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# Repeated-Measures and Mixed ANOVAs

DOI link for Repeated-Measures and Mixed ANOVAs

Repeated-Measures and Mixed ANOVAs book

# Repeated-Measures and Mixed ANOVAs

DOI link for Repeated-Measures and Mixed ANOVAs

Repeated-Measures and Mixed ANOVAs book

## 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 10.1, to do the analysis, you will do a repeated-measures ANOVA, using the General Linear Model program (called GLM) in SPSS. In Problem 10.3, 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 10.2, 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 (except for those analyzed using the “within-subjects” or “repeatedmeasures” factor), normality, and homogeneity of variances. Variances are deviations of each person’s score on a single measure from the mean for that measure, multiplied by themselves (squared). In addition to variances, because the repeated-measures design includes multiple measures (the same measure at more than one time point), it includes covariances. Covariances are the same as variances, except that instead of being calculated by multiplying each deviation score by itself, they involve multiplying each person’s deviation scores on two variables, in this case the same variable at different times (repeated measures), paired across time. Thus, 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 within-subjects 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. For example, if you have three groups (Group A, Group B, and Group C) sphericity is assessing the following: varianceA-B varianceA-C varianceB-C. 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 (other than the dependency that is dealt with by the within-subjects variable). 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 through the Multivariate General Linear Model.