In Wyss et al.,1 the authors use simulations to evaluate the use of the Collaborative Controlled Lasso (CCL) to undersmooth propensity score (PS) models for high-dimensional confounding control. In this letter, we describe how the implementation of the CCL in Wyss et al. may have given an overly optimistic view of the benefits that are likely to be seen in most practical settings. We suggest changes to how the CCL was implemented and present some modified results based on these changes.
The Lasso PS model estimates coefficients based on the following penalized likelihood:
where n is the sample size, A is a binary treatment, W is an m-dimensional vector of covariates, βis an m-dimensional vector of coefficient parameters, and λ is the regularization tuning parameter. The optimal choice for λ depends on the purpose for which the Lasso model is used. If the goal is to optimize out-of-sample prediction, then λ is often chosen using cross-validation, which we will refer to as λCV. Previous work has shown, however, that when using Lasso for estimating PSs, less regularization is beneficial to minimize bias in causal estimators.2,3 This is referred to as undersmoothing the Lasso model and corresponds to choosing a value for λ that is less than λCV so that the model selects more covariates from W compared to the model using λCV.
Undersmoothing can reduce bias in estimated treatment effects by including more confounders in the fitted model. However, too much undersmoothing can eventually harm the accuracy of the fitted model to a degree where the benefit of including more confounder information is outweighed by the cost of a poorly fit model producing unstable predictions. The CCL is one approach that has been proposed by Ju et al.2 to determine the degree of undersmoothing when fitting Lasso PS models.