1 of 23 Solving Linear Regression 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 Multiple regression requires that the dependent variable be interval and the independent variables be interval or dichotomous. If one of the variables 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. These problems use the rule of thumb from Tabachnick and Fidell that the required number of cases should be the larger of the number of independent variables x 8 + 50 or the number of independent variables + 105. 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. In many of problems we have worked, failing to meet sample size implies that it is an inappropriate application of the statistic and we halted all further work on the problem. We will not apply that policy to these problems. If our sample size is less than the minimum requirement, we leave the check box unmarked and continue with the problem, mention the sample size issues as a limitation for the analysis.2 of 23 The Assumption of Normality Regression assumes that the residual are normally distributed. We will meet this assumption if each of the interval variables 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 sample size is large. The problems evaluate normality based on the criteria that the skewness and kurtosis of each variable falls within the range from -1.0 to +1.0. If the variables satisfies these criteria for skewness and kurtosis, the check box “The skewness and kurtosis of the variables 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 variables. The Assumption of Homoscedasticity Regression assumes that the variance of the residuals is homogeneous across predicted values of the dependent variable. SPSS does not compute Levene’s test for equality of variance when all of the variables are interval (or ordinal treated as interval). The check box “The regression analysis satisfies assumption of homoscedasticity” will remain unchecked for these problems. The Assumption of Linearity The assumption of linearity is tested with the lack of fit test in the Univariate General Linear Model procedure. If the test is significant, it implies that there is a non-linear component that should be added to the model. If the test is not significant, we assume that a linear model is present and is an adequate representation of the relationship between the dependent and independent variables. If the lack of fit test is not significant at the alpha level for diagnositic statistics, the check box “The regression analysis satisfies the assumption of linearity” is marked. The Assumption of Independence of Errors SPSS does not compute the Durbin-Watson statistic in the Univariate General Linear Model procedure. In these problems, we will acknowledge that fact and not mark the check box “The regression analysis satisfies the assumption of independence of errors”. The Assumption of Independence of Variables SPSS does not compute tolerance for VIF in the Univariate General Linear Model procedure. In these problems, we will acknowledge that fact and not mark the check box “The regression analysis satisfies the assumption of independence of variables”. I have included the complete list of assumptions in the list of possible answers even though some will not ever be marked in this assignment because of limitations in the univariate general linear procedure. In the future, we will develop a strategy for testing all of the assumptions.3 of 23 Interpretation of the Overall Relationship The presence of overall relationship between the dependent variable and the independent variables is represented by the statement that both predictors together have a relationship to the dependent variable. If the ANOVA test of the overall relationship (“Corrected Model” in the table of “Tests of Between Subjects Factors”) is not statistically significant, this statement is not marked as a correct finding. If the overall relationship is not statistically significant, we will not interpret the individual relationships. If the overall relationship is statistically significant, we should examine the adjective describing the strength of the relationship. 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). If the adjective describing the strength of the relationship is not correct, the check box for the overall relationship is not marked. Interpretation of the Individual Relationships Determination of the correctness of statements about individual relationships is a two stage process. First, it is required that the relationship be statistically significant (the test of the slope in the table of “Parameter Estimates”). Second, it is required that the statement be correct a correct interpretation of the direction of the relationship with the