Accessible Requires Authentication Published by De Gruyter October 28, 2015

Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments

Joel A. Middleton and Peter M. Aronow

Abstract

Many estimators of the average treatment effect, including the difference-in-means, may be biased when clusters of units are allocated to treatment. This bias remains even when the number of units within each cluster grows asymptotically large. In this paper, we propose simple, unbiased, location-invariant, and covariate-adjusted estimators of the average treatment effect in experiments with random allocation of clusters, along with associated variance estimators. We then analyze a cluster-randomized field experiment on voter mobilization in the US, demonstrating that the proposed estimators have precision that is comparable, if not superior, to that of existing, biased estimators of the average treatment effect.


Corresponding author: Peter M. Aronow, Department of Political Science, Yale University, New Haven, CT, USA, e-mail:

Acknowledgments

The authors acknowledge support from the Yale University Faculty of Arts and Sciences High Performance Computing facility and staff. The authors would also like to thank Allison Carnegie, Adam Dynes, Don Green, Jennifer Hill, Mary McGrath, David Nickerson, Cyrus Samii and two helpful reviewers for helpful comments. The authors thank Kyle Peyton for research assistance and manuscript preparation. Any errata are the sole responsibility of the authors.

Appendix

A Proof of Non-invariance of the Horvitz-Thompson Estimator

To prove that the HT estimator is not invariant to location shifts, we need only replace YjT with its linear transformation:

ΔHT^=MN[1mtjJ1YjT1mcjJ0YjT]=MN[1mtjJ1i=1njYij1mcjJ0i=1njYij]=MN[1mtjJ1i=1nj(b0+b1Yij)1mcjJ0i=1nj(b0+b1Yij)]=MN[1mtjJ1(njb0+i=1njb1Yij)1mcjJ0(njb0+i=1njb1Yij)]=b0MN[1mtjJ1nj1mcjJ0nj]+b1MN[1mtjJ1i=1njYij1mcjJ0i=1njYij]=b0MN[1mtjJ1nj1mcjJ0nj]+b1Δ^HTHT.

B Bias from Estimating k from Within-Sample Data

Consider the situation where one wishes to improve upon the HT estimator by adjusting for cluster size; in other words, one wishes to estimate k in equations 20 and 21 from the data to approximate the optimal value of k with an estimator k^. In this scenario, the expected value of equation 20 yields

(29)E[Y1,R1T^]=E[MmtjJ1(YjTk^(njN/M))]=Mmt(E[jJ1YjT]E[jJ1k^nj]+E[jJ1k^N/M])=Mmt(E[mtY1jT¯]E[k^mtntj¯]+E[k^mtN/M])=Y1TM(E[k^ntj¯]E[k^]E[ntj¯])=Y1TMCov (k^, ntj¯), (29)

where ntj¯ is the mean value of nj for clusters in the treatment condition in a given randomization. In the third line of equation 29, k^ moves outside the summation operator because it is a constant for a given randomization. Likewise,

(30)E [Y0,R1T^]=Y0TMCov (k^,ncj¯), (30)

where ncj¯ is the mean value of nj for units in the control condition in a given randomization. So the expected value of the estimator will be

(31)E[Y1,R1T^Y0,R1T^N]=Δ+MN(Cov (k^, ncj¯)Cov (k^, ncj¯)). (31)

The term on the right of equation 31 represents the bias. A special case with no bias is when the sharp null hypothesis of no treatment effect holds and treatment and control groups have equal numbers of clusters. We refer the reader to Williams (1961), Freedman (2008a) and Freedman (2008b) for additional reading on the particular bias associated with the regression adjustment of random samples and experimental data.

C Derivation of the Optimal Value of k

To identify a single optimal value of k, koptim*, we refer to the first line of equation 17,

(32)vV (ΔR1^)=cσ2(Uj0T)+tσ2(Uj1T)+2σ(Uj0T,Uj1T) (32)

where v=(M1)N2M2,c=Mmcmc, and t=Mmtmt. Now note that the terms σ2(Uj0T),σ2(Uj0T), and σ(Uj0T,Uj1T) in equation 32 can be written as follows:

(33)σ2(Uj1T)=σ2(Yj1T)+k2σ2(nj)2kσ(Yj1T,nj), (33)
(34)σ2(Uj0T)=σ2(Yj0T)+k2σ2(nj)2kσ(Yj0T,nj), (34)

and, defining δj=(njN/M),

(35)σ(Uj0T,Uj1T)=E [Uj0TUj1T]U0T¯U1T¯=E [Yj0Tkδj](Yj1Tkδj)Y0T¯Y1T¯=E [Yj0TYj1TYj0TkδjYj1Tkδj+k2δj2]Y0T¯Y1T¯=E[Yj0TYj1T]Y0T¯Y1T¯E[Yj0Tkδj]E[Yj1Tkδj]+E[k2δj2]=σ(Yj0T,Yj1T)k[σ(Yj0T,nj)+E[Yj0T]E[δj]]k[σ(Yj1T,nj)+E[Yj1T]E[δj]]+k2σ2(nj)=σ(Yj0T,Yj1T)k[σ(Yj0T,nj)+E[Yj0T]0]k[σ(Yj1T,nj)+E[Yj1T]0]+k2σ2(nj)=σ(Yj0T,Yj1T)kσ(Yj0T,nj)kσ(Yj1T,nj)+k2σ2(nj), (35)

respectively. Substituting equations 33, 34, and 35 into equation 32,

vV(ΔR1^)=c[σ2(Yj0T)+k2σ2(nj)2kσ(Yj0T,nj)]+t[σ2(Yj1T)+k2σ2(nj)2kσ(Yj1T,nj)]+2[σ(Yj0T,Yj1T)kσ(Yj0T,nj)kσ(Yj1T,nj)+k2σ2(nj)].

Setting the first derivative with respect to k equal to zero,

0=c[2koptimσ2(nj)2σ(Yj0T,nj)]+t[2koptimσ2(nj)2σ(Yj1T,nj)]+2[σ(Yj0T,nj)σ(Yj1T,nj)+2koptimσ2(nj)],

ckoptimσ2(nj)+tkoptimσ2(nj)+2koptimσ2(nj)=cσ(Yj0T,nj)+tσ(Yj1T,nj)+σ(Yj0T,nj)+σ(Yj1T,nj)

(Mmcmc+Mmtmt+mcmc+mtmt)koptimσ2(nj)=(Mmcmc+mcmc)σ(Yj0T,nj)+(Mmtmt+mtmt)σ(Yj1T,nj)

(Mmc+Mmt)koptimσ2(nj)=(Mmc)σ(Yj0T,nj)+(Mmt)σ(Yj1T,nj)

koptim=(1mc+1mt)1[(1mc)σ(Yj0T,nj)σ2(nj)+(1mt)σ(Yj1T,nj)σ2(nj)]

koptim=(1mc+1mt)1[(1mc)koptimc+(1mt)koptimt]

koptim=mtMkoptimc+mcMkoptimt.

The Des Raj estimator will be more efficient than the HT estimator when

(c+t+2)k2σ2(nj)<2k[(c+1)σ(Yj0T,nj)+(t+1)σ(Yj1T,nj)](c+t+2)k2<2k[(c+1)σ(Yj0T,nj)σ2(nj)+(t+1)σ(Yj1T,nj)σ2(nj)](c+t+2)k2<2k[(c+1)koptimc+(t+1)koptimt]k2<2k[mtMkoptimc+mcMkoptimt]k2<2kkoptim.

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Published Online: 2015-10-28
Published in Print: 2015-12-1

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