Course Mult Regression Topic Interaction Between Quantitative and Nominal Variables 1 Interpreting Interactions Between Quantitative and Nominal Variables An 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 a quantitative 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 VARIABLE To 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 exercisetherapy 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 have higher 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 two1 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 3level therapy variable as T1 and T2 Depending on our specific research question we can form T1 and 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 System Dummy Effects Contrast Therapy T1 T2 T1 T2 T1 T2 Smiling 1 0 1 0 1 3 Exercise 0 1 0 1 1 3 No therapy 0 0 1 1 2 3 0 For each coding system it is assumed that T1 and T2 are treated as a set and consequently each is partialled from the other In the context of dummy coding partialled T1 compares smiling with Course Mult Regression Topic Interaction Between Quantitative and Nominal Variables 2 no 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 T2 compares 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 Support Keep 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 CODING Because the coding systems test different hypotheses regarding the levels of the nominal variable the coding systems generate different betas for each parameter in the interaction model Let s use the following model to discuss the meaning of each beta for the three coding systems Depression B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support Interpretation in the Context of Dummy Coding Dummy coding compares each level of the nominal variable with the level that serves as the reference level which in the current example is no therapy In the context of an interaction model with dummy coding B0 is the mean level of depression in no therapy i e the reference group B1 indicates the amount by which the mean of smiling therapy differs from no therapy B2 indicates the amount by which the mean of depression in exercise therapy differs from the mean of depression in no therapy B3 is the social support slope in the reference group i e no therapy B4 which is part of the Therapy x Support interaction indicates the extent to which the support slope in smiling therapy differs from support slope in no therapy Likewise B5 which is also a part of the Therapy x Support interaction indicates the extent to which the support slope in exercise therapy differs from the support slope in no therapy Course Mult Regression Topic Interaction Between Quantitative and Nominal Variables 3 Interpretation in the Context of Effects Coding Effects coding compares each level of the nominal variable excluding
View Full Document
Unlocking...