Double Cross-fit Doubly Robust Estimators: Beyond Series Regression

Doubly robust estimators with cross-fitting have gained popularity in causal inference due to their favorable structure-agnostic error guarantees. However, when additional structure, such as H\"{o}lder smoothness, is available then more accurate "double cross-fit doubly robust" (DCDR) estimators can be constructed by splitting the training data and undersmoothing nuisance function estimators on independent samples. We study a DCDR estimator of the Expected Conditional Covariance, a functional of interest in causal inference and conditional independence testing, and derive a series of increasingly powerful results with progressively stronger assumptions. We first provide a structure-agnostic error analysis for the DCDR estimator with no assumptions on the nuisance functions or their estimators. Then, assuming the nuisance functions are H\"{o}lder smooth, but without assuming knowledge of the true smoothness level or the covariate density, we establish that DCDR estimators with several linear smoothers are semiparametric efficient under minimal conditions and achieve fast convergence rates in the non-$\sqrt{n}$ regime. When the covariate density and smoothnesses are known, we propose a minimax rate-optimal DCDR estimator based on undersmoothed kernel regression. Moreover, we show an undersmoothed DCDR estimator satisfies a slower-than-$\sqrt{n}$ central limit theorem, and that inference is possible even in the non-$\sqrt{n}$ regime. Finally, we support our theoretical results with simulations, providing intuition for double cross-fitting and undersmoothing, demonstrating where our estimator achieves semiparametric efficiency while the usual "single cross-fit" estimator fails, and illustrating asymptotic normality for the undersmoothed DCDR estimator.

Continuous Treatment Effects with Surrogate Outcomes

In many real-world causal inference applications, the primary outcomes (labels) are often partially missing, especially if they are expensive or difficult to collect. If the missingness depends on covariates (i.e., missingness is not completely at random), analyses based on fully-observed samples alone may be biased. Incorporating surrogates, which are fully observed post-treatment variables related to the primary outcome, can improve estimation in this case. In this paper, we study the role of surrogates in estimating continuous treatment effects and propose a doubly robust method to efficiently incorporate surrogates in the analysis, which uses both labeled and unlabeled data and does not suffer from the above selection bias problem. Importantly, we establish asymptotic normality of the proposed estimator and show possible improvements on the variance compared with methods that solely use labeled data. Extensive simulations show our methods enjoy appealing empirical performance.

The Fundamental Limits of Structure-Agnostic Functional Estimation

Many recent developments in causal inference, and functional estimation problems more generally, have been motivated by the fact that classical one-step (first-order) debiasing methods, or their more recent sample-split double machine-learning avatars, can outperform plugin estimators under surprisingly weak conditions. These first-order corrections improve on plugin estimators in a black-box fashion, and consequently are often used in conjunction with powerful off-the-shelf estimation methods. These first-order methods are however provably suboptimal in a minimax sense for functional estimation when the nuisance functions live in Holder-type function spaces. This suboptimality of first-order debiasing has motivated the development of "higher-order" debiasing methods. The resulting estimators are, in some cases, provably optimal over Holder-type spaces, but both the estimators which are minimax-optimal and their analyses are crucially tied to properties of the underlying function space. In this paper we investigate the fundamental limits of structure-agnostic functional estimation, where relatively weak conditions are placed on the underlying nuisance functions. We show that there is a strong sense in which existing first-order methods are optimal. We achieve this goal by providing a formalization of the problem of functional estimation with black-box nuisance function estimates, and deriving minimax lower bounds for this problem. Our results highlight some clear tradeoffs in functional estimation -- if we wish to remain agnostic to the underlying nuisance function spaces, impose only high-level rate conditions, and maintain compatibility with black-box nuisance estimators then first-order methods are optimal. When we have an understanding of the structure of the underlying nuisance functions then carefully constructed higher-order estimators can outperform first-order estimators.

An Efficient Doubly-Robust Test for the Kernel Treatment Effect

The average treatment effect, which is the difference in expectation of the counterfactuals, is probably the most popular target effect in causal inference with binary treatments. However, treatments may have effects beyond the mean, for instance decreasing or increasing the variance. We propose a new kernel-based test for distributional effects of the treatment. It is, to the best of our knowledge, the first kernel-based, doubly-robust test with provably valid type-I error. Furthermore, our proposed algorithm is efficient, avoiding the use of permutations.

Doubly Robust Counterfactual Classification

We study counterfactual classification as a new tool for decision-making under hypothetical (contrary to fact) scenarios. We propose a doubly-robust nonparametric estimator for a general counterfactual classifier, where we can incorporate flexible constraints by casting the classification problem as a nonlinear mathematical program involving counterfactuals. We go on to analyze the rates of convergence of the estimator and provide a closed-form expression for its asymptotic distribution. Our analysis shows that the proposed estimator is robust against nuisance model misspecification, and can attain fast $\sqrt{n}$ rates with tractable inference even when using nonparametric machine learning approaches. We study the empirical performance of our methods by simulation and apply them for recidivism risk prediction.

The role of the geometric mean in case-control studies

Historically used in settings where the outcome is rare or data collection is expensive, outcome-dependent sampling is relevant to many modern settings where data is readily available for a biased sample of the target population, such as public administrative data. Under outcome-dependent sampling, common effect measures such as the average risk difference and the average risk ratio are not identified, but the conditional odds ratio is. Aggregation of the conditional odds ratio is challenging since summary measures are generally not identified. Furthermore, the marginal odds ratio can be larger (or smaller) than all conditional odds ratios. This so-called non-collapsibility of the odds ratio is avoidable if we use an alternative aggregation to the standard arithmetic mean. We provide a new definition of collapsibility that makes this choice of aggregation method explicit, and we demonstrate that the odds ratio is collapsible under geometric aggregation. We describe how to partially identify, estimate, and do inference on the geometric odds ratio under outcome-dependent sampling. Our proposed estimator is based on the efficient influence function and therefore has doubly robust-style properties.

Doubly robust confidence sequences for sequential causal inference

This paper derives time-uniform confidence sequences (CS) for causal effects in experimental and observational settings. A confidence sequence for a target parameter $\psi$ is a sequence of confidence intervals $(C_t)_{t=1}^\infty$ such that every one of these intervals simultaneously captures $\psi$ with high probability. Such CSs provide valid statistical inference for $\psi$ at arbitrary stopping times, unlike classical fixed-time confidence intervals which require the sample size to be fixed in advance. Existing methods for constructing CSs focus on the nonasymptotic regime where certain assumptions (such as known bounds on the random variables) are imposed, while doubly-robust estimators of causal effects rely on (asymptotic) semiparametric theory. We use sequential versions of central limit theorem arguments to construct large-sample CSs for causal estimands, with a particular focus on the average treatment effect (ATE) under nonparametric conditions. These CSs allow analysts to update statistical inferences about the ATE in lieu of new data, and experiments can be continuously monitored, stopped, or continued for any data-dependent reason, all while controlling the type-I error rate. Finally, we describe how these CSs readily extend to other causal estimands and estimators, providing a new framework for sequential causal inference in a wide array of problems.

Median Optimal Treatment Regimes

Optimal treatment regimes are personalized policies for making a treatment decision based on subject characteristics, with the policy chosen to maximize some value. It is common to aim to maximize the mean outcome in the population, via a regime assigning treatment only to those whose mean outcome is higher under treatment versus control. However, the mean can be an unstable measure of centrality, resulting in imprecise statistical procedures, as well as unfair decisions that can be overly influenced by a small fraction of subjects. In this work, we propose a new median optimal treatment regime that instead treats individuals whose conditional median is higher under treatment. This ensures that optimal decisions for individuals from the same group are not overly influenced either by (i) a small fraction of the group (unlike the mean criterion), or (ii) unrelated subjects from different groups (unlike marginal median/quantile criteria). We introduce a new measure of value, the Average Conditional Median Effect (ACME), which summarizes across-group median treatment outcomes of a policy, and which the optimal median treatment regime maximizes. After developing key motivating examples that distinguish median optimal treatment regimes from mean and marginal median optimal treatment regimes, we give a nonparametric efficiency bound for estimating the ACME of a policy, and propose a new doubly robust-style estimator that achieves the efficiency bound under weak conditions. Finite-sample properties of the estimator are explored via numerical simulations and the proposed algorithm is illustrated using data from a randomized clinical trial in patients with HIV.

Counterfactual Predictions under Runtime Confounding

Algorithms are commonly used to predict outcomes under a particular decision or intervention, such as predicting whether an offender will succeed on parole if placed under minimal supervision. Generally, to learn such counterfactual prediction models from observational data on historical decisions and corresponding outcomes, one must measure all factors that jointly affect the outcomes and the decision taken. Motivated by decision support applications, we study the counterfactual prediction task in the setting where all relevant factors are captured in the historical data, but it is either undesirable or impermissible to use some such factors in the prediction model. We refer to this setting as runtime confounding. We propose a doubly-robust procedure for learning counterfactual prediction models in this setting. Our theoretical analysis and experimental results suggest that our method often outperforms competing approaches. We also present a validation procedure for evaluating the performance of counterfactual prediction methods.

Counterfactual Risk Assessments, Evaluation, and Fairness

Algorithmic risk assessments are increasingly used to help humans make decisions in high-stakes settings, such as medicine, criminal justice and education. In each of these cases, the purpose of the risk assessment tool is to inform actions, such as medical treatments or release conditions, often with the aim of reducing the likelihood of an adverse event such as hospital readmission or recidivism. Problematically, most tools are trained and evaluated on historical data in which the outcomes observed depend on the historical decision-making policy. These tools thus reflect risk under the historical policy, rather than under the different decision options that the tool is intended to inform. Even when tools are constructed to predict risk under a specific decision, they are often improperly evaluated as predictors of the target outcome. Focusing on the evaluation task, in this paper we define counterfactual analogues of common predictive performance and algorithmic fairness metrics that we argue are better suited for the decision-making context. We introduce a new method for estimating the proposed metrics using doubly robust estimation. We provide theoretical results that show that only under strong conditions can fairness according to the standard metric and the counterfactual metric simultaneously hold. Consequently, fairness-promoting methods that target parity in a standard fairness metric may --- and as we show empirically, do --- induce greater imbalance in the counterfactual analogue. We provide empirical comparisons on both synthetic data and a real world child welfare dataset to demonstrate how the proposed method improves upon standard practice.