Rejoinder to response from Yamana, et al.

On behalf of the authors, we appreciate the comments by Yamana, et al. and agree that it is impossible to rule out the potential of unmeasured confounders in our study. However, we have to say the logic behind their critiques is flawed.

First, they criticized that the model for propensity score calculation is inadequate since the discriminatory ability, also called as c-statistic, was not quantified and the distributions of the propensity scores were identical in the exposed and non-exposed groups. However, overlap in propensity scores does not provide any evidence of unmeasured confounders. For example, true values of propensity scores of individuals are identical in a randomized clinical trial, but there are no unmeasured confounders in such a study.

Second, the goal of a propensity score model is to reduce bias, not to predict exposure. It is widely accepted that a high c-statistic is neither necessary nor sufficient for control of confounding and reliance on the c-statistic may provide false confidence that the model is adequate for confounding control [1]. Consider a propensity score model with a perfect discriminatory ability. It implies lack of positivity and the comparison is therefore invalid.

Third, they suggested that alcohol consumption of parents could improve the model. Note that we already adjusted for alcohol consumption during pregnancy. Therefore they seem to have recommended including alcohol consumption of parents between birth and 3 years of age but we should not include any post-exposure variables into the model [2].

Fourth, they suggested a per-protocol analysis which takes smoking cessation after 4 months into account. However, to handle time-dependent exposure, structural mean models or marginal structural models rather than a per-protocol analysis, which cannot adjust for time-dependent confounding, are necessary.

In summary, the arguments of Yamana, et al. go against the theoretical results of the literature on causal inference [1, 2]. We also guarantee that the covariates listed in Potential Confounders Panel are all relevant pre-exposure variables we obtained.

1. Westreich D, Cole SR, Funk MJ, Brookhart MA, Sturmer T. The role of the c-statistic in variable selection for propensity score models. Pharmacoepidemiol Drug Saf 2011;20(3):317-20.
2. VanderWeele TJ, Shpitser I. A new criterion for confounder selection. Biometrics 2011;67(4):1406-13.