ANOVA
Problem
You want to compare multiple groups using an ANOVA.
Solution
Suppose this is your data:
data <- read.table(header=TRUE, text='
subject sex age before after
1 F old 9.5 7.1
2 M old 10.3 11.0
3 M old 7.5 5.8
4 F old 12.4 8.8
5 M old 10.2 8.6
6 M old 11.0 8.0
7 M young 9.1 3.0
8 F young 7.9 5.2
9 F old 6.6 3.4
10 M young 7.7 4.0
11 M young 9.4 5.3
12 M old 11.6 11.3
13 M young 9.9 4.6
14 F young 8.6 6.4
15 F young 14.3 13.5
16 F old 9.2 4.7
17 M young 9.8 5.1
18 F old 9.9 7.3
19 F young 13.0 9.5
20 M young 10.2 5.4
21 M young 9.0 3.7
22 F young 7.9 6.2
23 M old 10.1 10.0
24 M young 9.0 1.7
25 M young 8.6 2.9
26 M young 9.4 3.2
27 M young 9.7 4.7
28 M young 9.3 4.9
29 F young 10.7 9.8
30 M old 9.3 9.4
')
# Make sure subject column is a factor, so that it's not treated as a continuous
# variable.
data$subject <- factor(data$subject)
One way between ANOVA
# One way between:
# IV: sex
# DV: before
aov1 <- aov(before ~ sex, data=data)
summary(aov1)
#> Df Sum Sq Mean Sq F value Pr(>F)
#> sex 1 1.53 1.529 0.573 0.455
#> Residuals 28 74.70 2.668
# Show the means
model.tables(aov1, "means")
#> Tables of means
#> Grand mean
#>
#> 9.703333
#>
#> sex
#> F M
#> 10 9.532
#> rep 11 19.000
Two way between ANOVA
# 2x2 between:
# IV: sex
# IV: age
# DV: after
# These two calls are equivalent
aov2 <- aov(after ~ sex*age, data=data)
aov2 <- aov(after ~ sex + age + sex:age, data=data)
summary(aov2)
#> Df Sum Sq Mean Sq F value Pr(>F)
#> sex 1 16.08 16.08 4.038 0.0550 .
#> age 1 38.96 38.96 9.786 0.0043 **
#> sex:age 1 89.61 89.61 22.509 6.6e-05 ***
#> Residuals 26 103.51 3.98
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Show the means
model.tables(aov2, "means")
#> Tables of means
#> Grand mean
#>
#> 6.483333
#>
#> sex
#> F M
#> 7.445 5.926
#> rep 11.000 19.000
#>
#> age
#> old young
#> 7.874 5.556
#> rep 12.000 18.000
#>
#> sex:age
#> age
#> sex old young
#> F 6.260 8.433
#> rep 5.000 6.000
#> M 9.157 4.042
#> rep 7.000 12.000
Tukey HSD post-hoc test
TukeyHSD(aov2)
#> Tukey multiple comparisons of means
#> 95% family-wise confidence level
#>
#> Fit: aov(formula = after ~ sex + age + sex:age, data = data)
#>
#> $sex
#> diff lwr upr p adj
#> M-F -1.519139 -3.073025 0.03474709 0.0549625
#>
#> $age
#> diff lwr upr p adj
#> young-old -2.31785 -3.846349 -0.7893498 0.0044215
#>
#> $`sex:age`
#> diff lwr upr p adj
#> M:old-F:old 2.8971429 -0.3079526 6.1022384 0.0869856
#> F:young-F:old 2.1733333 -1.1411824 5.4878491 0.2966111
#> M:young-F:old -2.2183333 -5.1319553 0.6952887 0.1832890
#> F:young-M:old -0.7238095 -3.7691188 2.3214997 0.9138789
#> M:young-M:old -5.1154762 -7.7187601 -2.5121923 0.0000676
#> M:young-F:young -4.3916667 -7.1285380 -1.6547953 0.0008841
ANOVAs with within-subjects variables
For ANOVAs with within-subjects variables, the data must be in long format. The data supplied above is in wide format, so we have to convert it first. (See ../../Manipulating data/Converting data between wide and long format for more information.)
Also, for ANOVAs with a within-subjects variable, there must be an identifier column. In this case, it is subject. This identifier variable must be a factor. If it is a numeric type, the function will interpret it incorrectly and it won’t work properly.
library(tidyr)
# This is what the original (wide) data looks like
# subject sex age before after
# 1 F old 9.5 7.1
# 2 M old 10.3 11.0
# 3 M old 7.5 5.8
# Convert it to long format
data_long <- gather(data, time, value, before:after)
# Look at first few rows
head(data_long)
#> subject sex age time value
#> 1 1 F old before 9.5
#> 2 2 M old before 10.3
#> 3 3 M old before 7.5
#> 4 4 F old before 12.4
#> 5 5 M old before 10.2
#> 6 6 M old before 11.0
# Make sure subject column is a factor
data_long$subject <- factor(data_long$subject)
One-way within ANOVA
First, convert the data to long format and make sure subject is a factor, as shown above. If subject is a numeric column, and not a factor, your results will be wrong!
# IV (within): time
# DV: value
aov_time <- aov(value ~ time + Error(subject/time), data=data_long)
summary(aov_time)
#>
#> Error: subject
#> Df Sum Sq Mean Sq F value Pr(>F)
#> Residuals 29 261.2 9.009
#>
#> Error: subject:time
#> Df Sum Sq Mean Sq F value Pr(>F)
#> time 1 155.53 155.53 71.43 2.59e-09 ***
#> Residuals 29 63.14 2.18
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# This won't work here for some reason (?)
model.tables(aov_time, "means")
#> Tables of means
#> Grand mean
#>
#> 8.093333
#>
#> time
#> time
#> after before
#> 6.483 9.703
Mixed design ANOVA
First, convert the data to long format and make sure subject is a factor, as shown above.
# 2x2 mixed:
# IV between: age
# IV within: time
# DV: value
aov_age_time <- aov(value ~ age*time + Error(subject/time), data=data_long)
summary(aov_age_time)
#>
#> Error: subject
#> Df Sum Sq Mean Sq F value Pr(>F)
#> age 1 24.44 24.440 2.89 0.1
#> Residuals 28 236.81 8.457
#>
#> Error: subject:time
#> Df Sum Sq Mean Sq F value Pr(>F)
#> time 1 155.53 155.53 98.14 1.18e-10 ***
#> age:time 1 18.77 18.77 11.84 0.00184 **
#> Residuals 28 44.37 1.58
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# This won't work here because the data is unbalanced
model.tables(aov_age_time, "means")
#> Tables of means
#> Grand mean
#>
#> 8.093333
#>
#> age
#> old young
#> 8.875 7.572
#> rep 24.000 36.000
#>
#> time
#> after before
#> 6.483 9.703
#> rep 30.000 30.000
#>
#> age:time
#> time
#> age after before
#> old 7.950 9.800
#> rep 12.000 12.000
#> young 5.506 9.639
#> rep 18.000 18.000
More ANOVAs with within-subjects variables
These examples don’t operate on the data above, but they should illustrate how to do things.
First, convert the data to long format and make sure subject is a factor, as shown above.
# Two within variables
aov.ww <- aov(y ~ w1*w2 + Error(subject/(w1*w2)), data=data_long)
# One between variable and two within variables
aov.bww <- aov(y ~ b1*w1*w2 + Error(subject/(w1*w2)) + b1, data=data_long)
# Two between variables and one within variables
aov.bww <- aov(y ~ b1*b2*w1 + Error(subject/(w1)) + b1*b2, data=data_long)
For more information on ANOVA in R, see: