# ONE FACTOR INDEPENDENT MEASURES ANOVA
hinton.anagram.altformat = data.frame(time=c(15,20,14,13,18,16,13,12,18,11,21,25,29,18,26,22,26,24,28,21,28,30,32,
28,26,30,25,36,20,25),condition=c("first","first","first","first","first","first",
"first","first","first","first","last","last","last","last","last","last","last",
"last","last","last","none","none","none","none","none","none","none","none","none",
"none"))
bwplot(time ~ condition,data=hinton.anagram.altformat)
# To work through this we'll use a slightly different format....
hinton.anagram = data.frame(
first.letter=c(15,20,14,13,18,16,13,12,18,11),
last.letter=c(21,25,29,18,26,22,26,24,28,21),
no.letter= c(28,30,32,28,26,30,25,36,20,25))
numberofdatapoints = 30
numberofconditions = 3
numberofdatapointspercondition = 10
# We need to calculate the total sum of squares of the data
SS_total = sum((hinton.anagram - (sum(hinton.anagram)/numberofdatapoints))^2)
# We need to calculate the between conditions sum of squares
SS_between = sum((mean(hinton.anagram) - (sum(hinton.anagram)/numberofdatapoints))^2*numberofdatapointspercondition)
# We can then use this to calculate to SS of the error
SS_err = SS_total - SS_between
# We need to consider the degrees of freedom. We divide by the df to give the "mean
# squares" in each case.
df_total = 30 - 1
df_between = 3 - 1
df_err = df_total - df_between
MS_between= SS_between/df_between
MS_error = SS_err/df_err
F = MS_between/MS_error
1 - pf(F,df_between,df_err)
# We can do this in R without all the steps
anagram.lm = lm(time ~ condition,data=hinton.anagram.altformat)
anova(anagram.lm)
# Wait! This look familiar! Yes it is exactly what we were doing for model comparison.
# There we simply calculated the SS of the error directly from the model residuals.
anagram_SSres = sum(residuals(anagram.lm)^2)
SStot = sum((hinton.anagram.altformat$time - mean(hinton.anagram.altformat$time))^2)
F = (SStot - anagram_SSres / 1) / (anagram_SSres/8)
1-pf(F,1,8)
# An alternative syntax
anagram.aov = aov(time ~ condition,data=hinton.anagram.altformat)
summary(anagram.aov)
# Anova tells you that the factor explains more variance that expected by chance, but
# it doesn't tell you where the differences lie. For that we need other tests.
pairwise.t.test(hinton.anagram.altformat$time,hinton.anagram.altformat$condition,
p.adj="bonferroni")
TukeyHSD(anagram.aov)
###
# ONE FACTOR REPEATED MEASURES ANOVA
####
hinton.keyboard.altformat = data.frame(errors=c(5,1,0,2,6,2,4,4,10,3,5,6),kb=c("one","one","one","one","two","two",
"two","two","three","three","three","three"),part=c("a","b","c","d","a","b","c","d",
"a","b","c","d"))
bwplot(errors ~ as.factor(kb),data=hinton.keyboard.altformat)
bwplot(errors ~ as.factor(kb)|part,data=hinton.keyboard.altformat)
#Again we'll use a slightly different format to work through this
hinton.keyboard = data.frame(
kb1=c(5,1,0,2),kb2=c(6,2,4,4),kb3=c(10,3,5,6)
)
numberofdatapoints = 12
numberofconditions = 3
numberofparticipants = 4
df_total = numberofdatapoints - 1
df_between = numberofconditions - 1
df_err = (numberofparticipants-1)*(numberofconditions-1)
SS_total = sum((hinton.keyboard - (sum(hinton.keyboard)/numberofdatapoints))^2)
SS_between_conditions = sum((mean(hinton.keyboard) - (sum(hinton.keyboard)/numberofdatapoints))^2*numberofparticipants)
SS_within_conditions = SS_total - SS_between_conditions
SS_between_subjs = sum((rowMeans(hinton.keyboard) - mean(mean(hinton.keyboard)))^2)*numberofconditions
SS_error = SS_within_conditions - SS_between_subjs
MS_between_conditions= SS_between_conditions/df_between
MS_error = SS_error/df_err
F = MS_between_conditions/MS_error
1 - pf(F,df_between,df_err)
# We can do this without all of the steps...
kb.aov = aov(errors ~ kb + Error(part/kb),data=hinton.keyboard.altformat)
#And to relate this back to regression again....The following command gives us the
# same thing!
anova(lm(errors ~ part,data=hinton.keyboard.altformat),lm(errors ~ part + kb,data=hinton.keyboard.altformat))
#And again we need additional tests to locate the differences..
pairwise.t.test(hinton.keyboard.altformat$errors,hinton.keyboard.altformat$kb,paired=T,
p.adj="bonferroni")
# You can add factors and interactions between factors to the ANOVA in the same way you
# added them to the regression models
# For example, if you had an additional factor font color in the anagram task, you
# could do the following:
hinton.anagram.altformat2 = data.frame(time=c(15,20,14,13,18,16,13,12,18,11,21,25,29,18,26,22,26,24,28,21,28,30,32,
28,26,30,25,36,20,25),condition=c("first","first","first","first","first","first",
"first","first","first","first","last","last","last","last","last","last","last",
"last","last","last","none","none","none","none","none","none","none","none","none",
"none"),color=c("blue","blue","blue","blue","blue","red","red","red","red","red",
"blue","blue","blue","blue","blue","red","red","red","red","red","blue","blue","blue",
"blue","blue","red","red","red","red","red"))
#For a simple two factor analysis:
anagram.2factor.aov = aov(time ~ condition + color,data=hinton.anagram.altformat2)
# For a two factor analysis with interaction
anagram.2factorwithinteraction.aov = aov(time ~ condition * color,data=hinton.anagram.altformat2)
# For repeated measures multi-factor, you just do the same and then add the multiple
# factors to the error partitioning, so if you were to have had an additional factor
# time_of_day in the keyboard data you could have written
# aov(errors ~ kb + time_of_day +
# Error(part/(kb+time_of_day)),data=hinton.keyboard.altformat)