26 One-way Repeated Measures ANOVA

The one-way repeated measures analysis of variance (also known as a within-subjects ANOVA) is an extension of the paired t-test designed to assess whether there are significant differences in the means of three or more related groups, such as comparing the difference between three or more time points.

Learning objectives

Applying hypothesis testing
Compare three or more dependent samples applying repeated ANOVA
Perform post-hoc tests
Interpret the results

26.1 Research question and Hypothesis Testing

Participants used margarine for 12 weeks. Their blood total cholesterol (TCH; in mmol/L) was measured before the special diet, after 6 weeks and after 12 weeks.

Null hypothesis and alternative hypothesis for the main research question

\(H_0\): all related group means are equal (the means of cholesterol at the three time points are equal; \(\mu_{1} = \mu_{2} = \mu_{3}\))
\(H_1\): at least one related group mean differs from the others (there is at least one time point at which the mean cholesterol level differs from the others)

26.2 Packages we need

We need to load the following packages:

library(rstatix)
library(superb)
library(ggpubr)
library(ggprism)
library(ggsci)
library(ggpubr)
library(ggrain)

library(PupillometryR)
library(gtsummary)
library(dabestr)

library(here)
library(tidyverse)

26.3 Preparing the data

We import the data cholesterol in R:

library(readxl)
dat_TCH <- read_excel(here("data", "cholesterol.xlsx"))

Figure 26.1: Table with data from “cholesterol” file.

We inspect the data and the type of variables:

glimpse(dat_TCH)

Rows: 18
Columns: 3
$ week0  <dbl> 6.42, 6.76, 6.56, 4.80, 8.43, 7.49, 8.05, 5.05, 5.77, 3.91, 6.7…
$ week6  <dbl> 5.83, 6.20, 5.83, 4.27, 7.71, 7.12, 7.25, 4.63, 5.31, 3.70, 6.1…
$ week12 <dbl> 5.75, 6.13, 5.71, 4.15, 7.67, 7.05, 7.10, 4.67, 5.33, 3.66, 5.9…

26.4 Assumptions

Check if the following assumptions are satisfied

The data are normally distributed in all time points.
The variances among the differences between all possible pairs of time points are equal (sphericity assumption).

A. Explore the characteristics of distribution for each time point and check for normality

The distributions can be explored visually with appropriate plots. Additionally, summary statistics and significance tests to check for normality (e.g., Shapiro-Wilk test) can be used.

Graphs

We can visualize the distribution of cholesterol for the three time points:

dat_TCH_long <- dat_TCH |> 
  mutate(id = row_number()) |> 
  pivot_longer(cols = -id, names_to = "time", values_to = "cholesterol") |> 
  mutate(time = factor(time, levels = c("week0", "week6", "week12")))

ggplot(dat_TCH_long, aes(x= time, y = cholesterol, fill = time)) +
  geom_rain(likert= TRUE, seed = 123, point.args = list(alpha = 0.3)) +
  #theme_prism(base_size = 14, base_line_size = 0.4, palette = "office") +
  labs(title = "Grouped Raincloud Plot: Cholesterol by time point") +
  scale_fill_jco() +
  theme(legend.position = "none")

ggqqplot(dat_TCH_long, "cholesterol", color = "time", conf.int = F) +
  #theme_prism(base_size = 14, base_line_size = 0.4, palette = "office") +
  scale_color_jco() +
  facet_wrap(~ time) + 
  theme(legend.position = "none")

The above figures show that the data are close to symmetry and the assumption of a normal distribution is reasonable.

Summary statistics

The cholesterol summary statistics for each time point are:

Summary statistics by time point

dplyr
dlookr

cholesterol_summary <- dat_TCH_long %>%
  group_by(time) %>%
  dplyr::summarise(
    n = n(),
    na = sum(is.na(cholesterol)),
    min = min(cholesterol, na.rm = TRUE),
    q1 = quantile(cholesterol, 0.25, na.rm = TRUE),
    median = quantile(cholesterol, 0.5, na.rm = TRUE),
    q3 = quantile(cholesterol, 0.75, na.rm = TRUE),
    max = max(cholesterol, na.rm = TRUE),
    mean = mean(cholesterol, na.rm = TRUE),
    sd = sd(cholesterol, na.rm = TRUE),
    skewness = EnvStats::skewness(cholesterol, na.rm = TRUE),
    kurtosis= EnvStats::kurtosis(cholesterol, na.rm = TRUE)
  ) %>%
  ungroup()

cholesterol_summary

# A tibble: 3 × 12
  time      n    na   min    q1 median    q3   max  mean    sd skewness kurtosis
  <fct> <int> <int> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>    <dbl>    <dbl>
1 week0    18     0  3.91  5.74   6.5   7.22  8.43  6.41  1.19   -0.311   -0.248
2 week6    18     0  3.7   5.18   5.83  6.73  7.71  5.84  1.12   -0.171   -0.708
3 week…    18     0  3.66  5.21   5.73  6.69  7.67  5.78  1.10   -0.188   -0.561

dat_TCH_long |> 
  group_by(time) |> 
  dlookr::describe(cholesterol) |> 
  select(described_variables,  time, n, mean, sd, p25, p50, p75, skewness, kurtosis) |> 
  ungroup() |> 
  print(width = 100)

Registered S3 methods overwritten by 'dlookr':
  method          from  
  plot.transform  scales
  print.transform scales

# A tibble: 3 × 10
  described_variables time       n  mean    sd   p25   p50   p75 skewness
  <chr>               <fct>  <int> <dbl> <dbl> <dbl> <dbl> <dbl>    <dbl>
1 cholesterol         week0     18  6.41  1.19  5.74  6.5   7.22   -0.311
2 cholesterol         week6     18  5.84  1.12  5.18  5.83  6.73   -0.171
3 cholesterol         week12    18  5.78  1.10  5.21  5.73  6.69   -0.188
  kurtosis
     <dbl>
1   -0.248
2   -0.708
3   -0.561

The means are close to medians and the standard deviations are also similar. Moreover, both skewness and (excess) kurtosis falls into the acceptable range of [-1, 1] indicating approximately normal distributions for all time points.

Normality test

dat_TCH_long |> 
  group_by(time) |> 
  shapiro_test(cholesterol) |>  
  ungroup()

# A tibble: 3 × 4
  time   variable    statistic     p
  <fct>  <chr>           <dbl> <dbl>
1 week0  cholesterol     0.982 0.967
2 week6  cholesterol     0.977 0.912
3 week12 cholesterol     0.977 0.918

The tests of normality suggest that the data for the cholesterol in all time points are normally distributed (p > 0.05).

In our example, the data at each time point are approximately normally distributed; a repeated ANOVA analysis can be performed.

B. Sphericity test for equality of variances of the differences

In addition, the assumption of sphericity must be met for accurate interpretation of the results of repeated ANOVA. This assumption is usually checked with the Mauchly’s sphericity test, wherein null hypothesis states that the variances of the differences are equal.

MauchlySphericityTest(dat_TCH)

[1] 0.0004439621

Here the assumption of sphericity has not been met (p < 0.001). In this case, we have to correct the degrees of freedom in repeated ANOVA analysis.

26.5 Run the one-way repeated ANOVA test

First, let’s run the repeated ANOVA test without any correction:

repeated_anova <- dat_TCH_long |>
  anova_test(dv = cholesterol, wid = id, within = time)

repeated_anova[1]

$ANOVA
  Effect DFn DFd       F        p p<.05   ges
1   time   2  34 212.321 6.17e-20     * 0.061

However, we must adjust the results of the repeated ANOVA analysis. The correction involves multiplying the degrees of freedom DFn=2 and DFd=34 by a quantity e, which measures the extent to which the data deviates from ideal sphericity (e ranges between 0 and 1, where 1 indicates no departure from sphericity). Two methods are commonly used for calculating e:

the correction of Greenhouse-Geisser (GGe).
the correction of Huynh-Feldt (HFe).

In R, we can calculate both GGe and HFe, as follows:

repeated_anova[3]

$`Sphericity Corrections`
  Effect   GGe   DF[GG]    p[GG] p[GG]<.05   HFe      DF[HF]    p[HF] p[HF]<.05
1   time 0.618 1.24, 21 3.89e-13         * 0.642 1.28, 21.82 1.44e-13         *

The general recommendation is to use the Greenhouse-Geisser correction when GGe is less than 0.75; otherwise, we should use the Huynh-Feldt correction HFe.

As the GGe value is less than 0.75, we use the Greenhouse-Geisser adjustment of 0.618. The corrected degrees of freedom are:

\(DF[GG]_n=2*0.618=1.24\)

and

\(DF[GG]_d=34*0.618=21\).

The new p-value (p[GG]) is available next to the corrected degrees of freedom (DF[GG]). In this example, as p<0.001 there is evidence of a difference between at least two time points.

get_anova_table()

By utilizing the get_anova_table() function to extract the ANOVA table, the Greenhouse-Geisser sphericity correction is automatically applied when the assumption of sphericity is violated.

get_anova_table(repeated_anova)

ANOVA Table (type III tests)

  Effect  DFn DFd       F        p p<.05   ges
1   time 1.24  21 212.321 3.89e-13     * 0.061