Mixed BetweenWithin Subjects ANOVA

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Mixed betweenwithin subjects ANOVA (also known as a splitplot ANOVA) combines two different types of oneway ANOVA into one study: betweengroups ANOVA and withinsubjects ANOVA. Thus, in a mixeddesign ANOVA model, one categorical independent variable is a betweensubjects variable and the other categorical independent variable is a withinsubjects variable. The mixeddesign 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 betweengroups ANOVA and withinsubjects ANOVA is in order:
Oneway betweengroups 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.
Oneway withinsubjects ANOVA, also known as repeatedmeasures 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 withinsubjects 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 betweenwithin ANOVA design.
Since this mixedtype ANOVA involves two independent variables, it is a type of twoway ANOVA. Twoway (or two factor) 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. Twoway ANOVA introduces a concept not known in oneway 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 betweenwithin subjects ANOVA is a study investigating the impact of an intervention on participants’ depression symptoms (using pretest and posttest design), but also investigating whether the impact varies for gender (males and females). In this case, there are two independent variables: gender (betweensubjects variable) and time (withinsubjects variable). The researcher would perform the intervention on both groups of males and females, and then measure their depression symptoms over time (Time 1 = preintervention and Time 2 = after the intervention).
Assumptions: same assumptions as with ttests and oneway ANOVA, plus homogeneity of intercorrelations.
There are three null hypotheses in mixed betweenwithin subjects ANOVA. F statistics and pvalues are used to test hypotheses about the main effects and the interaction. An Fstatistic is computed to test for betweensubject effect. Another Fstatistic is computed to test for withinsubjects 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 main effects. Plots are a useful aid.
The effect size for mixed betweenwithin 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 betweenwithin subjects ANOVA, he or she needs to include the following information: verification of parametric assumptions; verification of homogeneity of intercorrelations (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, Fratio 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 BetweenWithin ANOVA (Pallant, 2007, p. 274)
A mixed betweenwithin 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 (preintervention, postintervention and 3mth followup). 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 < .0005, partial eta squared = .66, with both groups showing a reduction in Fear of Statistics Test scores across the three time periods (see Table 1). The main effect comparing the two types of intervention was not significant, F (1, 28) = .059, p = .81, partial eta squared = .002, suggesting no difference in the effectiveness of the two teaching approaches.
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: McGrawHill Education.
Polit, D. F., & Beck, C. T. (2008). Nursing research: Generating and assessing evidence for nursing practice (8th ed.). Philadelphia: Wolters Kluwer Health.Do you like this Article? Click Like? 
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