Estimating long‐term treatment effects in observational data: A comparison of the performance of different methods under real‐world uncertainty

In the presence of time‐dependent confounding, there are several methods available to estimate treatment effects. With correctly specified models and appropriate structural assumptions, any of these methods could provide consistent effect estimates, but with real‐world data, all models will be misspecified and it is difficult to know if assumptions are violated. In this paper, we investigate five methods: inverse probability weighting of marginal structural models, history‐adjusted marginal structural models, sequential conditional mean models, g‐computation formula, and g‐estimation of structural nested models. This work is motivated by an investigation of the effects of treatments in cystic fibrosis using the UK Cystic Fibrosis Registry data focussing on two outcomes: lung function (continuous outcome) and annual number of days receiving intravenous antibiotics (count outcome). We identified five features of this data that may affect the performance of the methods: misspecification of the causal null, long‐term treatment effects, effect modification by time‐varying covariates, misspecification of the direction of causal pathways, and censoring. In simulation studies, under ideal settings, all five methods provide consistent estimates of the treatment effect with little difference between methods. However, all methods performed poorly under some settings, highlighting the importance of using appropriate methods based on the data available. Furthermore, with the count outcome, the issue of non‐collapsibility makes comparison between methods delivering marginal and conditional effects difficult. In many situations, we would recommend using more than one of the available methods for analysis, as if the effect estimates are very different, this would indicate potential issues with the analyses.     

A tutorial on propensity score estimation for multiple treatments using generalized boosted models

The use of propensity scores to control for pretreatment imbalances on observed variables in non‐randomized or observational studies examining the causal effects of treatments or interventions has become widespread over the past decade. For settings with two conditions of interest such as a treatment and a control, inverse probability of treatment weighted estimation with propensity scores estimated via boosted models has been shown in simulation studies to yield causal effect estimates with desirable properties. There are tools (e.g., the twang package in R) and guidance for implementing this method with two treatments. However, there is not such guidance for analyses of three or more treatments. The goals of this paper are twofold: (1) to provide step‐by‐step guidance for researchers who want to implement propensity score weighting for multiple treatments and (2) to propose the use of generalized boosted models (GBM) for estimation of the necessary propensity score weights. We define the causal quantities that may be of interest to studies of multiple treatments and derive weighted estimators of those quantities. We present a detailed plan for using GBM to estimate propensity scores and using those scores to estimate weights and causal effects. We also provide tools for assessing balance and overlap of pretreatment variables among treatment groups in the context of multiple treatments. A case study examining the effects of three treatment programs for adolescent substance abuse demonstrates the methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian methods for meta‐analysis of causal relationships estimated using genetic instrumental variables

Genetic markers can be used as instrumental variables, in an analogous way to randomization in a clinical trial, to estimate the causal relationship between a phenotype and an outcome variable. Our purpose is to extend the existing methods for such Mendelian randomization studies to the context of multiple genetic markers measured in multiple studies, based on the analysis of individual participant data. First, for a single genetic marker in one study, we show that the usual ratio of coefficients approach can be reformulated as a regression with heterogeneous error in the explanatory variable. This can be implemented using a Bayesian approach, which is next extended to include multiple genetic markers. We then propose a hierarchical model for undertaking a meta‐analysis of multiple studies, in which it is not necessary that the same genetic markers are measured in each study. This provides an overall estimate of the causal relationship between the phenotype and the outcome, and an assessment of its heterogeneity across studies. As an example, we estimate the causal relationship of blood concentrations of C‐reactive protein on fibrinogen levels using data from 11 studies. These methods provide a flexible framework for efficient estimation of causal relationships derived from multiple studies. Issues discussed include weak instrument bias, analysis of binary outcome data such as disease risk, missing genetic data, and the use of haplotypes. Copyright © 2010 John Wiley & Sons, Ltd.     

Mendelian randomization analysis of a time‐varying exposure for binary disease outcomes using functional data analysis methods

A Mendelian randomization (MR) analysis is performed to analyze the causal effect of an exposure variable on a disease outcome in observational studies, by using genetic variants that affect the disease outcome only through the exposure variable. This method has recently gained popularity among epidemiologists given the success of genetic association studies. Many exposure variables of interest in epidemiological studies are time varying, for example, body mass index (BMI). Although longitudinal data have been collected in many cohort studies, current MR studies only use one measurement of a time‐varying exposure variable, which cannot adequately capture the long‐term time‐varying information. We propose using the functional principal component analysis method to recover the underlying individual trajectory of the time‐varying exposure from the sparsely and irregularly observed longitudinal data, and then conduct MR analysis using the recovered curves. We further propose two MR analysis methods. The first assumes a cumulative effect of the time‐varying exposure variable on the disease risk, while the second assumes a time‐varying genetic effect and employs functional regression models. We focus on statistical testing for a causal effect. Our simulation studies mimicking the real data show that the proposed functional data analysis based methods incorporating longitudinal data have substantial power gains compared to standard MR analysis using only one measurement. We used the Framingham Heart Study data to demonstrate the promising performance of the new methods as well as inconsistent results produced by the standard MR analysis that relies on a single measurement of the exposure at some arbitrary time point.     

Instrumental variable analysis of multiplicative models with potentially invalid instruments

Instrumental variable (IV) methods have potential to consistently estimate the causal effect of an exposure on an outcome in the presence of unmeasured confounding. However, validity of IV methods relies on strong assumptions, some of which cannot be conclusively verified from observational data. One such assumption is that the effect of the proposed instrument on the outcome is completely mediated by the exposure. We consider the situation where this assumption is violated, but the remaining IV assumptions hold; that is, the proposed IV (1) is associated with the exposure and (2) has no unmeasured causes in common with the outcome. We propose a method to estimate multiplicative structural mean models of binary outcomes in this scenario in the presence of unmeasured confounding. We also extend the method to address multiple scenarios, including mediation analysis. The method adapts the asymptotically efficient G‐estimation approach that was previously proposed for additive structural mean models, and it can be carried out using off‐the‐shelf software for generalized method of moments. Monte Carlo simulation studies show that the method has low bias and accurate coverage. We applied the method to a case study of circulating vitamin D and depressive symptoms using season of blood collection as a (potentially invalid) instrumental variable. Potential applications of the proposed method include randomized intervention studies as well as Mendelian randomization studies with genetic variants that affect multiple phenotypes, possibly including the outcome. Published 2016. This article is a U.S. Government work and is in the public domain in the USA     

Combining information on multiple instrumental variables in Mendelian randomization: comparison of allele score and summarized data methods

Mendelian randomization is the use of genetic instrumental variables to obtain causal inferences from observational data. Two recent developments for combining information on multiple uncorrelated instrumental variables (IVs) into a single causal estimate are as follows: (i) allele scores, in which individual‐level data on the IVs are aggregated into a univariate score, which is used as a single IV, and (ii) a summary statistic method, in which causal estimates calculated from each IV using summarized data are combined in an inverse‐variance weighted meta‐analysis. To avoid bias from weak instruments, unweighted and externally weighted allele scores have been recommended. Here, we propose equivalent approaches using summarized data and also provide extensions of the methods for use with correlated IVs. We investigate the impact of different choices of weights on the bias and precision of estimates in simulation studies. We show that allele score estimates can be reproduced using summarized data on genetic associations with the risk factor and the outcome. Estimates from the summary statistic method using external weights are biased towards the null when the weights are imprecisely estimated; in contrast, allele score estimates are unbiased. With equal or external weights, both methods provide appropriate tests of the null hypothesis of no causal effect even with large numbers of potentially weak instruments. We illustrate these methods using summarized data on the causal effect of low‐density lipoprotein cholesterol on coronary heart disease risk. It is shown that a more precise causal estimate can be obtained using multiple genetic variants from a single gene region, even if the variants are correlated. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Estimating parsimonious models of longitudinal causal effects using regressions on propensity scores

Parsimony is important for the interpretation of causal effect estimates of longitudinal treatments on subsequent outcomes. One method for parsimonious estimates fits marginal structural models by using inverse propensity scores as weights. This method leads to generally large variability that is uncommon in more likelihood‐based approaches. A more recent method fits these models by using simulations from a fitted g‐computation, but requires the modeling of high‐dimensional longitudinal relations that are highly susceptible to misspecification. We propose a new method that, first, uses longitudinal propensity scores as regressors to reduce the dimension of the problem and then uses the approximate likelihood for the first estimates to fit parsimonious models. We demonstrate the methods by estimating the effect of anticoagulant therapy on survival for cancer and non‐cancer patients who have inferior vena cava filters. Copyright © 2013 John Wiley & Sons, Ltd.     

Joint mixed‐effects models for causal inference with longitudinal data

Causal inference with observational longitudinal data and time‐varying exposures is complicated due to the potential for time‐dependent confounding and unmeasured confounding. Most causal inference methods that handle time‐dependent confounding rely on either the assumption of no unmeasured confounders or the availability of an unconfounded variable that is associated with the exposure (eg, an instrumental variable). Furthermore, when data are incomplete, validity of many methods often depends on the assumption of missing at random. We propose an approach that combines a parametric joint mixed‐effects model for the study outcome and the exposure with g‐computation to identify and estimate causal effects in the presence of time‐dependent confounding and unmeasured confounding. G‐computation can estimate participant‐specific or population‐average causal effects using parameters of the joint model. The joint model is a type of shared parameter model where the outcome and exposure‐selection models share common random effect(s). We also extend the joint model to handle missing data and truncation by death when missingness is possibly not at random. We evaluate the performance of the proposed method using simulation studies and compare the method to both linear mixed‐ and fixed‐effects models combined with g‐computation as well as to targeted maximum likelihood estimation. We apply the method to an epidemiologic study of vitamin D and depressive symptoms in older adults and include code using SAS PROC NLMIXED software to enhance the accessibility of the method to applied researchers.     

A causal model for longitudinal randomised trials with time‐dependent non‐compliance

In the presence of non‐compliance, conventional analysis by intention‐to‐treat provides an unbiased comparison of treatment policies but typically under‐estimates treatment efficacy. With all‐or‐nothing compliance, efficacy may be specified as the complier‐average causal effect (CACE), where compliers are those who receive intervention if and only if randomised to it. We extend the CACE approach to model longitudinal data with time‐dependent non‐compliance, focusing on the situation in which those randomised to control may receive treatment and allowing treatment effects to vary arbitrarily over time. Defining compliance type to be the time of surgical intervention if randomised to control, so that compliers are patients who would not have received treatment at all if they had been randomised to control, we construct a causal model for the multivariate outcome conditional on compliance type and randomised arm. This model is applied to the trial of alternative regimens for glue ear treatment evaluating surgical interventions in childhood ear disease, where outcomes are measured over five time points, and receipt of surgical intervention in the control arm may occur at any time. We fit the models using Markov chain Monte Carlo methods to obtain estimates of the CACE at successive times after receiving the intervention. In this trial, over a half of those randomised to control eventually receive intervention. We find that surgery is more beneficial than control at 6months, with a small but non‐significant beneficial effect at 12months. © 2015 The Authors. Statistics in Medicine Published by JohnWiley & Sons Ltd.     

Causal Genetic Inference Using Haplotypes as Instrumental Variables

In genomic studies with both genotypes and gene or protein expression profile available, causal effects of gene or protein on clinical outcomes can be inferred through using genetic variants as instrumental variables (IVs). The goal of introducing IV is to remove the effects of unobserved factors that may confound the relationship between the biomarkers and the outcome. A valid inference under the IV framework requires pairwise associations and pathway exclusivity. Among these assumptions, the IV expression association needs to be strong for the casual effect estimates to be unbiased. However, a small number of single nucleotide polymorphisms (SNPs) often provide limited explanation of the variability in the gene or protein expression and can only serve as weak IVs. In this study, we propose to replace SNPs with haplotypes as IVs to increase the variant‐expression association and thus improve the casual effect inference of the expression. In the classical two‐stage procedure, we developed a haplotype regression model combined with a model selection procedure to identify optimal instruments. The performance of the new method was evaluated through simulations and compared with the IV approaches using observed multiple SNPs. Our results showed the gain of power to detect a causal effect of gene or protein on the outcome using haplotypes compared with using only observed SNPs, under either complete or missing genotype scenarios. We applied our proposed method to a study of the effect of interleukin‐1 beta (IL‐1β) protein expression on the 90‐day survival following sepsis and found that overly expressed IL‐1β is likely to increase mortality.     

Robust estimation of causal effects of binary treatments in unconfounded studies with dichotomous outcomes

The estimation of causal effects has been the subject of extensive research. In unconfounded studies with a dichotomous outcome, Y, Cangul, Chretien, Gutman and Rubin (2009) demonstrated that logistic regression for a scalar continuous covariate X is generally statistically invalid for testing null treatment effects when the distributions of X in the treated and control populations differ and the logistic model for Y given X is misspecified. In addition, they showed that an approximately valid statistical test can be generally obtained by discretizing X followed by regression adjustment within each interval defined by the discretized X. This paper extends the work of Cangul et al. 2009 in three major directions. First, we consider additional estimation procedures, including a new one that is based on two independent splines and multiple imputation; second, we consider additional distributional factors; and third, we examine the performance of the procedures when the treatment effect is non‐null. Of all the methods considered and in most of the experimental conditions that were examined, our proposed new methodology appears to work best in terms of point and interval estimation. Copyright © 2012 John Wiley & Sons, Ltd.     

Testing concordance of instrumental variable effects in generalized linear models with application to Mendelian randomization

Instrumental variable regression is one way to overcome unmeasured confounding and estimate causal effect in observational studies. Built on structural mean models, there has been considerable work recently developed for consistent estimation of causal relative risk and causal odds ratio. Such models can sometimes suffer from identification issues for weak instruments. This hampered the applicability of Mendelian randomization analysis in genetic epidemiology. When there are multiple genetic variants available as instrumental variables, and causal effect is defined in a generalized linear model in the presence of unmeasured confounders, we propose to test concordance between instrumental variable effects on the intermediate exposure and instrumental variable effects on the disease outcome, as a means to test the causal effect. We show that a class of generalized least squares estimators provide valid and consistent tests of causality. For causal effect of a continuous exposure on a dichotomous outcome in logistic models, the proposed estimators are shown to be asymptotically conservative. When the disease outcome is rare, such estimators are consistent because of the log‐linear approximation of the logistic function. Optimality of such estimators relative to the well‐known two‐stage least squares estimator and the double‐logistic structural mean model is further discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Semiparametric methods for estimation of a nonlinear exposure‐outcome relationship using instrumental variables with application to Mendelian randomization

Mendelian randomization, the use of genetic variants as instrumental variables (IV), can test for and estimate the causal effect of an exposure on an outcome. Most IV methods assume that the function relating the exposure to the expected value of the outcome (the exposure‐outcome relationship) is linear. However, in practice, this assumption may not hold. Indeed, often the primary question of interest is to assess the shape of this relationship. We present two novel IV methods for investigating the shape of the exposure‐outcome relationship: a fractional polynomial method and a piecewise linear method. We divide the population into strata using the exposure distribution, and estimate a causal effect, referred to as a localized average causal effect (LACE), in each stratum of population. The fractional polynomial method performs metaregression on these LACE estimates. The piecewise linear method estimates a continuous piecewise linear function, the gradient of which is the LACE estimate in each stratum. Both methods were demonstrated in a simulation study to estimate the true exposure‐outcome relationship well, particularly when the relationship was a fractional polynomial (for the fractional polynomial method) or was piecewise linear (for the piecewise linear method). The methods were used to investigate the shape of relationship of body mass index with systolic blood pressure and diastolic blood pressure.     

The use of prognostic scores for causal inference with general treatment regimes

In nonrandomised studies, inferring causal effects requires appropriate methods for addressing confounding bias. Although it is common to adopt propensity score analysis to this purpose, prognostic score analysis has recently been proposed as an alternative strategy. While both approaches were originally introduced to estimate causal effects for binary interventions, the theory of propensity score has since been extended to the case of general treatment regimes. Indeed, many treatments are not assigned in a binary fashion and require a certain extent of dosing. Hence, researchers may often be interested in estimating treatment effects across multiple exposures. To the best of our knowledge, the prognostic score analysis has not been yet generalised to this case. In this article, we describe the theory of prognostic scores for causal inference with general treatment regimes. Our methods can be applied to compare multiple treatments using nonrandomised data, a topic of great relevance in contemporary evaluations of clinical interventions. We propose estimators for the average treatment effects in different populations of interest, the validity of which is assessed through a series of simulations. Finally, we present an illustrative case in which we estimate the effect of the delay to Aspirin administration on a composite outcome of death or dependence at 6 months in stroke patients.     

Inverse odds ratio‐weighted estimation for causal mediation analysis

An important scientific goal of studies in the health and social sciences is increasingly to determine to what extent the total effect of a point exposure is mediated by an intermediate variable on the causal pathway between the exposure and the outcome. A causal framework has recently been proposed for mediation analysis, which gives rise to new definitions, formal identification results and novel estimators of direct and indirect effects. In the present paper, the author describes a new inverse odds ratio‐weighted approach to estimate so‐called natural direct and indirect effects. The approach, which uses as a weight the inverse of an estimate of the odds ratio function relating the exposure and the mediator, is universal in that it can be used to decompose total effects in a number of regression models commonly used in practice. Specifically, the approach may be used for effect decomposition in generalized linear models with a nonlinear link function, and in a number of other commonly used models such as the Cox proportional hazards regression for a survival outcome. The approach is simple and can be implemented in standard software provided a weight can be specified for each observation. An additional advantage of the method is that it easily incorporates multiple mediators of a categorical, discrete or continuous nature. Copyright © 2013 John Wiley & Sons, Ltd.     

A Bayesian approach for instrumental variable analysis with censored time‐to‐event outcome

Instrumental variable (IV) analysis has been widely used in economics, epidemiology, and other fields to estimate the causal effects of covariates on outcomes, in the presence of unobserved confounders and/or measurement errors in covariates. However, IV methods for time‐to‐event outcome with censored data remain underdeveloped. This paper proposes a Bayesian approach for IV analysis with censored time‐to‐event outcome by using a two‐stage linear model. A Markov chain Monte Carlo sampling method is developed for parameter estimation for both normal and non‐normal linear models with elliptically contoured error distributions. The performance of our method is examined by simulation studies. Our method largely reduces bias and greatly improves coverage probability of the estimated causal effect, compared with the method that ignores the unobserved confounders and measurement errors. We illustrate our method on the Women's Health Initiative Observational Study and the Atherosclerosis Risk in Communities Study. Copyright © 2014 John Wiley & Sons, Ltd.     

Incorporating prognostic factors into causal estimators: A comparison of methods for randomised controlled trials with a time‐to‐event outcome

In randomised controlled trials, the effect of treatment on those who comply with allocation to active treatment can be estimated by comparing their outcome to those in the comparison group who would have complied with active treatment had they been allocated to it. We compare three estimators of the causal effect of treatment on compliers when this is a parameter in a proportional hazards model and quantify the bias due to omitting baseline prognostic factors. Causal estimates are found directly by maximising a novel partial likelihood; based on a structural proportional hazards model; and based on a ‘corrected dataset’ derived after fitting a rank‐preserving structural failure time model. Where necessary, we extend these methods to incorporate baseline covariates. Comparisons use simulated data and a real data example. Analysing the simulated data, we found that all three methods are accurate when an important covariate was included in the proportional hazards model (maximum bias 5.4%). However, failure to adjust for this prognostic factor meant that causal treatment effects were underestimated (maximum bias 11.4%), because estimators were based on a misspecified marginal proportional hazards model. Analysing the real data example, we found that adjusting causal estimators is important to correct for residual imbalances in prognostic factors present between trial arms after randomisation. Our results show that methods of estimating causal treatment effects for time‐to‐event outcomes should be extended to incorporate covariates, thus providing an informative compliment to the corresponding intention‐to‐treat analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome

In clinical research, investigators are interested in inferring the average causal effect of a treatment. However, the causal parameter that can be used to derive the average causal effect is not well defined for ordinal outcomes. Although some definitions have been proposed, they are limited in that they are not identical to the well‐defined causal risk for a binary outcome, which is the simplest ordinal outcome. In this paper, we propose the use of a causal parameter for an ordinal outcome, defined as the proportion that a potential outcome under one treatment condition would not be smaller than that under the other condition. For a binary outcome, this proportion is identical to the causal risk. Unfortunately, the proposed causal parameter cannot be identified, even under randomization. Therefore, we present a numerical method to calculate the sharp nonparametric bounds within a sample, reflecting the impact of confounding. When the assumption of independent potential outcomes is included, the causal parameter can be identified when randomization is in play. Then, we present exact tests and the associated confidence intervals for the relative treatment effect using the randomization‐based approach, which are an extension of the existing methods for a binary outcome. Our methodologies are illustrated using data from an emetic prevention clinical trial.     

Causal mediation analysis with multiple mediators in the presence of treatment noncompliance

Randomized experiments are often complicated because of treatment noncompliance. This challenge prevents researchers from identifying the mediated portion of the intention‐to‐treated (ITT) effect, which is the effect of the assigned treatment that is attributed to a mediator. One solution suggests identifying the mediated ITT effect on the basis of the average causal mediation effect among compliers when there is a single mediator. However, considering the complex nature of the mediating mechanisms, it is natural to assume that there are multiple variables that mediate through the causal path. Motivated by an empirical analysis of a data set collected in a randomized interventional study, we develop a method to estimate the mediated portion of the ITT effect when both multiple dependent mediators and treatment noncompliance exist. This enables researchers to make an informed decision on how to strengthen the intervention effect by identifying relevant mediators despite treatment noncompliance. We propose a nonparametric estimation procedure and provide a sensitivity analysis for key assumptions. We conduct a Monte Carlo simulation study to assess the finite sample performance of the proposed approach. The proposed method is illustrated by an empirical analysis of JOBS II data, in which a job training intervention was used to prevent mental health deterioration among unemployed individuals.     

Extending the MR‐Egger method for multivariable Mendelian randomization to correct for both measured and unmeasured pleiotropy

Methods have been developed for Mendelian randomization that can obtain consistent causal estimates while relaxing the instrumental variable assumptions. These include multivariable Mendelian randomization, in which a genetic variant may be associated with multiple risk factors so long as any association with the outcome is via the measured risk factors (measured pleiotropy), and the MR‐Egger (Mendelian randomization‐Egger) method, in which a genetic variant may be directly associated with the outcome not via the risk factor of interest, so long as the direct effects of the variants on the outcome are uncorrelated with their associations with the risk factor (unmeasured pleiotropy). In this paper, we extend the MR‐Egger method to a multivariable setting to correct for both measured and unmeasured pleiotropy. We show, through theoretical arguments and a simulation study, that the multivariable MR‐Egger method has advantages over its univariable counterpart in terms of plausibility of the assumption needed for consistent causal estimation and power to detect a causal effect when this assumption is satisfied. The methods are compared in an applied analysis to investigate the causal effect of high‐density lipoprotein cholesterol on coronary heart disease risk. The multivariable MR‐Egger method will be useful to analyse high‐dimensional data in situations where the risk factors are highly related and it is difficult to find genetic variants specifically associated with the risk factor of interest (multivariable by design), and as a sensitivity analysis when the genetic variants are known to have pleiotropic effects on measured risk factors.