Adaptive Centering with Random Effects in Studies of Time-Varying Treatments

1. Adaptive Centering with Random Effects in Studies of Time-Varying Treatments Stephen W. Raudenbush University of Chicago December 11, 2006

2. Adaptive Centering with Random Effects in Studies of Time-Varying Treatmentsby Stephen W. RaudenbushUniversity of ChicagoAbstract Of widespread interest in education are observational studies in which children are exposed to interventions as they pass through classrooms and schools. The interventions might include instructional approaches, levels of teacher qualifications, or school organization. As in all observational studies, the non-randomized assignment of treatments poses challenges to valid causal inference. An attractive feature of panel studies with time-varying treatments, however, is that the design makes it possible to remove the influence of unobserved time-invariant confounders in assessing the impact of treatments. The removal of such confounding is typically achieved by including fixed effects of children and/or schools. In this paper, I introduce an alternative procedure: adaptive centering of treatment variables with random effects. I demonstrate how this alternative procedure can be specified to replicate the popular fixed effects approach in any dimension. I then argue that this alternative approach offers a number of important advantages: appropriately incorporating clustering in standard errors, modeling heterogeneity of treatment effects, improved estimation of unit-specific effects, and computational simplicity.

3. Claims • Adaptive centering with random effects can replicate the fixed effects analysis of time-varying treatments in any dimension of clustering. • Adaptive centering with random effects has several advantages • Incorporating multiple sources of uncertainty • Modeling heterogeneity • Modeling multi-level treatments • Improved estimates of unit-specific effects • Computational simplicity

4. Table 1. Outcome data for 20 hypothetical kids by 9 teachers nested with 3 schools

5. Table 2 Correlations

6. 1. “True model” Estimates of Fixed Effects

7. Methods of Estimation • OLS – no control • Child random effects • Child fixed effects: • Child random effects, within-child centering • Child and school random effects • Child and school fixed effects • Child and school random effects, two-way centering • Without teacher random effects • With teacher random effects*

8. OLS : No Control Estimates of Fixed Effects

9. Child random effects “as if randomized” Estimates of Fixed Effects

10. One-Dimensional Control: OLS Fixed Child Effects Estimates of Covariance Parameters

11. One-Dimensional Control:Child random effects with person-mean centered x Note this gives the same coefficient, standard error, and residual variance estimate as the student fixed effects model. Estimates of Fixed Effects Estimate of Covariance Parameters

12. Table 3. Treatment Received

13. Random child and school effects with x “as if randomized” Estimates of Fixed Effects Estimates of Covariance Parameters

14. Two dimensional controls: OLS fixed child and school effects Estimates of Covariance Parameters

15. Two-Dimensional Controls: Random child and school effects with interaction-contrast centering Estimates of Fixed Effects Estimates of Covariance Parameters

16. Two-Dimensional Controls:fixed school effects, random kid effects, person-mean centered x. Estimates of Fixed Effects Estimates of Covariance Parameters

17. Claims • For studying time-varying treatments, adaptive centering with random effects replicates fixed effects analysis in any dimension • Adaptive centering with random effects is generally the preferable approach

18. a. A natural way to incorporate uncertainty as a function of clustering Note we are incorporating uncertainty associated with classrooms, which cannot be done using fixed effects if the treatment is at that level.

19. Two-dimensional controls (kids and schools)random effects of kids, teachers within schools, schoolsinteraction contrast for treatment Estimates of Fixed Effects

20. b. A natural framework for modeling heterogeneity * Heterogeneity is interesting; * A failure to incorporate heterogeneity leads to biased standard errors.

21. c. We can easily study multilevel treatment and their interaction

22. d. Improved estimates of unit-specific effects • Fixed Effects Approach via OLS

23. Random Effects ApproachEmpirical BayesStep 1: Estimate

24. Random Effects Approach • Step 2: Compute

25. Results • Correlation • Mean Squared Error • Relative Efficiency

26. Role of reliability • Reliability of OLS Fixed Effects • In large samples,efficiency of OLS relative to EB is approximately equal to the reliability (Raudenbush, 1988, Journal of Educational Statistics).

27. e. Computational Ease We don’t need dummy variables to represent kids, teachers, or schools.