1ANALYZING SETS OF VARIABLES2Variable Set-A collection of related predictors E.g.,Demographic Set Ability/Achievement SetSex GPA Age SAT SES # of Gold Medals-Can apply principles of hierarchical and simultaneous regression to the analysis of setsDoes the Ability set predict popularity independent of the Demographic set?3Organization of Lecture-Further examine the simultaneous & hierarchical strategies-Extend simultaneous & hierarchical strategies to sets4Simultaneous & Hierarchical Analysis-Simultaneous-DV is simultaneously regressed on all of the predictorsSalary = 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 predictorsSalary = 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 R22,,2,22,, PhDnspublicatiocitationssalaryPhDnspublicatiosalaryPhDsalaryCitationsnsPublicatioPhDsalarysrsrrRSimultaneous Analysis of Salary Data5proc reg;model salary = phd pubs citations / scorr2;run;Not additiveTest against model with no predictorsPartialled betas6Model Comparison Test in Output7-F(3, 11) = 2.98, p = .0779-Compares current (full) model with a (reduce) model with no predictorsFFullFRstrictedFulldfRdfdfRRF)1()(22Re2 = )1()1()()(22Re2FFullRFstrictedFullknRkkRRn = # of observations and kF and kR are the number of predictor variables (excluding the Y-intercept) in the full and restricted models, respectively.-current example: n = 15, 3 predictors, & R2 = .448698.2)1315()4486.1()03()04486(.FModel Comparison Test)1()1()()(22Re2FFullRFstrictedFullknRkkRRF8-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 variablesHierarchical 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 Publicationsproc reg;9model salary = phd / scorr2;model salary = phd pubs / scorr2;model salary = phd pubs citations / scorr2;run;1011R2 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., 2nspublicatiosr)-Diff in R2 of Model 2 an 3 is the proportion of variability in salary explained by citations beyond pubs & PhD (i.e., 2citationssr)12R2 Differences in Hierarchical AnalysisModel 1: Salary = PhD…………………………….R2 = .3824Model 2: Salary = PhD Publications ..…………….R2 = .3852Model 3: Salary = PhD Publications Citations…….R2 = .4486-2phdr=.3824unpartialled PhD accounts for 38% of variability in salary-2pubssr=.3852 - .3824 = .0028, pubs accounts for 0.3% of the variability in salary beyond PhD-2citationssr= .4486 - .3852 = .0634citations accounts for 6% of the variability in salary beyond PhD & pubs.3824 + .0028 + .0634 = .4486Do Sequential Variables13Significantly Add to Prediction?(Is sr2 > 0 in the population)-F-test for Sequential models)1()1()()(22Re2FFullRFstrictedFullknRkkRRF-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 CitationsTest of Sequential Effect of Publications-Does Pubs add to prediction beyond PhD (i.e., 2pubssr>0)?14Model 1: Salary = PhD…………………………….R2 = .3824Model 2: Salary = PhD Publications ..…………….R2 = .3852F =(RFull2- RRestricted2)(kF- kR)(1- RFull2)(n- kF- 1)=(.3852 - .3824)(2 - 1)(1- .3852)(15 - 2 - 1)=0.055-No, F(1, 12) = 0.055, p > .05Pubs does not significantly increase prediction of salary beyond PhDTest of Sequential Effect of Citations-Does Citations add to prediction beyond Pubs & PhD (i.e., 2citationssr>0)?15Model 2: Salary = PhD Publications ..…………….R2 = .3852Model 3: Salary = PhD Publications Citations…….R2 = .4486F =(RFull2- RRestricted2)(kF- kR)(1- RFull2)(n - kF- 1)=(.4486 - .3852)(3 - 2)(1- .4486)(15 - 3 - 1)=1.26-No, F(1, 11) = 1.26, p > .05Citations does not significantly increase prediction of salary beyond Pubs & PhDAPA Table of Hierarchical Analysis16-Use Model 1 :Pubs & Citations did not significantly add beyond PhDAnalysis of Variable Sets-Why Sets?17-Hierarchical Analysis of Sets-Simultaneous Analysis of Sets-Model I and Model II ErrorWhy Sets?-Nominal scale variable with more than 2-levelsEg., Therapy (smiling, exercise, or no-therapy)182-contrasts are needed to explain effect of therapy (1) no-therapy vs smiling&exercise (2) smiling vs exerciseIn regression each contrast is entered as a separate predictorGroup together the two predictor variables as a set to account for the effect of therapyWhy Sets?-Quantitative scale variableE.g., degree of Anxiety19-typically interested in linear effect of predictor on DV-might be interested in non-linear effectsquadratic and cubic effects of anxietyGroup together linear, quadratic, and cubic components of anxiety as a set to account for AnxietyWhy Sets?-Conceptually related predictorsmight have measures of different styles of attachmentsecure, anxious, avoidant20-Group together ratings of each attachment style as a set to account for AttachmentAn 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
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