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Register a new userCausal Inference on Non-linear Spaces: Distribution Functions and Beyond
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Linbo Wang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Understanding causal relationships is one of the most important goals of modern science. So far, the causal inference literature has focused almost exclusively on outcomes coming from a linear space, most commonly the Euclidean space. However, it is increasingly common that complex datasets collected through electronic sources, such as wearable devices and medical imaging, cannot be represented as data points from linear spaces. In this paper, we present a formal definition of causal effects for outcomes from non-linear spaces, with a focus on the Wasserstein space of cumulative distribution functions. We develop doubly robust estimators and associated asymptotic theory for these causal effects. Our framework extends to outcomes from certain Riemannian manifolds. As an illustration, we use our framework to quantify the causal effect of marriage on physical activity patterns using wearable device data collected through the National Health and Nutrition Examination Survey.
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Frequentist inference has a well-established supporting theory for doubly robust causal inference based on the potential outcomes framework, which is realized via outcome regression (OR) and propensity score (PS) models. The Bayesian counterpart, however, is not obvious as the PS model loses its balancing property in joint modeling. In this paper, we propose a natural and formal Bayesian solution by bridging loss-type Bayesian inference with a utility function derived from the notion of a pseudo-population via the change of measure. Consistency of the posterior distribution is shown with correctly specified and misspecified OR models. Simulation studies suggest that our proposed method can estimate the true causal effect more efficiently and achieve the frequentist coverage if either the OR model is correctly specified or fit with a flexible function of the confounders, compared to the previous Bayesian approach via the Bayesian bootstrap. Finally, we apply this novel Bayesian method to assess the impact of speed cameras on the reduction of car collisions in England.
Distribution function is essential in statistical inference, and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space, but without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric between random objects and a fixed location in metric spaces. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces that lie the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop homogeneity test and mutual independence test for non-Euclidean random objects, and present comprehensive empirical evidence to support the performance of our proposed methods.
Skepticism about the assumption of no unmeasured confounding, also known as exchangeability, is often warranted in making causal inferences from observational data; because exchangeability hinges on an investigators ability to accurately measure covariates that capture all potential sources of confounding. In practice, the most one can hope for is that covariate measurements are at best proxies of the true underlying confounding mechanism operating in a given observational study. In this paper, we consider the framework of proximal causal inference introduced by Tchetgen Tchetgen et al. (2020), which while explicitly acknowledging covariate measurements as imperfect proxies of confounding mechanisms, offers an opportunity to learn about causal effects in settings where exchangeability on the basis of measured covariates fails. We make a number of contributions to proximal inference including (i) an alternative set of conditions for nonparametric proximal identification of the average treatment effect; (ii) general semiparametric theory for proximal estimation of the average treatment effect including efficiency bounds for key semiparametric models of interest; (iii) a characterization of proximal doubly robust and locally efficient estimators of the average treatment effect. Moreover, we provide analogous identification and efficiency results for the average treatment effect on the treated. Our approach is illustrated via simulation studies and a data application on evaluating the effectiveness of right heart catheterization in the intensive care unit of critically ill patients.
We propose novel estimators for categorical and continuous treatments by using an optimal covariate balancing strategy for inverse probability weighting. The resulting estimators are shown to be consistent and asymptotically normal for causal contrasts of interest, either when the model explaining treatment assignment is correctly specified, or when the correct set of bases for the outcome models has been chosen and the assignment model is sufficiently rich. For the categorical treatment case, we show that the estimator attains the semiparametric efficiency bound when all models are correctly specified. For the continuous case, the causal parameter of interest is a function of the treatment dose. The latter is not parametrized and the estimators proposed are shown to have bias and variance of the classical nonparametric rate. Asymptotic results are complemented with simulations illustrating the finite sample properties. Our analysis of a data set suggests a nonlinear effect of BMI on the decline in self reported health.
Propensity score methods have been shown to be powerful in obtaining efficient estimators of average treatment effect (ATE) from observational data, especially under the existence of confounding factors. When estimating, deciding which type of covariates need to be included in the propensity score function is important, since incorporating some unnecessary covariates may amplify both bias and variance of estimators of ATE. In this paper, we show that including additional instrumental variables that satisfy the exclusion restriction for outcome will do harm to the statistical efficiency. Also, we prove that, controlling for covariates that appear as outcome predictors, i.e. predict the outcomes and are irrelevant to the exposures, can help reduce the asymptotic variance of ATE estimation. We also note that, efficiently estimating the ATE by non-parametric or semi-parametric methods require the estimated propensity score function, as described in Hirano et al. (2003)cite{Hirano2003}. Such estimation procedure usually asks for many regularity conditions, Rothe (2016)cite{Rothe2016} also illustrated this point and proposed a known propensity score (KPS) estimator that requires mild regularity conditions and is still fully efficient. In addition, we introduce a linearly modified (LM) estimator that is nearly efficient in most general settings and need not estimation of the propensity score function, hence convenient to calculate. The construction of this estimator borrows idea from the interaction estimator of Lin (2013)cite{Lin2013}, in which regression adjustment with interaction terms are applied to deal with data arising from a completely randomized experiment. As its name suggests, the LM estimator can be viewed as a linear modification on the IPW estimator using known propensity scores. We will also investigate its statistical properties both analytically and numerically.
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