Z_i\trans Z_i$ as $\sum_{j=1}^n \bm J_{ij} \bm J_{ij}\trans \otimes \bm X_{ij} \bm X_{ij}\trans$. Because $\bm J_{ij}$ is a unit vector along a coordinate axis, the cross product $\bm J_{ij} \bm J_{ij}\trans$ is a $p_i\times p_i$ matrix of all zeros except for a single $1$ along the diagonal. Therefore, the cross products, $\bm X_{ij} \bm X_{ij}\trans$, will be added to one of the $\ell_i$ blocks of size $p_i\times p_i$ along the diagonal of $Z_i\trans Z_i$.} Note that this means that when $k=1$ (i.e., there is only one random-effects term, and $\bm Z_i = \bm Z$), $\bm Z\trans\bm Z$ will be block diagonal. These block-diagonal properties allow for more efficient sparse matrix computations (Section~\ref{sec:CSCmats}). The full random effects model matrix, $\bm Z$, is constructed from $k\ge 1$ blocks, \begin{equation} \label{eq:Z} \bm Z = \begin{bmatrix} \bm Z_1 & \bm Z_2 & \hdots & \bm Z_k \\ \end{bmatrix} \end{equation} By transposing Equation~\ref{eq:Z} and substituting in Equation~\ref{eq:Zi}, we may represent the structure of the transposed random effects model matrix as follows, \begin{equation} \label{eq:Zt} \bm Z\trans = \begin{blockarray}{ccccc} \text{sample 1} & \text{sample 2} & \hdots & \text{sample n} & \\ \begin{block}{[cccc]c} \bm J_{11} \otimes \bm X_{11} & \bm J_{12} \otimes \bm X_{12} & \hdots & \bm J_{1n} \otimes \bm X_{1n} & \text{term 1} \\ \bm J_{21} \otimes \bm X_{21} & \bm J_{22} \otimes \bm X_{22} & \hdots & \bm J_{2n} \otimes \bm X_{2n} & \text{term 2} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \end{block} \end{blockarray} \end{equation} Note that the proportion of elements of $Z\trans$ that are structural zeros is \begin{equation} \label{eq:ZtSparsity} \frac{\sum_{i=1}^k p_i(\ell_i - 1)}{\sum_{i=1}^k p_i} \qquad . \end{equation} Therefore, the sparsity of $\bm Z\trans$ increases with the number of grouping factor levels. As the number of levels is often large in practice, it is essential for speed and efficiency to take account of this sparsity, for example by using sparse matrix methods, when fitting mixed models (Section~\ref{sec:CSCmats}). \subsubsection{Constructing the relative covariance factor} \label{sec:mkLambdat} The $q\times q$ covariance factor, $\bLt$, is a block diagonal matrix whose $i$th diagonal block, $\bm\Lambda_i$, is of size $q_i,i=1,\dots,k$. We refer to $\bm\Lambda_i$ as the \emph{term-wise relative covariance factor}. Furthermore, $\bm\Lambda_i$ is a homogeneous block diagonal matrix with each of the $\ell_i$ lower-triangular blocks on the diagonal being a copy of a $p_i\times p_i$ lower-triangular \emph{template matrix}, $\bm T_i$. The covariance parameter vector, $\bm\theta$, of length $m_i =\binom{p_i+1}{2}$, consists of the elements in the lower triangle of $\bm T_i,i=1,\dots,k$. To provide a unique representation we require that the diagonal elements of the $\bm T_i,i=1,\dots,k$ be non-negative. The template, $\bm T_i$, can be constructed from the number $p_i$ alone. In \proglang{R} code we denote $p_i$ as \code{nc}. For example, if we set \code{nc <- 3}\Sexpr{nc <- 3}, we could create the template for term $i$ as, <>= nc <- 3 @ %% sequence() is equivalent to unlist(lapply(nvec, seq_len)) %% and (?sequence) ``mainly exists in reverence to the very early history of R'' %% scw: i like sequence, and in fact i never understood why that %% statement is there in the help file. <