A comparison of machine learning algorithms and covariate balance measures for propensity score matching and weighting

Propensity score matching (PSM) and propensity score weighting (PSW) are popular tools to estimate causal effects in observational studies. We address two open issues: how to estimate propensity scores and assess covariate balance. Using simulations, we compare the performance of PSM and PSW based on logistic regression and machine learning algorithms (CART; Bagging; Boosting; Random Forest; Neural Networks; naive Bayes). Additionally, we consider several measures of covariate balance (Absolute Standardized Average Mean (ASAM) with and without interactions; measures based on the quantile‐quantile plots; ratio between variances of propensity scores; area under the curve (AUC)) and assess their ability in predicting the bias of PSM and PSW estimators. We also investigate the importance of tuning of machine learning parameters in the context of propensity score methods. Two simulation designs are employed. In the first, the generating processes are inspired to birth register data used to assess the effect of labor induction on the occurrence of caesarean section. The second exploits more general generating mechanisms. Overall, among the different techniques, random forests performed the best, especially in PSW. Logistic regression and neural networks also showed an excellent performance similar to that of random forests. As for covariate balance, the simplest and commonly used metric, the ASAM, showed a strong correlation with the bias of causal effects estimators. Our findings suggest that researchers should aim at obtaining an ASAM lower than 10% for as many variables as possible. In the empirical study we found that labor induction had a small and not statistically significant impact on caesarean section.     

Directional penalties for optimal matching in observational studies

Multivariate matching in observational studies tends to view covariate differences symmetrically: a difference in age of 10 years is thought equally problematic whether the treated subject is older or younger than the matched control. If matching is correcting an imbalance in age, such that treated subjects are typically older than controls, then the situation in need of correction is asymmetric: a matched pair with a difference in age of 10 years is much more likely to have an older treated subject and a younger control than the opposite. Correcting the bias may be easier if matching tries to avoid the typical case that creates the bias. We describe several easily used, asymmetric, directional penalties and illustrate how they can improve covariate balance in a matched sample. The investigator starts with a matched sample built in a conventional way, then diagnoses residual covariate imbalances in need of reduction, and achieves the needed reduction by slightly altering the distance matrix with directional penalties, creating a new matched sample. Unlike penalties commonly used in matching, a directional penalty can go too far, reversing the direction of the bias rather than reducing the bias, so the magnitude of the directional penalty matters and may need adjustment. Our experience is that two or three adjustments, guided by balance diagnostics, can substantially improve covariate balance, perhaps requiring fifteen minutes effort sitting at the computer. We also explore the connection between directional penalties and a widely used technique in integer programming, namely Lagrangian relaxation of problematic linear side constraints in a minimum cost flow problem. In effect, many directional penalties are Lagrange multipliers, pushing a matched sample in the direction of satisfying a linear constraint that would not be satisfied without penalization. The method and example are in an R package DiPs at CRAN .     

Optimal matching with minimal deviation from fine balance in a study of obesity and surgical outcomes

In multivariate matching, fine balance constrains the marginal distributions of a nominal variable in treated and matched control groups to be identical without constraining who is matched to whom. In this way, a fine balance constraint can balance a nominal variable with many levels while focusing efforts on other more important variables when pairing individuals to minimize the total covariate distance within pairs. Fine balance is not always possible; that is, it is a constraint on an optimization problem, but the constraint is not always feasible. We propose a new algorithm that returns a minimum distance finely balanced match when one is feasible, and otherwise minimizes the total distance among all matched samples that minimize the deviation from fine balance. Perhaps we can come very close to fine balance when fine balance is not attainable; moreover, in any event, because our algorithm is guaranteed to come as close as possible to fine balance, the investigator may perform one match, and on that basis judge whether the best attainable balance is adequate or not. We also show how to incorporate an additional constraint. The algorithm is implemented in two similar ways, first as an optimal assignment problem with an augmented distance matrix, second as a minimum cost flow problem in a network. The case of knee surgery in the Obesity and Surgical Outcomes Study motivated the development of this algorithm and is used as an illustration. In that example, 2 of 47 hospitals had too few nonobese patients to permit fine balance for the nominal variable with 47 levels representing the hospital, but our new algorithm came very close to fine balance. Moreover, in that example, there was a shortage of nonobese diabetic patients, and incorporation of an additional constraint forced the match to include all of these nonobese diabetic patients, thereby coming as close as possible to balance for this important but recalcitrant covariate.