Nonparametric inference for interventional effects with multiple mediators

Understanding the pathways whereby an intervention has an effect on an outcome is a common scientific goal. A rich body of literature provides various decompositions of the total intervention effect into pathway-specific effects. Interventional direct and indirect effects provide one such decomposition. Existing estimators of these effects are based on parametric models with confidence interval estimation facilitated via the nonparametric bootstrap. We provide theory that allows for more flexible, possibly machine learning-based, estimation techniques to be considered. In particular, we establish weak convergence results that facilitate the construction of closed-form confidence intervals and hypothesis tests and prove multiple robustness properties of the proposed estimators. Simulations show that inference based on large-sample theory has adequate small-sample performance. Our work thus provides a means of leveraging modern statistical learning techniques in estimation of interventional mediation effects.

Novel bounds for causal effects based on sensitivity parameters on the risk difference scale

Unmeasured confounding is an important threat to the validity of observational studies. A common way to deal with unmeasured confounding is to compute bounds for the causal effect of interest, that is, a range of values that is guaranteed to include the true effect, given the observed data. Recently, bounds have been proposed that are based on sensitivity parameters, which quantify the degree of unmeasured confounding on the risk ratio scale. These bounds can be used to compute an E-value, that is, the degree of confounding required to explain away an observed association, on the risk ratio scale. We complement and extend this previous work by deriving analogous bounds, based on sensitivity parameters on the risk difference scale. We show that our bounds can also be used to compute an E-value, on the risk difference scale. We compare our novel bounds with previous bounds through a real data example and a simulation study.

Incremental intervention effects in studies with dropout and many timepoints#

Modern longitudinal studies collect feature data at many timepoints, often of the same order of sample size. Such studies are typically affected by dropout and positivity violations. We tackle these problems by generalizing effects of recent incremental interventions (which shift propensity scores rather than set treatment values deterministically) to accommodate multiple outcomes and subject dropout. We give an identifying expression for incremental intervention effects when dropout is conditionally ignorable (without requiring treatment positivity) and derive the nonparametric efficiency bound for estimating such effects. Then we present efficient nonparametric estimators, showing that they converge at fast parametric rates and yield uniform inferential guarantees, even when nuisance functions are estimated flexibly at slower rates. We also study the variance ratio of incremental intervention effects relative to more conventional deterministic effects in a novel infinite time horizon setting, where the number of timepoints can grow with sample size and show that incremental intervention effects yield near-exponential gains in statistical precision in this setup. Finally, we conclude with simulations and apply our methods in a study of the effect of low-dose aspirin on pregnancy outcomes.

Learning linear non-Gaussian graphical models with multidirected edges

In this article, we propose a new method to learn the underlying acyclic mixed graph of a linear non-Gaussian structural equation model with given observational data. We build on an algorithm proposed by Wang and Drton, and we show that one can augment the hidden variable structure of the recovered model by learning multidirected edges rather than only directed and bidirected ones. Multidirected edges appear when more than two of the observed variables have a hidden common cause. We detect the presence of such hidden causes by looking at higher order cumulants and exploiting the multi-trek rule. Our method recovers the correct structure when the underlying graph is a bow-free acyclic mixed graph with potential multidirected edges.

A fundamental measure of treatment effect heterogeneity

The stratum-specific treatment effect function is a random variable giving the average treatment effect (ATE) for a randomly drawn stratum of potential confounders a clinician may use to assign treatment. In addition to the ATE, the variance of the stratum-specific treatment effect function is fundamental in determining the heterogeneity of treatment effect values. We offer a non-parametric plug-in estimator, the targeted maximum likelihood estimator (TMLE) and the cross-validated TMLE (CV-TMLE), to simultaneously estimate both the average and variance of the stratum-specific treatment effect function. The CV-TMLE is preferable because it guarantees asymptotic efficiency under two conditions without needing entropy conditions on the initial fits of the outcome model and treatment mechanism, as required by TMLE. Particularly, in circumstances where data adaptive fitting methods are very important to eliminate bias but hold no guarantee of satisfying the entropy condition, we show that the CV-TMLE sampling distributions maintain normality with a lower mean squared error than TMLE. In addition to verifying the theoretical properties of TMLE and CV-TMLE through simulations, we highlight some of the challenges in estimating the variance of the treatment effect, which lack double robustness and might be biased if the true variance is small and sample size insufficient.

Designing experiments informed by observational studies

The increasing availability of passively observed data has yielded a growing interest in “data fusion” methods, which involve merging data from observational and experimental sources to draw causal conclusions. Such methods often require a precarious tradeoff between the unknown bias in the observational dataset and the often-large variance in the experimental dataset. We propose an alternative approach, which avoids this tradeoff: rather than using observational data for inference, we use it to design a more efficient experiment. We consider the case of a stratified experiment with a binary outcome and suppose pilot estimates for the stratum potential outcome variances can be obtained from the observational study. We extend existing results to generate confidence sets for these variances, while accounting for the possibility of unmeasured confounding. Then, we pose the experimental design problem as a regret minimization problem subject to the constraints imposed by our confidence sets. We show that this problem can be converted into a concave maximization and solved using conventional methods. Finally, we demonstrate the practical utility of our methods using data from the Women’s Health Initiative.

Two seemingly paradoxical results in linear models: the variance inflation factor and the analysis of covariance

A result from a standard linear model course is that the variance of the ordinary least squares (OLS) coefficient of a variable will never decrease when including additional covariates into the regression. The variance inflation factor (VIF) measures the increase of the variance. Another result from a standard linear model or experimental design course is that including additional covariates in a linear model of the outcome on the treatment indicator will never increase the variance of the OLS coefficient of the treatment at least asymptotically. This technique is called the analysis of covariance (ANCOVA), which is often used to improve the efficiency of treatment effect estimation. So we have two paradoxical results: adding covariates never decreases the variance in the first result but never increases the variance in the second result. In fact, these two results are derived under different assumptions. More precisely, the VIF result conditions on the treatment indicators but the ANCOVA result averages over them. Comparing the estimators with and without adjusting for additional covariates in a completely randomized experiment, I show that the former has smaller variance averaging over the treatment indicators, and the latter has smaller variance at the cost of a larger bias conditioning on the treatment indicators. Therefore, there is no real paradox.

Causal versions of maximum entropy and principle of insufficient reason

The principle of insufficient reason (PIR) assigns equal probabilities to each alternative of a random experiment whenever there is no reason to prefer one over the other. The maximum entropy principle (MaxEnt) generalizes PIR to the case where statistical information like expectations are given. It is known that both principles result in paradoxical probability updates for joint distributions of cause and effect. This is because constraints on the conditional P ( effect ∣ cause ) P\left({\rm{effect}}| {\rm{cause}}) result in changes of P ( cause ) P\left({\rm{cause}}) that assign higher probability to those values of the cause that offer more options for the effect, suggesting “intentional behavior.” Earlier work therefore suggested sequentially maximizing (conditional) entropy according to the causal order, but without further justification apart from plausibility on toy examples. We justify causal modifications of PIR and MaxEnt by separating constraints into restrictions for the cause and restrictions for the mechanism that generates the effect from the cause. We further sketch why causal PIR also entails “Information Geometric Causal Inference.” We briefly discuss problems of generalizing the causal version of MaxEnt to arbitrary causal DAGs.

Radical empiricism and machine learning research

I contrast the “data fitting” vs “data interpreting” approaches to data science along three dimensions: Expediency, Transparency, and Explainability. “Data fitting” is driven by the faith that the secret to rational decisions lies in the data itself. In contrast, the data-interpreting school views data, not as a sole source of knowledge but as an auxiliary means for interpreting reality, and “reality” stands for the processes that generate the data. I argue for restoring balance to data science through a task-dependent symbiosis of fitting and interpreting, guided by the Logic of Causation.

Estimating causal effects with the neural autoregressive density estimator

The estimation of causal effects is fundamental in situations where the underlying system will be subject to active interventions. Part of building a causal inference engine is defining how variables relate to each other, that is, defining the functional relationship between variables entailed by the graph conditional dependencies. In this article, we deviate from the common assumption of linear relationships in causal models by making use of neural autoregressive density estimators and use them to estimate causal effects within Pearl’s do-calculus framework. Using synthetic data, we show that the approach can retrieve causal effects from non-linear systems without explicitly modeling the interactions between the variables and include confidence bands using the non-parametric bootstrap. We also explore scenarios that deviate from the ideal causal effect estimation setting such as poor data support or unobserved confounders.