ABSTRACT
Repeated-measures and Mixed ANOVAs In this chapter, you will analyze a new data set that includes repeated measure data. These data allow you to compare four products (or these could be four instructional programs), each of which was evaluated by 12 consumers/judges (6 male and 6 female). The analysis requires statistical techniques for withinsubjects and mixed designs. In Problem 9.1, to do the analysis, you will do a repeated-measures ANOVA, using the General Linear Model program (called GLM) in SPSS. In Problem 9.3, you will use the same GLM program to do a mixed ANOVA, one that has a repeated-measures independent variable and a between-groups independent variable. In Problem 9.2, you will use a nonparametric statistic, the Friedman test, which is similar to the repeated-measures ANOVA. SPSS does not have a nonparametric equivalent to the mixed ANOVA. Chapter 5 provides several tables to help you decide what statistic to use with various types of difference statistics problems. Tables 5.1 and 5.3 include the statistics used in this chapter. Please refer back to Chapter 5 to see how these statistics fit into the big picture. Assumptions of Repeated-measures ANOVA The assumptions of repeated-measures ANOVA are similar to those for between-groups ANOVA, and include independence of observations (unless the dependent data comprise the “within-subjects” or “repeated-measures” factor), normality, and homogeneity of variances. However, in addition to variances, which involve deviations from the mean of each person’s score on one measure, the repeatedmeasures design includes more than one measure for each person. Thus, covariances, which involve deviations from the mean of each of two measures for each person, also exist, and these covariances need to meet certain assumptions as well. The homogeneity assumption for repeated-measures designs, known as sphericity, requires equal variances and covariances for each level of the within-subjects variable. Another way of thinking about sphericity is that, if one created new variables for each pair of withinsubjects variable levels by subtracting each person’s score for one level of the repeated-measures variable from that same person’s score for the other level of the within subject variable, the variances for all of these new difference scores would be equal. Unfortunately, it is rare for behavioral science data to meet the sphericity assumption, and violations of this assumption can seriously affect results. However, fortunately, there are good ways of dealing with this problem-either by adjusting the degrees of freedom or by using a multivariate approach to repeated measures. Both of these are discussed later in this chapter. One can test for the sphericity assumption using the Mauchly’s test, the Box test, the Greenhouse-Geisser test, and/or the Huynh-Feldt tests (see below). Even though the repeated-measures ANOVA is fairly robust to violations of normality, the dependent variable should be approximately normally distributed for each level of the independent variable. Assumptions of the Friedman Test There are two main assumptions of the Friedman test. First, all of the data must come from populations having the same continuous distribution. This is not as stringent as the assumption of normality, which is the common assumption for ANOVA tests. The assumption of a continuous distribution can be checked by creating histograms. The second assumption is independence of observations, which is a design issue. Assumptions of Mixed ANOVA The assumptions for mixed ANOVA are similar to those for repeated-measures ANOVA, except that the assumption of sphericity must hold for levels of the within-subjects variable at each level of betweensubjects variables. This can be tested using SPSS with Box’s M.
