Search

Researchers has long been focusing on causal inference with binary or categorical treatments, where causal estimands are well understood and inference tools are rich. However, causal problems involving continuous treatments are common in practice, yet a formal framework is scarce in the literature. We extend classic Fisher's randomization test to address an initial question: does a treatment have effect on the outcome of interest conditional on a set of covariates. Our theory starts from randomized experiments and generalizes to observational studies. Inference tools are developed to establish causal relationships and to verify underlying assumptions.

Meta-analysis can be a critical part of the research process, often serving as the primary analysis on which the practitioners, policymakers, and individuals base their decisions. However, current literature synthesis approaches to meta-analysis typically estimate a different quantity than what is implicitly intended; concretely, standard approaches estimate the average effect of a treatment for a population of imperfect studies, rather than the true scientific effect that would be measured in a population of hypothetical perfect studies. We advocate for an alternative method, called response-surface meta-analysis, which models the relationship between the quality of the study design as predictor variables and its reported estimated effect size as the outcome variable in order to estimate the effect size obtained by the hypothetical ideal study. The idea was first introduced by Rubin several decades ago, and here we provide a practical implementation. First, we reintroduce the idea of response-surface meta-analysis, highlighting its focus on a scientifically-motivated estimand while proposing a straightforward implementation. Then we compare the approach to traditional meta-analysis techniques used in practice. We then implement response-surface meta-analysis and contrast its results with existing literature-synthesis approaches on both simulated data and a real-world example published by the Cochrane Collaboration. We conclude by detailing the primary challenges in the implementation of response-surface meta-analysis and offer some suggestions to tackle these challenges., Comment: Oral presentation at Cochrane Colloquium 2023

Re-randomization has gained popularity as a tool for experiment-based causal inference due to its superior covariate balance and statistical efficiency compared to classic randomized experiments. However, the basic re-randomization method, known as ReM, and many of its extensions have been deemed sub-optimal as they fail to prioritize covariates that are more strongly associated with potential outcomes. To address this limitation and design more efficient re-randomization procedures, a more precise quantification of covariate heterogeneity and its impact on the causal effect estimator is in a great appeal. This work fills in this gap with a Bayesian criterion for re-randomization and a series of novel re-randomization procedures derived under such a criterion. Both theoretical analyses and numerical studies show that the proposed re-randomization procedures under the Bayesian criterion outperform existing ReM-based procedures significantly in effectively balancing covariates and precisely estimating the unknown causal effect.

Catalytic prior distributions provide general, easy-to-use, and interpretable specifications of prior distributions for Bayesian analysis. They are particularly beneficial when the observed data are inadequate to stably estimate a complex target model. A catalytic prior distribution is constructed by augmenting the observed data with synthetic data that are sampled from the predictive distribution of a simpler model estimated from the observed data. We illustrate the usefulness of the catalytic prior approach using an example from labor economics. In the example, the resulting Bayesian inference reflects many important aspects of the observed data, and the estimation accuracy and predictive performance of the inference based on the catalytic prior are superior to, or comparable to, that of other commonly used prior distributions. We further explore the connection between the catalytic prior approach and a few popular regularization methods. We expect the catalytic prior approach to be useful in many applications.

Consider a situation with two treatments, the first of which is randomized but the second is not, and the multifactor version of this. Interest is in treatment effects, defined using standard factorial notation. We define estimators for the treatment effects and explore their properties when there is information about the nonrandomized treatment assignment and when there is no information on the assignment of the nonrandomized treatment. We show when and how hidden treatments can bias estimators and inflate their sampling variances.

Mahalanobis distance between treatment group and control group covariate means is often adopted as a balance criterion when implementing a rerandomization strategy. However, this criterion may not work well for high-dimensional cases because it balances all orthogonalized covariates equally. Here, we propose leveraging principal component analysis (PCA) to identify proper subspaces in which Mahalanobis distance should be calculated. Not only can PCA effectively reduce the dimensionality for high-dimensional cases while capturing most of the information in the covariates, but it also provides computational simplicity by focusing on the top orthogonal components. We show that our PCA rerandomization scheme has desirable theoretical properties on balancing covariates and thereby on improving the estimation of average treatment effects. We also show that this conclusion is supported by numerical studies using both simulated and real examples.

The weaponization of digital communications and social media to conduct disinformation campaigns at immense scale, speed, and reach presents new challenges to identify and counter hostile influence operations (IOs). This paper presents an end-to-end framework to automate detection of disinformation narratives, networks, and influential actors. The framework integrates natural language processing, machine learning, graph analytics, and a novel network causal inference approach to quantify the impact of individual actors in spreading IO narratives. We demonstrate its capability on real-world hostile IO campaigns with Twitter datasets collected during the 2017 French presidential elections, and known IO accounts disclosed by Twitter over a broad range of IO campaigns (May 2007 to February 2020), over 50,000 accounts, 17 countries, and different account types including both trolls and bots. Our system detects IO accounts with 96% precision, 79% recall, and 96% area-under-the-PR-curve, maps out salient network communities, and discovers high-impact accounts that escape the lens of traditional impact statistics based on activity counts and network centrality. Results are corroborated with independent sources of known IO accounts from U.S. Congressional reports, investigative journalism, and IO datasets provided by Twitter., Comment: Proc. Natl. Acad. Sciences U.S.A. Vol. 118, No. 4, e2011216118

With many pretreatment covariates and treatment factors, the classical factorial experiment often fails to balance covariates across multiple factorial effects simultaneously. Therefore, it is intuitive to restrict the randomization of the treatment factors to satisfy certain covariate balance criteria, possibly conforming to the tiers of factorial effects and covariates based on their relative importances. This is rerandomization in factorial experiments. We study the asymptotic properties of this experimental design under the randomization inference framework without imposing any distributional or modeling assumptions of the covariates and outcomes. We derive the joint asymptotic sampling distribution of the usual estimators of the factorial effects, and show that it is symmetric, unimodal, and more "concentrated" at the true factorial effects under rerandomization than under the classical factorial experiment. We quantify this advantage of rerandomization using the notions of "central convex unimodality" and "peakedness" of the joint asymptotic sampling distribution. We also construct conservative large-sample confidence sets for the factorial effects.

Estimating influence on social media networks is an important practical and theoretical problem, especially because this new medium is widely exploited as a platform for disinformation and propaganda. This paper introduces a novel approach to influence estimation on social media networks and applies it to the real-world problem of characterizing active influence operations on Twitter during the 2017 French presidential elections. The new influence estimation approach attributes impact by accounting for narrative propagation over the network using a network causal inference framework applied to data arising from graph sampling and filtering. This causal framework infers the difference in outcome as a function of exposure, in contrast to existing approaches that attribute impact to activity volume or topological features, which do not explicitly measure nor necessarily indicate actual network influence. Cram\'er-Rao estimation bounds are derived for parameter estimation as a step in the causal analysis, and used to achieve geometrical insight on the causal inference problem. The ability to infer high causal influence is demonstrated on real-world social media accounts that are later independently confirmed to be either directly affiliated or correlated with foreign influence operations using evidence supplied by the U.S. Congress and journalistic reports., Comment: 5 pages, 4 figures, 1 table

The health effects of environmental exposures have been studied for decades, typically using standard regression models to assess exposure-outcome associations found in observational non-experimental data. We propose and illustrate a different approach to examine causal effects of environmental exposures on health outcomes from observational data. Our strategy attempts to structure the observational data to approximate data from a hypothetical, but realistic, randomized experiment. This approach, based on insights from classical experimental design, involves four stages, and relies on modern computing to implement the effort in two of the four stages.More specifically, our strategy involves: 1) a conceptual stage that involves the precise formulation of the causal question in terms of a hypothetical randomized experiment where the exposure is assigned to units; 2) a design stage that attempts to reconstruct (or approximate) a randomized experiment before any outcome data are observed, 3) a statistical analysis comparing the outcomes of interest in the exposed and non-exposed units of the hypothetical randomized experiment, and 4) a summary stage providing conclusions about statistical evidence for the sizes of possible causal effects of the exposure on outcomes. We illustrate our approach using an example examining the effect of parental smoking on children's lung function collected in families living in East Boston in the 1970's. To complement the traditional purely model-based approaches, our strategy, which includes outcome free matched-sampling, provides workable tools to quantify possible detrimental exposure effects on human health outcomes especially because it also includes transparent diagnostics to assess the assumptions of the four-stage statistical approach being applied.

This article considers causal inference for treatment contrasts from a randomized experiment using potential outcomes in a finite population setting. Adopting a Neymanian repeated sampling approach that integrates such causal inference with finite population survey sampling, an inferential framework is developed for general mechanisms of assigning experimental units to multiple treatments. This framework extends classical methods by allowing the possibility of randomization restrictions and unequal replications. Novel conditions that are "milder" than strict additivity of treatment effects, yet permit unbiased estimation of the finite population sampling variance of any treatment contrast estimator, are derived. The consequences of departures from such conditions are also studied under the criterion of minimax bias, and a new justification for using the Neymanian conservative sampling variance estimator in experiments is provided. The proposed approach can readily be extended to the case of treatments with a general factorial structure.

Although complete randomization ensures covariate balance on average, the chance for observing significant differences between treatment and control covariate distributions increases with many covariates. Rerandomization discards randomizations that do not satisfy a predetermined covariate balance criterion, generally resulting in better covariate balance and more precise estimates of causal effects. Previous theory has derived finite sample theory for rerandomization under the assumptions of equal treatment group sizes, Gaussian covariate and outcome distributions, or additive causal effects, but not for the general sampling distribution of the difference-in-means estimator for the average causal effect. To supplement existing results, we develop asymptotic theory for rerandomization without these assumptions, which reveals a non-Gaussian asymptotic distribution for this estimator, specifically a linear combination of a Gaussian random variable and a truncated Gaussian random variable. This distribution follows because rerandomization affects only the projection of potential outcomes onto the covariate space but does not affect the corresponding orthogonal residuals. We also demonstrate that, compared to complete randomization, rerandomization reduces the asymptotic sampling variances and quantile ranges of the difference-in-means estimator. Moreover, our work allows the construction of accurate large-sample confidence intervals for the average causal effect, thereby revealing further advantages of rerandomization over complete randomization.

Data analyses typically rely upon assumptions about missingness mechanisms that lead to observed versus missing data. When the data are missing not at random, direct assumptions about the missingness mechanism, and indirect assumptions about the distributions of observed and missing data, are typically untestable. We explore an approach, where the joint distribution of observed data and missing data is specified through non-standard conditional distributions. In this formulation, which traces back to a factorization of the joint distribution, apparently proposed by J.W. Tukey, the modeling assumptions about the conditional factors are either testable or are designed to allow the incorporation of substantive knowledge about the problem at hand, thereby offering a possibly realistic portrayal of the data, both missing and observed. We apply Tukey's conditional representation to exponential family models, and we propose a computationally tractable inferential strategy for this class of models. We illustrate the utility of this approach using high-throughput biological data with missing data that are not missing at random., Comment: 37 pages, 9 figures, 1 table

Factorial designs are widely used in agriculture, engineering, and the social sciences to study the causal effects of several factors simultaneously on a response. The objective of such a design is to estimate all factorial effects of interest, which typically include main effects and interactions among factors. To estimate factorial effects with high precision when a large number of pre-treatment covariates are present, balance among covariates across treatment groups should be ensured. We propose utilizing rerandomization to ensure covariate balance in factorial designs. Although both factorial designs and rerandomization have been discussed before, the combination has not. Here, theoretical properties of rerandomization for factorial designs are established, and empirical results are explored using an application from the New York Department of Education.

We examine the possible consequences of a change in law school admissions in the United States from an affirmative action system based on race to one based on socioeconomic class. Using data from the 1991-1996 Law School Admission Council Bar Passage Study, students were reassigned attendance by simulation to law school tiers by transferring the affirmative action advantage for black students to students from low socioeconomic backgrounds. The hypothetical academic outcomes for the students were then multiply-imputed to quantify the uncertainty of the resulting estimates. The analysis predicts dramatic decreases in the numbers of black students in top law school tiers, suggesting that class-based affirmative action is insufficient to maintain racial diversity in prestigious law schools. Furthermore, there appear to be no statistically significant changes in the graduation and bar passage rates of students in any demographic group. The results thus provide evidence that, other than increasing their representation in upper tiers, current affirmative action policies relative to a socioeconomic-based system neither substantially help nor harm minority academic outcomes, contradicting the predictions of the "mismatch" hypothesis, which asserts otherwise., Comment: Published at http://dx.doi.org/10.1214/15-STS514 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Many outcomes of interest in the social and health sciences, as well as in modern applications in computational social science and experimentation on social media platforms, are ordinal and do not have a meaningful scale. Causal analyses that leverage this type of data, termed ordinal non-numeric, require careful treatment, as much of the classical potential outcomes literature is concerned with estimation and hypothesis testing for outcomes whose relative magnitudes are well defined. Here, we propose a class of finite population causal estimands that depend on conditional distributions of the potential outcomes, and provide an interpretable summary of causal effects when no scale is available. We formulate a relaxation of the Fisherian sharp null hypothesis of constant effect that accommodates the scale-free nature of ordinal non-numeric data. We develop a Bayesian procedure to estimate the proposed causal estimands that leverages the rank likelihood. We illustrate these methods with an application to educational outcomes in the General Social Survey.

The Neyman-Fisher controversy considered here originated with the 1935 presentation of Jerzy Neyman's Statistical Problems in Agricultural Experimentation to the Royal Statistical Society. Neyman asserted that the standard ANOVA F-test for randomized complete block designs is valid, whereas the analogous test for Latin squares is invalid in the sense of detecting differentiation among the treatments, when none existed on average, more often than desired (i.e., having a higher Type I error than advertised). However, Neyman's expressions for the expected mean residual sum of squares, for both designs, are generally incorrect. Furthermore, Neyman's belief that the Type I error (when testing the null hypothesis of zero average treatment effects) is higher than desired, whenever the expected mean treatment sum of squares is greater than the expected mean residual sum of squares, is generally incorrect. Simple examples show that, without further assumptions on the potential outcomes, one cannot determine the Type I error of the F-test from expected sums of squares. Ultimately, we believe that the Neyman-Fisher controversy had a deleterious impact on the development of statistics, with a major consequence being that potential outcomes were ignored in favor of linear models and classical statistical procedures that are imprecise without applied contexts., Comment: Published in at http://dx.doi.org/10.1214/13-STS454 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In educational research, outcome measures are often estimated across separate studies or across schools, districts, or other subgroups to assess the overall causal effect of an active treatment versus a control treatment. Students may be partitioned into such strata or blocks by experimental design, or separated into studies within a meta-analysis. In non-randomized studies, students may be partitioned into subclasses based on key covariates or estimated propensity scores to improve observed covariate balance across treatment groups (e.g., Rosenbaum & Rubin, 1983). Procedures designed to estimate any estimand in the presence of strata, including a simple t-test for the difference in mean outcomes (Neyman, Iwaszkiewicz, & Kolodziejczyk, 1935), rely on implicit assumptions about the unknowable correlation between potential outcomes under active treatment and control treatment. For binary outcomes, the standard procedures used to estimate overall odds ratios in the presence of strata were introduced by Cochran (1954), who first proposed a hypothesis test for the difference in proportions across strata. Mantel and Haenszel (1959) proposed a very similar test and introduced an estimator for a common odds ratio. Consider the following hypothetical studies designed to estimate the causal effect of an existing program on high school graduation: (1) Within each of several school districts, half of the schools are randomized to participate in the program, and half are randomized not to participate in the program. (2) Within each of several cities, schools participating in the program are compared to schools not participating in the program, though participation was not randomized. (3) Several separate evaluations of the program are collected, to be combined in a meta-analysis. In each hypothetical study, a binary outcome (graduation) must be measured over strata (1. school districts, 2. cities, 3. evaluations). The effect of the program on graduation rates may be measured by a difference in proportions, odds ratio, or some other quantity. A hypothesis test will be conducted and a confidence interval constructed for the chosen estimand. The authors propose tests and intervals that can be more powerful for any finite population estimand than traditional tests and intervals, including t-tests for the difference in means or Cochran-Mantel- Haenszel procedures for the odds ratio.

A framework for causal inference from two-level factorial designs is proposed. The framework utilizes the concept of potential outcomes that lies at the center stage of causal inference and extends Neyman's repeated sampling approach for estimation of causal effects and randomization tests based on Fisher's sharp null hypothesis to the case of 2-level factorial experiments. The framework allows for statistical inference from a finite population, permits definition and estimation of estimands other than "average factorial effects" and leads to more flexible inference procedures than those based on ordinary least squares estimation from a linear model., Comment: Preliminary version; comments welcome. Added figures and a table

Randomized experiments are the "gold standard" for estimating causal effects, yet often in practice, chance imbalances exist in covariate distributions between treatment groups. If covariate data are available before units are exposed to treatments, these chance imbalances can be mitigated by first checking covariate balance before the physical experiment takes place. Provided a precise definition of imbalance has been specified in advance, unbalanced randomizations can be discarded, followed by a rerandomization, and this process can continue until a randomization yielding balance according to the definition is achieved. By improving covariate balance, rerandomization provides more precise and trustworthy estimates of treatment effects., Comment: Published in at http://dx.doi.org/10.1214/12-AOS1008 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

For obtaining causal inferences that are objective, and therefore have the best chance of revealing scientific truths, carefully designed and executed randomized experiments are generally considered to be the gold standard. Observational studies, in contrast, are generally fraught with problems that compromise any claim for objectivity of the resulting causal inferences. The thesis here is that observational studies have to be carefully designed to approximate randomized experiments, in particular, without examining any final outcome data. Often a candidate data set will have to be rejected as inadequate because of lack of data on key covariates, or because of lack of overlap in the distributions of key covariates between treatment and control groups, often revealed by careful propensity score analyses. Sometimes the template for the approximating randomized experiment will have to be altered, and the use of principal stratification can be helpful in doing this. These issues are discussed and illustrated using the framework of potential outcomes to define causal effects, which greatly clarifies critical issues., Comment: Published in at http://dx.doi.org/10.1214/08-AOAS187 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Causal inference is best understood using potential outcomes. This use is particularly important in more complex settings, that is, observational studies or randomized experiments with complications such as noncompliance. The topic of this lecture, the issue of estimating the causal effect of a treatment on a primary outcome that is ``censored'' by death, is another such complication. For example, suppose that we wish to estimate the effect of a new drug on Quality of Life (QOL) in a randomized experiment, where some of the patients die before the time designated for their QOL to be assessed. Another example with the same structure occurs with the evaluation of an educational program designed to increase final test scores, which are not defined for those who drop out of school before taking the test. A further application is to studies of the effect of job-training programs on wages, where wages are only defined for those who are employed. The analysis of examples like these is greatly clarified using potential outcomes to define causal effects, followed by principal stratification on the intermediated outcomes (e.g., survival)., Comment: This paper commented in: [math.ST/0612785], [math.ST/0612786], [math.ST/0612788]. Rejoinder in [math.ST/0612789]. Published at http://dx.doi.org/10.1214/088342306000000114 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Rejoinder on Causal Inference Through Potential Outcomes and Principal Stratification: Application to Studies with ``Censoring'' Due to Death by D. B. Rubin [math.ST/0612783], Comment: Published at http://dx.doi.org/10.1214/088342306000000303 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In observational studies designed to estimate the effects of interventions or exposures, such as cigarette smoking, it is desirable to try to control background differences between the treated group (e.g., current smokers) and the control group (e.g., never smokers) on covariates $X$ (e.g., age, education). Matched sampling attempts to effect this control by selecting subsets of the treated and control groups with similar distributions of such covariates. This paper examines the consequences of matching using affinely invariant methods when the covariate distributions are ``discriminant mixtures of proportional ellipsoidally symmetric'' (DMPES) distributions, a class herein defined, which generalizes the ellipsoidal symmetry class of Rubin and Thomas [Ann. Statist. 20 (1992) 1079--1093]. The resulting generalized results help indicate why earlier results hold quite well even when the simple assumption of ellipsoidal symmetry is not met [e.g., Biometrics 52 (1996) 249--264]. Extensions to conditionally affinely invariant matching with conditionally DMPES distributions are also discussed., Comment: Published at http://dx.doi.org/10.1214/009053606000000407 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Propensity score methods were proposed by Rosenbaum and Rubin [Biometrika 70 (1983) 41--55] as central tools to help assess the causal effects of interventions. Since their introduction more than two decades ago, they have found wide application in a variety of areas, including medical research, economics, epidemiology and education, especially in those situations where randomized experiments are either difficult to perform, or raise ethical questions, or would require extensive delays before answers could be obtained. In the past few years, the number of published applications using propensity score methods to evaluate medical and epidemiological interventions has increased dramatically. Nevertheless, thus far, we believe that there have been few applications of propensity score methods to evaluate marketing interventions (e.g., advertising, promotions), where the tradition is to use generally inappropriate techniques, which focus on the prediction of an outcome from background characteristics and an indicator for the intervention using statistical tools such as least-squares regression, data mining, and so on. With these techniques, an estimated parameter in the model is used to estimate some global ``causal'' effect. This practice can generate grossly incorrect answers that can be self-perpetuating: polishing the Ferraris rather than the Jeeps ``causes'' them to continue to win more races than the Jeeps $\Leftrightarrow$ visiting the high-prescribing doctors rather than the low-prescribing doctors ``causes'' them to continue to write more prescriptions. This presentation will take ``causality'' seriously, not just as a casual concept implying some predictive association in a data set, and will illustrate why propensity score methods are generally superior in practice to the standard predictive approaches for estimating causal effects., Comment: Published at http://dx.doi.org/10.1214/088342306000000259 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)