Dummy Coding, Centering, & Forming Product Terms in SASDummy Coding, Centering, & Forming Product Terms in SPSSModel Comparison for Testing Therapy x Support Using SPSSModel Comparison for Testing Therapy x Support Using SASFull & Restricted Models for Testing the Therapy x Support Interaction Using SASFull & Restricted Models for Testing the Therapy x Support Interaction Using SPSSTesting Dummy Coded Predictors 1 SD above Support Mean in SPSSTesting Dummy Coded Predictors 1 SD above Support Mean in SASTesting Dummy Coded Predictors 1 SD Below Support Mean in SASRe-coding with Smiling as Reference in SPSSRe-coding with Smiling as Reference in SASRe-coding with Exercise as Reference in SASRe-coding with Exercise as Reference in SPSSFORMING AN INTERACTION BETWEENA QUANTITATIVE AND NOMINAL VARIABLEInterpretation in the Context of Dummy CodingInterpretation in the Context of Effects CodingInterpretation in the Context of Contrast CodingInterpretation In the Special Case of a Two-Level Nominal VariableDECOMPOSING A QUANTITATIVE x NOMINAL INTERACTIONTesting & Decomposing an Interaction in The Context of Dummy CodingBivariate Correlations Among the PredictorsTesting & Decomposing an Interaction in The Context of Contrast CodingCourse: Mult Regression Topic: Interaction Between Quantitative and Nominal Variables 1Interpreting Interactions Between Quantitative and Nominal VariablesAn interaction between a quantitative and nominal variable indicates that the slope between the DV and the quantitative variable changes in direction and/or magnitude across the levels of the nominal variable. Conversely, the pattern of mean difference on the DV across levels of the nominal variable changes in direction and/or magnitude across values of the quantitative variable. Keep in mind that in regression, a G-level nominal variable must be represented with G-1 coded predictors and we can code the levels of the nominal variable using dummy coding, effects coding, or contrast coding. The interpretation of those coded predictors change depending on the method of coding. Likewise, when the nominal variable interacts with aquantitative predictor the interpretations of the lower order effect of the quantitative predictor and the interaction change depending on the method by which the nominal variable is coded. Today we will discuss how to (a) form an interaction between a quantitative and nominal variable, (b) interpret the main effects and interaction for the nominal and quantitative predictor for our three methods of coding (i.e., dummy, effects, and contrast), and (c) decompose the interaction between the quantitative and nominal variables.FORMING AN INTERACTION BETWEEN A QUANTITATIVE AND NOMINAL VARIABLETo make our discussion somewhat less abstract, let’s discuss the statistical procedures in the context of an example. Assume we have the post-treatment depression scores and subjective ratings of perceived social support for depressed patients who received smiling-therapy, exercise-therapy, or no-therapy. We are interested in the efficacy of smiling and exercise therapy as treatments for depression and whether the efficacies of those treatments vary as a function of perceived social support – perhaps the treatments are more or less effective for persons who havehigher or lower levels of perceived social support. Consequently, we are interested in testing whether the three level therapy variable interacts with perceived social support in the prediction of depression. Because the therapy variable is nominal and has two levels, we need to form a set of two-1 degree of freedom predictor variables that represent the therapy variable in the regression analysis. For the sake of notation, we’ll label the two predictors, which as a set represent the 3-level therapy variable, as T1 and T2. Depending on our specific research question, we can form T1and T2 using dummy, effects, or contrast coding. The following table is modified from our previous discussion of nominal variables and demonstrates how the 3-levels of therapy would be coded for each of the coding systems.Coding the Levels of Therapy for Each Coding SystemDummy Effects ContrastTherapy T1T2T1T2T1T2Smiling 1 0 1 0 1/3 ½Exercise 0 1 0 1 1/3 -½No therapy 0 0 -1 -1 -2/3 0For each coding system it is assumed that T1 and T2 are treated as a set and, consequently, each ispartialled from the other. In the context of dummy coding, partialled T1 compares smiling withCourse: Mult Regression Topic: Interaction Between Quantitative and Nominal Variables 2no-therapy and partialled T2 compares exercise with no-therapy. In the context of effects coding, partialled T1 compares smiling with the unweighted mean of all three treatments and partialled T2compares exercise with the unweighted mean of all three treatments. In the context of contrast coding, partialled T1 compares no-therapy with the unweighted mean of smiling and exercise therapy and partialled T2 compares smiling therapy with exercise therapy. Notice that the contrast-coded predictors were weighted with the “optimal weights” that enable a direct interpretation of the betas associated with T1 and T2 (see the lecture notes on nominal variables for details on deriving the optimal weights).To form an interaction between therapy and social support we would (a) center social support (to minimize collinearity between the 1st order predictors and the product term), (b) multiply T1 and T2, respectively, by centered social support, and (c) partial from the product term all lower order constituents of the product term. The following model contains the necessary predictors to test the interaction between therapy and social support:Depression = B0 + B1T1 + B2T2 + B3Support + B4T1*Support + B5T2*SupportKeep in mind that B4 and B5 together as a set represent the Therapy x Support interaction. Consequently, to test the interaction we need to perform a model comparison test between the above model and the following model that excludes B4 and B5:Depression = B0 + B1T1 + B2T2 + B3Support Before examining the process by which we can decompose the interaction, lets discuss the interpretation of the betas from the interaction model for dummy coding, effects coding, and contrast coding.INTERPRETING THE BETAS FROM THE INTERACTION MODEL FOR DUMMY, EFFECTS, AND CONTRAST CODINGBecause the coding systems test different hypotheses regarding the levels of the nominal variable, the coding systems generate different
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