1 ANALYZING SETS OF VARIABLES 2 Variable Set A collection of related predictors E g Demographic Set Sex Age SES Ability Achievement Set GPA SAT of Gold Medals Can apply principles of hierarchical and simultaneous regression to the analysis of sets Does the Ability set predict popularity independent of the Demographic set 3 Organization of Lecture Further examine the simultaneous hierarchical strategies Extend simultaneous hierarchical strategies to sets 4 Simultaneous Hierarchical Analysis Simultaneous DV is simultaneously regressed on all of the predictors Salary PhD Publications Citations Semi partial sr partial correlations pr partial from each predictor all of the other predictors sr2 are not additive i e are not linear components of R2 Hierarchical DV is sequentially regressed on an ordered hierarchy of predictors Salary PhD Salary PhD Publications Salary PhD Publications Citations sr pr partial from each predictor only those predictors that precede it in the sequential series sr2 are additive i e are linear components of R2 2 2 2 2 Rsalary PhD Publications Citations rsalary PhD srsalary publications PhD srsalary citations publications PhD Simultaneous Analysis of Salary Data 5 Test against model with no predictors proc reg model salary phd pubs citations scorr2 run Not additive Partialled betas 6 Model Comparison Test in Output 7 F 3 11 2 98 p 0779 Compares current full model with a reduce model with no predictors 2 RFull RRe2 stricted F 2 Full 1 R df R df F 2 RFull RRe2 stricted 2 Full 1 R df F k F k R n k F 1 n of observations and kF and kR are the number of predictor variables excluding the Yintercept in the full and restricted models respectively current example n 15 3 predictors R2 4486 4486 0 F 1 4486 3 0 2 98 15 3 1 Model Comparison Test 2 RFull RRe2 stricted F 2 1 RFull k F k R n k F 1 8 Can change restricted model to test other hypotheses Hierarchical analysis uses the model comparison test to test whether sequentially added variables predict beyond those of subsequently added variables Hierarchical Analysis of Salary Data Assume we want to test if and how much a Publications predicts beyond the effects of PhD b Citations predicts beyond the effects of Phd and Publications proc reg 9 model salary phd scorr2 model salary phd pubs scorr2 model salary phd pubs citations scorr2 run 10 11 2 R Differences in Hierarchical Analysis Purpose of hierarchical analysis is to determine additive contribution of each predictor R2 difference between sequential models indicates the proportion of variability in DV that is accounted for by the sequentially added variable s Model 1 Salary PhD Model 2 Salary PhD Publications Model 3 Salary PhD Publications Citations Diff in R2 of Model 1 an 2 is the proportion of variability in salary explained by publications beyond PhD i e sr 2 publications Diff in R2 of Model 2 an 3 is the proportion of variability in salary explained by citations beyond pubs PhD i e sr 2 citations 12 2 R Differences in Hierarchical Analysis Model 1 Salary PhD R2 3824 Model 2 Salary PhD Publications R2 3852 Model 3 Salary PhD Publications Citations R2 4486 r 3824 2 phd unpartialled PhD accounts for 38 of variability in salary sr 3852 3824 0028 2 pubs pubs accounts for 0 3 of the variability in salary beyond PhD sr 2 citations 4486 3852 0634 citations accounts for 6 of the variability in salary beyond PhD pubs 3824 0028 0634 4486 Do Sequential Variables 13 Significantly Add to Prediction Is sr2 0 in the population F test for Sequential models 2 RFull RRe2 stricted F 2 1 RFull k F k R n k F 1 Full Restricted are defined by sequential models To test whether publications significantly adds beyond Phd compare Model 1 Salary PhD Model 2 Salary PhD Publications To test whether citations significantly adds beyond pubs Phd compare Model 2 Salary PhD Publications Model 3 Salary PhD Publications Citations Test of Sequential Effect of Publications Does Pubs add to prediction beyond PhD i e 2 srpubs 0 14 Model 1 Salary PhD R2 3824 Model 2 Salary PhD Publications R2 3852 2 2 RFull RRestricted F 1 R 2 Full kF kR n kF 1 3852 3824 1 3852 2 1 0 055 15 2 1 No F 1 12 0 055 p 05 Pubs does not significantly increase prediction of salary beyond PhD Test of Sequential Effect of Citations Does Citations add to prediction beyond Pubs PhD i e sr 0 2 citations 15 Model 2 Salary PhD Publications R2 3852 Model 3 Salary PhD Publications Citations R2 4486 2 2 RFull RRestricted F 1 R 2 Full kF kR n kF 1 4486 3852 1 4486 3 2 1 26 15 3 1 No F 1 11 1 26 p 05 Citations does not significantly increase prediction of salary beyond Pubs PhD APA Table of Hierarchical Analysis 16 Use Model 1 Pubs Citations did not significantly add beyond PhD Analysis of Variable Sets Why Sets 17 Hierarchical Analysis of Sets Simultaneous Analysis of Sets Model I and Model II Error Why Sets Nominal scale variable with more than 2 levels Eg Therapy smiling exercise or no therapy 18 2 contrasts are needed to explain effect of therapy 1 no therapy vs smiling exercise 2 smiling vs exercise In regression each contrast is entered as a separate predictor Group together the two predictor variables as a set to account for the effect of therapy Why Sets Quantitative scale variable E g degree of Anxiety 19 typically interested in linear effect of predictor on DV might be interested in non linear effects quadratic and cubic effects of anxiety Group together linear quadratic and cubic components of anxiety as a set to account for Anxiety Why Sets Conceptually related predictors might have measures of different styles of attachment secure anxious avoidant 20 Group together ratings of each attachment style as a set to account for Attachment An Example of Analyzing Variable Sets Data are from an extensive study of adolescent dating violence Vangie Foshee s Safe Dates Project at UNC 21 Pretend we are interested in distinguishing between attachment style and exposure to parental violence as predictors of perpetration of dating violence Safe dates project has measures of attachment exposure and perpetration DATA Responses of 920 adolescents 529 F 391 M who have dating experience Perpetration 22 How frequently have you used physical force e g hitting kicking stabbing against a dating partner that was not used in self defense 1 never to 4 10 or more times 3 measures of exposure to parental violence 1 witness Have you ever witnessed your mom and dad hit each other 0 no 1 yes 2 Maternal victimization 3 paternal victimization Rated extent to which
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