Easy Anova with R
Christophe Pallier
2012-08-25
This document provides a few examples of Analyses of Variance for typical experimental designs. We recommend to use of the ez package to perform ANOVAs with R.
require(ez)## Loading required package: ezrequire(lattice)## Loading required package: latticerequire(ggplot2)## Loading required package: ggplot2S<G> Design (one-way ANOVA between subjects)
We start by analyzing a design where the factor Subject is nested within the factor Group, that is, we want to compare independent groups.
ns <- c(20, 30, 20, 15)
means <- c(100, 110, 100, 130)
sd <- c(20, 20, 30, 25)
dat1 <- data.frame(
subj=factor(1:sum(ns)),
group=factor(rep(1:4, ns)),
y=c(rnorm(ns[1], means[1], sd[1]),
rnorm(ns[2], means[2], sd[2]),
rnorm(ns[3], means[3], sd[3]),
rnorm(ns[4], means[4], sd[4])))Let us inspect the data:
str(dat1)## 'data.frame': 85 obs. of 3 variables:
## $ subj : Factor w/ 85 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ group: Factor w/ 4 levels "1","2","3","4": 1 1 1 1 1 1 1 1 1 1 ...
## $ y : num 75.9 108.9 148.1 80.3 63.4 ...head(dat1)## subj group y
## 1 1 1 75.85807
## 2 2 1 108.91287
## 3 3 1 148.09366
## 4 4 1 80.28648
## 5 5 1 63.41147
## 6 6 1 111.60601Visual inspection
Different graphics can be used, the best choiice depends on the number of data points. For example, dotplots are good when there are not too many points:
require(lattice)
with(dat1, dotplot(y ~ group))
with(dat1, boxplot(y ~ group))
require(ggplot2)
ggplot(dat1, aes(group, y)) +
geom_violin() + geom_dotplot(binaxis='y', stackdir='center', dotsize=.5)## `stat_bindot()` using `bins = 30`. Pick better value with `binwidth`.
require(ez)
ezPlot(dat1, y, wid=subj, between=group, x=.(group))## Warning: Data is unbalanced (unequal N per group). Make sure you specified
## a well-considered value for the type argument to ezANOVA().## Warning in ezStats(data = data, dv = dv, wid = wid, within = within,
## within_full = within_full, : Unbalanced groups. Mean N will be used in
## computation of FLSD
Anova
ezANOVA(dat1, dv=y, wid=subj, between=group)## Warning: Data is unbalanced (unequal N per group). Make sure you specified
## a well-considered value for the type argument to ezANOVA().## $ANOVA
## Effect DFn DFd F p p<.05 ges
## 1 group 3 81 3.409065 0.02139588 * 0.1121069
##
## $`Levene's Test for Homogeneity of Variance`
## DFn DFd SSn SSd F p p<.05
## 1 3 81 499.0771 20586.27 0.6545666 0.5823939To conduct post-hoc tests comparing all pairs of means:
hsd <- TukeyHSD(aov(y ~ group, data=dat1))
print(hsd)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = y ~ group, data = dat1)
##
## $group
## diff lwr upr p adj
## 2-1 0.6734241 -16.952321 18.29917 0.9996365
## 3-1 7.3623076 -11.945729 26.67034 0.7496736
## 4-1 22.1939720 1.338909 43.04903 0.0324969
## 3-2 6.6888835 -10.936861 24.31463 0.7523986
## 4-2 21.5205479 2.212512 40.82858 0.0227903
## 4-3 14.8316644 -6.023398 35.68673 0.2510137plot(hsd)
If you worry about heteroscedasticity, you can check the diagnostic plots of lm:
par(mfcol=c(2, 2))
plot(lm(y ~ group, data=dat1))
If heteroscedasticity is severe, it will be visible on the plots of the “residuals vs. fitted” plot.
S*T Design (one-way Anova within subjects)
In this design, the factor ‘Subject’ is crossed with a factor ‘Treatment’, that is, there is one data point per subject and treatment.
ns <- 20
means <- c(100, 110, 100, 130)
nt <- length(means)
sd <- 20
dat2 <- data.frame(
subj=gl(ns, nt),
treat=gl(nt, 1, nt*ns))
dat2$y <- rep(means, ns) + rep(rnorm(ns, sd=20), each=nt) + rnorm(nt*ns, sd=20)
str(dat2)## 'data.frame': 80 obs. of 3 variables:
## $ subj : Factor w/ 20 levels "1","2","3","4",..: 1 1 1 1 2 2 2 2 3 3 ...
## $ treat: Factor w/ 4 levels "1","2","3","4": 1 2 3 4 1 2 3 4 1 2 ...
## $ y : num 105.6 103.3 69.1 123 69.6 ...head(dat2)## subj treat y
## 1 1 1 105.59941
## 2 1 2 103.30880
## 3 1 3 69.06811
## 4 1 4 123.01079
## 5 2 1 69.58211
## 6 2 2 97.29196We first plot the raw data, per subject:
require(lattice)
xyplot(y ~ treat | subj, data=dat2, type='b')
Or on a single plot (in the right panel, the subject effects are removed):
par(mfcol=c(1,2))
with(dat2, interaction.plot(treat, subj, y))
with(dat2, interaction.plot(treat, subj, y-tapply(y, subj, mean)[subj]))
And we perform the Anova with ez:
ezPlot(dat2, dv=y, wid=subj, within=treat, x=.(treat))
ezANOVA(dat2, dv=y, wid=subj, within=treat)## $ANOVA
## Effect DFn DFd F p p<.05 ges
## 2 treat 3 57 14.39573 4.280537e-07 * 0.1902848
##
## $`Mauchly's Test for Sphericity`
## Effect W p p<.05
## 2 treat 0.4735923 0.02137733 *
##
## $`Sphericity Corrections`
## Effect GGe p[GG] p[GG]<.05 HFe p[HF] p[HF]<.05
## 2 treat 0.714377 1.257336e-05 * 0.808031 4.135472e-06 *S<A*B> Design (Two-way between subjects Anova)
Generation of a dataset S<A3*B2>
subject = factor(paste('sub', 1:30, sep=''))
A = gl(3 ,10, 30, labels=c('a1','a2','a3'))
B = gl(2, 5, 30, labels=c('b1', 'b2'))
x = rnorm(30, mean=10) + 1 * (A=='a1' & B=='b2')
dat3 = data.frame(subject, A, B, x)rm(subject, A, B, x)
attach(dat3)Classical R approach
table(A, B)## B
## A b1 b2
## a1 5 5
## a2 5 5
## a3 5 5tapply(x, list(A, B), mean)## b1 b2
## a1 9.614235 11.01789
## a2 10.133099 10.07080
## a3 10.000732 10.63608interaction.plot(A,B,x)
summary(aov(x ~ A * B, data=dat3))## Df Sum Sq Mean Sq F value Pr(>F)
## A 2 0.31 0.155 0.068 0.934
## B 1 3.26 3.256 1.435 0.243
## A:B 2 2.69 1.344 0.592 0.561
## Residuals 24 54.47 2.270Using ez
ezPlot(data=dat3, dv=.(x), wid=.(subject), between=.(A,B),
x=.(B), split=.(A))
ezANOVA(data=dat3, dv=x, wid=subject, between=c('A','B'))## $ANOVA
## Effect DFn DFd F p p<.05 ges
## 1 A 2 24 0.0680766 0.9343686 0.005641048
## 2 B 1 24 1.4346721 0.2426994 0.056406155
## 3 A:B 2 24 0.5922504 0.5609637 0.047032926
##
## $`Levene's Test for Homogeneity of Variance`
## DFn DFd SSn SSd F p p<.05
## 1 5 24 2.677601 27.35433 0.469852 0.7948862detach(dat3)S*A*B Design (Two-way within-subject Anova)
Same dataset but with A & B within subject
dat4 <- dat3
subject = gl(5, 1, 30, labels=paste('sub', 1:5, sep=''))
dat4$subject = subject
table(dat4$subject, dat4$A, dat4$B)## , , = b1
##
##
## a1 a2 a3
## sub1 1 1 1
## sub2 1 1 1
## sub3 1 1 1
## sub4 1 1 1
## sub5 1 1 1
##
## , , = b2
##
##
## a1 a2 a3
## sub1 1 1 1
## sub2 1 1 1
## sub3 1 1 1
## sub4 1 1 1
## sub5 1 1 1attach(dat4)## The following object is masked _by_ .GlobalEnv:
##
## subjectinteraction.plot(A:B, subject, x)
summary(aov(x ~ A*B + Error(subject/(A*B))))##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 4 5.321 1.33
##
## Error: subject:A
## Df Sum Sq Mean Sq F value Pr(>F)
## A 2 0.309 0.1545 0.113 0.894
## Residuals 8 10.922 1.3653
##
## Error: subject:B
## Df Sum Sq Mean Sq F value Pr(>F)
## B 1 3.256 3.256 1.755 0.256
## Residuals 4 7.421 1.855
##
## Error: subject:A:B
## Df Sum Sq Mean Sq F value Pr(>F)
## A:B 2 2.688 1.344 0.349 0.716
## Residuals 8 30.807 3.851summary(aov(x ~ A + Error(subject/A), data=dat4, subset= (B=='b1')))##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 4 8.045 2.011
##
## Error: subject:A
## Df Sum Sq Mean Sq F value Pr(>F)
## A 2 0.727 0.3634 0.155 0.859
## Residuals 8 18.713 2.3391summary(aov(x ~ A + Error(subject/A), data=dat4, subset= (B=='b2')))##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 4 4.698 1.174
##
## Error: subject:A
## Df Sum Sq Mean Sq F value Pr(>F)
## A 2 2.271 1.135 0.395 0.686
## Residuals 8 23.016 2.877for (a in levels(A))
{
print(paste("Effect of B for A =",a))
print(summary(aov(x ~ B + Error(subject/(B), data=dat4, subset=(A==a)))))
}## [1] "Effect of B for A = a1"
##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 4 5.321 1.33
##
## Error: subject:B
## Df Sum Sq Mean Sq F value Pr(>F)
## B 1 3.256 3.256 1.755 0.256
## Residuals 4 7.421 1.855
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 20 44.73 2.236
## [1] "Effect of B for A = a2"
##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 4 5.321 1.33
##
## Error: subject:B
## Df Sum Sq Mean Sq F value Pr(>F)
## B 1 3.256 3.256 1.755 0.256
## Residuals 4 7.421 1.855
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 20 44.73 2.236
## [1] "Effect of B for A = a3"
##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 4 5.321 1.33
##
## Error: subject:B
## Df Sum Sq Mean Sq F value Pr(>F)
## B 1 3.256 3.256 1.755 0.256
## Residuals 4 7.421 1.855
##
## Error: Within
## Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 20 44.73 2.236detach(dat4)ezPlot(data=dat4, dv=.(x), wid=.(subject), between=.(A), within=.(B),
x=.(B), split=.(A))## Warning: The column supplied as the wid variable contains non-unique values
## across levels of the supplied between-Ss variables. Automatically fixing
## this by generating unique wid labels.
ezANOVA(data=dat4,
dv=x,
wid=subject,
within=.(A, B))## $ANOVA
## Effect DFn DFd F p p<.05 ges
## 2 A 2 8 0.1131701 0.8944030 0.005641048
## 3 B 1 4 1.7550178 0.2558619 0.056406155
## 4 A:B 2 8 0.3490647 0.7155772 0.047032926
##
## $`Mauchly's Test for Sphericity`
## Effect W p p<.05
## 2 A 0.9922891 0.9884559
## 4 A:B 0.8754243 0.8190829
##
## $`Sphericity Corrections`
## Effect GGe p[GG] p[GG]<.05 HFe p[HF] p[HF]<.05
## 2 A 0.9923481 0.8931132 1.965828 0.8944030
## 4 A:B 0.8892242 0.6934962 1.551223 0.7155772S<B>*A Split-plot ANOVA (A within, B between)
dat5 <- dat3
subject = gl(10, 1, 30, labels=paste('sub', 1:10, sep=''))
dat5$subject = subject
table(dat5$subject, dat5$A, dat5$B)## , , = b1
##
##
## a1 a2 a3
## sub1 1 1 1
## sub2 1 1 1
## sub3 1 1 1
## sub4 1 1 1
## sub5 1 1 1
## sub6 0 0 0
## sub7 0 0 0
## sub8 0 0 0
## sub9 0 0 0
## sub10 0 0 0
##
## , , = b2
##
##
## a1 a2 a3
## sub1 0 0 0
## sub2 0 0 0
## sub3 0 0 0
## sub4 0 0 0
## sub5 0 0 0
## sub6 1 1 1
## sub7 1 1 1
## sub8 1 1 1
## sub9 1 1 1
## sub10 1 1 1table(dat5$subject, dat5$B:dat5$A)##
## b1:a1 b1:a2 b1:a3 b2:a1 b2:a2 b2:a3
## sub1 1 1 1 0 0 0
## sub2 1 1 1 0 0 0
## sub3 1 1 1 0 0 0
## sub4 1 1 1 0 0 0
## sub5 1 1 1 0 0 0
## sub6 0 0 0 1 1 1
## sub7 0 0 0 1 1 1
## sub8 0 0 0 1 1 1
## sub9 0 0 0 1 1 1
## sub10 0 0 0 1 1 1summary(aov(x ~ A*B + Error(subject/A), data=dat5))##
## Error: subject
## Df Sum Sq Mean Sq F value Pr(>F)
## B 1 3.256 3.256 2.044 0.191
## Residuals 8 12.742 1.593
##
## Error: subject:A
## Df Sum Sq Mean Sq F value Pr(>F)
## A 2 0.31 0.1545 0.059 0.943
## A:B 2 2.69 1.3442 0.515 0.607
## Residuals 16 41.73 2.6080ezANOVA(data=dat5, dv=x, wid=subject, within=.(A), between=.(B))## $ANOVA
## Effect DFn DFd F p p<.05 ges
## 2 B 1 8 2.04433366 0.1906423 0.056406155
## 3 A 2 16 0.05924289 0.9426836 0.005641048
## 4 B:A 2 16 0.51539916 0.6068478 0.047032926
##
## $`Mauchly's Test for Sphericity`
## Effect W p p<.05
## 3 A 0.9431864 0.8148755
## 4 B:A 0.9431864 0.8148755
##
## $`Sphericity Corrections`
## Effect GGe p[GG] p[GG]<.05 HFe p[HF] p[HF]<.05
## 3 A 0.9462406 0.9353259 1.230641 0.9426836
## 4 B:A 0.9462406 0.5975871 1.230641 0.6068478S<G>*A*B Design (Split-plot Anova with two within variables)
One can have both between and within-subject factors. We consider here the case of a S20<G2>*A4*B2 design where S=subject is nested within a factor Group and crossed with the factors A and B which are also crossed with each other.
dat6 = data.frame(
subj = gl(20, 8, 160),
group = gl(2, 80, 160),
a = gl(4, 1, 160),
b = gl(2, 4, 160),
y = rnorm(160, 100, 10))
str(dat6)## 'data.frame': 160 obs. of 5 variables:
## $ subj : Factor w/ 20 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ group: Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
## $ a : Factor w/ 4 levels "1","2","3","4": 1 2 3 4 1 2 3 4 1 2 ...
## $ b : Factor w/ 2 levels "1","2": 1 1 1 1 2 2 2 2 1 1 ...
## $ y : num 85.3 90.7 86.2 120.3 65.7 ...head(dat6)## subj group a b y
## 1 1 1 1 1 85.31750
## 2 1 1 2 1 90.68432
## 3 1 1 3 1 86.18480
## 4 1 1 4 1 120.25579
## 5 1 1 1 2 65.70271
## 6 1 1 2 2 88.61727with(dat6, table(subj, group))## group
## subj 1 2
## 1 8 0
## 2 8 0
## 3 8 0
## 4 8 0
## 5 8 0
## 6 8 0
## 7 8 0
## 8 8 0
## 9 8 0
## 10 8 0
## 11 0 8
## 12 0 8
## 13 0 8
## 14 0 8
## 15 0 8
## 16 0 8
## 17 0 8
## 18 0 8
## 19 0 8
## 20 0 8with(dat6, table(subj, a:b))##
## subj 1:1 1:2 2:1 2:2 3:1 3:2 4:1 4:2
## 1 1 1 1 1 1 1 1 1
## 2 1 1 1 1 1 1 1 1
## 3 1 1 1 1 1 1 1 1
## 4 1 1 1 1 1 1 1 1
## 5 1 1 1 1 1 1 1 1
## 6 1 1 1 1 1 1 1 1
## 7 1 1 1 1 1 1 1 1
## 8 1 1 1 1 1 1 1 1
## 9 1 1 1 1 1 1 1 1
## 10 1 1 1 1 1 1 1 1
## 11 1 1 1 1 1 1 1 1
## 12 1 1 1 1 1 1 1 1
## 13 1 1 1 1 1 1 1 1
## 14 1 1 1 1 1 1 1 1
## 15 1 1 1 1 1 1 1 1
## 16 1 1 1 1 1 1 1 1
## 17 1 1 1 1 1 1 1 1
## 18 1 1 1 1 1 1 1 1
## 19 1 1 1 1 1 1 1 1
## 20 1 1 1 1 1 1 1 1xyplot(y ~ a | subj, group=b, data=dat6, type='b')
signif(with(dat6, tapply(y, list(group=group, a=a, b=b), mean )), 3)## , , b = 1
##
## a
## group 1 2 3 4
## 1 103 101 98.0 106.0
## 2 102 103 96.5 95.4
##
## , , b = 2
##
## a
## group 1 2 3 4
## 1 98 96.2 101.0 100.0
## 2 95 104.0 97.3 95.8ezPlot(dat6, dv=y, wid=subj, within = .(a, b), between=group, x=.(a), split=b, col=group)## Warning: Mixed within-and-between-Ss effect requested; FLSD is only
## appropriate for within-Ss comparisons (see warning in ?ezStats or ?ezPlot).
par(mfcol=c(1, 2))
with(subset(dat6, group==1), interaction.plot(a, b, y))
with(subset(dat6, group==2), interaction.plot(a, b, y))
a1 = ezANOVA(dat6, dv=y, wid=subj, within = .(a, b), between=group)
print(a1)## $ANOVA
## Effect DFn DFd F p p<.05 ges
## 2 group 1 18 0.9087354 0.35307732 0.009630646
## 3 a 3 54 0.4772297 0.69945304 0.010606525
## 5 b 1 18 1.6447238 0.21594900 0.012420774
## 4 group:a 3 54 2.5441752 0.06567282 0.054061319
## 6 group:b 1 18 0.2229011 0.64251141 0.001701595
## 7 a:b 3 54 1.5048658 0.22368195 0.021706584
## 8 group:a:b 3 54 0.7631969 0.51966285 0.011127599
##
## $`Mauchly's Test for Sphericity`
## Effect W p p<.05
## 3 a 0.7545970 0.4532317
## 4 group:a 0.7545970 0.4532317
## 7 a:b 0.8348933 0.6977726
## 8 group:a:b 0.8348933 0.6977726
##
## $`Sphericity Corrections`
## Effect GGe p[GG] p[GG]<.05 HFe p[HF] p[HF]<.05
## 3 a 0.8324903 0.66549006 0.9773018 0.69520478
## 4 group:a 0.8324903 0.07803365 0.9773018 0.06722156
## 7 a:b 0.8901186 0.22804152 1.0597497 0.22368195
## 8 group:a:b 0.8901186 0.50641523 1.0597497 0.51966285