INTERPRETING INTERACTIONS BETWEEN QUANTITATIVE NOMINAL VARIABLES Interaction b w Quantitative Nominal Association between DV and quantitative variable changes in direction and or magnitude across levels of the nominal variable That is the slope of the quantitative variable is different across the levels of the nominal variable AND Pattern of differences among the levels of the nominal variable on the DV changes across values of the quantitative variable e g males may score higher on the DV than do females at low levels of Nuances of Nominal Variables G level nominal variable requires G 1 predictors for regression Predictors can be formed via dummy effects or contrast coding With an interaction in the model the interpretation of the betas for the g 1 predictors AND the beta for the quantitative predictor and product terms change across coding systems Organization of Today s Discussion How to 1 Form an interaction b w a nominal quantitative variable 2 Interpret betas for lower order and higher order effects across dummy effects and contrast coding 3 Decompose the interaction An Example Bogus Data Pretend we have the post treatment depression scores and ratings of perceived social support for 26 depressed patients who received either smiling therapy exercise therapy or notherapy We are interested in whether the efficacy of smiling and exercise therapy change depending on the patients perceived level of social support That is do therapy and social support interact to predict post treatment depression An Example Bogus Data Assume depression social support were assessed on valid and reliable scales that range from 1 7 with higher s indicating higher degrees of the measured construct Forming the Interaction Need to code the 3 level therapy variable for regression Form 2 predictors with dummy effects or contrast coding Forming Therapy x Support Interaction 1 Center Continuous predictor i e support 2 Form necessary product terms T1 Support T2 Support 3 Partial all lower order constituent effects Depress B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support B4 and B5 comprise the set that represents Therapy x Support Testing Therapy x Support Interaction Model comparison Depress B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support vs Depress B0 B1T1 B2T2 B3Support How To Interpret the B s from Full Model Depress B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support Context of Dummy Coding Depress B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support B0 mean level of depression in no therapy reference group B1 amount by which mean of smiling differs from mean of no therapy B2 amount by which mean of exercise differs from mean of no therapy B3 support slope in no therapy B4 amount support slope in smil differs from support slope in no therapy B5 amount support slope in exer differs from support slope in no therapy Context of Effects Coding Depress B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support B0 unweighted depression mean across 3 levels of therapy B1 amount smiling mean differs from unweighted mean of all 3 groups B2 amount exercise mean differs from unweighted mean of all 3 groups B3 unweighted mean of the support slopes across 3 levels of therapy B4 amount support slope in smil differs from unweighted mean of support slopes across all 3 groups B5 amount support slope in exer differs from unweighted mean of support slopes across all 3 groups Context of Contrast Coding Depress B0 B1T1 B2T2 B3Support B4T1 Support B5T2 Support B0 unweighted depression mean across 3 levels of therapy B1 amount no ther mean differs from unweighted mean of smil exer B2 amount smil mean differs from exercise mean B3 unweighted mean of the support slopes across 3 levels of therapy B4 amount support slope in no ther differs from unweighted mean of support slopes across smiling and exercise B5 amount support slope in smile differs from support slope in exercise Special Case of a 2 level Nominal Variable e g Smiling therapy and no therapy Need to form only 1 predictor variable contrast is effects coding both differ from dummy coding Contrast effects No ther 1 Smile 1 Dummy No ther 0 Smile 1 Depression B0 B1T1 B2Support B3T1 Support Special Case of a 2 level Nominal Variable Depression B0 B1T1 B2Support B3T1 Support Dummy Coding B0 mean level of depression in no therapy reference group B1 amount by which mean of smiling differs from mean of no therapy B2 support slope in no therapy B3 amount support slope in smil differs from support slope in no therapy Contrast effects Coding B0 unweighted depression mean across 2 levels of therapy B1 amount smiling mean differs from unweighted mean of both groups B2 unweighted mean of the support slopes across both levels of therapy B3 amount support slope in smil differs from unweighted mean of support slopes across both groups Decomposing a Quant x Nominal Interaction Dummy Coding Contrast Coding Decomposing with Dummy Coding Check Cor Among Predictors A more formal check of multi collinearity is to examine the tolerance of each predictor i e 1 R2 in which R2 is from models in which each predictor is regressed on the others Tolerance values 10 are troublesome and suggest that 10 of variability in predictor is unique from the others In SAS use tol option at end of the model statement Test the Interaction proc reg model depress de ds support de sup ds sup model depress de ds support run 2 RFull RRe2 stricted F 2 1 RFull k F k R n k F 1 8413 6469 1 8413 5 3 12 25 26 5 1 Therapy x Support is significant F 2 20 12 25 p 05 Decompose in Context of the Full Model Decompose Therapy x Support Depression 6 25 2 23De 2 58Ds 0 07Support 1 03De Support 0 97Ds Support Effect of therapy changes across values of support effect of support changes across levels of therapy Test dummy coded predictors at high and low value of support or Test simple support slope in each therapy group Test Dummies at High Low Support Dummies high support Re center centered support so it is centered 1 SD above mean form necessary product terms i e recentered support X dummies conduct regression in SAS Depress B0 B1T1 B2T2 B3SupHigh B4T1 SupHigh B5T2 SupHigh B0 depression mean of no therapy at high support B1 diff b w smile and no therapy at high support B2 diff b w exercise and no therapy at high support Dummies low support Re center centered support so it is centered 1 SD below mean form necessary product terms i e recentered support X dummies conduct regression in SAS Depress B0 B1T1 B2T2 B3SupLow B4T1 SupLow B5T2 SupLow B0
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