1. Stephen W. Raudenbush University of Chicago December 11, 2006
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 Treatments by Stephen W. Raudenbush University of Chicago Abstract
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 schools4 5 6 7 8 9 x -1 w
Child
2.4628
6.2245
3.6396
4.1441
2.1827
-3170
3.6596
4.8397
-.0727
1.6280
6.0525
1.4795
.2350
6.0839
7.5142
-.8803
3.5167
9.7337
5.8636 10
2.6814
7.6954 11
4.4966
9.5578 12
4.7195
8.2204 13
4.3609 14
4.7778 15
8.5264 16
8.6820 17
9.5595 18
5.6075
21.075 19
8.9094
20.049 20
6.3465
7.3268

5. Table 2 Correlations w x school id child id teacher id y
w = child covariate
-.23
.00
.43
.06 58
x = teacher covariate
.34
-.48
.57
.14
.97
.67
-.13
-.06
.62
.58

6. 1. “True model” Estimates of Fixed Effects Predictors β Std. Err. T p
(Constant)
-.415
.302
-1.375
.175 x
2.171
.200
10.866
.000 w
4.799
.278
17.294
schooled-2
3.970
.166
23.912
child id
.539
.027
20.001

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 Predictor β Std. Err. T p
(Constant)
7.963
.748
10.638
.000 X
1.001
.966
1.036
.305

9. Child random effects “as if randomized”
Estimates of Fixed Effects
Parameter
Estimate
Std. Err. df t
Sig.
Intercept
16.067
6.071
.000 x
47.768
5.330
Parameter
Estimate σ2 τ2

10. One-Dimensional Control: OLS Fixed Child Effects
Parameter
Estimate
Std. Error t
Sig.
Intercept
6.267
.000 x
6.350
[childid=1.00]
-5.140
[childid=2.00]
-3.492
.001
[childid=3.00]
-2.897
.006
[childid=4.00]
-4.596
[childid=5.00]
-3.769
[childid=6.00]
-4.126
[childid=7.00]
-3.576
[childid=8.00]
-3.242
.002
[childid=9.00]
-2.992
.005
[childid=10.00]
-3.741
[childid=11.00]
-.002
.998
[childid=12.00]
-.352
.727
[childid=13.00]
-.263
.794
[childid=14.00]
-.723
.474
[childid=15.00]
-.763
.450
[childid=16.00]
.157
.876
[childid=17.00]
-.222
.825
[childid=18.00]
.373
.711
[childid=19.00]
.089
.929
[childid=20.00]
0(a) .
Estimates of Covariance Parameters
Parameter
Estimate σ2

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
Parameter
Estimate
Std. Err. df t
Sig.
Intercept 19
8.661
.000 39
6.350
Estimate of Covariance Parameters
Parameter
Estimate
σ 2
τ 2

12. Table 3. Treatment Received
Teacher 1 2 3 4 5 6 7 8 9 x -1
Child
.6667
.3333 10 11
-.3333 12
-.6667 13 14 15 16 17 18 19 20
-0.25
0.45

13. Random child and school effects with x “as if randomized”
Estimates of Fixed Effects
Parameter
Estimate
Std. Err. df t
Sig.
Intercept
3.034
3.154
.050 x
38.494
8.658
.000
Estimates of Covariance Parameters
Parameter
Estimate σ2 τ2 ψ2

14. Two dimensional controls: OLS fixed child and school effects
Parameter
Estimate
Std. Error df T
Sig.
Intercept 37
23.229
.000 X
8.936
[childid=1.00]
[childid=2.00]
-6.756
[childid=3.00]
-4.729
[childid=4.00]
[childid=5.00]
-9.751
[childid=6.00]
-9.902
[childid=7.00]
-8.005
[childid=8.00]
-7.915
[childid=9.00]
-7.042
[childid=10.00]
-6.755
[childid=11.00]
-.008
.994
[childid=12.00]
-.055
.956
[childid=13.00]
1.404
.169
[childid=14.00]
.862
.394
[childid=15.00]
1.694
.099
[childid=16.00]
2.881
.007
[childid=17.00]
2.608
.013
[childid=18.00]
3.637
.001
[childid=19.00]
3.693
[childid=20.00]
0(a) .
[schoolid=1.00]
[schoolid=2.00]
-9.622
[schoolid=3.00]
Estimates of Covariance Parameters
Parameter
Estimate σ2

15. Two-Dimensional Controls: Random child and school effects with interaction-contrast centering
Estimates of Fixed Effects
Parameter
Estimate
Std. Err. t
Sig.
Intercept
2.816
.083
8.936
.000
Estimates of Covariance Parameters
Parameter
Estimate σ2 τ2 ψ2

16. Two-Dimensional Controls: fixed school effects, random kid effects, person-mean centered x.
Estimates of Fixed Effects
Parameter
Estimate
Std. Err. df T
Sig.
Intercept
21.032
12.302
.000
37.000
8.936
[schoolid=1.00]
[schoolid=2.00]
-9.622
[schoolid=3.00]
0(a) .
Estimates of Covariance Parameters
Parameter
Estimate σ2 τ2

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, schools interaction contrast for treatment
Estimates of Fixed Effects
Parameter
Estimate
Std. Err. t
Sig.
Intercept
3.296
0.170
9.049
.000
Parameter
Estimate σ2 τ2   2    ψ2

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 Approach Empirical Bayes Step 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.