Use the following commands to initialize R before answering the questions in this section.
options(contrasts=c("contr.sum","contr.poly") ) # set definition of contrasts
load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/3-way-data-6.rda") )An experiment examined how performance in three tasks was affected by
divided attention in three age groups of subjects using a 3
(group) x 3 (task) x 2
(attention) crossed-factorial design. The data are stored
in the data frame df1. The independent variables are
group, task, and attention, and
the dependent variable is Y
summary(df1)
## subject group task attention Y
## s1 : 1 g1:60 t1:60 focus :90 Min. :43.9
## s2 : 1 g2:60 t2:60 divide:90 1st Qu.:67.8
## s3 : 1 g3:60 t3:60 Median :74.8
## s4 : 1 Mean :75.0
## s5 : 1 3rd Qu.:82.3
## s6 : 1 Max. :98.2
## (Other):174
xtabs(~group+task+attention,df1)
## , , attention = focus
##
## task
## group t1 t2 t3
## g1 10 10 10
## g2 10 10 10
## g3 10 10 10
##
## , , attention = divide
##
## task
## group t1 t2 t3
## g1 10 10 10
## g2 10 10 10
## g3 10 10 10aov.01 <- aov(Y~group*task*attention,data=df1)
summary(aov.01)
## Df Sum Sq Mean Sq F value Pr(>F)
## group 2 1142 571 6.06 0.0029 **
## task 2 587 294 3.12 0.0468 *
## attention 1 513 513 5.44 0.0209 *
## group:task 4 197 49 0.52 0.7195
## group:attention 2 0 0 0.00 0.9979
## task:attention 2 61 31 0.32 0.7230
## group:task:attention 4 1646 411 4.37 0.0022 **
## Residuals 162 15250 94
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
library(afex)
aov.01.car <- aov_car(Y~group*task*attention+Error(subject),data=df1)
summary(aov.01.car)# The F test for the 3-way interaction evaluates the null hypothesis
# that i) the group x task interaction is the same at both levels of attention;
# ii) the attention x task interaction is the same in all groups;
# iii) the group x attention interaction is the same in all tasks.
levels(df1$group)
## [1] "g1" "g2" "g3"
# simple task x attention at g1:
aov.task.x.attention.g1 <- aov(Y~task*attention,data=subset(df1,group=="g1"))
summary(aov.task.x.attention.g1)
## Df Sum Sq Mean Sq F value Pr(>F)
## task 2 448 224 2.17 0.1242
## attention 1 162 162 1.57 0.2161
## task:attention 2 1095 547 5.30 0.0079 **
## Residuals 54 5582 103
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
levels(df1$attention)
## [1] "focus" "divide"
# simple simple main effect of task at "attention==focus" & group==g1
aov.task.focus.g1 <- aov(Y~task,data=subset(df1,attention=="focus"&group=="g1"))
summary(aov.task.focus.g1)
## Df Sum Sq Mean Sq F value Pr(>F)
## task 2 1437 719 7.86 0.002 **
## Residuals 27 2469 91
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# analyze sub-effect with TukeyHSD
TukeyHSD(aov.task.focus.g1,which="task")
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Y ~ task, data = subset(df1, attention == "focus" & group == "g1"))
##
## $task
## diff lwr upr p adj
## t2-t1 9.499 -1.104 20.102 0.0856
## t3-t1 -7.413 -18.016 3.190 0.2114
## t3-t2 -16.912 -27.515 -6.308 0.0014
# simple simple main effect of task at "attention==divide" & group==g1
aov.task.divide.g1 <- aov(Y~task,data=subset(df1,attention=="divide"&group=="g1"))
summary(aov.task.divide.g1)
## Df Sum Sq Mean Sq F value Pr(>F)
## task 2 106 53 0.46 0.64
## Residuals 27 3113 115
# simple task x attention at g2:
aov.task.x.attention.g2 <- aov(Y~task*attention,data=subset(df1,group=="g2"))
summary(aov.task.x.attention.g2)
## Df Sum Sq Mean Sq F value Pr(>F)
## task 2 233 116.3 1.49 0.24
## attention 1 184 184.2 2.36 0.13
## task:attention 2 131 65.7 0.84 0.44
## Residuals 54 4223 78.2
# simple task x attention at g3:
aov.task.x.attention.g3 <- aov(Y~task*attention,data=subset(df1,group=="g3"))
summary(aov.task.x.attention.g3)
## Df Sum Sq Mean Sq F value Pr(>F)
## task 2 103 51.5 0.51 0.6
## attention 1 167 166.8 1.65 0.2
## task:attention 2 481 240.4 2.39 0.1
## Residuals 54 5444 100.8t1
& t2) is the same in groups 1 and 3 (g1
& g3).# using aov:
levels(df1$task)
## [1] "t1" "t2" "t3"
t1.vs.t2 <- c(-1,1,0)
levels(df1$group)
## [1] "g1" "g2" "g3"
g1.vs.g3 <- c(-1,0,1)
contrasts(df1$task) <- cbind(t1.vs.t2)
contrasts(df1$group) <- cbind(g1.vs.g3)
aov.10 <- aov(Y~group*task*attention,data=df1)
summary(aov.10,split=list(group=list(g1.vs.g3=1,other=2),task=list(t1.vs.t2=1,other=2)))
## Df Sum Sq Mean Sq F value Pr(>F)
## group 2 1142 571 6.06 0.00289 **
## group: g1.vs.g3 1 1039 1039 11.03 0.00111 **
## group: other 1 103 103 1.09 0.29697
## task 2 587 294 3.12 0.04684 *
## task: t1.vs.t2 1 39 39 0.41 0.52186
## task: other 1 549 549 5.83 0.01690 *
## attention 1 513 513 5.44 0.02085 *
## group:task 4 197 49 0.52 0.71955
## group:task: g1.vs.g3.t1.vs.t2 1 99 99 1.05 0.30595
## group:task: other.t1.vs.t2 1 34 34 0.36 0.54938
## group:task: g1.vs.g3.other 1 63 63 0.66 0.41603
## group:task: other.other 1 1 1 0.01 0.92380
## group:attention 2 0 0 0.00 0.99791
## group:attention: g1.vs.g3 1 0 0 0.00 0.98913
## group:attention: other 1 0 0 0.00 0.94964
## task:attention 2 61 31 0.32 0.72305
## task:attention: t1.vs.t2 1 59 59 0.63 0.42831
## task:attention: other 1 2 2 0.02 0.88966
## group:task:attention 4 1646 411 4.37 0.00220 **
## group:task:attention: g1.vs.g3.t1.vs.t2 1 176 176 1.87 0.17328
## group:task:attention: other.t1.vs.t2 1 378 378 4.02 0.04665 *
## group:task:attention: g1.vs.g3.other 1 1088 1088 11.55 0.00085 ***
## group:task:attention: other.other 1 4 4 0.04 0.83667
## Residuals 162 15250 94
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# the (g1.vs.g3) X (t1.vs.t2) interaction is NOT significant (F=1.055, p=0.306)
# also, the value of that contrast does not differ significantly across attention (F=1.871, p=0.173)# Here is the same interaction contrast computed with emmeans
library(emmeans)
emm.10 <- emmeans(aov.10,specs=~task+group+attention)
emm.10 # note the order of the conditions!
## task group attention emmean SE df lower.CL upper.CL
## t1 g1 focus 78.3 3.07 162 72.3 84.4
## t2 g1 focus 87.8 3.07 162 81.8 93.9
## t3 g1 focus 70.9 3.07 162 64.8 77.0
## t1 g2 focus 79.4 3.07 162 73.3 85.4
## t2 g2 focus 78.4 3.07 162 72.3 84.5
## t3 g2 focus 75.6 3.07 162 69.6 81.7
## t1 g3 focus 72.4 3.07 162 66.3 78.4
## t2 g3 focus 71.5 3.07 162 65.4 77.5
## t3 g3 focus 75.6 3.07 162 69.5 81.7
## t1 g1 divide 77.4 3.07 162 71.3 83.4
## t2 g1 divide 73.1 3.07 162 67.0 79.2
## t3 g1 divide 76.7 3.07 162 70.7 82.8
## t1 g2 divide 72.4 3.07 162 66.3 78.5
## t2 g2 divide 78.7 3.07 162 72.6 84.7
## t3 g2 divide 71.8 3.07 162 65.7 77.9
## t1 g3 divide 74.0 3.07 162 67.9 80.0
## t2 g3 divide 71.2 3.07 162 65.1 77.2
## t3 g3 divide 64.3 3.07 162 58.3 70.4
##
## Confidence level used: 0.95
# create contrasts that compare conditions:
( t1.vs.t2 <- rep(c(-1,1,0),times=6) ) # task 1 vs task 2
## [1] -1 1 0 -1 1 0 -1 1 0 -1 1 0 -1 1 0 -1 1 0
( g1.vs.g3 <- rep(c(-1,0,1),each=3,times=2) ) # group 1 vs group 2
## [1] -1 -1 -1 0 0 0 1 1 1 -1 -1 -1 0 0 0 1 1 1
( focus.vs.divide <- rep(c(1,-1),each=9)) # focus vs divide
## [1] 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1
( int.contrast <- g1.vs.g3 * t1.vs.t2 ) # g1.vs.g3 x t1.vs.t2 (ignores attention)
## [1] 1 -1 0 0 0 0 -1 1 0 1 -1 0 0 0 0 -1 1 0
( int.contrast.2 <- g1.vs.g3 * t1.vs.t2 * focus.vs.divide ) # g1.vs.g3 x t1.vs.t2 x attention
## [1] 1 -1 0 0 0 0 -1 1 0 -1 1 0 0 0 0 1 -1 0
# p values match corresponding values in ANOVA table:
contrast(emm.10,
method=list(g1.vs.g3.x.t1.vs.t2=int.contrast,
g1.vs.g3.x.t1.vs.t2.x.attn=int.contrast.2))
## contrast estimate SE df t.ratio p.value
## g1.vs.g3.x.t1.vs.t2 -8.91 8.68 162 -1.027 0.3060
## g1.vs.g3.x.t1.vs.t2.x.attn -11.87 8.68 162 -1.368 0.1733# Here is another emmeans method for evaluating slightly different hypotheses
task.contrast <- c(-1,1,0)
con1 <- contrast(emm.10,method=list(task=task.contrast),by=c("group","attention"))
# following table doesn't include the *specific* group contrast used above
# So 1st & 3rd lines are closer to an omnibus F of variation across groups
# 1st line asks if task contrast differs IN ANY WAY across groups
# 2nd line asks if task contrast differs between levels of attention (this >>is<< the same as the anova result)
# 3rd line asks if task-contrast variation across groups DEPENDS on attention
joint_tests(con1)Use the following commands to initialize R before answering the questions in this section.
options(contrasts=c("contr.sum","contr.poly") ) # set definition of contrasts
load(url("http://pnb.mcmaster.ca/bennett/psy710/datasets/DfCD-2.rda") )An experiment was done to measure the effects of treatment
C and treatment D on a dependent variable,
y, using a crossed-factorial design. Six subjects were
assigned randomly to each condition; however, the data from two subjects
in one of the conditions were lost. The data are stored in the data
frame Df.CD.
# with(Df.CD,tapply(y,list(C,D),length))
xtabs(~C+D,data=Df.CD)
## D
## C b1 b2 b3
## a1 6 6 6
## a2 6 6 4C and D.options(contrasts=c("contr.sum","contr.poly"))
cd.aov.01 <- aov(y~C+D+C:D, data=Df.CD)
cd.aov.02 <- aov(y~D+C+D:C, data=Df.CD)
summary(cd.aov.01)
## Df Sum Sq Mean Sq F value Pr(>F)
## C 1 83.0 83.0 14.5 0.00069 ***
## D 2 275.0 137.5 24.1 8.2e-07 ***
## C:D 2 26.3 13.1 2.3 0.11853
## Residuals 28 159.8 5.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(cd.aov.02)
## Df Sum Sq Mean Sq F value Pr(>F)
## D 2 299.6 149.8 26.2 3.8e-07 ***
## C 1 58.4 58.4 10.2 0.0034 **
## D:C 2 26.3 13.1 2.3 0.1185
## Residuals 28 159.8 5.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
C_SS_1 <- 83.0 # C: Type I SS
C_SS_2 <- 58.4 # C: Type II SS
D_SS_1 <- 299.6 # D: Type I SS
D_SS_2 <- 275.0 # D: Type II SSC and D and
the C x D interaction using Type III sums of squares. What
null hypotheses are evaluated by these \(F\) tests?# using ANOVA:
require(car)
Anova(cd.aov.01,type="3")Anova(cd.aov.02,type="3") # order independent# using drop1:
drop1(cd.aov.01,.~.,test="F")# type III SS using aov_car & aov_ez:
library(afex)
N <- dim(Df.CD)[1]
Df.CD$id <- factor(x=1:N,labels="s",ordered=FALSE)
# type 3 sums of squares
acar.t3.01 <- aov_car(y~C*D+Error(id),data=Df.CD,type="3")
# type 3 sums of squares with aov_ez:
# aez.t3.01 <- aov_ez(id="id",dv="y",between=c("C","D"),data=Df.CD,type="3")
summary(acar.t3.01)# type II SS using aov_car:
# type 2 sums of squares
acar.t2.01 <- aov_car(y~C*D+Error(id),data=Df.CD,type="2")
summary(acar.t2.01) # same as Anova(cd.aov.01,type="3")# compare to Anova:
Anova(cd.aov.01,type="2")# using aov_ez:
# type 3 sums of squares with aov_ez:
aez.t3.01 <- aov_ez(id="id",dv="y",between=c("C","D"),data=Df.CD,type="3")
summary(aez.t3.01) # same as Anova(cd.aov.01,type="3")# type 2 sums of squares with aov_ez:
aez.t2.01 <- aov_ez(id="id",dv="y",between=c("C","D"),data=Df.CD,type="2")
summary(aez.t2.01) # same as Anova(cd.aov.01,type="2")Answer: The hypotheses for the main effects are that
the unweighted marginal means are equal. The hypothesis for the
interaction is that the effect of C does not vary across
the levels of D (or, equivalently, that the effect of
D does not vary across the levels of C).