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1. 17/03/2016 One­way ANOVA (CR­p design)
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2. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 2/12
kiel.de/psychologie/rexrepos/posts/anovaSStypes.html)
General Topics
Assess normality (http://www.uni­kiel.de/psychologie/rexrepos/posts/normality.html)
Assess variance homogeneity (http://www.uni­kiel.de/psychologie/rexrepos/posts/varianceHom.html)
Tags
ANOVA (http://www.uni­kiel.de/psychologie/rexrepos/tags.html#ANOVA­ref)
One­way ANOVA (CR­p design)
TODO
Install required packages
CR­ ANOVA
Simulate data
Using oneway.test()
Using aov()
Model comparisons using anova(lm())
Effect size estimates
Planned comparisons ­ a­priori
General contrasts using glht() from package multcomp
Pairwise ­tests
Planned comparisons ­ post­hoc
Scheffe tests
Tukey's simultaneous confidence intervals
Assess test assumptions
Normality
Variance homogeneity
Detach (automatically) loaded packages (if possible)
Get the article source from GitHub
TODO
link to normality, varianceHom, regressionDiag, regression for model comparison,
resamplingPerm, resamplingBootALM
Install required packages
car (http://cran.r­project.org/package=car), DescTools (http://cran.r­project.org/package=DescTools),
multcomp (http://cran.r­project.org/package=multcomp)
wants <‐ c("car", "DescTools", "multcomp")
has <‐ wants %in% rownames(installed.packages())
if(any(!has)) install.packages(wants[!has])
CR­ ANOVA
p
t
p

3. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 3/12
Simulate data
set.seed(123)
P <‐ 4
Nj <‐ c(41, 37, 42, 40)
muJ <‐ rep(c(‐1, 0, 1, 2), Nj)
dfCRp <‐ data.frame(IV=factor(rep(LETTERS[1:P], Nj)),
DV=rnorm(sum(Nj), muJ, 5))
plot.design(DV ~ IV, fun=mean, data=dfCRp, main="Group means")
plot of chunk rerAnovaCRp01
Using oneway.test()
Assuming variance homogeneity
oneway.test(DV ~ IV, data=dfCRp, var.equal=TRUE)
One‐way analysis of means
data: DV and IV
F = 2.0057, num df = 3, denom df = 156, p‐value = 0.1154

4. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 4/12
Generalized Welch­test without assumption of variance homogeneity
oneway.test(DV ~ IV, data=dfCRp, var.equal=FALSE)
One‐way analysis of means (not assuming equal variances)
data: DV and IV
F = 2.0203, num df = 3.000, denom df = 85.503, p‐value = 0.1171
Using aov()
aovCRp <‐ aov(DV ~ IV, data=dfCRp)
summary(aovCRp)
Df Sum Sq Mean Sq F value Pr(>F)
IV 3 133 44.35 2.006 0.115
Residuals 156 3450 22.11
model.tables(aovCRp, type="means")
Tables of means
Grand mean
0.4318522
IV
A B C D
‐0.8643 0.05185 1.042 1.471
rep 41.0000 37.00000 42.000 40.000
Model comparisons using anova(lm())
(anovaCRp <‐ anova(lm(DV ~ IV, data=dfCRp)))
Analysis of Variance Table
Response: DV
Df Sum Sq Mean Sq F value Pr(>F)
IV 3 133.1 44.353 2.0057 0.1154
Residuals 156 3449.7 22.113
anova(lm(DV ~ 1, data=dfCRp), lm(DV ~ IV, data=dfCRp))

5. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 5/12
Analysis of Variance Table
Model 1: DV ~ 1
Model 2: DV ~ IV
Res.Df RSS Df Sum of Sq F Pr(>F)
1 159 3582.8
2 156 3449.7 3 133.06 2.0057 0.1154
anovaCRp["Residuals", "Sum Sq"]
[1] 3449.703
Effect size estimates
dfSSb <‐ anovaCRp["IV", "Df"]
SSb <‐ anovaCRp["IV", "Sum Sq"]
MSb <‐ anovaCRp["IV", "Mean Sq"]
SSw <‐ anovaCRp["Residuals", "Sum Sq"]
MSw <‐ anovaCRp["Residuals", "Mean Sq"]
(etaSq <‐ SSb / (SSb + SSw))
[1] 0.03713889
library(DescTools) # for EtaSq()
EtaSq(aovCRp, type=1)
eta.sq eta.sq.part
IV 0.03713889 0.03713889
,
(omegaSq <‐ dfSSb * (MSb‐MSw) / (SSb + SSw + MSw))
[1] 0.01850809
(f <‐ sqrt(etaSq / (1‐etaSq)))
[1] 0.196396
Planned comparisons ­ a­priori
η
2^
ω
2^
f
2^

6. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 6/12
General contrasts using glht() from package multcomp
cntrMat <‐ rbind("A‐D" =c( 1, 0, 0, ‐1),
"1/3*(A+B+C)‐D"=c(1/3, 1/3, 1/3, ‐1),
"B‐C" =c( 0, 1, ‐1, 0))
library(multcomp) # for glht()
summary(glht(aovCRp, linfct=mcp(IV=cntrMat), alternative="less"),
test=adjusted("none"))
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: User‐defined Contrasts
Fit: aov(formula = DV ~ IV, data = dfCRp)
Linear Hypotheses:
Estimate Std. Error t value Pr(<t)
A‐D >= 0 ‐2.3351 1.0451 ‐2.234 0.0134 *
1/3*(A+B+C)‐D >= 0 ‐1.3941 0.8589 ‐1.623 0.0533 .
B‐C >= 0 ‐0.9906 1.0603 ‐0.934 0.1758
‐‐‐
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported ‐‐ none method)
Pairwise ­tests
pairwise.t.test(dfCRp$DV, dfCRp$IV, p.adjust.method="bonferroni")
Pairwise comparisons using t tests with pooled SD
data: dfCRp$DV and dfCRp$IV
A B C
B 1.00 ‐ ‐
C 0.40 1.00 ‐
D 0.16 1.00 1.00
P value adjustment method: bonferroni
Planned comparisons ­ post­hoc
Scheffe tests
library(DescTools) # for ScheffeTest()
ScheffeTest(aovCRp, which="IV", contrasts=t(cntrMat))
t

7. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 7/12
Posthoc multiple comparisons of means : Scheffe Test
95% family‐wise confidence level
Fit: aov(formula = DV ~ IV, data = dfCRp)
$IV
diff lwr.ci upr.ci pval
A‐D ‐2.3351002 ‐5.288758 0.6185575 0.1770
A,B,C‐D ‐1.3941211 ‐3.821531 1.0332885 0.4538
B‐C ‐0.9906183 ‐3.987210 2.0059738 0.8319
‐‐‐
Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
Tukey's simultaneous confidence intervals
(tHSD <‐ TukeyHSD(aovCRp))
Tukey multiple comparisons of means
95% family‐wise confidence level
Fit: aov(formula = DV ~ IV, data = dfCRp)
$IV
diff lwr upr p adj
B‐A 0.9161596 ‐1.8529795 3.685299 0.8257939
C‐A 1.9067779 ‐0.7743204 4.587876 0.2555117
D‐A 2.3351002 ‐0.3789061 5.049107 0.1185540
C‐B 0.9906183 ‐1.7628388 3.744075 0.7864641
D‐B 1.4189406 ‐1.3665697 4.204451 0.5497967
D‐C 0.4283223 ‐2.2696814 3.126326 0.9762890
plot(tHSD)

8. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 8/12
plot of chunk rerAnovaCRp02
Using glht() from package multcomp
library(multcomp) # for glht()
tukey <‐ glht(aovCRp, linfct=mcp(IV="Tukey"))
summary(tukey)
Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DV ~ IV, data = dfCRp)
Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|)
B ‐ A == 0 0.9162 1.0663 0.859 0.826
C ‐ A == 0 1.9068 1.0324 1.847 0.255
D ‐ A == 0 2.3351 1.0451 2.234 0.119
C ‐ B == 0 0.9906 1.0603 0.934 0.786
D ‐ B == 0 1.4189 1.0726 1.323 0.550
D ‐ C == 0 0.4283 1.0389 0.412 0.976
(Adjusted p values reported ‐‐ single‐step method)

9. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 9/12
confint(tukey)
Simultaneous Confidence Intervals
Multiple Comparisons of Means: Tukey Contrasts
Fit: aov(formula = DV ~ IV, data = dfCRp)
Quantile = 2.5972
95% family‐wise confidence level
Linear Hypotheses:
Estimate lwr upr
B ‐ A == 0 0.9162 ‐1.8533 3.6856
C ‐ A == 0 1.9068 ‐0.7746 4.5882
D ‐ A == 0 2.3351 ‐0.3792 5.0494
C ‐ B == 0 0.9906 ‐1.7632 3.7444
D ‐ B == 0 1.4189 ‐1.3669 4.2048
D ‐ C == 0 0.4283 ‐2.2700 3.1266
Assess test assumptions
Normality
Estud <‐ rstudent(aovCRp)
qqnorm(Estud, pch=20, cex=2)
qqline(Estud, col="gray60", lwd=2)

10. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 10/12
plot of chunk rerAnovaCRp03
shapiro.test(Estud)
Shapiro‐Wilk normality test
data: Estud
W = 0.9937, p‐value = 0.7149
Variance homogeneity
plot(Estud ~ dfCRp$IV, main="Residuals per group")

11. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 11/12
plot of chunk rerAnovaCRp04
library(car)
leveneTest(aovCRp)
Levene's Test for Homogeneity of Variance (center = median)
Df F value Pr(>F)
group 3 0.8551 0.4659
156
Detach (automatically) loaded packages (if possible)
try(detach(package:car))
try(detach(package:multcomp))
try(detach(package:survival))
try(detach(package:mvtnorm))
try(detach(package:splines))
try(detach(package:TH.data))
try(detach(package:DescTools))
Get the article source from GitHub

12. 17/03/2016 One­way ANOVA (CR­p design)
http://www.uni­kiel.de/psychologie/rexrepos/posts/anovaCRp.html 12/12
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