Both one-way ANOVA (Analysis of Variance) and repeated measures ANOVA employ the F statistic as a tool to evaluate the null hypothesis that there are no significant differences between the groups being studied.
Numerous other statistical tests utilize the F statistic, including regression, Levene’s and Bartlett’s tests for homogeneity of variance, MANOVA, ANCOVA, Wilks’ lambda test, and Mauchly’s test of sphericity.
The F statistic operates by conducting a comparative analysis of two estimated variances.
The first of these is the between-group variance, which is indicative of systematic differences that may exist across the groups. This variance captures the extent to which the groups differ from one another in a consistent, non-random manner.
The second variance examined is the within-groups variance, representing the random variability present within each group.
The more the between-group variability exceeds the within-group variability, the higher the F value becomes, indicating a greater likelihood that the groups are significantly different.
How is the F statistic calculated? Let’s explore this using simple examples with small numbers to illustrate how the F statistic is derived.
Let’s consider a scenario where we compare three groups of three patients each, subjected to different treatments (A, B, and C). The treatment results are as follows:
| Group | Results |
|---|---|
| A | 5, 6, 7 |
| B | 8, 9, 7 |
| C | 10, 10, 11 |
We can easily calculate the means of groups A, B, and C, as well as the overall mean:
The sum of squares between groups (SSB) quantifies the extent to which group means deviate from the overall mean.
SSB = 3(6-8.22)^2 + 3(8-8.22)^2 + 3(10.33-8.22)^2
SSB = 28.212
The sum of squares within groups (SSW) is calculated as follows:
Group A = (5−6)^2 + (6−6)^2 + (7−6)^2 = 2
Group B = (8−8)^2 + (9−8)^2 + (7−8)^2 = 2
Group C = (10−10.33)^2 + (10−10.33)^2 + (11−10.33)^2 = 0.67
SSW = 4.67
Let’s proceed with calculating the degrees of freedom (df):
Between groups = df_B = 3-1 = 2
Within groups =df_W = 9-3 = 6
Finally, we calculate the Mean Squares for Between Groups (MSB) and Within (MSW) Groups:
MSB = SSB/df_B = 28.212/2 = 14.106
MSW = SSW/df_W = 4.67/6 = 0.778
The F statistic is the ratio of MSB to MSW, thus
F = MSB/MSW = 14.106/0.778 = 18.13
In repeated measures ANOVA, the calculations follow a similar pattern to one-way ANOVA, but there’s a crucial difference in how the concept of “group” is applied. Instead of comparing different groups of subjects, repeated measures ANOVA focuses on the individual subject as the unit of analysis. In this context, each subject essentially becomes their own “group,” with the repeated measurements taken from that subject over time or under different conditions forming the data points within that group.
For instance, if we’re studying the effects of a new medication on blood pressure, we might measure each participant’s blood pressure at baseline, after one week of treatment, and after one month of treatment. In this case, each participant’s set of three measurements would constitute a “group” in the repeated measures ANOVA framework.