Mixed-design Analysis of Variance - Partitioning The Sums of Squares & The Logic of ANOVA

Partitioning The Sums of Squares & The Logic of ANOVA

Due to the fact that the mixed-design ANOVA uses both between-subject variables and within-subject variables (a.k.a repeated measures), it is necessary to partition out (or separate) the between-subject effects and the within-subject effects. It is as if you are running two separate ANOVAs with the same data set, except that it is possible to examine the interaction of the two effects in a mixed design. As can be seen in the source table provided below, the between-subject variables can be partitioned into the main effect of the first factor and into the error term. The within-subjects terms can be partitioned into three terms: the second (within-subjects) factor, the interaction term for the first and second factors, and the error term. The main difference between the sum of squares of the within-subject factors and between-subject factors is that within-subject factors have an interaction factor.

More specifically, the total sum of squares in a regular one-way ANOVA would consist of two parts: variance due to treatment or condition (SS_{between-subjects}) and variance due to error (SS_{within-subjects}). Normally the SS_{within-subjects} is a measurement of variance. In a mixed-design, you are taking repeated measures from the same participants and therefore the sum of squares can be broken down even further into three components: SS_{within-subjects} (variance due to being in different repeated measure conditions), SS_error (other variance), and SS_BT*WT (variance of interaction of between-subjects by within-subjects conditions).

Each effect has its own F value. Both the between-subject and within-subject factors have their own MS_error term which is used to calculate separate F values.

Between-subjects:

• F_{Between-subjects} = MS_{between-subjects}/MS_{Error(between-subjects)}

Within-subjects:

• F_{Within-subjects} = MS_{within-subjects}/MS_{Error(within-subjects)}

• F_BSXWS = MS_{between X within}/MS_{Error(within-subjects)}