1. Variance Partitions How to Slice a Pie (into Peachy Pieces)

2. Skill Set Why does the order of entry in a prediction equation change the incremental variance accounted for by a variable? What is commonality analysis? How is it used? How can a variable be important from an understanding point of view even if its unique proportion of variance is small?

3. Variance Partition Assigning variance in Y to a given X. No problem when Xs are uncorrelated. Serious problem if Xs are correlated. X1X2Y 224 335 112 446 Is the change in Y due to X1 or to X2? There is no way to tell because there is no case where X1 goes up as X2 goes down or vice versa.

4. Unique Variance Approach (1) GPA (Y) GREQGREVMATAR GPA (Y) 1 GREQ.6111 GREV.581.4681 MAT.604.267.4261 AR.621.508.405.5251 Mean3.313565.333575.33367.003.567 S.D..60048.61883.039.248.838 Recall data from Prediction lecture.

5. Unique Variance Approach (2) kR2R2 Variables in Model 1.385AR 1.384GREQ 1.365MAT 1.338GREV 2.583GREQ MAT 2.515GREV AR 2.503GREQ AR 2.493GREV MAT 2.492MAT AR 2.485GREQ GREV 3.617GREQ GREV MAT 3.610GREQ MAT AR 3.572GREV MAT AR 3.572GREQ GREV AR 4.640GREQ GREV MAT AR In hierarchical regression, we add predictors to the equation in a systematic way, examining the change in R 2 at each step. In the unique variance approach, we look at the contribution that each variable makes when entered last.

6. Unique Variance Approach (3) 4 Variable R 2 3 Variable R 2 In model R 2 forResult.640 -.617GREQ GREV MAT AR.023.640 -.610GREQ MAT AR GREV.030.640 -.572GREV MAT AR GREQ.068.640 -.572GREQ GREV AR MAT.068 We can find the unique variance in Y attributable to each X by comparing the R 2 with all 4 IVs to R 2 with 3 IVs. The difference is the proportion of unique variance. Unique Var Unique Var = (Type III SS) /( Reg SS). Test of sig of b = test of Type III SS = Test of Unique Var inc!

7. Hierarchical Regression Sequence When IVs are correlated, the sequence of entry in hierarchical regression will matter. R 2 change adds up to.64 but look at differences in usefulness. Use hierarchical regression for theory based tests only. Justify sequence. Variables inR2R2 R 2 change Useful-ness of AR.385 AR AR GREQ.503.118GREQ AR GREQ MAT.610.107MAT AR GREQ MAT GREV.640.030GREV Variables in EquationR2R2 R 2 changeUseful-ness of GREV.338 GREV GREV MAT.493.155MAT GREV MAT GREQ.617.124GREQ GREV MAT GREQ AR.640.023AR

8. Review Why does the order of entry in a prediction equation change the incremental variance accounted for by a variable? Authors of an article report a hierarchical regression. In step 1, variables 1, 2 and 3 are included and R-square is.20, p<.05. In step 2, variable 4 and 5 are entered and R-square increases to.25; the increase is significant. The b-weight for variable 4 is not significant and the authors conclude that variable 4 is not important. What about that? Reasonable?

9. Commonality Analysis Commonality analysis is a good way to partition the variance in Y. It shows the unique parts for each X, and then also shows the shared parts for each combination of X variables. Note C12 and C123, for example. Each area is defined mathematically. For example: Use this if you want to tell a story beyond that told by the combination of b and r. Always report b and r.

10. Unique Variance and Importance If a variable adds unique variance to a regression equation, then it is important in the sense that it helps prediction. However, it is possible for a variable to be theoretically important and not add much or even any unique variance to a regression equation. In Fig 1, if SAT doesn’t add unique variance to GPA, OK. In Fig 2, SAT acts to explain the influence of SES on GPA, so it is important theoretically even if it doesn’t add unique variance. SATGPA SES GPA SAT SES 1 2

11. Review What is commonality analysis? How is it used? How can a variable be important from an understanding point of view even if its unique proportion of variance is small?