UT SW 388R - Solving One-way ANOVA Problems as a General Linear Model

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1 of 26 Solving One-way ANOVA Problems as a General Linear Model Homework problems are multiple answer rather than multiple choice. The format for multiple answer questions is shown in the example below. The directions for the problems instruct you to mark the check boxes for all of the statements that are true. One or more answers must be marked for each problem. Full or partial credit is computed for each question. To receive full credit, you must mark all of the correct answers and not mark any of the incorrect answers. Partial credit is computed by summing the points for each correct response and subtracting points for each incorrect answer. If the computation for partial credit results in a negative number, zero credit is assigned. Level of Measurement Requirement and Sample Size Requirement In a one-way analysis of variance, the level of measurement for the independent variable can be any level that defines groups (dichotomous, nominal, ordinal, or grouped interval) and the dependent variable is required to be interval level. If the dependent variable is ordinal level, we will follow the common convention of treating ordinal variables as interval level, but we should consider noting the use of an ordinal variable as a limitation to our findings. I have imposed a minimum sample size requirement of 5 cases per category of the independent variable for these problems. This is a convention for these problems and is based on the needed to have a reasonably stable mean for each cell when analyzing observational data. If the sample size requirement (along with the level of measurement requirement) is satisfied, the check box “The level of measurement requirement and the sample size requirement are satisfied” should be marked. If the sample size requirement is not satisfied, the correct answer to the problem is “Inappropriate application of the statistic.” All other answers should be unmarked when the answer is “Inappropriate application of the statistic.”2 of 26 The Assumption of Normality Analysis of variance assumes that the dependent variable is normally distributed, but there is general consensus that violations of this assumption do not seriously affect the probabilities needed for statistical decision making, especially when the number of cases in each cell are equal. Our problems evaluate normality based on the criteria that the skewness and kurtosis of the dependent variable fall within the range from -1.0 to +1.0. If the dependent variable satisfies these criteria for skewness and kurtosis, the check box “The skewness and kurtosis of income satisfy the assumption of normality” should be marked. If the criteria for normality are not satisfied, the check box should remain unmarked and we should consider including a statement about the violation of this assumption in the discussion of our results. In these problems we will not test transformations or consider removing outliers to improve the normality of the variable. The Assumption of Homogeneity of Variance Analysis of variance assumes that the variance of the dependent variable is homogeneous across all of the cells formed by the factors (independent variable). We will use the significance of Levene’s test for equality of variance as our criteria for satisfying the assumption. SPSS computes the Levene test as part of the output for general linear models. Levene’s test is a diagnostic statistic that tests the null hypothesis that the variance is homogeneous or equal across all cells. The desired outcome, and support for satisfying the assumption, is to fail to reject the null hypothesis. If the significance for the Levene test is greater that the alpha for diagnostic statistics, we fail to reject the null hypothesis and the check box “The assumption of homogeneity of variance is supported by Levene's test for equality of variances” should be marked. If the criterion for homogeneity of variance is not satisfied, the check box should remain unmarked. Analysis of variance is robust to violations of the assumption of homogeneity of variances provided the ratio of the largest group variance is not more than 3 time the smallest group variance. If we violate this assumption, but the ratio is less than or equal to 3.0, we should consider including a statement about the violation of this assumption in the discussion of our results. If we violate this assumption and the ratio of largest to smallest variance is greater than 3.0, we should not use a one-way analysis of variance for the data for these variables and we mark the check box, “Inappropriate application of the statistic.” The check boxes for level of measurement and sample size, and the assumption of normality should remain marked if they were previously satisfied, even if the problem is found to be an inappropriate application of a statistic because of heterogeneity. Interpretation of the Relationship The statement of the relationship between the dependent and independent variable is a statement that the different categories of the independent variable are linked to different average scored on the dependent variable. The statement is correct if the relationship is statistically significant in the table of “Tests of Between-Subjects Effects.” Since there is only one independent variable in this analysis, the table entries for the “Corrected Model” and the variable will be identical.3 of 26 SPSS computes partial eta squared as a measure of effect. We characterize it as trivial, small, moderate, or large, applying Cohen's criteria for effect size (less than .01 = trivial; .01 up to 0.06 = small; .06 up to .14 = moderate; .14 or greater = large). Effect size should only be interpreted if the relationship is statistically significant. Determination of the correctness of statements about specific relationships is a two stage process. First, it is required that the relationship be statistically significant and the strength of the relationship be correctly described. Second, it is required that the statement be a correct comparison of the direction of the means, based on either a direct comparison of the group means when the factor contains two categories, or a post-hoc test when the factor includes three or more categories. We will use the Bonferroni test for multiple comparisons for these and future problems rather than the Tukey HSD or Games-Howell post test. The Bonferroni test is a set of t-tests of the


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UT SW 388R - Solving One-way ANOVA Problems as a General Linear Model

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