SAS Code for Simultaneous AnalysisSAS Code for Hierarchical AnalysisSPSS Code for Hierarchical AnalysisSIMULTANEOUS AND HIERARCHAL ANALYSIS OF VARIABLESSimultaneous Analysis of the Salary DataHierarchical Analysis of the Salary DataSIMULTANEOUS AND HIERARCHICAL ANALYSIS OF VARIABLE SETSAn Example of the Analysis of Variable SetsHierarchical Analysis of SetsSimultaneous Analysis of SetsModel I and Model II ErrorCourse: Multiple Regression Topic: Regression Diagnostics 1ANALYZING SETS OF VARIABLESThus far we have used multiple regression to partial from variables the effects of other variables. In our salary data, for example, we were able to partial from publications its shared variability with years since PhD to examine the unique effect of publications on salary. In this lecture we will extend the basic concepts of multiple regression to the analysis of variable sets. In general, a set is a collection of related variables. For example, imagine we have various demographic variables (age, sex, SES, etc.) and measures of achievement/ability (GPA, SAT, GRE, number of Olympic gold medals, etc). In addition to examining whether each of the latter variables (age, sex, SAT, Gold Medals, etc.) uniquely predicts, say, popularity, we could examinewhether the achievement variables as a group (i.e., as a set) predict popularity independent of (i.e., unique from) the group (i.e., set) of demographic variables. Before elaborating on the meaning of a set of variables and extending concepts of multiple regression to the analysis of sets, it would be helpful to further develop our understanding of the strategies of simultaneous and hierarchical regression analysis.SIMULTANEOUS AND HIERARCHAL ANALYSIS OF VARIABLESWe previously briefly discussed the distinction between the simultaneous and hierarchicalapproaches to regression analysis. In the simultaneous approach, all of the predictor variables areentered into a single regression model in which the dependent variable is simultaneously regressed onto the predictor variables. In our academic salary data, for example, salary is simultaneously regressed onto PhD, Publications, and Citations:Salary = PhD + Publications + CitationsSuch a model produces regression coefficients, semi-partial correlations, and partial-correlations in which each predictor is fully partialled from all of the other predictors in the model. The squared semi-partial correlations from this model, however, are not additive. That is, the squared semi-partial correlation of each predictor do not sum to the R2 of the model.In the hierarchical approach, the multiple predictor variables are entered sequentially to a series of regression models. For example, we can sequentially add PhD, Publications, and Citations to models predicting salary as follows:Model 1: Salary = PhDModel 2: Salary = PhD + PublicationsModel 3: Salary = PhD + Publications + CitationsNotice that the final model of the hierarchical approach is identical to the single model used in the simultaneous approach. Consequently, the partial betas in this 3rd model are identical to the partial betas in the simultaneous approach. The benefit of the hierarchical approach, however, is that it can be used to determine the percentage of variation in the dependent variable that is accounted for by each of the predictor variables. In other words, it can be used to generate squared semi-partial correlations for each predictor that sum to the R2 of the fullest model (e.g., model 3). In particular, the difference between the R2 of each successive model is equivalent to the percentage of variability in the dependent variable that is uniquely associated with the variable that is most recently added across models (i.e., successive semi-partial correlation). TheSAS Code for Simultaneous Analysisproc reg;model salary = phd pubs citations /scorr2;run;Course: Multiple Regression Topic: Regression Diagnostics 2caveat to keep in mind is that the unique partitioning of the total R2 is specific to the order in which the variables are entered across models. A different partitioning of the R2 would be obtained if Citations were added in Model 2 as opposed to being added in Model 3. Let’s look at some examples to make these abstract concepts a bit more concrete.Simultaneous Analysis of the Salary DataThe following table contains the SAS code for simultaneously regressing salary onto PhD, Publications, and Citations.The following table contains the SAS output.SPSS Code for Simultaneous AnalysisREGRESSION /STATISTICS COEFF OUTS R ANOVA ZPP /DEPENDENT = salary /METHOD = ENTER phd pubs citations.Note: The statements in the “STATISTICS” subcommand are used to request the various tables provided in the SAS output.Course: Multiple Regression Topic: Regression Diagnostics 3The R2 indicates that the model explains roughly 45% of the variability in salary. Despite this relatively impressive R2, none of the partial betas for the predictors are significant – this is due tothe fact that we have low power due to our small sample of 15 observations. In this simultaneousregression, each predictor has partialled from it the effects of the other predictors. Consistent with our previous discussion, notice that the semi-partial correlations do not sum to the total R2 (i.e., .086 + .003 + .063 .45). Finally, we should note that the F-test of the model indicates that our model with PhD, Publications, and Citations does not predict better than does a model without PhD, Publications and Citations, F(3, 11) = 2.98, p = .0779 – again this null effect can beattributed to low power afforded by the small sample size. Because we are going to utilize the concept of model comparison throughout the lecture itwill be beneficial to examine the R2 version of the F-test which was presented in a previous lecture. That is, the F-test in the above SAS output directly compares the R2 of the current model with the R2 of a reduced model that excludes the predictors contained in the current model. The formula for the test can be stated as:FFullFRstrictedFulldfRdfdfRRF)1()(22Re2---=, where df refer to the dferror in SAS for the competing modelsAn algebraically equivalent and more convenient expression of the formula is:SPSS Output for Simultaneous RegressionModel SummaryModel R R SquareAdjusted RSquareStd. Error of theEstimate1 .670(a) .449 .298 5286.553 a Predictors:
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