1. Multilevel Modeling: Other Topics
David A. Kenny

2. Presumed Background Multilevel Modeling Nested Design*
Presumed Background
Multilevel Modeling
Nested Design
Growth Curve Models *

3. Outline Centering and the Three Effects Multiple Correlation
Tau Matrix
Modeling Nonindependence with Overtime Data
Significance Testing
Non-normal Outcomes
GEE

4. 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

5. 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.

6. 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 centering

7. 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.
sE2 from the empty model; sM2 from the model
(sE2 - sM2)/sE2
If negative, report as zero.
Sometimes called pseudo R2.

8. Illustration: Doctor-Patient
Variances >
Terma Empty Model Model R2 DD DP PD PP
aDP implies that the respondent is the doctor and that level is that of the patient.

9. Tau Matrix
Whenever there is more than one random effect at level 2, 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.

10. 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.

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

12. Modeling Nonindependence with Overtime Data
k overtime measurements
There are k(k + 1)/2 variances and covariances.
That makes k(k + 1)/2 potential parameters that could be estimated.
Note that with a growth curve model only 4 parameters are estimated or 5 with autoregressive errors.

13. We Should Test
Fit of a growth-curve model can be compared to the fit of a repeated measures model that exactly fits all variances and covariances (saturated error model).
If the growth-curve model fits as well as the saturated model, the simpler model (growth-curve model) is preferred).

14. 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.

15. 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

16. 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.

17. References
References (pdf)