REPRESENTING NOMINAL PREDICTOR VARIABLES Nominal Variables Predictors in a regression equation do not have to be quantitative variables Predictors can be nominal variables E g Male vs Female American German Japanese Italian persons Different conditions of an experiment Analysis Strategy for Nominal Variables Typically the effect of a nominal variable is assessed with ANOVA ANOVA is a special case of linear regression We can assess nominal variables with OLS regression Nominal Variable in Regression Beta assesses change in DV per unit change in IV Need to represent the levels of the nominal variable with numbers to estimate amount DV changes with shifts from one level of a nominal variable to another level regression doesn t understand male or female Multiple comparisons are necessary to extract all of the info from a nominal variable with more than 2 levels A variable with G levels requires g 1 comparisons Each g 1 comparison is represented by a separate predictor Three Coding Systems Dummy Coding Effects Coding Contrast Coding Similarity Among Coding Systems When g 1 predictors are simultaneously included in the regression equation the 3 coding systems produce the same results for the overall model i e R2 and F value So when g 1 predictors are treated as a set to represent the nominal variable the 3 coding systems produce the same conclusion regarding the omnibus effect of the nominal variable Difference Among Coding Systems Each coding system tests different hypotheses regarding the comparisons among the levels of the nominal variable So the coding systems produce different estimated regression parameters B for the g 1 predictors and statistical tests of those predictors i e t When the nominal variable has only 2 levels the effects coding and contrast coding systems are identical both result in the same statistical test i e t p value of the regression parameter as does dummy coding However the value of the regression parameter will differ for the dummy and contrast effects coding A Data Set Randomly assign depressed patients to one of three therapy conditions Subsequently assess depression 1 8 higher numbers greater depression No Smiling Exercise Therapy Therapy Therapy 7 2 3 7 1 1 6 1 2 6 2 2 x 6 50 x 1 60 x 2 00 n 4 n 5 n 3 unequal n is not a problem for the regression analysis may compromise causal inference Data In SAS To Compare ANOVA Regression A one factor ANOVA indicated that the omnibus effect of therapy on depression was significant F 2 9 64 97 p 0001 Two orthogonal contrasts revealed that depression was greater in the no therapy condition than in the mean of the smiling and exercise therapy conditions F 1 9 123 4 p 0001 and the latter conditions did not differ F 1 9 0 64 p 4433 Therapy F 2 9 64 97 p 0001 No therapy vs Smiling Exercise F 1 9 123 4 p 0001 Smiling vs Exercise F 1 9 0 64 p 4433 Dummy Coding One of the G levels of the nominal variable is treated as a reference level g 1 predictors compare other levels to the reference level only when g 1 predictors are fully partialled E g Therapy has 3 levels so need 2 predictors treat no therapy as reference level predictor1 smiling vs no therapy predictor2 exercise vs no therapy Creating g 1 Dummy Coded Predictors Participant receives either 0 or 1 on each g 1 predictor Receive a 1 if in the condition being compared to the reference level Receive a 0 if not in the condition being compared to the reference level Receive a 0 if in the condition that serves as reference level Dummy Coding as a Function of Condition Unpartialled Partialled Predictors Unpartialled X1 compares smiling with not smiling i e a weighted mean of exercise and no therapy Unpartialled X2 compares exercise with not exercise i e a weighted mean of smiling and no therapy X1 and X2 contain redundant information and are correlated Partialling X2 from X1 removes exercise info and creates a comparison of smiling with no therapy Partialling X1 from X2 removes smiling info and creates a comparison of exercise with no therapy Partialling yields unique meaning to dummy coding Creating Dummy Coded Predictors in SAS Testing Therapy in SAS The set of g 1 predictors contain the effect of Therapy depression d smil d exer vs depression Difference between models yields the effect of Therapy proc reg model depress d smil d exer run SAS Output Model comparison reveals same results as ANOVA Omnibus effect of Therapy F 2 9 64 97 p 0001 Interpretation of Regression Parameters Condition Mean No therapy Smiling Exercise 6 50 1 60 2 00 Depression 6 50 4 90 d smil 4 50 d exer Y intercept mean of the reference level B 4 90 difference between smiling and no therapy d smil Mean level of depression is 4 9 points less in smiling therapy condtion than no therapy condition i e 1 60 6 50 4 90 B 4 50 difference between exercise and no therapy d exer Mean level of depression is 4 5 points less in exercise therapy condtion than no therapy condition i e 2 00 6 50 4 50 Recovering Condition Means from Model Plug 0 1 codes for each condition in the model No therapy coded 0 for d smile and 0 for d exer Depression 6 50 4 90 0 4 50 0 6 50 Smiling therapy coded 1 for d smile and 0 for d exer Depression 6 50 4 90 1 4 50 0 1 60 Exercise therapy coded 0 for d smile and 1 for d exer Depression 6 50 4 90 0 4 50 1 2 00 2 sr for Dummy Coded Predictors The unique interpretation of the dummy coded predictors are obtained by partialling each from the other Therefore the sr2 are from a simultaneous regression and they do not sum to the R2 of the full model Because the g 1 predictors are a set i e Therapy it doesn t make sense to enter them hierarchically can be entered as a set in a hierarchical model to partial out effects of other variables sr2 for d smil indicates the proportion of variability in depression that is explained by the difference between smiling therapy and no therapy Effects Coding Partialled effects coded predictor compares the mean of a given level of nominal variable with the unweighted mean of the means of all levels of the nominal variable E g X exercise X no therapy X X smiling smiling 3 Derived from an ANOVA framework in which the effect of a given treatment is assessed as the difference between the mean of the treatment condition versus the unweighted mean of all the other conditions i e To what extent does a treatment change an outcome relative to the average outcome of all treatment groups Effects Coding With effects coding can determine the effect of g 1 levels In therapy
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