Multilevel Modeling: Other Topics David A. Kenny January 25, 2014 Presumed Background • Multilevel Modeling • Nested Design • Growth Curve Models Outline • • • • Centering and the Three Effects Multiple Correlation Tau Matrix Modeling Nonindependence with Overtime Data • Significance Testing • Non-normal Outcomes • GEE 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 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 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 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. – sE2 from the empty model; sM2 from the model – (sE2 - sM2)/sE2 – If negative, report as zero. – Sometimes called pseudo R2. 7 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 .000 aD implies that the respondent is the doctor P and that level is that of the patient. 8 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. 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 Example: Growth Curve Model with Indistinguishable Dyad Members Slope P1 (1) Int. P1 (2) Slope P2 (3) Int. P2 (4) a c b d e a e f c b (1) (2) (3) (4) Letters symbolize different elements of the tau matrix, some of which are set equal. 11 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. 12 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). 13 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. 14 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 15 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. 16 References References (pdf) 17