1. HLM Models

2. General Analysis Strategy Baseline Model - No Predictors Model 1- Level 1 Predictors Model 2 – Level 2 Predictors of Group Mean Model 3 – Variance of Level 1 Predictors Model 4 – Predictors of Level 1 Slopes (cross-level interactions)

3. Model Evaluation Variance Partitioning –Do scores vary at individual vs. group level? –ICC, reliability Tests of Fixed Effects –Is each of the  significantly different from 0? –T-test Tests of Variance Components –Is there significant variance in the parameter across groups? –Chi-square test Explained Variance –How well do predictors in each equation account for the outcome variable? –R 2

4. Testing Fixed Effects  is distributed approximately as t with df = # groups - # level 2 predictors - 1 Confidence Interval Wald Test

5. Test on Variance Component Chi-Square Test df = # groups - # level 2 predictors - 1

6. Baseline Model –Combined model –Analyses: Compute ICC Chi-Square Test on unconditional level-2 variance

7. Model 1: Level 1 Explanatory Variables with Fixed Slopes Analyses –Wald tests on average slope (  10 ) –Individual-Level R 2

8. Interpretation of level-1 intercept and level-2 variance (  00 ) Uncentered –predicted Y when X=0 Group Centered –group mean on Y Grand Centered –group mean adjusted for level-1 predictors Any other value (L) –predicted Y when X=L

9. Model 2: Level 2 Explanatory Variables Combined Equation Analysis –Wald tests on slopes for level 2 predictors (  01 ) –Group-Level R 2 –Chi-square test on residual level 2 variance

10. Level 2 Explanatory Variables: Contextual or Incremental Effects Does the environmental context matter? Does the group’s level on variable X influence behavior, beyond the influence of X as an individual difference? Include the same variable as a predictor at level 1 and level 2.

11. Contextual Effect Group Centered –  01 = total group-level relationship –combination of individual and group- level effects Grand Centered –  01 = unique group- level relationship –e.g., contextual effect

12. Recommendations for Centering Level-1 Predictors Separate Models at different levels –Group Center Incremental or Contextual Effects –Grand Center Cross-Level Interactions –Group Center If you grand center, cross level interaction will be confounded with the level-2 interaction

13. Model 3: Random Slopes Combined Model Analyses –  2 test on random slope (  11 ) –Can remove u’s if not significant

14. Model 4: Cross-Level Interactions (predictors of slopes) Combined Model Analyses –Wald test on cross-level interaction (  11 ) –R 2 for prediction of slope –  2 test on random slope (  11 )

15. Model 4: Level-2 Centering W2 Uncentered –  10 = level-1 slope of X for a group with W2=0 W2 Grand Centered –  10 = average level-1 slope of X  11 not affected by centering of W2

16. HLM Analysis Options Multiple Parameter Test Deviance Test Testing Homogeneity of Level 1 Variance Modeling Heterogeneous Level 1 Variance Create Residual File