``````## Chapter 10: Multi-level Models, and Repeated Measures

## Sec 10.1: A One-Way Random Effects Model
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") ``````

``````## ss 10.1.1: Analysis with {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 10.1.2: A More Formal Approach
##       Relations between variance components and mean squares
##               Interpretation of variance components
##                      Intra-class correlation
## ss 10.1.3: Analysis using {lmer()}
library(lme4) ``````
``````## Loading required package: Matrix
``````ant111b.lmer <- lmer(harvwt ~ 1 + (1 | site), data=ant111b)

## Output is from version 0.999375-28 of lmer
## Note that there is no degrees of freedom information.
ant111b.lmer ``````
``````## Linear mixed model fit by REML ['lmerMod']
## Formula: harvwt ~ 1 + (1 | site)
##    Data: ant111b
## REML criterion at convergence: 94.42
## Random effects:
##  Groups   Name        Std.Dev.
##  site     (Intercept) 1.54
##  Residual             0.76
## Number of obs: 32, groups:  site, 8
## Fixed Effects:
## (Intercept)
##        4.29``````
``````##              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
CI95 <- confint(profile(ant111b.lmer), level=0.95)
rbind("sigmaL^2"=CI95[1,]^2, "sigma^2"=exp(CI95[2,])^2,
"(Intercept)"=CI95[3,]) ``````
``````##              2.5 % 97.5 %
## sigmaL^2    0.7965  6.936
## sigma^2     3.2337  7.981
## (Intercept) 3.1277  5.456``````
``````##         Handling more than two levels of random variation
## house.lmer <- lmer(price ~ 1 + (1|city) + (1|city:suburb))

## Sec 10.2: 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")
attach(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) ``````

``````detach(classmeans)

## ss 10.2.1: Alternative models
science.lmer <- lmer(like ~ sex + PrivPub + (1 | school) +
(1 | school:class), data = science,
na.action=na.exclude)
print(summary(science.lmer), show.resid=FALSE) ``````
``````## Linear mixed model fit by REML ['lmerMod']
## Formula: like ~ sex + PrivPub + (1 | school) + (1 | school:class)
##    Data: science
##
## REML criterion at convergence: 5546
##
## Random effects:
##  Groups       Name        Variance Std.Dev.
##  school:class (Intercept) 0.321    0.566
##  school       (Intercept) 0.000    0.000
##  Residual                 3.052    1.747
## Number of obs: 1383, groups:  school:class, 66; school, 41
##
## Fixed effects:
##               Estimate Std. Error t value
## (Intercept)     4.7218     0.1624   29.07
## sexm            0.1823     0.0982    1.86
## PrivPubpublic   0.4117     0.1857    2.22
##
## Correlation of Fixed Effects:
##             (Intr) sexm
## sexm        -0.309
## PrivPubpblc -0.795  0.012``````
``````## Footnote Code
## The variances are included in the output from VarCorr()
VarCorr(science.lmer)  # Displayed output differs slightly ``````
``````##  Groups       Name        Std.Dev.
##  school:class (Intercept) 0.566
##  school       (Intercept) 0.000
##  Residual                 1.747``````
`````` # The between students (Residual) component of variance is
# attr(VarCorr(science.lmer),"sc")^2

science1.lmer <- lmer(like ~ sex + PrivPub + (1 | school:class),
data = science, na.action=na.exclude)
print(summary(science1.lmer), show.resid=FALSE) ``````
``````## Linear mixed model fit by REML ['lmerMod']
## Formula: like ~ sex + PrivPub + (1 | school:class)
##    Data: science
##
## REML criterion at convergence: 5546
##
## Random effects:
##  Groups       Name        Variance Std.Dev.
##  school:class (Intercept) 0.321    0.566
##  Residual                 3.052    1.747
## Number of obs: 1383, groups:  school:class, 66
##
## Fixed effects:
##               Estimate Std. Error t value
## (Intercept)     4.7218     0.1624   29.07
## sexm            0.1823     0.0982    1.86
## PrivPubpublic   0.4117     0.1857    2.22
##
## Correlation of Fixed Effects:
##             (Intr) sexm
## sexm        -0.309
## PrivPubpblc -0.795  0.012``````
``````# mcmcsamp() did noi work reliably, & is no longer available
# set.seed(41)
# science1.mcmc <- mcmcsamp(science1.lmer, n=1000)
# samps <- VarCorr(science1.mcmc, type="varcov")
# colnames(samps) <- c("sigma_Class^2", "sigma^2")
# signif(HPDinterval(samps, prob=0.95), 3)

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)
par(mfrow=c(2,2), pty="s")
## 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)
## Within plot 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)
## Normal probability plot of site effects
qqnorm(ranf, ylab="Ordered site effects", main="")
## Normal probability plot of residuals
qqnorm(res, ylab="Ordered w/i class residuals", main="") ``````

``````par(mfrow=c(1,1), pty="m")

## ss 10.2.2: 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
## The numerical values can be extracted from
VarCorr(science2.lmer)  # The within schools (Residual) component of variance ``````
``````##  Groups   Name        Std.Dev.
##  school   (Intercept) 0.407
##  Residual             1.794``````
``````                        # is the square of the scale parameter

##             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 10.2.3: Predictive accuracy

## Sec 10.3: A Multi-level Experimental Design
## ss 10.3.1: The anova table
## Analysis of variance: data frame kiwishade (DAAG)

``````##
## Error: block
##           Df Sum Sq Mean Sq F value Pr(>F)
## Residuals  2    172    86.2
##
##           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 10.3.2: Expected values of mean squares
``````## Tables of means
## Grand mean
##
## 96.53
##
##    none Aug2Dec Dec2Feb Feb2May
##  100.20  103.23   89.92   92.77``````
``````## Footnote Code
## Calculate treatment means
``````##    none Aug2Dec Dec2Feb Feb2May
##  100.20  103.23   89.92   92.77``````
``````## ss 10.3.3: * 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)
Mean=kiwimeans, kiwivines-kiwimeans)
kiwivines <- with(kiwivines, kiwivines[order(block, shade), ])
mean(kiwimeans)      # the grand mean ``````
``## [1] 96.53``
``````## ss 10.3.4: The variance components
## ss 10.3.5: The mixed model analysis
# block:shade is an alternative to block:plot

``````## Linear mixed model fit by REML ['lmerMod']
## Formula: yield ~ shade + (1 | block) + (1 | block:plot)
## REML criterion at convergence: 252
## Random effects:
##  Groups     Name        Std.Dev.
##  block:plot (Intercept) 1.48
##  block      (Intercept) 2.02
##  Residual               3.49
## Number of obs: 48, groups:  block:plot, 12; block, 3
## Fixed Effects:
##       100.20          3.03        -10.28         -7.43``````
``````##                  Residuals and estimated effects
## Footnote Code
## Simplified version of plot
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),

``````## Footnote Code
## Simplified version of graph that shows the plot effects
qqmath(ploteff, xlab="Normal quantiles", ylab="Plot effect estimates",
scales=list(tck=0.5), aspect=1) ``````

``````## Footnote Code
## Overlaid normal probability plots of 2 sets of simulated effects
## To do more simulations, change nsim as required, and re-execute
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 10.3.6: Predictive accuracy

## Sec 10.4: Within and Between Subject Effects
##                       Model fitting criteria
## ss 10.4.1: Model selection
## Change initial letters of levels of tinting\$agegp to upper case
library(R.utils)``````
``````## Loading required package: R.oo
## 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
par(mfrow=c(2,2))
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)) ``````