# Propensity score matching and unbalanced co-variables in IPWT

When using propensity score (PS) for calculating inverse probability weighting (IPW) in an average treatment effect (ATE) approach, is it valid to remove from the PS those co-variables that remain unbalanced, and then, when performing the final analyses using them as co-variables together with the treatment variable?

EDIT: Since the original text may be confusing, I would try to clarify it.

I have used ps() function from twang R package, which implements GBM models. Once executed, there are some co-variables that have a higher absolute standardized effect size in the balanced data than in the unweighted original.

My question was, Does it make sense to remove the co-variables that will be unbalanced after calculating ps(), and then, when performing a logistic regression with the weighted data add them as co-variables?

From @Noah answer: Is statistical valid (and also makes sense), to use all co-variables for calculating the weights, and then used unbalanced co-variables in the final logistic regression analyses?

The goal of IPTW is to achieve balance. If balance is not achieved by your IPTW specification, can you try to respecify the model or you can use regression in the weighted sample with the imbalanced covariates included to adjust for confounding by those covariates. This is not necessarily the best way to proceed, though. Failing to balance a covariate with the weights means that you are placing the entire burden of adjusting for the covariate onto the outcome regression model. If that model is wrong (and it almost certainly is), confounding will remain. The point of balancing is to make it so that the confounding that remains after covariate adjustment by an incorrect model is as minimal as possible. This is the thesis of Ho, Imai, King, and Stuart (2007).

It doesn't make much sense to remove a covariate from a propensity score model. If that model fails to balance a covariate, you should want to add that covariate into the model in multiple different ways (e.g., squared terms, log terms, interactions, subclasses) to achieve balance, not drop it from the model because the model with it in is doing poorly. Surely a model without the covariate will balance the covariate even worse.

Ideally, you should combine IPTW with an outcome regression model so that the remaining imbalance is accounted for by the outcome regression model and the misspecification of the outcome regression model is mitigated by the balance. There several estimators that combine a propensity score and outcome model; these are called "doubly robust" estimators, and outcome regression in an IPTW-weighted sample is one of them, but there are others.

You should also consider using either optimization-based approaches like entropy balancing, which guarantee balance on the covariate means and have good efficiency properties, or machine learning methods like generalized boosted modeling (GBM) or Bayesian additive regression trees (BART), which attempt to flexibly model the propensity score. These are available in the R package WeightIt (which I developed). There has been so much work done on new, robust methods with excellent statistical properties that one should not be using the simple methods developed 20 years ago.

• I have used ps() function from twang R package, which uses GBM models. However, for some co-variables, the absolute standardized effect size, after computing the ps is higher in the es/ks balanced, than in the unweighted. That's why I use the approach of the question. I removed those variables, compute again the ps and then I perform a logistic regression model considering the treatment + the unbalanced variables Oct 2, 2020 at 12:37
• I have edit the original text to clarify it. From your answer: Do you mean that is plausible to, to use all co-variables for calculating the weights, and then use remaining unbalanced co-variables in the final logistic regression analyses? Oct 6, 2020 at 10:35
• You can do whatever you want, basically. As long as covariates are adjusted for by weighting or regression adjustment, you are reducing the bias due to that covariate. There is no theory that requires both models to have the same covariates. Just know that failing to achieve balance means that more burden is placed on the regression model to adjust for the covariate, and there is a good chance that model will be wrong, leaving bias. Instead of trying to improve the balance after GBM, why don't you try other methods of estimating the propensity score?
– Noah
Oct 6, 2020 at 23:45
• You are right. I'll have a look at WeightIt package and try other methods. Thank you Oct 7, 2020 at 6:59