1. Multilevel Modeling: Other Topics David A. Kenny January 7, 2014

2. 2 Outline Centering and the Three Effects Multiple Correlation Convergence issues in SPSS Tau matrix Significance Testing Non-normal outcomes GEE

3. 3 Centering and the Three Effects The Three Effects of X (a level 1 variable) on Y –Within: effect of X on Y estimated for each level 2 unit and then averaged –Between: effect of mean X on Y –Pooled: an “average” of the two

4. 4 Example: Effect of Daily Stress on Mood Within: the effect of daily stress on mood computed for each person and then averaged Between: Stress is averaged for each person and then average stress is used to predict mood. Pooled: Stress is used to predict mood using all people and all days.

5. 5 Types of Centering and the Effect Grand mean centering –X effect: Pooled estimate Grand mean centering with mean X as a predictor –X effect: Within estimate –Mean X effect: Between minus within estimate Group (or person) mean –X effect: Within estimate

6. 6 Multiple Correlation Not outputted by any MLM program. Estimate a second model without any fixed effects besides the intercept, the empty model. Measure the relative changes in variances with predictors in and out of the model. –s E 2 from the empty model; s E 2 from the model –(s E 2 - s M 2 )/s E 2 –If negative, report as zero. –Sometimes called pseudo R 2.

7. 7 Illustration: Légaré Variances > Term a Empty Model Model R 2 D D 0.105 0.102.040 D P 0.131 0.131.004 P D 0.009 0.008.054 P P 0.184 0.184.000 a D P implies that the respondent is the doctor and that level is that of the patient.

8. 8 Tau Matrix Whenever there is more than one random effect, there is a variance- covariance matrix of random effects. That matrix is called the “tau matrix” in the program HLM. Different programs make different restriction on this matrix.

9. 9 Programs HLM: Unstructured only SPSS and R’s nlme: Allows various possibilities but not any matrix. SAS and MLwiN: User can enter own matrix which gives maximal flexibility.

10. 10 Example: Growth Curve Model with Indistinguishable Dyad Members Slope P1 (1) a Int. P1 (2) c b Slope P2 (3) d e a Int. 2 (4) e f c b (1) (2) (3) (4) Letters symbolize different elements of the tau matrix, some of which are set equal.

11. 11 Significance Testing SPSS uses the Wald test for variances. The likelihood ratio test involving deviance differences is used by other programs and provides a more powerful and accurate test of significance.

12. 12 Non-normal Outcomes Types –Dichotomous or binary outcomes –Counts For these cases, the error variance is not an additional parameter. Basic model is often multiplicative. Can access in SPSS: Mixed Models, Generalized Linear

13. 13 GEE: Generalized Estimating Equations An alternative to MLM Does not test variance components, but rather using a “working model.” Weaker assumptions about the distribution of random variables. Used often in medical research. Used also with non-normal outcomes.