We develop a novel and general framework for reduced-bias $M$-estimation from asymptotically unbiased estimating functions. The framework relies on an empirical approximation of the bias by a function of derivatives of estimating function contributions. Reduced-bias $M$-estimation operates either implicitly, by solving empirically-adjusted estimating equations, or explicitly, by subtracting the estimated bias from the original $M$-estimates, and applies to models that are partially- or fully-specified, with either likelihoods or other surrogate objectives. Automatic differentiation can be used to abstract away the only algebra required to implement reduced-bias $M$-estimation. As a result, the bias reduction methods we introduce have markedly broader applicability with more straightforward implementation and less algebraic or computational effort than other established bias-reduction methods that require resampling or evaluation of expectations of products of log-likelihood derivatives. If $M$-estimation is by maximizing an objective, then there always exists a bias-reducing penalized objective. That penalized objective relates closely to information criteria for model selection, and can be further enhanced with plug-in penalties to deliver reduced-bias $M$-estimates with extra properties, like finiteness in models for categorical data. The reduced-bias $M$-estimators have the same asymptotic distribution as the original $M$-estimators, and, hence, standard procedures for inference and model selection apply unaltered with the improved estimates. We demonstrate and assess the properties of reduced-bias $M$-estimation in well-used, prominent modelling settings of varying complexity.
We develop and evaluate point and interval estimates for the random effects $θ_i$, having made observations $y_i|θ_i\stackrel{\m athit{ind}}{\sim}N[θ_i,V_i],i=1,...,k$ that follow a two-level Normal hierarchical model. Fitting this model requires assessing the Level-2 variance $A\equiv\operatorname {Var}(θ_i)$ to estimate shrinkages $B_i\equiv V_i/(V_i+A)$ toward a (possibly estimated) subspace, with $B_i$ as the target because the conditional means and variances of $θ_i$ depend linearly on $B_i$, not on $A$. Adjustment for density maximization, ADM, can do the fitting for any smooth prior on …
The accuracy of a diagnostic test is typically characterised using the receiver operating characteristic (ROC) curve. Summarising indexes such as the area under the ROC curve (AUC) are used to compare different tests as well as to measure the difference between two populations. Often additional information is available on some of the covariates which are known to influence the accuracy of such measures. We propose nonparametric methods for covariate adjustment of the AUC. Models with normal errors and non-normal errors are discussed and analysed separately. Nonparametric regression is used for…
In this paper we consider the estimation of population size from one-source capture--recapture data, that is, a list in which individuals can potentially be found repeatedly and where the question is how many individuals are missed by the list. As a typical example, we provide data from a drug user study in Bangkok from 2001 where the list consists of drug users who repeatedly contact treatment institutions. Drug users with 1, 2, 3$,...$ contacts occur, but drug users with zero contacts are not present, requiring the size of this group to be estimated. Statistically, these data can be consider…
When estimating a Global Average Treatment Effect (GATE) under network interference, units can have widely different relationships to the treatment depending on a combination of the structure of their network neighborhood, the structure of the interference mechanism, and how the treatment was distributed in their neighborhood. In this work, we introduce a sequential procedure to generate and select graph- and treatment-based covariates for GATE estimation under regression adjustment. We show that it is possible to simultaneously achieve low bias and considerably reduce variance with such a pro…
Power Under Multiplicity Project (PUMP): Estimating Power, Minimum Detectable Effect Size, and Sample Size When Adjusting for Multiple Outcomes in Multi-level Experiments
The purpose of this work is to improve the efficiency in estimating the average causal effect (ACE) on the survival scale where right-censoring exists and high-dimensional covariate information is available. We propose new estimators using regularized survival regression and survival random forests (SRF) to make the adjustment for the high dimensional covariates to improve efficiency. We study the behavior of the adjusted estimator under mild assumptions and show theoretical guarantees that the proposed estimators are more efficient than the unadjusted ones asymptotically when using SRF for ad…
Towards Best Practices for Covariate Adjustment in Regulatory Trials: From Fixed to Data-Adaptive Approaches
University of California Berkeley（加州大学伯克利分校）；Division of Biometrics IV, OB/OTS/CDER/FDA（美国食品药品监督管理局药物评价与研究中心生物统计学第四处）；Harvard T.H. Chan School of Public Health（哈佛大学陈曾熙公共卫生学院）；Beth Israel Deaconess Hospital（贝斯以色列女执事医疗中心）；Novo Nordisk A/S（诺和诺德公司）；Pfizer Inc.（辉瑞公司）；Northeastern University（东北大学）；Gilead Sciences（吉利德科学公司）；National Center for the Advancement of Translational Science, National Institutes of Health（美国国立卫生研究院转化科学促进国家中心）；Section of Epidemiology, Biostatistics and Biodemography, University of Southern Denmark（南丹麦大学流行病学、生物统计学和生物人口学系）；Forum for Collaborative Research, School of Public Health, University of California Berkeley（加州大学伯克利分校公共卫生学院合作研究论坛）；OncoBrain Inc（OncoBrain 公司）；Moffitt Cancer Center（莫菲特癌症中心）；Department of Biostatistics, University of Michigan（密歇根大学生物统计学系）；Department of Mathematics, Computer Science and Statistics, Ghent University（根特大学数学、计算机科学和统计系）；Department of Mathematics and Data Science, Vrije Universiteit Brussel（布鲁塞尔自由大学数学与数据科学系）
Identifying effects of actions (treatments) on outcome variables from observational data and causal assumptions is a fundamental problem in causal inference. This identification is made difficult by the presence of confounders which can be related to both treatment and outcome variables. Confounders are often handled, both in theory and in practice, by adjusting for covariates, in other words considering outcomes conditioned on treatment and covariate values, weighed by probability of observing those covariate values. In this paper, we give a complete graphical criterion for covariate adjustme…
Propensity score methods are common for estimating a binary treatment effect when treatment assignment is not randomized. When exposure is measured on an ordinal scale (i.e., low - medium - high), however, propensity score inference requires extensions which have received limited attention. Estimands of possible interest with an ordinal exposure are the average treatment effects between each pair of exposure levels. Using these estimands, it is possible to determine an optimal exposure level. Traditional methods, including dichotomization of the exposure or a series of binary propensity score …
The situation frequently arises where working with the likelihood function is problematic. This can happen for several reasons---perhaps the likelihood is prohibitively computationally expensive, perhaps it lacks some robustness property, or perhaps it is simply not known for the model under consideration. In these cases, it is often possible to specify alternative functions of the parameters and the data that can be maximized to obtain asymptotically normal estimates. However, these scenarios present obvious problems if one is interested in applying Bayesian techniques. Here we describe open-…
Standard estimators of the global average treatment effect can be biased in the presence of interference. This paper proposes regression adjustment estimators for removing bias due to interference in Bernoulli randomized experiments. We use a fitted model to predict the counterfactual outcomes of global control and global treatment. Our work differs from standard regression adjustments in that the adjustment variables are constructed from functions of the treatment assignment vector, and that we allow the researcher to use a collection of any functions correlated with the response, turning the…
A key challenge in estimating the infection fatality rate (IFR) -- and its relation with various factors of interest -- is determining the total number of cases. The total number of cases is not known because not everyone is tested, but also, more importantly, because tested individuals are not representative of the population at large. We refer to the phenomenon whereby infected individuals are more likely to be tested than non-infected individuals, as "preferential testing." An open question is whether or not it is possible to reliably estimate the IFR without any specific knowledge about th…
The Clean Air Act mandates that the National Ambient Air Quality Standards (NAAQS) must be routinely assessed to protect populations based on the latest science. Therefore, researchers should continue to address whether exposure to levels of air pollution below the NAAQS is harmful to human health. The contentious nature surrounding environmental regulations urges us to cast this question within a causal inference framework. Parametric and semi-parametric regression approaches have been used to estimate the exposure-response (ER) curve between ambient air pollution and health outcomes. Most of…
Complete randomization allows for consistent estimation of the average treatment effect based on the difference in means of the outcomes without strong modeling assumptions on the outcome-generating process. Appropriate use of the pretreatment covariates can further improve the estimation efficiency. However, missingness in covariates is common in experiments and raises an important question: should we adjust for covariates subject to missingness, and if so, how? The unadjusted difference in means is always unbiased. The complete-covariate analysis adjusts for all completely observed covariate…
Institute for Analytical Sociology（分析社会学研究所）；Linköping University（林雪平大学）；Chalmers University of Technology（查尔姆斯理工大学）
Observational studies of causal effects require adjustment for confounding factors. In the tabular setting, where these factors are well-defined, separate random variables, the effect of confounding is well understood. However, in public policy, ecology, and in medicine, decisions are often made in non-tabular settings, informed by patterns or objects detected in images (e.g., maps, satellite or tomography imagery). Using such imagery for causal inference presents an opportunity because objects in the image may be related to the treatment and outcome of interest. In these cases, we rely on the…
We consider the estimation of average and counterfactual treatment effects, under two settings: back-door adjustment and front-door adjustment. The goal in both cases is to recover the treatment effect without having an access to a hidden confounder. This objective is attained by first estimating the conditional mean of the desired outcome variable given relevant covariates (the "first stage" regression), and then taking the (conditional) expectation of this function as a "second stage" procedure. We propose to compute these conditional expectations directly using a regression function to the …
The Consent-to-Contact (C2C) registry at the University of California, Irvine collects data from community participants to aid in the recruitment to clinical research studies. Self-selection into the C2C likely leads to bias due in part to enrollees having more years of education relative to the US general population. Salazar et al. (2020) recently used the C2C to examine associations of race/ethnicity with participant willingness to be contacted about research studies. To address questions about generalizability of estimated associations we estimate propensity for self-selection into the conv…
Nonresponse weighting adjustment using the response propensity score is a popular tool for handling unit nonresponse. Statistical inference after the nonresponse weighting adjustment is complicated because the effect of estimating the propensity model parameter needs to be incorporated. In this paper, we propose an approximate Bayesian approach to handle unit nonresponse with parametric model assumptions on the response probability, but without model assumptions for the outcome variable. The proposed Bayesian method is calibrated to the frequentist inference in that the credible region obtaine…
Respondent-driven sampling (RDS) is a popular approach to study marginalized or hard-to-reach populations. It collects samples from a networked population by incentivizing participants to refer their friends into the study. One major challenge in analyzing RDS samples is seed bias. Seed bias refers to the fact that when the social network is divided into multiple communities (or blocks), the RDS sample might not provide a balanced representation of the different communities in the population, and such unbalance is correlated with the initial participant (or the seed). In this case, the distrib…
We revisit the problem of estimating the local average treatment effect (LATE) and the local average treatment effect on the treated (LATT) when control variables are available, either to render the instrumental variable (IV) suitably exogenous or to improve precision. Unlike previous approaches, our doubly robust (DR) estimation procedures use quasi-likelihood methods weighted by the inverse of the IV propensity score - so-called inverse probability weighted regression adjustment (IPWRA) estimators. By properly choosing models for the propensity score and outcome models, fitted values are ens…
Estimating causal effects from randomized experiments is central to clinical research. Reducing the statistical uncertainty in these analyses is an important objective for statisticians. Registries, prior trials, and health records constitute a growing compendium of historical data on patients under standard-of-care that may be exploitable to this end. However, most methods for historical borrowing achieve reductions in variance by sacrificing strict type-I error rate control. Here, we propose a use of historical data that exploits linear covariate adjustment to improve the efficiency of trial…
Choosing the Right Approach at the Right Time: A Comparative Analysis of Causal Effect Estimation using Confounder Adjustment and Instrumental Variables
We present a general framework for using existing data to estimate the efficiency gain from using a covariate-adjusted estimator of a marginal treatment effect in a future randomized trial. We describe conditions under which it is possible to define a mapping from the distribution that generated the existing external data to the relative efficiency of a covariate-adjusted estimator compared to an unadjusted estimator. Under conditions, these relative efficiencies approximate the ratio of sample size needed to achieve a desired power. We consider two situations where the outcome is either fully…
Design of experiments and estimation of treatment effects in large-scale networks, in the presence of strong interference, is a challenging and important problem. Most existing methods' performance deteriorates as the density of the network increases. In this paper, we present a novel strategy for accurately estimating the causal effects of a class of treatments in a dense large-scale network. First, we design an approximate randomized controlled experiment by solving an optimization problem to allocate treatments in the presence of competition among neighboring nodes. Then we apply an importa…
A robust score test in g-computation for covariate adjustment in randomized clinical trials leveraging different variance estimators via influence functions
Data Sciences and Analytics, Pfizer Inc（辉瑞公司数据科学与分析部）；Division of Biostatistics and Health Data Science, University of Minnesota（明尼苏达大学生物统计学与健康数据科学系）；Institute of Natural Sciences, MOE-LSC, School of Mathematical Sciences, CMA-Shanghai, and SJTU-Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University（上海交通大学自然科学研究院、教育部拉曼森数学中心、数学科学学院、上海数学中心及上交-耶鲁生物统计与数据科学联合中心）
Bias reduction in g-computation for covariate adjustment in randomized clinical trials
Data Sciences and Analytics, Pfizer Inc.（辉瑞公司数据科学与分析部）；Institute of Natural Sciences, MOE–LSC, School of Mathematical Sciences, CMA–Shanghai, and SJTU–Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University（上海交通大学数学科学学院自然科学与工程教育部重点实验室、上海中心数学与应用数学研究所及上海交大-耶鲁生物统计与数据科学联合中心）；Division of Biostatistics and Health Data Science, University of Minnesota Twin Cities（明尼苏达大学双城分校生物统计与健康数据科学系）
Estimating Causal Effects with Double Machine Learning -- A Method Evaluation
School of Business and Economics, University of Tübingen（蒂宾根大学商学院与经济学院）；Hertie Institute for AI in Brain Health, University of Tübingen（蒂宾根大学赫蒂脑健康人工智能研究所）
We describe our framework, deployed at Facebook, that accounts for interference between experimental units through cluster-randomized experiments. We document this system, including the design and estimation procedures, and detail insights we have gained from the many experiments that have used this system at scale. We introduce a cluster-based regression adjustment that substantially improves precision for estimating global treatment effects as well as testing for interference as part of our estimation procedure. With this regression adjustment, we find that imbalanced clusters can better acc…
