# Mixed-design ANOVA

The mixed-model design ANOVA gets its name because there are two types of variables involved, that is at least one:

• between-subjects variable
• within-subjects variable

## Design

The mixed-design ANOVA model (also known as Split-plot ANOVA (SPANOVA)) tests for mean differences between two or more independent groups while subjecting participants to repeated measures. Thus, there is at least one between-subjects variable and at least one within-subjects variable.

For example, are there differences in males' and females' happiness on weekdays and weekends?

1. Gender (male or female) is the between-subjects variable
2. Happiness Day (weekday or weekend) is the within-subjects variable
3. Of interest are the main effects for Gender and Happiness Day, and the Gender-Happiness Day interaction effect.
4. This could be described as a 2 x (2) mixed-design ANOVA

The results are interest in a two-variable mixed-design ANOVA are:

1. Main effect for the within-subject variable
2. Main effect for the between-subject variable
3. Interaction between the within- and between-subject variable

## Assumption testing

1. Design:
1. One or more within-subject variables e.g., day (weekday and weekend)
2. One or more between-subject variables e.g., gender
2. Sample size - ideally, at least 20 cases per cell
3. Normality - Distribution of the DV (e.g., pulse rate) for each cell is normal
4. Independence: Each participants' responses are sampled independently from each other participants' responses (e.g., this can be satisfied by random selection).
5. Homogeneity of variance: Cells have similar variances.
6. Sphericity: Population variances of the repeated measurements are equal and the population correlations among all pairs of measures are equal. Tested by Mauchly's. Violation increases Type I error rate. If violated, interpret adjusted results (e.g., Greenhouse-Geisser).
7. Homogeneity of inter-correlations: Tested by Box's M: "The assumption ... is that the vector of the dependent variables follow a multivariate normal distribution, and the variance-covariance matrices are equal across the cells formed by the between-subjects effects." (SPSS 14 Help - Tutorial)