Recall the way that the ANOVA is formatted:
onefact <-
data.frame(groups = rep(c("Control","Treatment 1", "Treatment 2"), each = 20),
score = c(rnorm(20,10),rnorm(20,15),rnorm(20,22)))
head(onefact) groups score
1 Control 10.053947
2 Control 10.705475
3 Control 10.503910
4 Control 8.875389
5 Control 8.802727
6 Control 10.602503We have three independent variables, or conditions, control, treatment 1 and treatment 2. We have one dependent variable, some idea of “score” or our dependent variabe.
The ANOVA is analyzed through the use of the aov function. Remember to save the analysis as a model so you can use it later if you need to do any post-hoc tests/unplanned comparisons.
onefact.mod <- aov(score~groups, data=onefact)
summary(onefact.mod) Df Sum Sq Mean Sq F value Pr(>F)
groups 2 1552.4 776.2 719.7 <2e-16 ***
Residuals 57 61.5 1.1
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This produces a sum of squares and mean squares for the between subjects factor, groups. It also produces a sum of squares and mean squares for the within subjects factor, also called the residuals, or the error.
From this we get one F-value.
The current design will not help us too much so we need to move on to another design, the two factor between subjects ANOVA.
The set-up is mostly the same:
twofact.df <-
data.frame(Gender=c(rep("Male", 30), rep("Female",30)),
Diet= c(rep(c("Diet 1", "Diet 2"), 30)),
Count=c(rnorm(30,28,2),rnorm(30,35,2)))
twofact.df Gender Diet Count
1 Male Diet 1 29.28565
2 Male Diet 2 27.77193
3 Male Diet 1 28.77947
4 Male Diet 2 27.13819
5 Male Diet 1 27.68155
6 Male Diet 2 32.14686
7 Male Diet 1 26.74604
8 Male Diet 2 26.48680
9 Male Diet 1 27.70064
10 Male Diet 2 24.20712
11 Male Diet 1 28.64474
12 Male Diet 2 27.62731
13 Male Diet 1 29.02422
14 Male Diet 2 26.43496
15 Male Diet 1 26.09912
16 Male Diet 2 25.31223
17 Male Diet 1 27.34079
18 Male Diet 2 28.09060
19 Male Diet 1 29.34636
20 Male Diet 2 31.89925
21 Male Diet 1 27.50672
22 Male Diet 2 24.67519
23 Male Diet 1 28.41077
24 Male Diet 2 26.41232
25 Male Diet 1 30.45810
26 Male Diet 2 27.96039
27 Male Diet 1 31.47558
28 Male Diet 2 27.71842
29 Male Diet 1 29.11803
30 Male Diet 2 25.88075
31 Female Diet 1 36.56940
32 Female Diet 2 37.10854
33 Female Diet 1 39.75562
34 Female Diet 2 33.67428
35 Female Diet 1 36.03668
36 Female Diet 2 34.20005
37 Female Diet 1 34.42411
38 Female Diet 2 31.63277
39 Female Diet 1 36.83795
40 Female Diet 2 35.80431
41 Female Diet 1 36.30694
42 Female Diet 2 30.80953
43 Female Diet 1 34.73127
44 Female Diet 2 36.05144
45 Female Diet 1 33.65283
46 Female Diet 2 33.32218
47 Female Diet 1 36.15832
48 Female Diet 2 33.17340
49 Female Diet 1 34.93337
50 Female Diet 2 32.42822
51 Female Diet 1 33.02687
52 Female Diet 2 34.26264
53 Female Diet 1 37.21966
54 Female Diet 2 38.13742
55 Female Diet 1 35.71552
56 Female Diet 2 37.01317
57 Female Diet 1 38.69807
58 Female Diet 2 36.73326
59 Female Diet 1 37.52918
60 Female Diet 2 37.74209We can see that there are two levels of the independent variable (gender) and two levels of the independent variable (Diet) and one dependent variable, count.
From this we will generate four sums of squares and four mean squares. - Sum of Squares for Factor A - Sum of Squares for Factor B - - Sum of Squares for the interaction between A and B - Sum of Squares within, or Error, or residual.
Analyzing the data just uses one more term:
twofact.mod<-aov(Count~Gender*Diet,data = twofact.df)
summary(twofact.mod) Df Sum Sq Mean Sq F value Pr(>F)
Gender 1 853.6 853.6 218.647 <2e-16 ***
Diet 1 23.3 23.3 5.958 0.0178 *
Gender:Diet 1 0.0 0.0 0.012 0.9147
Residuals 56 218.6 3.9
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1What do these results look like?
In order to plot these relationships we can use interaction.plot.
interaction.plot(twofact.df$Gender,twofact.df$Diet, twofact.df$Count)
df <-
data.frame(iv1 = rep(c("Level 1","Level 2"),each=2,20),
iv2 = rep(c("Level 1","Level 2"),each=1,40),
scored = rnorm(80,c(15,20,30,40)))
length(df$iv1)[1] 80length(df$iv2)[1] 80length(df$scored)[1] 80summary(aov(scored~iv1*iv2,data = df)) Df Sum Sq Mean Sq F value Pr(>F)
iv1 1 5989 5989 4487.65 < 2e-16 ***
iv2 1 1157 1157 866.93 < 2e-16 ***
iv1:iv2 1 129 129 96.52 3.57e-15 ***
Residuals 76 101 1
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1interaction.plot(df$iv1,df$iv2,df$scored)
In a Two Factor Between Subjects ANOVA, a particular participant is only ever in one condition or group or treatment level.
In a Two Factor Within Subjects ANOVA, a particular participant is in each condition or group or treatment level.
Setting this data up uses the same principles as we have learned before.
The one major difference in the set-up of the data is that there is now a variable of the subject itself.
When we had a between subjects design, each participant was unique, with a within deisgn, each participant experiences every aspect of the experiment so it is reasonable that they may have an effect on the experiment itself.
Here is a sample dataset:
set.seed(1234)
within_anova <-
data.frame(subject=rep(1:10,4),
groups=rep(c("pre","post","post 1", "post 2"),each = 10),
score=rnorm(40,c(20,30,40,70)))
head(within_anova) subject groups score
1 1 pre 18.79293
2 2 pre 30.27743
3 3 pre 41.08444
4 4 pre 67.65430
5 5 pre 20.42912
6 6 pre 30.50606It would be helpful if we could see the means of each group, and plot those means.
For this, we will use a new function called tapply. This applies a specified function to whatever variables you provide in the arguments.
To get the means:
within_means <-
tapply(within_anova$score,within_anova$groups,mean)
within_means post post 1 post 2 pre
42.88183 36.61205 42.23381 36.61684 To get the standard deviations:
within_sd <-
tapply(within_anova$score,within_anova$groups,sd)
within_sd post post 1 post 2 pre
20.47817 18.96593 20.24584 18.49391 For now, we will only use tapply in order to get the means.
Here is how we will plot the means:
plot(within_means, pch=19,
xlab = "Treatments",
ylab = "Survey Score",
main = "Means of Survey Score by Treatment")
Performing the ANOVA requires two new terms: Error and the Subject factor.
The model that the ANOVA should resemble looks like this:
aov(DV~IV+Error(Subjects))
From here it is important to note that your IV *must* be a factor.
With this in mind, you should run the function levels(Grouping Variable) or factor(Grouping Variable.)
If the first function returns NULL, use the second function.
Putting all of this together gives us the following:
within_anova.mod <- aov(score~groups+Error(subject),data = within_anova)
summary(within_anova.mod)
Error: subject
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 1 113.9 113.9
Error: Within
Df Sum Sq Mean Sq F value Pr(>F)
groups 3 355 118.4 0.303 0.823
Residuals 35 13665 390.4 When reporting these findings, you will ignore the subject variable and report the F-value for the grouping variable.
This particular finding should be reported as follows:
\(F(3,35) = .303, p >.05\)
The results of a one-factor within subjects ANOVA revealed that there does not appear to be any effect of treatment on survey score.
Most times you will want to report effect sizes for your experiment. Effect sizes help to tell you how much of your effect is due to your manipulation.
In this example, there does not appear to be an effect at all, but we will compute an effect size anyways.
We will be using the \(\omega^2\) effect size estimate.
The formula is as follows:
\[\omega^2 = \frac{SS_B-(k-1)(MS_W)}{SS_T-MS_W}\]
or
\[\omega^2 = \frac{SS_{Effect}-(k-1)(MS_{Within})}{SS_{Total}-MS_{Within}}\]
There is no R function that reports \(\omega^2\), so we will do this by hand.
\(\omega^2 = \frac{355-(4-1)(390.4)}{14020-390.4}\)
\(\omega^2 = -.059\)
This effect size is negative because our effect was not significant, but it is still important for you to see how to get these numbers!