1 of 30 Mixed Models ANOVA While the term “mixed models ANOVA” can be used to describe the inclusion of a variety of different combinations of independent variables, we will use it to imply a repeated measures model that also includes a factor variable. To distinguish the different roles of the variables, the repeated measures factor is referred to as a “within-subjects” factor, while the factor comparing groups or categories is the “between-subjects” factor. Specifically, our analysis will look at how a between-subjects factor affects, or interacts with, the within-subjects factor (the repeated measures factor). Our goal in this analysis is to test and interpret a model which focuses on the interaction between the between-subjects factor and the repeated measures factor. Our research hypothesis is that there is a significant interaction effect, and the subjects have a different pattern of change over time depending on which category of the between-subjects factor they are associated with. We will use the Omaha.sav data base which contains data on the feelings and attitudes of victims of domestic. We will look at how these feelings and attitudes change over time, and whether or not the pattern is different for victims who are employed compare to those who are not employed. These analyses could test a theory that employment changes the responses to domestic violence. Victims who have a job have a broader social network whose support offsets the loss of self-esteem, the tenuous feelings of control, and the fears experienced of victims. (PLEASE NOTE: I am making this theory up to fit the data that is available; I do not know enough about domestic violence to know whether or not the “theory” would be supported by experts.) If the presence of an interaction between the repeated measures factor and the repeated measures factor is supported by our analysis, we will refine our interpretation by using the syntax features of SPSS to force it to do the Bonferroni post hoc test for the interaction effect. The interpretation of the interaction will be similar to the statements we made about interactions in the two-factor ANOVA problems, bolstered by a test of significance. If the interaction effect between the repeated measures factor and the between-subjects factor is not significant, we revert to the interpretation of the repeated measures, as we did for the repeated measures problems without any additional factor. While we could also interpret the main effect for the between subjects factor, we will not because it complicates our interpretation without providing additional useful information. Interpreting the employment factor would be answering questions of whether or not victims differed at a specific time, rather than testing whether or not they experienced a different pattern over time. In combining repeated measures ANOVA and factorial ANOVA, we will be required to satisfy the requirements and assumptions of both: • Mixed models analysis of variance requires that the repeated measure variables be interval level and the between-subject factor be any level that defines groups (dichotomous, nominal, ordinal, or grouped interval) • There is a minimum sample size both for the total number of subjects (10 + the number of time periods making up the within-subjects factor) and the minimum number in each cell (5) • The assumption of normality for the repeated measures • The assumption of sphericity for the within-subjects factor for the repeated measures • The assumption of homogeneity of variance for the between-subjects factor2 of 30 Solving Mixed Models Problems in SPSS Statements about requirements and assumptions Note that we use a more conservative alpha (.01) for diagnostic statistics than we do for the statistics that answer our research questions. The introductory statement identifies the variables for the analysis and the significance levels to use. The repeated measures and the factor are listed in the problem statement. The first block of statements addresses level of measurement, sample size, and assumptions required for the use of the statistic.3 of 30 Statements about the interaction effect Statements about the main effect of change over time The next block of statements is about the presence and interpretation of the interaction effect. If there is an interaction effect, we will mark the first check box, and the check boxes below it that are the correct interpretation of the interaction. If there is a significant interaction effect, we will not do a separate interpretation of the main effect of change over time. The final block of questions interprets the main effect for change over time when the interaction effect is not significant. We mark the first check box if there is a significant main effect for the repeated measures, and any post hoc effects that are statistically significant.4 of 30 Level of Measurement The Sample Size Requirement The variables "At times I fear for my life (1 week)" [fear1_4],"At times I fear for my life (6 months)" [fear6_4] or "At times I fear for my life (12 months)" [fear12_4] are ordinal level. However, we will follow the common convention of treating ordinal variables as interval level. We should consider including the use of this convention as a limitation of the analysis. "The victim's employment status" [employed] is dichotomous satisfying the requirement for the factor. To check sample size requirements, we run the univariate general linear model procedure. This procedure will give us the total number of cases and the correct number of cases in each cell, removing any missing data for the three time period variables and the between-subjects factor. The minimum sample size requirement that we will use for mixed models ANOVA is 10 + the number of dependent variables, adapted from Tabachnick and Fidell. We will also require that the smallest cell have 5 or more cases. Mixed-model analysis of variance requires that the repeated measure variables be interval level and the between-subjects factor be any level that defines groups (dichotomous, nominal, ordinal, or grouped interval).5 of 30 Using Univariate General Linear Model for Mixed Models - 1 Using Univariate General Linear Model for Mixed Models - 2 Select General Linear Model > Repeated Measures from the Analyze menu. Third, click on the Add button to add this factor to the list. First, change the name of the Within-Subject Factor