Balancing vs modeling approaches to weighting in practice

Abstract

There are two seemingly unrelated approaches to weighting in observational studies. One of them maximizes the fit of a model for treatment assignment to then derive weights-we call this the modeling approach. The other directly optimizes certain features of the weights-we call this the balancing approach. The implementations of these two approaches are related: the balancing approach implicitly models the propensity score, while instances of the modeling approach impose balance conditions on the covariates used to estimate the propensity score. In this article, we review and compare these two approaches to weighting. Previous review papers have focused on the modeling approach, emphasizing the importance of checking covariate balance. However, as we discuss, the dispersion of the weights is another important aspect of the weights to consider, in addition to the representativeness of the weighted sample and the sample boundedness of the weighted estimator. In particular, the dispersion of the weights is important because it translates into a measure of effective sample size, which can be used to select between alternative weighting schemes. In this article, we examine the balancing approach to weighting, discuss recent methodological developments, and compare instances of the balancing and modeling approaches in a simulation study and an empirical study. In practice, unless the treatment assignment model is known, we recommend using the balancing approach to weighting, as it systematically results in better covariate balance with weights that are minimally dispersed. As a result, effect estimates tend to be more accurate and stable.

Entropy Balancing for Causal Generalization with Target Sample Summary Information

“…Since ( 5) is equivalent to modeling ρ(x)/(1 − ρ(x)) with exponential tilting, the weights constructed in this way constitute estimates of w * (a, x). However, this approach is vulnerable to estimation error in the estimated propensity score and can result in extreme weights as one may encounter in the conventional inverse propensity score weights (Kang et al, 2007;Chattopadhyay et al, 2020). In Section 5, we systematically compare this method with the proposed weights (8), and the results suggest that our entropy balancing weighting approach leads to more favorable performance.…”

On the implied weights of linear regression for causal inference

“…(see, e.g., Chattopadhyay et al 2020). As a special case, when both m 1 and m 0 are linear in X i , balancing the mean of X i relative to X suffices to remove the bias of T .…”

“…Negative weights are difficult to interpret and, moreover, they can produce estimates that are an extrapolation outside (instead of an interpolation inside) of the support of the available data. In other words, negative weights can produce estimates that are not sample bounded in the sense of Robins et al (2007) (see also Chattopadhyay et al 2020). In some settings, there is no alternative to using negative weights in order to adjust for or balance certain features of the distributions of the observed covariates.…”

Optimal transport weights for causal inference

“…This will reduce the variance of the weights in exchange for the addition of some bias (Owen, 2013a), but there is evidence that this leads to a lower mean-squared error for the target estimand (Wang and Zubizarreta, 2019a;Huling and Mak, 2020;Chattopadhyay et al, 2020).…”

“…where k is the number of basis functions, or ‹ can be selected by first running a stable balancing weights tuning algorithm (Wang and Zubizarreta, 2019a;Chattopadhyay et al, 2020) and then using the selected ‹.…”