# Mixed Between-Within Subjects ANOVA

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Mixed between-within subjects ANOVA (also known as a split-plot ANOVA) combines two different types of one-way ANOVA into one study: between-groups ANOVA and within-subjects ANOVA. Thus, in a mixed-design ANOVA model, one categorical independent variable is a between-subjects variable and the other categorical independent variable is a within-subjects variable. The mixed-design ANOVA model is used to test for differences between two or more independent groups while subjecting participants to repeated measures. The dependent variable is continuous (measured at the ratio or interval level) and is measured for each group across each level of the repeated factor.

The mixed ANOVA design is unique because there are two factors, one of which is repeated. Since the mixed design employs both types of ANOVA, a brief review of between-groups ANOVA and within-subjects ANOVA is in order:

One-way between-groups ANOVA consists of different subjects or cases in each group - an independent group design. There is one independent (grouping) variable with three or more levels (groups) and one dependent continuous variable. There is only one independent categorical variable with different subjects or cases in each of the groups.

One-way within-subjects ANOVA, also known as repeated-measures ANOVA, measures the same subjects at different points of time or under different conditions and is a dependent group design. This type of ANOVA is used when the subjects encounter repeated measures (i.e., the same subjects are used for each treatment). All subjects participate in all conditions of the research experiment. Each subject responds to every level of the repeated factor, but to only one level of the nonrepeated factor. The participants serve as their own control because they are involved in both the treatment and control groups. The within-subjects design should only be used when the two sets of scores represent measures of exactly the same thing. Therefore exactly the same test needs to be given at both times or under both conditions to all participants.

These two different approaches, of course, could be calculated separately. Often it is more efficient to combine both types of ANOVA into one analysis and study the two factors simultaneously rather than separately. Interactions between factors can also be investigated with this mixed between-within ANOVA design.

Since this mixed-type ANOVA involves two independent variables, it is a type of two-way ANOVA. Two-way (or two factors) ANOVA is used to test the relationship between two categorical independent variables and one continuous dependent variable. With two independent variables, three hypotheses, or main effects, are being tested. Two-way ANOVA introduces a concept not known in a one-way analysis: interaction. Interaction refers to the way in which a category of one independent variable combines with a category of the other independent variable to produce an effect on the dependent variable that goes beyond the sum of the separate effects. It questions whether the effect of one independent variable is consistent for all levels of a second independent variable. Interaction is a feature common in both experimental and observational studies.

An example of mixed between-within subjects ANOVA is a study investigating the impact of an intervention on participants' depression symptoms (using a pre-test and post-test design), but also investigating whether the impact varies for gender (males and females). In this case, there are two independent variables: gender (between-subjects variable) and time (within-subjects variable). The researcher would perform the intervention on both groups of males and females, and then measure their depression symptoms over time (Time 1 = pre-intervention and Time 2 = after the intervention).

Assumptions: same assumptions as with t-tests and one-way ANOVA, plus homogeneity of inter-correlations.

There are three null hypotheses in mixed between-within subjects ANOVA. F statistics and p-values are used to test hypotheses about the main effects and the interaction. An F-statistic is computed to test for between-subject effect. Another F-statistic is computed to test for within-subjects effect or time factor. This statistic indicates whether, across the groups, the dependent variable differs over time. Finally, an interaction effect is tested to determine whether group differences vary across time. The test for interaction should be examined first, since the presence of a strong interaction may influence the interpretation of the main effects. Plots are a useful aid.

The effect size for mixed between-within subjects ANOVA is calculated by the partial eta squared statistic:

Small effect .01

Moderate effect .06

Large effect .14

When a researcher reports the results from a mixed between-within subjects ANOVA, he or she needs to include the following information: verification of parametric assumptions; verification of homogeneity of inter-correlations (Box's M statistic); verification of homogeneity of variances (Levene's Test of Equality of Error Variances); interaction effect (Wilks' Lambda); dependent variable scores; independent variables, levels; statistical data: significance, F-ratio scores, probability, means measured for each group across each level of the repeated factor, group standard deviations, and effect size. An example is below:

Presenting the Results from Mixed Between-Within ANOVA (Pallant, 2007, p. 274)

A mixed between-within subjects analysis of variance was conducted to assess the impact of two different interventions (Math skills, Confidence Building) on participants' scores on the Fear of Statistics Test, across three time periods (pre-intervention, post-intervention, and 3-mth follow-up). There was no significant interaction between program type and time, Wilks Lambda = .87, F (2, 27) = 2.03, p = .15, partial eta squared = .13. There was a substantial main effect for time, Wilks Lambda = .34, F (2, 27) = 26.59, p

References

Moore, D. S., & McCabe, G. P. (2003). Introduction to the practice of statistics (4th ed.). New York: W. H. Freeman and Company.

Pallant, J. (2007). SPSS survival manual. New York: McGraw-Hill Education.

Polit, D. F., & Beck, C. T. (2008). Nursing research: Generating and assessing evidence for nursing practice (8th ed.). Philadelphia: Wolters Kluwer Health.

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