Abstract
Propensity score (PS) matching is widely used for studying treatment effects in observational studies. This article proposes the method of matching weights (MWs) as an analog to one-to-one pair matching without replacement on the PS with a caliper. Compared with pair matching, the proposed method offers more efficient estimation, more accurate variance calculation, better balance, and simpler asymptotic analysis. A statistical test for the misspecification of the PS model is proposed for balance checking purposes. An augmented version of the MW estimator is developed that has the double robust property, that is, the estimator is consistent, if either the outcome model or the PS model is correct. The proposed method is studied in simulations and illustrated through a real data example.
Appendix
Proposition 1 Suppose that the PS can only take finitely many values at 
, 
. 
and 
. If we do a one-to-one exact matching on the PS without replacement and choose randomly when multiple matched pairs are available, then the matching estimator has the same asymptotic limit as the MW estimator as 
. In addition, the effective sample size of the MW estimator is asymptotically equivalent to the expected number of matched pairs.
Proof Denote 
by 
, and let 
(
) be the set of matched subjects from the treatment (control) group with PS 
. The matching estimator is
Similarly,
Therefore, as 
,
which is the same asymptotic limit as the MW estimator 
. The effective sample size of the MW estimator is 
for the treatment group and 
for the control group. The number of matched pairs is 
. These quantities are asymptotically equivalent from the derivation above and 
. □
Proof of Theorem 1: If 
and 
are correctly specified, then 
.
Since 
and 
,
Since 
and
Similarly, 
. Summarizing the results above and the expression of 
, we have 
when 
and 
are correctly specified.
Next, we assume that the PS model is correctly specified, that is, 
s are correct. We can rearrange the terms in eq. [6] of the article and write 
as
![[10]](/document/doi/10.1515/ijb-2012-0030/asset/graphic/ijb-2012-0030_eq10.png)
The first term is 
, which converges to 
. The second term equals to
Since 
and 
, the second term converges to 0. Similarly, the third term in eq. [10] also converges to 0. Therefore, 
when the PS model is correctly specified. □
Proof of Theorem 2: When the PS model is known, 
. The MW estimator approximately equals to
![[11]](/document/doi/10.1515/ijb-2012-0030/asset/graphic/ijb-2012-0030_eq11.png)
where 
. We can define 
and 
similarly and view 
as the new data and 
as the new potential outcomes. It is obvious that if the SUTVA assumption and unconfoundedness assumption hold, then similar assumptions hold for the new data and new potential outcomes as well: 
and 
.
Expression (11) suggests that 
can be viewed as an inverse probability weighting estimator, if we think of 
as the outcome variable. Therefore, the semiparametric theory in §13.5 of Tsiatis [44], originally developed for inverse probability weighting method, can be applied to show that the class of influence functions of regular asymptotically linear estimators of 
is given by
where 
for any function 
. The efficient influence function in this class is uniquely given by
as in eq. [6] of the article is the estimator corresponding to this efficient influence function. □
Acknowledgement
We greatly appreciate the helpful comments from the editor, associate editor and referees. This work was carried out while Liang Li was a faculty biostatistician in the Department of Quantitative Health Sciences at Cleveland Clinic.
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