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Thomas S. Richardson

Publications and source records attributed to Thomas S. Richardson.

At least 37 records · Page 2Linked to original sources
arXiv2020

Empirical Bayes for Large-scale Randomized Experiments: a Spectral Approach

Large-scale randomized experiments, sometimes called A/B tests, are increasingly prevalent in many industries. Though such experiments are often analyzed via frequentist $t$-tests, arguably such analyses are deficient: $p$-values are hard to interpret and not easily incorporated into decision-making. As an alternative, we propose an empirical Bayes approach, which assumes that the treatment effects are realized from a "true prior". This requires inferring the prior from previous experiments. Following Robbins, we estimate a family of marginal densities of empirical effects, indexed by the noise scale. We show that this family is characterized by the heat equation. We develop a spectral maximum likelihood estimate based on a Fourier series representation, which can be efficiently computed via convex optimization. In order to select hyperparameters and compare models, we describe two model selection criteria. We demonstrate our method on simulated and real data, and compare posterior inference to that under a Gaussian mixture model of the prior.

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arXiv2020

On Testing Marginal versus Conditional Independence

We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are non-nested and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback-Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such rate, we show that the asymptotic distribution of the likelihood ratio depends on where and how the local alternatives approach the intersection. To deal with this non-uniformity, we study a class of "envelope" distributions by taking pointwise suprema over asymptotic cumulative distribution functions. We show that these envelope distributions are well-behaved and lead to model selection procedures with rate-free uniform error guarantees and near-optimal power. To control the error even when the two models are indistinguishable, rather than insist on a dichotomous choice, the proposed procedure will choose either or both models.

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arXiv2018

Identification and Estimation of Causal Effects with Outcomes Truncated by Death

It is common in medical studies that the outcome of interest is truncated by death, meaning that a subject has died before the outcome could be measured. In this case, restricted analysis among survivors may be subject to selection bias. Hence, it is of interest to estimate the survivor average causal effect, defined as the average causal effect among the subgroup consisting of subjects who would survive under either exposure. In this paper, we consider the identification and estimation problems of the survivor average causal effect. We propose to use a substitution variable in place of the latent membership in the always-survivor group. The identification conditions required for a substitution variable are conceptually similar to conditions for a conditional instrumental variable, and may apply to both randomized and observational studies. We show that the survivor average causal effect is identifiable with use of such a substitution variable, and propose novel model parameterizations for estimation of the survivor average causal effect under our identification assumptions. Our approaches are illustrated via simulation studies and a data analysis.

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arXiv2017

Smooth, identifiable supermodels of discrete DAG models with latent variables

We provide a parameterization of the discrete nested Markov model, which is a supermodel that approximates DAG models (Bayesian network models) with latent variables. Such models are widely used in causal inference and machine learning. We explicitly evaluate their dimension, show that they are curved exponential families of distributions, and fit them to data. The parameterization avoids the irregularities and unidentifiability of latent variable models. The parameters used are all fully identifiable and causally-interpretable quantities.

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arXiv2016

Causal analysis of ordinal treatments and binary outcomes under truncation by death

It is common that in multiarm randomized trials, the outcome of interest is "truncated by death," meaning that it is only observed or well defined conditioning on an intermediate outcome. In this case, in addition to pairwise contrasts, the joint inference for all treatment arms is also of interest. Under a monotonicity assumption we present methods for both pairwise and joint causal analyses of ordinal treatments and binary outcomes in presence of truncation by death. We illustrate via examples the appropriateness of our assumptions in different scientific contexts.

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arXiv2016

On falsification of the binary instrumental variable model

Instrumental variables are widely used for estimating causal effects in the presence of unmeasured confounding. The discrete instrumental variable model has testable implications on the law of the observed data. However, current assessments of instrumental validity are typically based solely on subject-matter arguments rather than these testable implications, partly due to a lack of formal statistical tests with known properties. In this paper, we develop simple procedures for testing the binary instrumental variable model. Our methods are based on existing approaches for comparing two treatments, such as the t-test and the Gail--Simon test. We illustrate the importance of testing the instrumental variable model by evaluating the exogeneity of college proximity using the National Longitudinal Survey of Young Men.

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arXiv2016

On Modeling and Estimation for the Relative Risk and Risk Difference

A common problem in formulating models for the relative risk and risk difference is the variation dependence between these parameters and the baseline risk, which is a nuisance model. We address this problem by proposing the conditional log odds-product as a preferred nuisance model. This novel nuisance model facilitates maximum-likelihood estimation, but also permits doubly-robust estimation for the parameters of interest. Our approach is illustrated via simulations and a data analysis.

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arXiv2016

Estimating the effect of joint interventions from observational data in sparse high-dimensional settings

We consider the estimation of joint causal effects from observational data. In particular, we propose new methods to estimate the effect of multiple simultaneous interventions (e.g., multiple gene knockouts), under the assumption that the observational data come from an unknown linear structural equation model with independent errors. We derive asymptotic variances of our estimators when the underlying causal structure is partly known, as well as high-dimensional consistency when the causal structure is fully unknown and the joint distribution is multivariate Gaussian. We also propose a generalization of our methodology to the class of nonparanormal distributions. We evaluate the estimators in simulation studies and also illustrate them on data from the DREAM4 challenge.

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arXiv2015

Causal Etiology of the Research of James M. Robins

This issue of Statistical Science draws its inspiration from the work of James M. Robins. Jon Wellner, the Editor at the time, asked the two of us to edit a special issue that would highlight the research topics studied by Robins and the breadth and depth of Robins' contributions. Between the two of us, we have collaborated closely with Jamie for nearly 40 years. We agreed to edit this issue because we recognized that we were among the few in a position to relate the trajectory of his research career to date.

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arXiv2014

Markovian acyclic directed mixed graphs for discrete data

Acyclic directed mixed graphs (ADMGs) are graphs that contain directed ($\rightarrow$) and bidirected ($\leftrightarrow$) edges, subject to the constraint that there are no cycles of directed edges. Such graphs may be used to represent the conditional independence structure induced by a DAG model containing hidden variables on its observed margin. The Markovian model associated with an ADMG is simply the set of distributions obeying the global Markov property, given via a simple path criterion (m-separation). We first present a factorization criterion characterizing the Markovian model that generalizes the well-known recursive factorization for DAGs. For the case of finite discrete random variables, we also provide a parameterization of the model in terms of simple conditional probabilities, and characterize its variation dependence. We show that the induced models are smooth. Consequently, Markovian ADMG models for discrete variables are curved exponential families of distributions.

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arXiv2014

A factorization criterion for acyclic directed mixed graphs

Acyclic directed mixed graphs, also known as semi-Markov models represent the conditional independence structure induced on an observed margin by a DAG model with latent variables. In this paper we present a factorization criterion for these models that is equivalent to the global Markov property given by (the natural extension of) d-separation.

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arXiv2013

Sparse Nested Markov models with Log-linear Parameters

Hidden variables are ubiquitous in practical data analysis, and therefore modeling marginal densities and doing inference with the resulting models is an important problem in statistics, machine learning, and causal inference. Recently, a new type of graphical model, called the nested Markov model, was developed which captures equality constraints found in marginals of directed acyclic graph (DAG) models. Some of these constraints, such as the so called `Verma constraint', strictly generalize conditional independence. To make modeling and inference with nested Markov models practical, it is necessary to limit the number of parameters in the model, while still correctly capturing the constraints in the marginal of a DAG model. Placing such limits is similar in spirit to sparsity methods for undirected graphical models, and regression models. In this paper, we give a log-linear parameterization which allows sparse modeling with nested Markov models. We illustrate the advantages of this parameterization with a simulation study.

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arXiv2013

Causal Inference in the Presence of Latent Variables and Selection Bias

We show that there is a general, informative and reliable procedure for discovering causal relations when, for all the investigator knows, both latent variables and selection bias may be at work. Given information about conditional independence and dependence relations between measured variables, even when latent variables and selection bias may be present, there are sufficient conditions for reliably concluding that there is a causal path from one variable to another, and sufficient conditions for reliably concluding when no such causal path exists.

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arXiv2013

A Discovery Algorithm for Directed Cyclic Graphs

Directed acyclic graphs have been used fruitfully to represent causal strucures (Pearl 1988). However, in the social sciences and elsewhere models are often used which correspond both causally and statistically to directed graphs with directed cycles (Spirtes 1995). Pearl (1993) discussed predicting the effects of intervention in models of this kind, so-called linear non-recursive structural equation models. This raises the question of whether it is possible to make inferences about causal structure with cycles, form sample data. In particular do there exist general, informative, feasible and reliable precedures for inferring causal structure from conditional independence relations among variables in a sample generated by an unknown causal structure? In this paper I present a discovery algorithm that is correct in the large sample limit, given commonly (but often implicitly) made plausible assumptions, and which provides information about the existence or non-existence of causal pathways from one variable to another. The algorithm is polynomial on sparse graphs.

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arXiv2013

A Polynomial-Time Algorithm for Deciding Markov Equivalence of Directed Cyclic Graphical Models

Although the concept of d-separation was originally defined for directed acyclic graphs (see Pearl 1988), there is a natural extension of he concept to directed cyclic graphs. When exactly the same set of d-separation relations hold in two directed graphs, no matter whether respectively cyclic or acyclic, we say that they are Markov equivalent. In other words, when two directed cyclic graphs are Markov equivalent, the set of distributions that satisfy a natural extension of the Global Directed Markov condition (Lauritzen et al. 1990) is exactly the same for each graph. There is an obvious exponential (in the number of vertices) time algorithm for deciding Markov equivalence of two directed cyclic graphs; simply chech all of the d-separation relations in each graph. In this paper I state a theorem that gives necessary and sufficient conditions for the Markov equivalence of two directed cyclic graphs, where each of the conditions can be checked in polynomial time. Hence, the theorem can be easily adapted into a polynomial time algorithm for deciding the Markov equivalence of two directed cyclic graphs. Although space prohibits inclusion of correctness proofs, they are fully described in Richardson (1994b).

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arXiv2013

General theory for interactions in sufficient cause models with dichotomous exposures

The sufficient-component cause framework assumes the existence of sets of sufficient causes that bring about an event. For a binary outcome and an arbitrary number of binary causes any set of potential outcomes can be replicated by positing a set of sufficient causes; typically this representation is not unique. A sufficient cause interaction is said to be present if within all representations there exists a sufficient cause in which two or more particular causes are all present. A singular interaction is said to be present if for some subset of individuals there is a unique minimal sufficient cause. Empirical and counterfactual conditions are given for sufficient cause interactions and singular interactions between an arbitrary number of causes. Conditions are given for cases in which none, some or all of a given set of causes affect the outcome monotonically. The relations between these results, interactions in linear statistical models and Pearl's probability of causation are discussed.

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