Multilevel Modeling:Other Topics

David A. Kenny

January 25, 2014

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– X effect: Within estimate

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.– sE

2 from the empty model; sM2 from the model

– (sE2 - sM

2)/sE2

– If negative, report as zero.

– Sometimes called pseudo R2.

8

Illustration: Doctor-Patient Variances >

Terma Empty Model Model R2

DD 0.105 0.102 .040

DP 0.131 0.131 .004

PD 0.009 0.008 .054

PP 0.184 0.184 .000aDP 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

ReferencesReferences (pdf)