Average Adjusted Association: Efficient Estimation with High Dimensional Confounders

The log odds ratio is a common parameter to measure association between (binary) outcome and exposure variables. Much attention has been paid to its parametric but robust estimation, or its nonparametric estimation as a function of confounders. However, discussion on how to use a summary statistic by averaging the log odds ratio function is surprisingly difficult to find despite the popularity and importance of averaging in other contexts such as estimating the average treatment effect. We propose a couple of efficient double/debiased machine learning (DML) estimators of the average log odds ratio, where the odds ratios are adjusted for observed (potentially high dimensional) confounders and are averaged over them. The estimators are built from two equivalent forms of the efficient influence function. The first estimator uses a prospective probability of the outcome conditional on the exposure and confounders; the second one employs a retrospective probability of the exposure conditional on the outcome and confounders. Our framework encompasses random sampling as well as outcome-based or exposure-based sampling. Finally, we illustrate how to apply the proposed estimators using real data.

Causal Inference in Case-Control Studies

We investigate identification of causal parameters in case-control and related studies. The odds ratio in the sample is our main estimand of interest and we articulate its relationship with causal parameters under various scenarios. It turns out that the odds ratio is generally a sharp upper bound for counterfactual relative risk under some monotonicity assumptions, without resorting to strong ignorability, nor to the rare-disease assumption. Further, we propose semparametrically efficient, easy-to-implement, machine-learning-friendly estimators of the aggregated (log) odds ratio by exploiting an explicit form of the efficient influence function. Using our new estimators, we develop methods for causal inference and illustrate the usefulness of our methods by a real-data example.