RM Anova

Introduction

The repeated measures ANOVA makes the following assumptions about the data:

• No significant outliers in any cell of the design. This can be checked by visualizing the data using box plot methods and by using the function identify_outliers() [rstatix package].

• Normality: the outcome (or dependent) variable should be approximately normally distributed in each cell of the design. This can be checked using the Shapiro-Wilk normality test (shapiro_test() [rstatix]) or by visual inspection using QQ plot (ggqqplot() [ggpubr package]).

• Assumption of sphericity: the variance of the differences between groups should be equal. This can be checked using the Mauchly’s test of sphericity, which is automatically reported when using the R function anova_test() [rstatix package].

Note that, if the above assumptions are not met there are a non-parametric alternative (Friedman test) to the one-way repeated measures ANOVA!

Unfortunately, there are no non-parametric alternatives to the two-way and the three-way repeated measures ANOVA. Thus, in the situation where the assumptions are not met, you could consider running the two-way/three-way repeated measures ANOVA on the transformed and non-transformed data to see if there are any meaningful differences.

If both tests lead you to the same conclusions, you might not choose to transform the outcome variable and carry on with the two-way/three-way repeated measures ANOVA on the original data.

It’s also possible to perform robust ANOVA test using the WRS2 R package.

No matter your choice, you should report what you did in your results.

RM Anova in R

Key R functions:

• anova_test() [rstatix package], a wrapper around car::Anova() for making easy the computation of repeated measures ANOVA. Key arguments for performing repeated measures ANOVA:

  • data: data frame

  • dv: (numeric) the dependent (or outcome) variable name.

  • wid: variable name specifying the case/sample identifier.

  • within: within-subjects factor or grouping variable

• get_anova_table() [rstatix package]. Extracts the ANOVA table from the output of anova_test(). It returns ANOVA table that is automatically corrected for eventual deviation from the sphericity assumption. The default is to apply automatically the Greenhouse-Geisser sphericity correction to only within-subject factors violating the sphericity assumption (i.e., Mauchly’s test p-value is significant, p <= 0.05). Read more in Chapter @ref(mauchly-s-test-of-sphericity-in-r).

1-way RM Anova

The dataset “selfesteem” contains 10 individuals’ self-esteem score on three time points during a specific diet to determine whether their self-esteem improved.

## # A tibble: 3 x 4
##      id    t1    t2    t3
##   <int> <dbl> <dbl> <dbl>
## 1     1  4.01  5.18  7.11
## 2     2  2.56  6.91  6.31
## 3     3  3.24  4.44  9.78

The one-way repeated measures ANOVA can be used to determine whether the means self-esteem scores are significantly different between the three time points. So let’s convert this data frame into long format:

## # A tibble: 3 x 3
##   id    time  score
##   <fct> <fct> <dbl>
## 1 1     t1     4.01
## 2 2     t1     2.56
## 3 3     t1     3.24

## # A tibble: 3 x 5
##   time  variable     n  mean    sd
##   <fct> <chr>    <dbl> <dbl> <dbl>
## 1 t1    score       10  3.14 0.552
## 2 t2    score       10  4.93 0.863
## 3 t3    score       10  7.64 1.14

## # A tibble: 2 x 5
##   time  id    score is.outlier is.extreme
##   <fct> <fct> <dbl> <lgl>      <lgl>     
## 1 t1    6      2.05 TRUE       FALSE     
## 2 t2    2      6.91 TRUE       FALSE

## # A tibble: 3 x 4
##   time  variable statistic     p
##   <fct> <chr>        <dbl> <dbl>
## 1 t1    score        0.967 0.859
## 2 t2    score        0.876 0.117
## 3 t3    score        0.923 0.380

## ANOVA Table (type III tests)
## 
## $ANOVA
##   Effect DFn DFd      F        p p<.05   ges
## 1   time   2  18 55.469 2.01e-08     * 0.829
## 
## $`Mauchly's Test for Sphericity`
##   Effect     W     p p<.05
## 1   time 0.551 0.092      
## 
## $`Sphericity Corrections`
##   Effect  GGe      DF[GG]    p[GG] p[GG]<.05   HFe      DF[HF]    p[HF]
## 1   time 0.69 1.38, 12.42 2.16e-06         * 0.774 1.55, 13.94 6.03e-07
##   p[HF]<.05
## 1         *

## ANOVA Table (type III tests)
## 
##   Effect DFn DFd      F        p p<.05   ges
## 1   time   2  18 55.469 2.01e-08     * 0.829

## # A tibble: 3 x 10
##   .y.   group1 group2    n1    n2 statistic    df           p p.adj p.adj.signif
## * <chr> <chr>  <chr>  <int> <int>     <dbl> <dbl>       <dbl> <dbl> <chr>       
## 1 score t1     t2        10    10     -4.97     9 0.000772     2e-3 **          
## 2 score t1     t3        10    10    -13.2      9 0.000000334  1e-6 ****        
## 3 score t2     t3        10    10     -4.87     9 0.000886     2e-3 **

2-way RM Anova

For Two-Way Repeated Measures ANOVA, “Two-way” means that there are two factors in the experiment, for example, different treatments and different conditions. “Repeated-measures” means that the same subject received more than one treatment and/or more than one condition. Similar to two-way ANOVA, two-way repeated measures ANOVA can be employed to test for significant differences between the factor level means within a factor and for interactions between factors.

Using a standard ANOVA in this case is not appropriate because it fails to model the correlation between the repeated measures, and the data violates the ANOVA assumption of independence. Two-Way Repeated Measures ANOVA designs can be two repeated measures factors, or one repeated measures factor and one non-repeated factor. If any repeated factor is present, then the repeated measures ANOVA should be used.

Please apply Two-way RM-ANOVA to analyze if any significant interactions (between time and music, time and image, music and image, or music and time and image)! Use the following data set:

set.seed(5250)
myData <- data.frame(PID = rep(seq(from = 1,
                               to = 60, by = 1), 20),
                     stress = sample(x = 1:100,
                                     size = 1200,
                                     replace = TRUE),
                     image = sample(c("Happy", "Angry"),
                                    size = 1200,
                                    replace = TRUE),
                     music = sample(c("Disney", "Horror"),
                                    size = 1200,
                                    replace = TRUE)
)
myData <- within(myData, {
  PID   <- factor(PID)
  image <- factor(image)
  music <- factor(music)
})
myData <- myData[order(myData$PID), ]
head(myData)

##     PID stress image  music
## 1     1     90 Happy Horror
## 61    1      7 Angry Disney
## 121   1     31 Happy Disney
## 181   1     68 Angry Disney
## 241   1      6 Happy Disney
## 301   1     80 Angry Horror

## # A tibble: 4 x 6
##   image music  variable     n  mean    sd
##   <fct> <fct>  <chr>    <dbl> <dbl> <dbl>
## 1 Angry Disney stress     310  48.9  29.4
## 2 Angry Horror stress     305  53.0  28.4
## 3 Happy Disney stress     289  49.5  29.2
## 4 Happy Horror stress     296  47.3  28.4

## [1] image      music      PID        stress     is.outlier is.extreme
## <0 wierszy> (lub 'row.names' o zerowej długości)

## # A tibble: 4 x 5
##   image music  variable statistic             p
##   <fct> <fct>  <chr>        <dbl>         <dbl>
## 1 Angry Disney stress       0.944 0.00000000171
## 2 Angry Horror stress       0.957 0.0000000924 
## 3 Happy Disney stress       0.952 0.0000000403 
## 4 Happy Horror stress       0.957 0.000000124

## Coefficient covariances computed by hccm()

## ANOVA Table (type II tests)
## 
##        Effect DFn  DFd     F     p p<.05      ges
## 1       image   1 1196 2.332 0.127       0.002000
## 2       music   1 1196 0.398 0.528       0.000332
## 3 image:music   1 1196 3.597 0.058       0.003000