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     

Mediation analysis via potential outcomes models

This paper develops a causal or manipulation model framework for mediation analysis based on the concept of potential outcome. Using this framework, we provide new definitions and measures of mediation. Effects of manipulations are modeled via the linear structural model. Corresponding structural equation models (SEMs), in conjunction with two-stage least-squares estimation and the delta method, are used to perform inference. The methods are applied to data from a study of nursing interventions for postoperative pain. We address the cases of more than two treatment groups, and an interaction among mediators. For the latter, a sensitivity analysis approach to handle unidentified parameters is described. Interpretative advantages of the potential outcomes framework for mediation are emphasized.     

Good Research Practices for Comparative Effectiveness Research: Analytic Methods to Improve Causal Inference from Nonrandomized Studies of Treatment Effects Using Secondary Data Sources: The ISPOR Good Research Practices for Retrospective Database Analysis Task Force Report—Part III

Objectives:  Most contemporary epidemiologic studies require complex analytical methods to adjust for bias and confounding. New methods are constantly being developed, and older more established methods are yet appropriate. Careful application of statistical analysis techniques can improve causal inference of comparative treatment effects from nonrandomized studies using secondary databases. A Task Force was formed to offer a review of the more recent developments in statistical control of confounding.
Methods:  The Task Force was commissioned and a chair was selected by the ISPOR Board of Directors in October 2007. This Report, the third in this issue of the journal, addressed methods to improve causal inference of treatment effects for nonrandomized studies.
Results:  The Task Force Report recommends general analytic techniques and specific best practices where consensus is reached including: use of stratification analysis before multivariable modeling, multivariable regression including model performance and diagnostic testing, propensity scoring, instrumental variable, and structural modeling techniques including marginal structural models, where appropriate for secondary data. Sensitivity analyses and discussion of extent of residual confounding are discussed.
Conclusions:  Valid findings of causal therapeutic benefits can be produced from nonrandomized studies using an array of state-of-the-art analytic techniques. Improving the quality and uniformity of these studies will improve the value to patients, physicians, and policymakers worldwide.     

Causal inference for community‐based multi‐layered intervention study

Estimating causal treatment effect for randomized controlled trials under post‐treatment confounding, that is, noncompliance and informative dropouts, is becoming an important problem in intervention/prevention studies when the treatment exposures are not completely controlled. When confounding is present in a study, the traditional intention‐to‐treat approach could underestimate the treatment effect because of insufficient exposure of treatment. In the recent two decades, many papers have been published to address such confounders to investigate the causal relationship between treatment and outcome of interest based on different modeling strategies. Most of the existing approaches, however, are suitable only for standard experiments. In this paper, we propose a new class of structural functional response model to address post‐treatment confounding in complex multi‐layered intervention studies within a longitudinal data setting. The new approach offers robust inference and is readily implemented. We illustrate and assess the performance of the proposed structural functional response model using both real and simulated data. Copyright © 2014 John Wiley & Sons, Ltd.     

Simpson’s Paradox is suppression,but Lord’s Paradox is neither: clarification of and correction to Tu,Gunnell, and Gilthorpe (2008)

The relationship between collapsibility and confounding has been subject to an extensive and ongoing discussion in the methodological literature. We discuss two subtly different definitions of collapsibility, and show that by considering causal effect measures based on counterfactual variables (rather than measures of association based on observed variables) it is possible to separate out the component of non-collapsibility which is due to the mathematical properties of the effect measure, from the components that are due to structural bias such as confounding. We provide new weights such that the causal risk ratio is collapsible over arbitrary baseline covariates. In the absence of confounding, these weights may be used for standardization of the risk ratio.

     

Severity of bias of a simple estimator of the causal odds ratio in Mendelian randomization studies

Mendelian randomization studies estimate causal effects using genetic variants as instruments. Instrumental variable methods are straightforward for linear models, but epidemiologists often use odds ratios to quantify effects. Also, odds ratios are often the quantities reported in meta‐analyses. Many applications of Mendelian randomization dichotomize genotype and estimate the population causal log odds ratio for unit increase in exposure by dividing the genotype‐disease log odds ratio by the difference in mean exposure between genotypes. This ‘Wald‐type’ estimator is biased even in large samples, but whether the magnitude of bias is of practical importance is unclear. We study the large‐sample bias of this estimator in a simple model with a continuous normally distributed exposure, a single unobserved confounder that is not an effect modifier, and interpretable parameters. We focus on parameter values that reflect scenarios in which we apply Mendelian randomization, including realistic values for the degree of confounding and strength of the causal effect. We evaluate this estimator and the causal odds ratio using numerical integration and obtain approximate analytic expressions to check results and gain insight. A small simulation study examines finite sample bias and mild violations of the normality assumption. For our simple data‐generating model, we find that the Wald estimator is asymptotically biased with a bias of around 10% in fairly typical Mendelian randomization scenarios but which can be larger in more extreme situations. Recently developed methods such as structural mean models require fewer untestable assumptions and we recommend their use when the individual‐level data they require are available. The Wald‐type estimator may retain a role as an approximate method for meta‐analysis based on summary data. Copyright © 2012 John Wiley & Sons, Ltd.     

Using instrumental variables to estimate a Cox’s proportional hazards regression subject to additive confounding

The estimation of treatment effects is one of the primary goals of statistics in medicine. Estimation based on observational studies is subject to confounding. Statistical methods for controlling bias due to confounding include regression adjustment, propensity scores and inverse probability weighted estimators. These methods require that all confounders are recorded in the data. The method of instrumental variables (IVs) can eliminate bias in observational studies even in the absence of information on confounders. We propose a method for integrating IVs within the framework of Cox’s proportional hazards model and demonstrate the conditions under which it recovers the causal effect of treatment. The methodology is based on the approximate orthogonality of an instrument with unobserved confounders among those at risk. We derive an estimator as the solution to an estimating equation that resembles the score equation of the partial likelihood in much the same way as the traditional IV estimator resembles the normal equations. To justify this IV estimator for a Cox model we perform simulations to evaluate its operating characteristics. Finally, we apply the estimator to an observational study of the effect of coronary catheterization on survival.     

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.     

Boosting the precision of mediation analyses of randomised experiments through covariate adjustment

Analyses of randomised experiments frequently include attempts to decompose the intention‐to‐treat effect into a direct and indirect effect, mediated by given intermediaries, with the aim to shed light onto the treatment mechanism. Methods from causal mediation analysis have facilitated this by allowing for arbitrary models for the outcome and the mediator. They thereby generalise the traditional approach to direct and indirect effects, which is essentially limited to linear models. The default maximum likelihood methods make use of a model for the conditional distribution of the mediator, given treatment and baseline covariates, but are prone to bias when that model is misspecified. In randomised experiments, specification of such model can be easily avoided, but at the expense of a sometimes major efficiency loss when those baseline covariates are predictive of the mediator. In this article, we develop a compromise approach: it makes use of a model for the mediator to optimally extract information from the baseline covariate data but is insulated from the impact of misspecification of that model; it achieves this by exploiting the known randomisation probabilities. Simulation studies and the analysis of a randomised study show major efficiency gains and confirm our theoretical findings that the default methods from causal mediation analysis are sometimes, although not always, reasonably robust to model misspecification. Copyright © 2017 John Wiley & Sons, Ltd.     

Causal logistic models for non-compliance under randomized treatment with univariate binary response

We propose a method for estimating the marginal causal log-odds ratio for binary outcomes under treatment non-compliance in placebo-randomized trials. This estimation method is a marginal alternative to the causal logistic approach by Nagelkerke et al. (2000) that conditions on partially unknown compliance (that is, adherence to treatment) status, and also differs from previous approaches that estimate risk differences or ratios in subgroups defined by compliance status. The marginal causal method proposed in this paper is based on an extension of Robins' G-estimation approach for fitting linear or log-linear structural nested models to a logistic model. Comparing the marginal and conditional causal log-odds ratio estimates provides a way of assessing the magnitude of unmeasured confounding of the treatment effect due to treatment non-adherence. More specifically, we show through simulations that under weak confounding, the conditional and marginal procedures yield similar estimates, whereas under stronger confounding, they behave differently in terms of bias and confidence interval coverage. The parametric structures that represent such confounding are not identifiable. Hence, the proof of consistency of causal estimators and corresponding simulations are based on two different models that fully identify the causal effects being estimated. These models differ in the way that compliance is related to potential outcomes, and thus differ in the way that the causal effect is identified. The simulations also show that the proposed marginal causal estimation approach performs well in terms of bias under the different levels of confounding due to non-adherence and under different causal logistic models. We also provide results from the analyses of two data sets further showing how a comparison of the marginal and conditional estimators can help evaluate the magnitude of confounding due to non-adherence.     

Estimating a marginal causal odds ratio in a case‐control design: analyzing the effect of low birth weight on the risk of type 1 diabetes mellitus

Estimation of marginal causal effects from case‐control data has two complications: (i) confounding due to the fact that the exposure under study is not randomized, and (ii) bias from the case‐control sampling scheme. In this paper, we study estimators of the marginal causal odds ratio, addressing these issues for matched and unmatched case‐control designs when utilizing the knowledge of the known prevalence of being a case. The estimators are implemented in simulations where their finite sample properties are studied and approximations of their variances are derived with the delta method. Also, we illustrate the methods by analyzing the effect of low birth weight on the risk of type 1 diabetes mellitus using data from the Swedish Childhood Diabetes Register, a nationwide population‐based incidence register. Copyright © 2013 John Wiley & Sons, Ltd.     

Confounding of indirect effects: a sensitivity analysis exploring the range of bias due to a cause common to both the mediator and the outcome

Several investigators have demonstrated that the assessment of indirect and direct effects is biased in the presence of a cause that is common to both the mediator and the outcome if one has not controlled for this variable in the analysis. However, little work has been done to quantify the bias caused by this type of unmeasured confounding and determine whether this bias will materially affect conclusions regarding mediation. The author developed a sensitivity analysis program to address this crucial issue. Data from 2 well-known studies in the methodological literature on mediation were reanalyzed using this program. The results of mediation analyses were found not to be as vulnerable to the impact of confounding as previously described; however, these findings varied sharply between the 2 studies. Although the indirect effect observed in one study could potentially be due to a cause common to both the mediator and the outcome, such confounding could not feasibly explain the results of the other study. These disparate results demonstrate the utility of the current sensitivity analysis when assessing mediation.     

Estimation of controlled direct effects in time‐varying treatments using structural nested mean models: application to a primary prevention trial for coronary events with pravastatin

For the estimation of controlled direct effects (i.e., direct effects controlling intermediates that are set at a fixed level for all members of the population) without bias, two fundamental assumptions must hold: the absence of unmeasured confounding factors for treatment and outcome and for intermediate variables and outcome. Even if these assumptions hold, one would nonetheless fail to estimate direct effects using standard methods, for example, stratification or regression modeling, when the treatment influences confounding factors. For such situations, the sequential g‐estimation method for structural nested mean models has been developed for estimating controlled direct effects in point‐treatment situations. In this study, we demonstrate that this method can be applied to longitudinal data with time‐varying treatments and repeatedly measured intermediate variables. We sequentially estimate the parameters in two structural nested mean models: one for a repeatedly measured intermediate and the other one for direct effects of a time‐varying treatment. The method was applied to data from a large primary prevention trial for coronary events, in which pravastatin was used to lower the cholesterol levels in patients with moderate hypercholesterolemia. Copyright © 2014 John Wiley & Sons, Ltd.     

The use of propensity scores and observational data to estimate randomized controlled trial generalizability bias

Although randomized controlled trials are considered the ‘gold standard’ for clinical studies, the use of exclusion criteria may impact the external validity of the results. It is unknown whether estimators of effect size are biased by excluding a portion of the target population from enrollment. We propose to use observational data to estimate the bias due to enrollment restrictions, which we term generalizability bias. In this paper, we introduce a class of estimators for the generalizability bias and use simulation to study its properties in the presence of non‐constant treatment effects. We find the surprising result that our estimators can be unbiased for the true generalizability bias even when all potentially confounding variables are not measured. In addition, our proposed doubly robust estimator performs well even for mis‐specified models. Copyright © 2013 John Wiley & Sons, Ltd.     

Quantifying biases in causal models: classical confounding vs collider-stratification bias

It has long been known that stratifying on variables affected by the study exposure can create selection bias. More recently it has been shown that stratifying on a variable that precedes exposure and disease can induce confounding, even if there is no confounding in the unstratified (crude) estimate. This paper examines the relative magnitudes of these biases under some simple causal models in which the stratification variable is graphically depicted as a collider (a variable directly affected by two or more other variables in the graph). The results suggest that bias from stratifying on variables affected by exposure and disease may often be comparable in size with bias from classical confounding (bias from failing to stratify on a common cause of exposure and disease), whereas other biases from collider stratification may tend to be much smaller.     

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.     

The impact of residual and unmeasured confounding in epidemiologic studies: a simulation study

Measurement error in explanatory variables and unmeasured confounders can cause considerable problems in epidemiologic studies. It is well recognized that under certain conditions, nondifferential measurement error in the exposure variable produces bias towards the null. Measurement error in confounders will lead to residual confounding, but this is not a straightforward issue, and it is not clear in which direction the bias will point. Unmeasured confounders further complicate matters. There has been discussion about the amount of bias in exposure effect estimates that can plausibly occur due to residual or unmeasured confounding. In this paper, the authors use simulation studies and logistic regression analyses to investigate the size of the apparent exposure-outcome association that can occur when in truth the exposure has no causal effect on the outcome. The authors consider two cases with a normally distributed exposure and either two or four normally distributed confounders. When the confounders are uncorrelated, bias in the exposure effect estimate increases as the amount of residual and unmeasured confounding increases. Patterns are more complex for correlated confounders. With plausible assumptions, effect sizes of the magnitude frequently reported in observational epidemiologic studies can be generated by residual and/or unmeasured confounding alone.     

Bayesian estimation of the average treatment effect on the treated using inverse weighting

We develop a Bayesian approach to estimate the average treatment effect on the treated in the presence of confounding. The approach builds on developments proposed by Saarela et al in the context of marginal structural models, using importance sampling weights to adjust for confounding and estimate a causal effect. The Bayesian bootstrap is adopted to approximate posterior distributions of interest and avoid the issue of feedback that arises in Bayesian causal estimation relying on a joint likelihood. We present results from simulation studies to estimate the average treatment effect on the treated, evaluating the impact of sample size and the strength of confounding on estimation. We illustrate our approach using the classic Right Heart Catheterization data set and find a negative causal effect of the exposure on 30-day survival, in accordance with previous analyses of these data. We also apply our approach to the data set of the National Center for Health Statistics Birth Data and obtain a negative effect of maternal smoking during pregnancy on birth weight.