Variance Partitions How to Slice a Pie (into Peachy Pieces)
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? Skill Set
X1 X2 Y 2 2 4 3 3 5 1 1 2 4 4 6 Variance Partition • Assigning variance in Y to a given X. • No problem when Xs are uncorrelated. • Serious problem if Xs are correlated. 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.
GPA (Y) GREQ GREV MAT AR GPA (Y) 1 GREQ .611 1 GREV .581 .468 1 MAT .604 .267 .426 1 AR .621 .508 .405 .525 1 Mean 3.313 565.333 575.333 67.00 3.567 S.D. .600 48.618 83.03 9.248 .838 Unique Variance Approach (1) Recall data from Prediction lecture.
k R2 Variables in Model 1 .385 AR 1 .384 GREQ 1 .365 MAT 1 .338 GREV 2 .583 GREQ MAT 2 .515 GREV AR 2 .503 GREQ AR 2 .493 GREV MAT 2 .492 MAT AR 2 .485 GREQ GREV 3 .617 GREQ GREV MAT 3 .610 GREQ MAT AR 3 .572 GREV MAT AR 3 .572 GREQ GREV AR 4 .640 GREQ GREV MAT AR Unique Variance Approach (2) In hierarchical regression, we add predictors to the equation in a systematic way, examining the change in R2 at each step. In the unique variance approach, we look at the contribution that each variable makes when entered last.
4 Variable R2 3 Variable R2 In model R2 for Result .640 - .617 GREQ GREV MAT AR .023 .640 - .610 GREQ MAT AR GREV .030 .640 - .572 GREV MAT AR GREQ .068 .640 - .572 GREQ GREV AR MAT .068 Unique Variance Approach (3) Unique Var We can find the unique variance in Y attributable to each X by comparing the R2 with all 4 IVs to R2 with 3 IVs. The difference is the proportion of unique variance. Unique Var = (Type III SS) /( Reg SS). Test of sig of b = test of Type III SS = Test of Unique Var inc!
Variables in Equation Variables in R2 R2 R2 change R2 change Useful-ness of Useful-ness of GREV .338 .338 GREV AR .385 .385 AR GREV MAT .493 .155 MAT AR GREQ .503 .118 GREQ GREV MAT GREQ .617 .124 GREQ AR GREQ MAT .610 .107 MAT GREV MAT GREQ AR .640 .023 AR AR GREQ MAT GREV .640 .030 GREV Hierarchical Regression Sequence When IVs are correlated, the sequence of entry in hierarchical regression will matter. R2 change adds up to .64 but look at differences in usefulness. Use hierarchical regression for theory based tests only. Justify sequence.
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?
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.
Unique Variance and Importance SAT GPA GPA SAT GPA SES 1 2 SES 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.
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?