### Introduction

_{something}denotes the variability explained by something known: e.g., SS

_{time}, SS

_{group}, and SS

_{subject}. The total sum of squares, SS

_{total}, is the sum of all the SS components of a dataset. If we have groups A, B, and C, then the notations can be simplified as SS

_{A}, SS

_{B}, and SS

_{C}.

_{total}.

_{error}is called the F value.

### Major Differences between ANOVA and RMANOVA

### Total Uncertainty Explained by No Factors

_{T}) = the SS of the error (SS

_{E}) and is computed by:

_{i,j}denotes the distance on the j

^{th}occasion in the i

^{th}subject and X denotes the mean distance. SS

_{T}is also computed using a simple ANOVA table that includes "nothing" as an explanatory variable (Table 2a).

### ANOVA Model of the Effect of Age

_{age}; this approach would be incorrect unless treated as a within-subjects effect. In this model, SS

_{T}is given by the sum of SS

_{age}= 50.65 and SS

_{E}= 196.7, such that SS

_{T}= 247.29 of the null model (Table 2b).

### ANOVA Model of the Effects of Age, Gender, and Their Interactive Effect

_{T}is given by the sum of SS

_{age}, SS

_{gender}, and SS

_{age : gender}(Table 3a). The models estimated thus far all exclude the effect of subject. Because the measurements for each subject were repeated four times, the SS values should have comprised SS

_{within-subject}and SS

_{between-subject}. Statistics are not correct here for effects that are repeated within-subjects, such as age and the age : gender interaction. The value of SS

_{T}= 917.7 after summing all of the SS components.

### RMANOVA Model of the Effects of Age, Gender, and Their Interactive Effect

_{age}= 237.19), the age : gender interaction (SS

_{age : gender}= 13.99), and its error term (SS

_{w}= 148.13) comprise the within-subject variability (SS

_{within-subject}). The effects of gender (SS

_{gender}= 140.5) and its error term (SS

_{between}= 377.9) comprise the between-subject variability (SS

_{between-subject}).

_{T}is always constant within a dataset. Changes in F values affect the calculation of P values. In the final RMANOVA model, the result of this is that the P values are either lower or higher than those listed in Table 3A, which indicates that, if RMANOVA is not used, a simple ANOVA will inflate Type I error (false-positives) in between-subject effects and Type II error (false-negative decision) in within-subject effects. A graphical approach may aid the reader in understanding the concept that total variability is comprised of several different sources of variability, denoted by the areas of the rectangles (SS; Fig. 1).

### Sphericity Assumption

*a priori*assessment of the data) allows us to interpret the results of RMANOVA. Returning to the girls dataset, six pairwise differences were calculated (Table 4): 10-8, 12-8, ... , 14-12. The variances of the pairwise differences ranged from 0.60 to 1.74, which appears relatively wide; however, the Mauchly statistic (W) = 0.69, and the estimated P = 0.67, indicating that the girls dataset satisfies the sphericity assumption. A favorable result was expected for this dataset because there was no reasonable basis on which to assume the presence of another factor aside from age over the 2-year periods.