3-way Anova
3-Way ANOVA
The three-way ANOVA is an extension of the two-way ANOVA for assessing whether there is an interaction effect between three independent categorical variables on a continuous outcome variable.
We’ll use the headache dataset [datarium package], which contains the measures of migraine headache episode pain score in 72 participants treated with three different treatments. The participants include 36 males and 36 females. Males and females were further subdivided into whether they were at low or high risk of migraine.
We want to understand how each independent variable (type of treatments, risk of migraine and gender) interact to predict the pain score.
Descriptive statistics
Mean + standard deviation
## # A tibble: 12 x 7
## gender risk treatment variable n mean sd
## <fct> <fct> <fct> <chr> <dbl> <dbl> <dbl>
## 1 male high X pain_score 6 92.7 5.12
## 2 male high Y pain_score 6 82.3 5.00
## 3 male high Z pain_score 6 79.7 4.05
## 4 male low X pain_score 6 76.1 3.86
## 5 male low Y pain_score 6 73.1 4.76
## 6 male low Z pain_score 6 74.5 4.89
## 7 female high X pain_score 6 78.9 5.32
## 8 female high Y pain_score 6 81.2 4.62
## 9 female high Z pain_score 6 81.0 3.98
## 10 female low X pain_score 6 74.2 3.69
## 11 female low Y pain_score 6 68.4 4.08
## 12 female low Z pain_score 6 69.8 2.72Visualization with histogram
Visualization with boxplot
Assumptions
Outliers
## # A tibble: 4 x 7
## gender risk treatment id pain_score is.outlier is.extreme
## <fct> <fct> <fct> <int> <dbl> <lgl> <lgl>
## 1 female high X 57 68.4 TRUE TRUE
## 2 female high Y 62 73.1 TRUE FALSE
## 3 female high Z 67 75.0 TRUE FALSE
## 4 female high Z 71 87.1 TRUE FALSEOne extreme outlier is identified - female at high risk treated with X.
Normality
## # A tibble: 1 x 3
## variable statistic p.value
## <chr> <dbl> <dbl>
## 1 residuals(model) 0.982 0.398Normality is proved, as in the QQ plot all the points follow the reference line and Shapiro-Wilk test gives p-value bigger than 0.5 (0.398).
Now, we can test normality by groups:
## # A tibble: 12 x 6
## gender risk treatment variable statistic p
## <fct> <fct> <fct> <chr> <dbl> <dbl>
## 1 male high X pain_score 0.958 0.808
## 2 male high Y pain_score 0.902 0.384
## 3 male high Z pain_score 0.955 0.784
## 4 male low X pain_score 0.982 0.962
## 5 male low Y pain_score 0.920 0.507
## 6 male low Z pain_score 0.924 0.535
## 7 female high X pain_score 0.714 0.00869
## 8 female high Y pain_score 0.939 0.654
## 9 female high Z pain_score 0.971 0.901
## 10 female low X pain_score 0.933 0.600
## 11 female low Y pain_score 0.927 0.555
## 12 female low Z pain_score 0.958 0.801One group failed the normality check - female with high risk treated with X, as their p-value is lower than 0.05 (around 0.009).
QQ plot confirms the outcomes we got from Shapiro-Wilk test.
Homogeneity of variance
## # A tibble: 1 x 4
## df1 df2 statistic p
## <int> <int> <dbl> <dbl>
## 1 11 60 0.179 0.998Levene’s test shows that we can assume homogeneity of variances, as p-value is higher than 0.05 (0.998).
Anova
## ANOVA Table (type II tests)
##
## Effect DFn DFd F p p<.05 ges
## 1 gender 1 60 16.196 0.000163000000000 * 0.213
## 2 risk 1 60 92.699 0.000000000000088 * 0.607
## 3 treatment 2 60 7.318 0.001000000000000 * 0.196
## 4 gender:risk 1 60 0.141 0.708000000000000 0.002
## 5 gender:treatment 2 60 3.338 0.042000000000000 * 0.100
## 6 risk:treatment 2 60 0.713 0.494000000000000 0.023
## 7 gender:risk:treatment 2 60 7.406 0.001000000000000 * 0.198As we see, there is a significant interaction between gender, risk and treatment (p-value = 0.001 < 0.05), so we’ve found three-way interaction.
Post-hoc tests
If there is a significant 3-way interaction effect, you can decompose it into:
- Simple two-way interaction: run two-way interaction at each level of third variable,
- Simple simple main effect: run one-way model at each level of second variable,
- Simple simple pairwise comparisons: run pairwise or other post-hoc comparisons if necessary.
If you do not have a statistically significant three-way interaction, you need to determine whether you have any statistically significant two-way interaction from the ANOVA output. You can follow up a significant two-way interaction by simple main effects analyses and pairwise comparisons between groups if necessary.
Two-way interactions
## # A tibble: 6 x 8
## gender Effect DFn DFd F p `p<.05` ges
## * <fct> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <dbl>
## 1 male risk 1 60 50.0 0.00000000187 "*" 0.455
## 2 male treatment 2 60 10.2 0.000157 "*" 0.253
## 3 male risk:treatment 2 60 5.25 0.008 "*" 0.149
## 4 female risk 1 60 42.8 0.000000015 "*" 0.416
## 5 female treatment 2 60 0.482 0.62 "" 0.016
## 6 female risk:treatment 2 60 2.87 0.065 "" 0.087There is two-way interaction between risk and treatment for males (p-value = 0.008 < 0.025 [as we use Bonferroni method]). Therefore we can conclude that risk has an effect on efficiency of treatment.
Main effects
## # A tibble: 4 x 9
## gender risk Effect DFn DFd F p `p<.05` ges
## * <fct> <fct> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <dbl>
## 1 male high treatment 2 60 14.8 0.0000061 "*" 0.33
## 2 male low treatment 2 60 0.66 0.521 "" 0.022
## 3 female high treatment 2 60 0.52 0.597 "" 0.017
## 4 female low treatment 2 60 2.83 0.067 "" 0.086We should take only men under consideration, but I couldn’t make filter function work. There’s a significant effect of treatment on men with high risk (p-value = 0.000006 < 0.0001). This means the treatment was more effective for such people and it didn’t work that well for men at low risk.
Pairwise comparisons
## # A tibble: 3 x 8
## gender risk term .y. group1 group2 p.adj p.adj.signif
## <chr> <chr> <chr> <chr> <chr> <chr> <dbl> <chr>
## 1 male high treatment pain_score X Y 0.000386 ***
## 2 male high treatment pain_score X Z 0.00000942 ****
## 3 male high treatment pain_score Y Z 0.897 nsWe can see that there were significant differences between male with high risk treated with X vs Y and X vs Z.