Estimation and inference for the causal effect of receiving treatment on a multinomial outcome

Summary .  This article considers the analysis of two-arm randomized trials with noncompliance, which have a multinomial outcome. We first define the causal effect in these trials as some function of outcome distributions of compliers with and without treatment (e.g., the complier average causal effect, the measure of stochastic superiority of treatment over control for compliers), then estimate the causal effect with the likelihood method. Next, based on the likelihood-ratio (LR) statistic, we test those functions of or the equality of the outcome distributions of compliers with and without treatment. Although the corresponding LR statistic follows a chi-squared  (χ²)  distribution asymptotically when the true values of parameters are in the interior of the parameter space under the null, its asymptotic distribution is not  χ²  when the true values of parameters are on the boundary of the parameter space under the null. Therefore, we propose a bootstrap/double bootstrap version of a LR test for the causal effect in these trials. The methods are illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

Identification of causal effects with latent confounding and classical additive errors in treatment

In this paper, we discuss the identifiability and estimation of causal effects of a continuous treatment on a binary response when the treatment is measured with errors and there exists a latent categorical confounder associated with both treatment and response. Under some widely used parametric models, we first discuss the identifiability of the causal effects and then propose an approach for estimation and inference. Our approach can eliminate the biases induced by latent confounding and measurement errors by using only a single instrumental variable. Based on the identification results, we give guidelines for determining the existence of a latent categorical confounder and for selecting the number of levels of the latent confounder. We apply the proposed approach to a data set from the Framingham Heart Study to evaluate the effect of the systolic blood pressure on the coronary heart disease.     

Principal stratification in causal inference

Many scientific problems require that treatment comparisons be adjusted for posttreatment variables, but the estimands underlying standard methods are not causal effects. To address this deficiency, we propose a general framework for comparing treatments adjusting for posttreatment variables that yields principal effects based on principal stratification. Principal stratification with respect to a posttreatment variable is a cross-classification of subjects defined by the joint potential values of that posttreatment variable tinder each of the treatments being compared. Principal effects are causal effects within a principal stratum. The key property of principal strata is that they are not affected by treatment assignment and therefore can be used just as any pretreatment covariate. such as age category. As a result, the central property of our principal effects is that they are always causal effects and do not suffer from the complications of standard posttreatment-adjusted estimands. We discuss briefly that such principal causal effects are the link between three recent applications with adjustment for posttreatment variables: (i) treatment noncompliance, (ii) missing outcomes (dropout) following treatment noncompliance. and (iii) censoring by death. We then attack the problem of surrogate or biomarker endpoints, where we show, using principal causal effects, that all current definitions of surrogacy, even when perfectly true, do not generally have the desired interpretation as causal effects of treatment on outcome. We go on to forrmulate estimands based on principal stratification and principal causal effects and show their superiority.     

Causal inference with outcomes truncated by death in multiarm studies

It is challenging to evaluate causal effects when the outcomes of interest suffer from truncation-by-death in many clinical studies; that is, outcomes cannot be observed if patients die before the time of measurement. To address this problem, it is common to consider average treatment effects by principal stratification, for which, the identifiability results and estimation methods with a binary treatment have been established in previous literature. However, in multiarm studies with more than two treatment options, estimation of causal effects becomes more complicated and requires additional techniques. In this article, we consider identification, estimation, and bounds of causal effects with multivalued ordinal treatments and the outcomes subject to truncation-by-death. We define causal parameters of interest in this setting and show that they are identifiable either using some auxiliary variable or based on linear model assumption. We then propose a semiparametric method for estimating the causal parameters and derive their asymptotic results. When the identification conditions are invalid, we derive sharp bounds of the causal effects by use of covariates adjustment. Simulation studies show good performance of the proposed estimator. We use the estimator to analyze the effects of a four-level chronic toxin on fetal developmental outcomes such as birth weight in rats and mice, with data from a developmental toxicity trial conducted by the National Toxicology Program. Data analyses demonstrate that a high dose of the toxin significantly reduces the weights of pups.     

Multiple imputation methods for treatment noncompliance and nonresponse in randomized clinical trials

Summary .  Randomized clinical trials are a powerful tool for investigating causal treatment effects, but in human trials there are oftentimes problems of noncompliance which standard analyses, such as the intention-to-treat or as-treated analysis, either ignore or incorporate in such a way that the resulting estimand is no longer a causal effect. One alternative to these analyses is the complier average causal effect (CACE) which estimates the average causal treatment effect among a subpopulation that would comply under any treatment assigned. We focus on the setting of a randomized clinical trial with crossover treatment noncompliance (e.g., control subjects could receive the intervention and intervention subjects could receive the control) and outcome nonresponse. In this article, we develop estimators for the CACE using multiple imputation methods, which have been successfully applied to a wide variety of missing data problems, but have not yet been applied to the potential outcomes setting of causal inference. Using simulated data we investigate the finite sample properties of these estimators as well as of competing procedures in a simple setting. Finally we illustrate our methods using a real randomized encouragement design study on the effectiveness of the influenza vaccine.     

The Sign of the Unmeasured Confounding Bias under Various Standard Populations

Unmeasured confounders are a common problem in drawing causal inferences in observational studies. VanderWeele (Biometrics 2008, 64, 702–706) presented a theorem that allows researchers to determine the sign of the unmeasured confounding bias when monotonic relationships hold between the unmeasured confounder and the treatment, and between the unmeasured confounder and the outcome. He showed that his theorem can be applied to causal effects with the total group as the standard population, but he did not mention the causal effects with treated and untreated groups as the standard population. Here, we extend his results to these causal effects, and apply our theorems to an observational study. When researchers have a sense of what the unmeasured confounder may be, conclusions can be drawn about the sign of the bias.     

Does a dose-response relationship reduce sensitivity to hidden bias?

It is often said that an important consideration in judging whether an association between treatment and response is causal is the presence or absence of a dose-response relationship, that is, larger ostensible treatment effects when doses of treatment are larger. This criterion is widely discussed in textbooks and is often mentioned in empirical papers. At the same time, it is well known through both important examples and elementary theory that a treatment may cause dramatic effects with no dose-response relationship, and hidden biases may produce a dose-response relationship when the treatment is without effect. What does a dose-response relationship say about causality? It is observed here that a dose-response relationship may or may not reduce sensitivity to hidden bias, and whether it has or has not can be determined by a suitable analysis using the data at hand. Moreover, a study without a dose-response relationship may or may not be less sensitive to hidden bias than another study with such a relationship, and this, too, can be determined from the data at hand. An example concerning cytogenetic damage among professional painters is used to illustrate.     

The sign of the bias of unmeasured confounding

Summary .   Unmeasured confounding variables are a common problem in drawing causal inferences in observational studies. A theorem is given which in certain circumstances allows the researcher to draw conclusions about the sign of the bias of unmeasured confounding. Specifically, it is possible to determine the sign of the bias when monotonicity relationships hold between the unmeasured confounding variable and the treatment, and between the unmeasured confounding variable and the outcome. Some discussion is given to the conditions under which the theorem applies and the strengths and limitations of using the theorem to assess the sign of the bias of unmeasured confounding.     

Polydesigns and causal inference

In an increasingly common class of studies, the goal is to evaluate causal effects of treatments that are only partially controlled by the investigator. In such studies there are two conflicting features: (1) a model on the full cohort design and data can identify the causal effects of interest, but can be sensitive to extreme regions of that design's data, where model specification can have more impact; and (2) models on a reduced design (i.e., a subset of the full data), for example, conditional likelihood on matched subsets of data, can avoid such sensitivity, but do not generally identify the causal effects. We propose a framework to assess how inference is sensitive to designs by exploring combinations of both the full and reduced designs. We show that using such a "polydesign" framework generates a rich class of methods that can identify causal effects and that can also be more robust to model specification than methods using only the full design. We discuss implementation of polydesign methods, and provide an illustration in the evaluation of a needle exchange program.     

On estimation of the survivor average causal effect in observational studies when important confounders are missing due to death

Summary .  We focus on estimation of the causal effect of treatment on the functional status of individuals at a fixed point in time t * after they have experienced a catastrophic event, from observational data with the following features: (i) treatment is imposed shortly after the event and is nonrandomized, (ii) individuals who survive to t * are scheduled to be interviewed, (iii) there is interview nonresponse, (iv) individuals who die prior to t * are missing information on preevent confounders, and (v) medical records are abstracted on all individuals to obtain information on postevent, pretreatment confounding factors. To address the issue of survivor bias, we seek to estimate the survivor average causal effect (SACE), the effect of treatment on functional status among the cohort of individuals who would survive to t * regardless of whether or not assigned to treatment. To estimate this effect from observational data, we need to impose untestable assumptions, which depend on the collection of all confounding factors. Because preevent information is missing on those who die prior to t *, it is unlikely that these data are missing at random. We introduce a sensitivity analysis methodology to evaluate the robustness of SACE inferences to deviations from the missing at random assumption. We apply our methodology to the evaluation of the effect of trauma center care on vitality outcomes using data from the National Study on Costs and Outcomes of Trauma Care.     

Evaluating candidate principal surrogate endpoints

SUMMARY: Frangakis and Rubin (2002, Biometrics 58, 21-29) proposed a new definition of a surrogate endpoint (a "principal" surrogate) based on causal effects. We introduce an estimand for evaluating a principal surrogate, the causal effect predictiveness (CEP) surface, which quantifies how well causal treatment effects on the biomarker predict causal treatment effects on the clinical endpoint. Although the CEP surface is not identifiable due to missing potential outcomes, it can be identified by incorporating a baseline covariate(s) that predicts the biomarker. Given case-cohort sampling of such a baseline predictor and the biomarker in a large blinded randomized clinical trial, we develop an estimated likelihood method for estimating the CEP surface. This estimation assesses the "surrogate value" of the biomarker for reliably predicting clinical treatment effects for the same or similar setting as the trial. A CEP surface plot provides a way to compare the surrogate value of multiple biomarkers. The approach is illustrated by the problem of assessing an immune response to a vaccine as a surrogate endpoint for infection.     

Testing and estimating gene-environment interactions in family-based association studies

Summary .   We propose robust and efficient tests and estimators for gene–environment/gene–drug interactions in family-based association studies in which haplotypes, dichotomous/quantitative phenotypes, and complex exposure/treatment variables are analyzed. Using causal inference methodology, we show that the tests and estimators are robust against unmeasured confounding due to population admixture and stratification, provided that Mendel's law of segregation holds and that the considered exposure/treatment variable is not affected by the candidate gene under study. We illustrate the practical relevance of our approach by an application to a chronic obstructive pulmonary disease study. The data analysis suggests a gene–environment interaction between a single nucleotide polymorphism in the Serpine2 gene and smoking status/pack-years of smoking. Simulation studies show that the proposed methodology is sufficiently powered for realistic sample sizes and that it provides valid tests and effect size estimators in the presence of admixture and stratification.     

Structural Nested Mean Models for Assessing Time-Varying Effect Moderation

Summary .  This article considers the problem of assessing causal effect moderation in longitudinal settings in which treatment (or exposure) is time varying and so are the covariates said to moderate its effect.  Intermediate causal effects  that describe time-varying causal effects of treatment conditional on past covariate history are introduced and considered as part of Robins' structural nested mean model. Two estimators of the intermediate causal effects, and their standard errors, are presented and discussed: The first is a proposed two-stage regression estimator. The second is Robins' G-estimator. The results of a small simulation study that begins to shed light on the small versus large sample performance of the estimators, and on the bias–variance trade-off between the two estimators are presented. The methodology is illustrated using longitudinal data from a depression study.     

Identifiability of causal effects for binary variables with baseline data missing due to death

We discuss identifiability and estimation of causal effects of a treatment in subgroups defined by a covariate that is sometimes missing due to death, which is different from a problem with outcomes censored by death. Frangakis et al. (2007, Biometrics 63, 641-662) proposed an approach for estimating the causal effects under a strong monotonicity (SM) assumption. In this article, we focus on identifiability of the joint distribution of the covariate, treatment and potential outcomes, show sufficient conditions for identifiability, and relax the SM assumption to monotonicity (M) and no-interaction (NI) assumptions. We derive expectation-maximization algorithms for finding the maximum likelihood estimates of parameters of the joint distribution under different assumptions. Further we remove the M and NI assumptions, and prove that signs of the causal effects of a treatment in the subgroups are identifiable, which means that their bounds do not cover zero. We perform simulations and a sensitivity analysis to evaluate our approaches. Finally, we apply the approaches to the National Study on the Costs and Outcomes of Trauma Centers data, which are also analyzed by Frangakis et al. (2007) and Xie and Murphy (2007, Biometrics 63, 655-658).     

A causal proportional hazards estimator for the effect of treatment actually received in a randomized trial with all-or-nothing compliance

Survival data from randomized trials are most often analyzed in a proportional hazards (PH) framework that follows the intention-to-treat (ITT) principle. When not all the patients on the experimental arm actually receive the assigned treatment, the ITT-estimator mixes its effect on treatment compliers with its absence of effect on noncompliers. The structural accelerated failure time (SAFT) models of Robins and Tsiatis are designed to consistently estimate causal effects on the treated, without direct assumptions about the compliance selection mechanism. The traditional PH-model, however, has not yet led to such causal interpretation. In this article, we examine a PH-model of treatment effect on the treated subgroup. While potential treatment compliance is unobserved in the control arm, we derive an estimating equation for the Compliers PROPortional Hazards Effect of Treatment (C-PROPHET). The jackknife is used for bias correction and variance estimation. The method is applied to data from a recently finished clinical trial in cancer patients with liver metastases.     

Causal diagrams for empirical research

The primary aim of this paper is to show how graphical modelscan be used as a mathematical language for integrating statisticaland subject-matter information. In particular, the paper developsa principled, nonparametric framework for causal inference,in which diagrams are queried to determine if the assumptionsavailable are sufficient for identifying causal effects fromnonexperimental data. If so the diagrams can be queried to producemathematical expressions for causal effects in terms of observeddistributions; otherwise, the diagrams can be queried to suggestadditional observations or auxiliary experiments from whichthe desired inferences can be obtained.     

Sensitivity analysis for the assessment of causal vaccine effects on viral load in HIV vaccine trials

Vaccines with limited ability to prevent HIV infection may positively impact the HIV/AIDS pandemic by preventing secondary transmission and disease in vaccine recipients who become infected. To evaluate the impact of vaccination on secondary transmission and disease, efficacy trials assess vaccine effects on HIV viral load and other surrogate endpoints measured after infection. A standard test that compares the distribution of viral load between the infected subgroups of vaccine and placebo recipients does not assess a causal effect of vaccine, because the comparison groups are selected after randomization. To address this problem, we formulate clinically relevant causal estimands using the principal stratification framework developed by Frangakis and Rubin (2002, Biometrics 58, 21-29), and propose a class of logistic selection bias models whose members identify the estimands. Given a selection model in the class, procedures are developed for testing and estimation of the causal effect of vaccination on viral load in the principal stratum of subjects who would be infected regardless of randomization assignment. We show how the procedures can be used for a sensitivity analysis that quantifies how the causal effect of vaccination varies with the presumed magnitude of selection bias.     

A new criterion for confounder selection

Summary We propose a new criterion for confounder selection when the underlying causal structure is unknown and only limited knowledge is available. We assume all covariates being considered are pretreatment variables and that for each covariate it is known (i) whether the covariate is a cause of treatment, and (ii) whether the covariate is a cause of the outcome. The causal relationships the covariates have with one another is assumed unknown. We propose that control be made for any covariate that is either a cause of treatment or of the outcome or both. We show that irrespective of the actual underlying causal structure, if any subset of the observed covariates suffices to control for confounding then the set of covariates chosen by our criterion will also suffice. We show that other, commonly used, criteria for confounding control do not have this property. We use formal theory concerning causal diagrams to prove our result but the application of the result does not rely on familiarity with causal diagrams. An investigator simply need ask, “Is the covariate a cause of the treatment?” and “Is the covariate a cause of the outcome?” If the answer to either question is “yes” then the covariate is included for confounder control. We discuss some additional covariate selection results that preserve unconfoundedness and that may be of interest when used with our criterion.     

Efficient nonparametric estimation of causal effects in randomized trials with noncompliance

Causal approaches based on the potential outcome framework providea useful tool for addressing noncompliance problems in randomizedtrials. We propose a new estimator of causal treatment effectsin randomized clinical trials with noncompliance. We use theempirical likelihood approach to construct a profile randomsieve likelihood and take into account the mixture structurein outcome distributions, so that our estimator is robust toparametric distribution assumptions and provides substantialfinite-sample efficiency gains over the standard instrumentalvariable estimator. Our estimator is asymptotically equivalentto the standard instrumental variable estimator, and it canbe applied to outcome variables with a continuous, ordinal orbinary scale. We apply our method to data from a randomizedtrial of an intervention to improve the treatment of depressionamong depressed elderly patients in primary care practices.     

Sensitivity analyses comparing outcomes only existing in a subset selected post-randomization, conditional on covariates, with application to HIV vaccine trials

In many experiments, researchers would like to compare between treatments and outcome that only exists in a subset of participants selected after randomization. For example, in preventive HIV vaccine efficacy trials it is of interest to determine whether randomization to vaccine causes lower HIV viral load, a quantity that only exists in participants who acquire HIV. To make a causal comparison and account for potential selection bias we propose a sensitivity analysis following the principal stratification framework set forth by Frangakis and Rubin (2002, Biometrics58, 21-29). Our goal is to assess the average causal effect of treatment assignment on viral load at a given baseline covariate level in the always infected principal stratum (those who would have been infected whether they had been assigned to vaccine or placebo). We assume stable unit treatment values (SUTVA), randomization, and that subjects randomized to the vaccine arm who became infected would also have become infected if randomized to the placebo arm (monotonicity). It is not known which of those subjects infected in the placebo arm are in the always infected principal stratum, but this can be modeled conditional on covariates, the observed viral load, and a specified sensitivity parameter. Under parametric regression models for viral load, we obtain maximum likelihood estimates of the average causal effect conditional on covariates and the sensitivity parameter. We apply our methods to the world's first phase III HIV vaccine trial.