Balance diagnosis, that is, checking whether holds, is an attractive feature of the PS analysis in comparison with direct regression on the outcome [20, 31]. In the PS matching framework, it is often recommended that balance be assessed by calculating the absolute value of the standardized difference of each covariate after matching [2, 17]:

where and are the sample mean and standard deviation of the matched treated subjects, respectively, and and are those of the controls. Let denote the covariate of interest, then , , , and , where the summation is over the matched subjects.

With MWs, we can develop an analog to the above by using weighted means and standard deviations, i.e. , , , and . If the are continuous, we can also compare the weighted cumulative distribution functions of between the treated and controls to assess balance in the entire distributions of the covariates.

The PS model should be formulated so that the absolute value of the standardized difference of all covariates be minimized “without limit” [18]. As mentioned in Section 1.3, it is necessary to develop well-justified criteria on whether balance is achieved in the propensity matched or propensity weighted data set. A formal hypothesis test of balance is often viewed as being undesirable in the matching framework, because “balance” is a property of the matched sample but not the underlying population, and because matched data have reduced sample size, which could inflate *p*-value by itself [13, 18]. We agree with this view. However, from another perspective, we can formulate the null and alternative hypotheses as follows:

*H*_{0}: the PS model is correctly specified

*H*_{A}: the PS model is misspecified.

Let be a vector of covariates whose balance we want to examine. *V* may include some or all of *X*, or their transformations. Let

[8]where is a vector of monotone smooth transformations applied to the corresponding element of its input vector. The identity transformation is often used. However, when *V* is a binary covariate with very low prevalence, defining as the logit transformation improves finite-sample performance [32]. Similarly, when *V* is highly skewed, we may use the log transformation. If the null hypothesis is true, we can prove that the expectation of is zero, that is, the covariates are balanced after being weighted by the MWs; otherwise, may deviate from zero, indicating that the covariates are not balanced.

In order to properly adjust for the estimated PS, we view this statistic as a solution to the following estimating equations for :

We can think of and as parameters representing and , respectively, and , with , where the first two blocks are identity and negative identity matrices of a dimension equal to the length of . Under the null hypothesis, as , . An estimator of the variance matrix , , can be obtained from the estimating equations above using the sandwich method, similar to Section 2. The test statistic is proposed as

[9]Under the null, this statistic has an asymptotically central distribution with degrees of freedom . The *p*-value of the test is .

In a randomized clinical trial, as long as the randomization protocol is properly designed and closely followed, the baseline covariates are expected to be balanced (only subject to chance imbalance) between randomized groups and a test of balance is unnecessary [33]. From another perspective, checking for covariate balance in a randomized trial is still useful, because if many covariates are unbalanced, that raises questions on whether the randomization protocol has been followed appropriately during the conduct of the trial. The PS model to an observational study is analogous to the randomization protocol to a randomized clinical trial. As long as the PS model is correctly specified, all covariates should be balanced after weighting by the MWs (subject only to chance imbalances). Therefore, the test above should be interpreted as a test for the misspecification of the PS model, but not as a test for covariate balance. However, the test statistic and its associated *p*-value may serve as an index of covariate balance, which may complement existing indices such as the standardized difference.

The idea of casting balance diagnosis as a test of the misspecification of the PS has been advocated by some researchers [34–36]. The test statistic (9) was previously proposed by Hansen and Bowers [34] as a metric of balance for general balance checking problems, not necessarily for PS analysis. No numerical result on this test was presented in that article. We developed an adaptation here, by incorporating MWs and adjustment for the uncertainty in estimated PSs.

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