## Chapter 10: Multi-level Models, and Repeated Measures
## Corn yield measurements example
library(lattice); library(DAAG)
Site <- with(ant111b, reorder(site, harvwt, FUN=mean))
stripplot(Site ~ harvwt, data=ant111b, scales=list(tck=0.5),
xlab="Harvest weight of corn")
## Sec 10.1: Corn Yield Data --- Analysis Using {aov()}
library(DAAG)
ant111b.aov <- aov(harvwt ~ 1 + Error(site), data=ant111b)
summary(ant111b.aov)
##
## Error: site
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 7 70.3 10.1
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 24 13.9 0.578
## Interpreting the mean squares
## Details of the calculations
## Practical use of the analysis of variance results
## Random effects vs. fixed effects
## Nested factors -- a variety of applications
## ss 10.1.1: A More Formal Approach
## Relations between variance components and mean squares
## Interpretation of variance components
## Intra-class correlation
## Sec 10.2: Analysis using {lmer()}, from the { lme4} package
library(lme4)
## Loading required package: Matrix
## Loading required package: Rcpp
ant111b.lmer <- lmer(harvwt ~ 1 + (1 | site), data=ant111b)
## Note that there is no degrees of freedom information.
print(ant111b.lmer, ranef.comp="Variance", digits=3)
## Linear mixed model fit by REML ['lmerMod']
## Formula: harvwt ~ 1 + (1 | site)
## Data: ant111b
## REML criterion at convergence: 94.42
## Random effects:
## Groups Name Variance
## site (Intercept) 2.368
## Residual 0.578
## Number of obs: 32, groups: site, 8
## Fixed Effects:
## (Intercept)
## 4.29
## The processing of output from {lmer()}
print(coef(summary(ant111b.lmer)), digits=3)
## Estimate Std. Error t value
## (Intercept) 4.29 0.56 7.66
## Fitted values and residuals in {lmer()}
s2W <- 0.578; s2L <- 2.37; n <- 4
sitemeans <- with(ant111b, sapply(split(harvwt, site), mean))
grandmean <- mean(sitemeans)
shrinkage <- (n*s2L)/(n*s2L+s2W)
grandmean + shrinkage*(sitemeans - grandmean)
## DBAN LFAN NSAN ORAN OVAN TEAN WEAN WLAN
## 4.851 4.212 2.217 6.764 4.801 3.108 5.455 2.925
##
## More directly, use fitted() with the lmer object
unique(fitted(ant111b.lmer))
## [1] 4.851 4.212 2.217 6.764 4.801 3.108 5.455 2.925
##
## Compare with site means
sitemeans
## DBAN LFAN NSAN ORAN OVAN TEAN WEAN WLAN
## 4.885 4.207 2.090 6.915 4.833 3.036 5.526 2.841
## *Uncertainty in the parameter estimates --- profile likelihood and alternatives
prof.lmer <- profile(ant111b.lmer)
CI95 <- confint(prof.lmer, level=0.95)
rbind("sigmaL^2"=CI95[1,]^2, "sigma^2"=CI95[2,]^2)
## 2.5 % 97.5 %
## sigmaL^2 0.7965 6.936
## sigma^2 0.3444 1.079
CI95[3,]
## 2.5 % 97.5 %
## 3.128 5.456
library(lattice)
print(xyplot(prof.lmer, conf=c(50, 80, 95, 99)/100,
aspect=0.8, between=list(x=0.35)))
## Handling more than two levels of random variation
## Sec 10.3: Survey Data, with Clustering
## Footnote Code
## Means of like (data frame science: DAAG), by class
classmeans <- with(science,
aggregate(like, by=list(PrivPub, Class), mean) )
# NB: Class identifies classes independently of schools
# class identifies classes within schools
names(classmeans) <- c("PrivPub", "Class", "avlike")
with(classmeans, {
## Boxplots: class means by Private or Public school
boxplot(split(avlike, PrivPub), horizontal=TRUE, las=2,
xlab = "Class average of score", boxwex = 0.4)
rug(avlike[PrivPub == "private"], side = 1)
rug(avlike[PrivPub == "public"], side = 3)
})
## ss 10.3.1: Alternative models
science.lmer <- lmer(like ~ sex + PrivPub + (1 | school) +
(1 | school:class), data = science,
na.action=na.exclude)
print(VarCorr(science.lmer), comp="Variance", digits=3)
## Groups Name Variance
## school:class (Intercept) 0.321
## school (Intercept) 0.000
## Residual 3.052
print(coef(summary(science.lmer)), digits=2)
## Estimate Std. Error t value
## (Intercept) 4.72 0.162 29.1
## sexm 0.18 0.098 1.9
## PrivPubpublic 0.41 0.186 2.2
summary(science.lmer)$ngrps
## school:class school
## 66 41
science1.lmer <- lmer(like ~ sex + PrivPub + (1 | school:class),
data = science, na.action=na.exclude)
print(VarCorr(science1.lmer), comp="Variance", digits=3)
## Groups Name Variance
## school:class (Intercept) 0.321
## Residual 3.052
print(coef(summary(science1.lmer)), digits=2)
## Estimate Std. Error t value
## (Intercept) 4.72 0.162 29.1
## sexm 0.18 0.098 1.9
## PrivPubpublic 0.41 0.186 2.2
library(afex)
## Loading required package: car
##
## Attaching package: 'car'
##
## The following object is masked from 'package:DAAG':
##
## vif
##
## Loading required package: pbkrtest
## Loading required package: MASS
##
## Attaching package: 'MASS'
##
## The following object is masked from 'package:DAAG':
##
## hills
##
## Loading required package: parallel
## Loading required package: reshape2
## ************
## Welcome to afex. Important notes:
##
## Due to popular demand, afex doesn't change the contrasts globally anymore.
## To set contrasts globally to contr.sum run set_sum_contrasts().
## To set contrasts globally to the default (treatment) contrasts run set_default_contrasts().
##
## All afex functions are unaffected by global contrasts and use contr.sum as long as check.contr = TRUE (which is the default).
## ************
mixed(like ~ sex + PrivPub + (1 | school:class),
data = na.omit(science), method="KR")
## Contrasts set to contr.sum for the following variables: sex, PrivPub, school, class
## Fitting 3 (g)lmer() models:
## [...]
## Obtaining 2 p-values:
## [..]
## Effect F ndf ddf F.scaling p.value
## 1 sex 3.44 1 1379.49 1.00 .06
## 2 PrivPub 4.91 1 60.44 1.00 .03
## More detailed examination of the output
## Use profile likelihood
pp <- profile(science1.lmer, which="theta_")
# which="theta_": all random parameters
# which="beta_": fixed effect parameters
var95 <- confint(pp, level=0.95)^2
# Square to get variances in place of SDs
rownames(var95) <- c("sigma_Class^2", "sigma^2")
signif(var95, 3)
## 2.5 % 97.5 %
## sigma_Class^2 0.178 0.511
## sigma^2 2.830 3.300
science1.lmer <- lmer(like ~ sex + PrivPub + (1 | school:class),
data = science, na.action=na.omit)
ranf <- ranef(obj = science1.lmer, drop=TRUE)[["school:class"]]
flist <- science1.lmer@flist[["school:class"]]
privpub <- science[match(names(ranf), flist), "PrivPub"]
num <- unclass(table(flist)); numlabs <- pretty(num)
opar <- par(mfrow=c(2,2), pty="s", mgp=c(2.25,0.5,0), mar=c(3.6,3.6,2.1, 0.6))
## Panel A: Plot effect estimates vs numbers
plot(sqrt(num), ranf, xaxt="n", pch=c(1,3)[as.numeric(privpub)],
xlab="# in class (square root scale)",
ylab="Estimate of class effect")
lines(lowess(sqrt(num[privpub=="private"]),
ranf[privpub=="private"], f=1.1), lty=2)
lines(lowess(sqrt(num[privpub=="public"]),
ranf[privpub=="public"], f=1.1), lty=3)
axis(1, at=sqrt(numlabs), labels=paste(numlabs))
res <- residuals(science1.lmer)
vars <- tapply(res, INDEX=list(flist), FUN=var)*(num-1)/(num-2)
## Panel B: Within class variance estimates vs numbers
plot(sqrt(num), vars, pch=c(1,3)[unclass(privpub)])
lines(lowess(sqrt(num[privpub=="private"]),
as.vector(vars)[privpub=="private"], f=1.1), lty=2)
lines(lowess(sqrt(num[privpub=="public"]),
as.vector(vars)[privpub=="public"], f=1.1), lty=3)
## Panel C: Normal probability plot of site effects
qqnorm(ranf, ylab="Ordered site effects", main="")
## Panel D: Normal probability plot of residuals
qqnorm(res, ylab="Ordered w/i class residuals", main="")
par(opar)
## ss 10.3.2: Instructive, though faulty, analyses
## Ignoring class as the random effect
science2.lmer <- lmer(like ~ sex + PrivPub + (1 | school),
data = science, na.action=na.exclude)
science2.lmer
## Linear mixed model fit by REML ['lmerMod']
## Formula: like ~ sex + PrivPub + (1 | school)
## Data: science
## REML criterion at convergence: 5584
## Random effects:
## Groups Name Std.Dev.
## school (Intercept) 0.407
## Residual 1.794
## Number of obs: 1383, groups: school, 41
## Fixed Effects:
## (Intercept) sexm PrivPubpublic
## 4.738 0.197 0.417
## Footnote Code
## Ignoring the random structure in the data
## Faulty analysis, using lm
science.lm <- lm(like ~ sex + PrivPub, data=science)
summary(science.lm)$coef
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.7402 0.09955 47.616 1.545e-293
## sexm 0.1509 0.09860 1.531 1.261e-01
## PrivPubpublic 0.3951 0.10511 3.759 1.779e-04
## ss 10.3.3: Predictive accuracy
## Sec 10.4: A Multi-level Experimental Design
## ss 10.4.1: The anova table
## Analysis of variance: data frame kiwishade (DAAG)
kiwishade.aov <- aov(yield ~ shade + Error(block/shade),
data=kiwishade)
summary(kiwishade.aov)
##
## Error: block
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 2 172 86.2
##
## Error: block:shade
## Df Sum Sq Mean Sq F value Pr(>F)
## shade 3 1395 465 22.2 0.0012
## Residuals 6 126 21
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 36 439 12.2
## ss 10.4.2: Expected values of mean squares
model.tables(kiwishade.aov, type="means")
## Tables of means
## Grand mean
##
## 96.53
##
## shade
## shade
## none Aug2Dec Dec2Feb Feb2May
## 100.20 103.23 89.92 92.77
## Footnote Code
## Calculate treatment means
with(kiwishade, sapply(split(yield, shade), mean))
## none Aug2Dec Dec2Feb Feb2May
## 100.20 103.23 89.92 92.77
## ss 10.4.3: * The analysis of variance sums of squares breakdown
## Footnote Code
## For each plot, calculate mean, and differences from the mean
vine <- paste("vine", rep(1:4, 12), sep="")
vine1rows <- seq(from=1, to=45, by=4)
kiwivines <- unstack(kiwishade, yield ~ vine)
kiwimeans <- apply(kiwivines, 1, mean)
kiwivines <- cbind(kiwishade[vine1rows, c("block","shade")],
Mean=kiwimeans, kiwivines-kiwimeans)
kiwivines <- with(kiwivines, kiwivines[order(block, shade), ])
mean(kiwimeans) # the grand mean
## [1] 96.53
## ss 10.4.4: The variance components
## ss 10.4.5: The mixed model analysis
kiwishade.lmer <- lmer(yield ~ shade + (1|block) + (1|block:plot),
data=kiwishade)
# block:shade is an alternative to block:plot
print(kiwishade.lmer, ranef.comp="Variance", digits=3)
## Linear mixed model fit by REML ['lmerMod']
## Formula: yield ~ shade + (1 | block) + (1 | block:plot)
## Data: kiwishade
## REML criterion at convergence: 252
## Random effects:
## Groups Name Variance
## block:plot (Intercept) 2.19
## block (Intercept) 4.08
## Residual 12.18
## Number of obs: 48, groups: block:plot, 12; block, 3
## Fixed Effects:
## (Intercept) shadeAug2Dec shadeDec2Feb shadeFeb2May
## 100.20 3.03 -10.28 -7.43
## Residuals and estimated effects
## Footnote Code
## Simplified version of plot
xyplot(residuals(kiwishade.lmer) ~ fitted(kiwishade.lmer)|block, data=kiwishade,
groups=shade, layout=c(3,1), par.strip.text=list(cex=1.0),
xlab="Fitted values (Treatment + block + plot effects)",
ylab="Residuals", pch=1:4, grid=TRUE, aspect=1,
scales=list(x=list(alternating=FALSE), tck=0.5),
key=list(space="top", points=list(pch=1:4),
text=list(labels=levels(kiwishade$shade)),columns=4))
## Footnote Code
## Simplified version of graph that shows the plot effects
ploteff <- ranef(kiwishade.lmer, drop=TRUE)[[1]]
qqmath(ploteff, xlab="Normal quantiles", ylab="Plot effect estimates",
aspect=1, scales=list(tck=0.5))
## Footnote Code
## Overlaid normal probability plots of 2 sets of simulated effects
## To do more simulations, change nsim as required, and re-execute
simvals <- simulate(kiwishade.lmer, nsim=2)
simeff <- apply(simvals, 2, function(y) scale(ranef(refit(kiwishade.lmer, y),
drop=TRUE)[[1]]))
simeff <- data.frame(v1=simeff[,1], v2=simeff[,2])
qqmath(~ v1+v2, data=simeff, xlab="Normal quantiles",
ylab="Simulated plot effects\n(2 sets, standardized)",
scales=list(tck=0.5), aspect=1)
## ss 10.4.6: Predictive accuracy
## Sec 10.5: Within and Between Subject Effects
## Model fitting criteria
## ss 10.5.1: Model selection
## Change initial letters of levels of tinting$agegp to upper case
library(R.utils)
## Loading required package: R.oo
## Loading required package: R.methodsS3
## R.methodsS3 v1.6.1 (2014-01-04) successfully loaded. See ?R.methodsS3 for help.
## R.oo v1.18.0 (2014-02-22) successfully loaded. See ?R.oo for help.
##
## Attaching package: 'R.oo'
##
## The following objects are masked from 'package:methods':
##
## getClasses, getMethods
##
## The following objects are masked from 'package:base':
##
## attach, detach, gc, load, save
##
## R.utils v1.32.4 (2014-05-14) successfully loaded. See ?R.utils for help.
##
## Attaching package: 'R.utils'
##
## The following object is masked from 'package:utils':
##
## timestamp
##
## The following objects are masked from 'package:base':
##
## cat, commandArgs, getOption, inherits, isOpen, parse, warnings
levels(tinting$agegp) <- capitalize(levels(tinting$agegp))
## Fit all interactions: data frame tinting (DAAG)
it3.lmer <- lmer(log(it) ~ tint*target*agegp*sex + (1 | id),
data=tinting, REML=FALSE)
it2.lmer <- lmer(log(it) ~ (tint+target+agegp+sex)^2 + (1 | id),
data=tinting, REML=FALSE)
it1.lmer <- lmer(log(it)~(tint+target+agegp+sex) + (1 | id),
data=tinting, REML=FALSE)
anova(it1.lmer, it2.lmer, it3.lmer)
## Data: tinting
## Models:
## it1.lmer: log(it) ~ (tint + target + agegp + sex) + (1 | id)
## it2.lmer: log(it) ~ (tint + target + agegp + sex)^2 + (1 | id)
## it3.lmer: log(it) ~ tint * target * agegp * sex + (1 | id)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## it1.lmer 8 1.14 26.8 7.43 -14.9
## it2.lmer 17 -3.74 50.7 18.87 -37.7 22.88 9 0.0065
## it3.lmer 26 8.15 91.5 21.93 -43.9 6.11 9 0.7288
## Footnote Code
## Code that gives the first four such plots, for the observed data
opar <- par(mfrow=c(2,2), pty="s", mgp=c(2.25,0.5,0), mar=c(3.6,3.6,2.1, 0.6))
interaction.plot(tinting$tint, tinting$agegp, log(tinting$it))
interaction.plot(tinting$target, tinting$sex, log(tinting$it))
interaction.plot(tinting$tint, tinting$target, log(tinting$it))
interaction.plot(tinting$tint, tinting$sex, log(tinting$it))
par(opar)
## ss 10.5.2: Estimates of model parameters
it2.reml <- update(it2.lmer, REML=TRUE)
print(coef(summary(it2.reml)), digits=2)
## Estimate Std. Error t value
## (Intercept) 3.6191 0.130 27.82
## tint.L 0.1609 0.044 3.64
## tint.Q 0.0210 0.045 0.46
## targethicon -0.1181 0.042 -2.79
## agegpOlder 0.4712 0.233 2.02
## sexm 0.0821 0.233 0.35
## tint.L:targethicon -0.0919 0.046 -2.00
## tint.Q:targethicon -0.0072 0.048 -0.15
## tint.L:agegpOlder 0.1308 0.049 2.66
## tint.Q:agegpOlder 0.0697 0.052 1.34
## tint.L:sexm -0.0979 0.049 -1.99
## tint.Q:sexm 0.0054 0.052 0.10
## targethicon:agegpOlder -0.1389 0.058 -2.38
## targethicon:sexm 0.0779 0.058 1.33
## agegpOlder:sexm 0.3316 0.326 1.02
# NB: The final column in the text, giving degrees of freedom, is not in the
# summary output. It is our addition.
## Footnote Code
## Sec 10.6: A Generalized Linear Mixed Model
moths$transect <- 1:41 # Each row is from a different transect
moths$habitat <- relevel(moths$habitat, ref="Lowerside")
A.glmer <- glmer(A~habitat+sqrt(meters)+(1|transect),
family=poisson(link=sqrt), data=moths)
print(summary(A.glmer), show.resid=FALSE, correlation=FALSE)
## Generalized linear mixed model fit by maximum likelihood (Laplace
## Approximation) [glmerMod]
## Family: poisson ( sqrt )
## Formula: A ~ habitat + sqrt(meters) + (1 | transect)
## Data: moths
##
## AIC BIC logLik deviance df.resid
## 212.6 229.7 -96.3 192.6 31
##
## Random effects:
## Groups Name Variance Std.Dev.
## transect (Intercept) 0.319 0.564
## Number of obs: 41, groups: transect, 41
##
## Fixed effects:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 1.7322 0.3513 4.93 8.2e-07
## habitatBank -2.0415 0.9377 -2.18 0.029
## habitatDisturbed -1.0359 0.4071 -2.54 0.011
## habitatNEsoak -0.7319 0.4323 -1.69 0.090
## habitatNWsoak 2.6787 0.5101 5.25 1.5e-07
## habitatSEsoak 0.1178 0.3923 0.30 0.764
## habitatSWsoak 0.3900 0.5260 0.74 0.458
## habitatUpperside -0.3135 0.7549 -0.42 0.678
## sqrt(meters) 0.0675 0.0631 1.07 0.285
## Footnote Code
## Mixed models with a binomial error and logit link
## Sec 10.7: Repeated Measures in Time
## The theory of repeated measures modeling
## *Correlation structure
## Different approaches to repeated measures analysis
## ss 10.7.1: Example -- random variation between profiles
## Footnote Code
## Plot points and fitted lines (panel A)
library(lattice)
xyplot(o2 ~ wattsPerKg, groups=id, data=humanpower1,
panel=function(x,y,subscripts,groups,...){
yhat <- fitted(lm(y ~ groups*x))
panel.superpose(x, y, subscripts, groups, pch=1:5)
panel.superpose(x, yhat, subscripts, groups, type="l")
},
xlab="Watts per kilogram",
ylab=expression("Oxygen intake ("*ml.min^{-1}*.kg^{-1}*")"))
## Separate lines for different athletes
## Calculate intercepts and slopes; plot Slopes vs Intercepts
## Uses the function lmList() from the lme4 package
library(lme4)
hp.lmList <- lmList(o2 ~ wattsPerKg | id, data=humanpower1)
coefs <- coef(hp.lmList)
names(coefs) <- c("Intercept", "Slope")
plot(Slope ~ Intercept, data=coefs)
abline(lm(Slope~Intercept, data=coefs))
## A random coefficients model
hp.lmer <- lmer(o2 ~ wattsPerKg + (wattsPerKg | id),
data=humanpower1)
print(summary(hp.lmer), digits=3)
## Linear mixed model fit by REML ['lmerMod']
## Formula: o2 ~ wattsPerKg + (wattsPerKg | id)
## Data: humanpower1
##
## REML criterion at convergence: 124.2
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.9947 -0.4878 -0.0835 0.4616 2.1212
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## id (Intercept) 50.73 7.12
## wattsPerKg 7.15 2.67 -1.00
## Residual 4.13 2.03
## Number of obs: 28, groups: id, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 2.09 3.78 0.55
## wattsPerKg 13.91 1.36 10.23
##
## Correlation of Fixed Effects:
## (Intr)
## wattsPerKg -0.992
hat <- fitted(hp.lmer)
lmhat <- with(humanpower1, fitted(lm(o2 ~ id*wattsPerKg)))
panelfun <-
function(x, y, subscripts, groups, ...){
panel.superpose(x, hat, subscripts, groups, type="l",lty=2)
panel.superpose(x, lmhat, subscripts, groups, type="l",lty=1)
}
xyplot(o2 ~ wattsPerKg, groups=id, data=humanpower1, panel=panelfun,
xlab="Watts",
ylab=expression("Oxygen intake ("*ml.min^{-1}*"."*kg^{-1}*")"))
## Footnote Code
## Plot of residuals
xyplot(resid(hp.lmer) ~ wattsPerKg, groups=id, type="b", data=humanpower1)
## Footnote Code
## Derive the sd from the data frame coefs that was calculated above
sd(coefs$Slope)
## [1] 3.278
## ss 10.7.2: Orthodontic measurements on chlldren
## Preliminary data exploration
## Footnote Code
## Plot showing pattern of change for each of the 25 individuals
library(MEMSS)
##
## Attaching package: 'MEMSS'
##
## The following objects are masked from 'package:datasets':
##
## CO2, Orange, Theoph
xyplot(distance ~ age | Subject, groups=Sex, data=Orthodont,
scales=list(y=list(log=2)), type=c("p","r"), layout=c(11,3))
## Use lmList() to find the slopes
ab <- coef(lmList(distance ~ age | Subject, Orthodont))
names(ab) <- c("a", "b")
## Obtain the intercept at x=mean(x)
## (For each subject, this is independent of the slope)
ab$ybar <- ab$a + ab$b*11 # mean age is 11, for each subject.
sex <- substring(rownames(ab), 1 ,1)
plot(ab[, 3], ab[, 2], col=c(F="gray40", M="black")[sex],
pch=c(F=1, M=3)[sex], xlab="Intercept", ylab="Slope")
extremes <- ab$ybar %in% range(ab$ybar) |
ab$b %in% range(ab$b[sex=="M"]) |
ab$b %in% range(ab$b[sex=="F"])
text(ab[extremes, 3], ab[extremes, 2], rownames(ab)[extremes], pos=4, xpd=TRUE)
## The following makes clear M13's difference from other points
opar <- par(mfrow=c(1,2), pty="s", mgp=c(2.25,0.5,0), mar=c(4.1,3.6,2.1, 0.6))
qqnorm(ab$b, main="QQ -- Slope in lm(dist~age)")
Orthodont$logdist <- log(Orthodont$distance)
## Now repeat, with logdist replacing distance
ablog <- coef(lmList(logdist ~ age | Subject, Orthodont))
names(ablog) <- c("a", "b")
qqnorm(ablog$b, main="QQ -- Slope in lm(logist~age)")
par(opar)
## Footnote Code
## Compare males slopes with female slopes
Orthodont$logdist <- log(Orthodont$distance)
ablog <- coef(lmList(logdist ~ age | Subject, Orthodont))
names(ablog) <- c("a", "b")
## Obtain the intercept at mean age (= 11), for each subject
## (For each subject, this is independent of the slope)
ablog$ybar <- with(ablog, a + b*11)
extreme.males <- rownames(ablog) %in% c("M04","M13")
sex <- substring(rownames(ab), 1, 1)
with(ablog,
t.test(b[sex=="F"], b[sex=="M" & !extreme.males], var.equal=TRUE))
##
## Two Sample t-test
##
## data: b[sex == "F"] and b[sex == "M" & !extreme.males]
## t = -2.32, df = 23, p-value = 0.02957
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.0160529 -0.0009191
## sample estimates:
## mean of x mean of y
## 0.02115 0.02963
# Specify var.equal=TRUE, to allow comparison with anova output
## A random coefficients model
keep <- !(Orthodont$Subject%in%c("M04","M13"))
orthdiff.lmer <- lmer(logdist ~ Sex * I(age-11) + (I(age-11) | Subject),
data=Orthodont, subset=keep, REML=FALSE)
print(summary(orthdiff.lmer), digits=3)
## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: logdist ~ Sex * I(age - 11) + (I(age - 11) | Subject)
## Data: Orthodont
## Subset: keep
##
## AIC BIC logLik deviance df.resid
## -247.1 -226.2 131.5 -263.1 92
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.370 -0.482 0.004 0.534 3.993
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Subject (Intercept) 5.79e-03 0.076124
## I(age - 11) 7.71e-07 0.000878 1.00
## Residual 2.31e-03 0.048109
## Number of obs: 100, groups: Subject, 25
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 3.11451 0.02407 129.4
## SexMale 0.09443 0.03217 2.9
## I(age - 11) 0.02115 0.00325 6.5
## SexMale:I(age - 11) 0.00849 0.00435 2.0
##
## Correlation of Fixed Effects:
## (Intr) SexMal I(-11)
## SexMale -0.748
## I(age - 11) 0.078 -0.058
## SxMl:I(-11) -0.058 0.078 -0.748
orthsame.lmer <- lmer(logdist ~ Sex + I(age-11) + (I(age-11) | Subject),
data=Orthodont, subset=keep, REML=FALSE)
print(anova(orthsame.lmer, orthdiff.lmer)[2, "Pr(>Chisq)"], digits=3)
## [1] 0.054
orthdiffr.lmer <- update(orthdiff.lmer, REML=TRUE)
print(summary(orthdiffr.lmer), digits=3)
## Linear mixed model fit by REML ['lmerMod']
## Formula: logdist ~ Sex * I(age - 11) + (I(age - 11) | Subject)
## Data: Orthodont
## Subset: keep
##
## REML criterion at convergence: -232.2
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.325 -0.477 0.001 0.522 3.939
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Subject (Intercept) 6.33e-03 0.079581
## I(age - 11) 8.42e-07 0.000918 1.00
## Residual 2.38e-03 0.048764
## Number of obs: 100, groups: Subject, 25
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 3.11451 0.02510 124.1
## SexMale 0.09443 0.03354 2.8
## I(age - 11) 0.02115 0.00330 6.4
## SexMale:I(age - 11) 0.00849 0.00441 1.9
##
## Correlation of Fixed Effects:
## (Intr) SexMal I(-11)
## SexMale -0.748
## I(age - 11) 0.080 -0.060
## SxMl:I(-11) -0.060 0.080 -0.748
## Correlation between successive times
res <- resid(orthdiffr.lmer)
Subject <- factor(Orthodont$Subject[keep])
orth.acf <- t(sapply(split(res, Subject),
function(x)acf(x, lag=4, plot=FALSE)$acf))
## Calculate respective proportions of Subjects for which
## autocorrelations at lags 1, 2 and 3 are greater than zero.
apply(orth.acf[,-1], 2, function(x)sum(x>0)/length(x))
## [1] 0.20 0.32 0.36
## *The variance for the difference in slopes
## Sec 10.8: Further Notes on Multi-level and Other Models with CorrelatedErrors
## ss 10.8.1: Different sources of variance -- complication or focus of interest?
## ss 10.8.2: Predictions from models with a complex error structure
## Consequences from assuming an overly simplistic error structure
## ss 10.8.3: An historical perspective on multi-level models
## ss 10.8.4: Meta-analysis
## ss 10.8.5: Functional data analysis
## ss 10.8.6: Error structure in explanatory variables
## Sec 10.9: Recap
## Sec 10.10: Further Reading
## References
## References for further reading
## Analysis of variance with multiple error terms
## Multi-level models and repeated measures
## Meta-analysis
## References
## References for further reading
n.omit <- 2
take <- rep(TRUE, 48)
take[sample(1:48,2)] <- FALSE
kiwishade.lmer <- lmer(yield ~ shade + (1|block) + (1|block:plot),
data = kiwishade,subset=take)
vcov <- VarCorr(kiwishade.lmer)
print(vcov, comp="Variance")
## Groups Name Variance
## block:plot (Intercept) 2.42
## block (Intercept) 4.00
## Residual 12.42
cult.lmer <- lmer(ct ~ Cultivar + Dose + factor(year) +
(-1 + Dose | gp), data = sorption,
REML=TRUE)
cultdose.lmer <- lmer(ct ~ Cultivar/Dose + factor(year) +
(-1 + Dose | gp), data = sorption,
REML=TRUE)