Bayesian sensitivity analysis for unmeasured confounding in observational studies

We consider Bayesian sensitivity analysis for unmeasured confounding in observational studies where the association between a binary exposure, binary response, measured confounders and a single binary unmeasured confounder can be formulated using logistic regression models. A model for unmeasured confounding is presented along with a family of prior distributions that model beliefs about a possible unknown unmeasured confounder. Simulation from the posterior distribution is accomplished using Markov chain Monte Carlo. Because the model for unmeasured confounding is not identifiable, standard large-sample theory for Bayesian analysis is not applicable. Consequently, the impact of different choices of prior distributions on the coverage probability of credible intervals is unknown. Using simulations, we investigate the coverage probability when averaged with respect to various distributions over the parameter space. The results indicate that credible intervals will have approximately nominal coverage probability, on average, when the prior distribution used for sensitivity analysis approximates the sampling distribution of model parameters in a hypothetical sequence of observational studies. We motivate the method in a study of the effectiveness of beta blocker therapy for treatment of heart failure.     

A sensitivity analysis using information about measured confounders yielded improved uncertainty assessments for unmeasured confounding

OBJECTIVE: In the analysis of observational data, the argument is sometimes made that if adjustment for measured confounders induces little change in the treatment-outcome association, then there is less concern about the extent to which the association is driven by unmeasured confounding. We quantify this reasoning using Bayesian sensitivity analysis (BSA) for unmeasured confounding. Using hierarchical models, the confounding effect of a binary unmeasured variable is modeled as arising from the same distribution as that of measured confounders. Our objective is to investigate the performance of the method compared to sensitivity analysis, which assumes that there is no relationship between measured and unmeasured confounders. STUDY DESIGN AND SETTING: We apply the method in an observational study of the effectiveness of beta-blocker therapy in heart failure patients. RESULTS: BSA for unmeasured confounding using hierarchical prior distributions yields an odds ratio (OR) of 0.72, 95% credible interval (CrI): 0.56, 0.93 for the association between beta-blockers and mortality, whereas using independent priors yields OR=0.72, 95% CrI: 0.45, 1.15. CONCLUSION: If the confounding effect of a binary unmeasured variable is similar to that of measured confounders, then conventional sensitivity analysis may give results that overstate the uncertainty about bias.     

Evaluating the Impact of Unmeasured Confounding with Internal Validation Data: An Example Cost Evaluation in Type 2 Diabetes

The quantitative assessment of the potential influence of unmeasured confounders in the analysis of observational data is rare, despite reliance on the “no unmeasured confounders” assumption. In a recent comparison of costs of care between two treatments for type 2 diabetes using a health care claims database, propensity score matching was implemented to adjust for selection bias though it was noted that information on baseline glycemic control was not available for the propensity model. Using data from a linked laboratory file, data on this potential “unmeasured confounder” were obtained for a small subset of the original sample. By using this information, we demonstrate how Bayesian modeling, propensity score calibration, and multiple imputation can utilize this additional information to perform sensitivity analyses to quantitatively assess the potential impact of unmeasured confounding. Bayesian regression models were developed to utilize the internal validation data as informative prior distributions for all parameters, retaining information on the correlation between the confounder and other covariates. While assumptions supporting the use of propensity score calibration were not met in this sample, the use of Bayesian modeling and multiple imputation provided consistent results, suggesting that the lack of data on the unmeasured confounder did not have a strong impact on the original analysis, due to the lack of strong correlation between the confounder and the cost outcome variable. Bayesian modeling with informative priors and multiple imputation may be useful tools for unmeasured confounding sensitivity analysis in these situations. Further research to understand the operating characteristics of these methods in a variety of situations, however, remains.     

Use of instrumental variables in the analysis of generalized linear models in the presence of unmeasured confounding with applications to epidemiological research

A major, often unstated, concern of researchers carrying out epidemiological studies of medical therapy is the potential impact on validity if estimates of treatment are biased due to unmeasured confounders. One technique for obtaining consistent estimates of treatment effects in the presence of unmeasured confounders is instrumental variables analysis (IVA). This technique has been well developed in the econometrics literature and is being increasingly used in epidemiological studies. However, the approach to IVA that is most commonly used in such studies is based on linear models, while many epidemiological applications make use of non-linear models, specifically generalized linear models (GLMs) such as logistic or Poisson regression. Here we present a simple method for applying IVA within the class of GLMs using the generalized method of moments approach. We explore some of the theoretical properties of the method and illustrate its use within both a simulation example and an epidemiological study where unmeasured confounding is suspected to be present. We estimate the effects of beta-blocker therapy on one-year all-cause mortality after an incident hospitalization for heart failure, in the absence of data describing disease severity, which is believed to be a confounder.     

Sensitivity analyses for unmeasured confounding assuming a marginal structural model for repeated measures

Robins introduced marginal structural models (MSMs) and inverse probability of treatment weighted (IPTW) estimators for the causal effect of a time-varying treatment on the mean of repeated measures. We investigate the sensitivity of IPTW estimators to unmeasured confounding. We examine a new framework for sensitivity analyses based on a nonidentifiable model that quantifies unmeasured confounding in terms of a sensitivity parameter and a user-specified function. We present augmented IPTW estimators of MSM parameters and prove their consistency for the causal effect of an MSM, assuming a correct confounding bias function for unmeasured confounding. We apply the methods to assess sensitivity of the analysis of Hernán et al., who used an MSM to estimate the causal effect of zidovudine therapy on repeated CD4 counts among HIV-infected men in the Multicenter AIDS Cohort Study. Under the assumption of no unmeasured confounders, a 95 per cent confidence interval for the treatment effect includes zero. We show that under the assumption of a moderate amount of unmeasured confounding, a 95 per cent confidence interval for the treatment effect no longer includes zero. Thus, the analysis of Hernán et al. is somewhat sensitive to unmeasured confounding. We hope that our research will encourage and facilitate analyses of sensitivity to unmeasured confounding in other applications.     

Sensitivity analysis for interactions under unmeasured confounding

We develop a sensitivity analysis technique to assess the sensitivity of interaction analyses to unmeasured confounding. We give bias formulas for sensitivity analysis for interaction under unmeasured confounding on both additive and multiplicative scales. We provide simplified formulas in the case in which either one of the two factors does not interact with the unmeasured confounder in its effects on the outcome. An interesting consequence of the results is that if the two exposures of interest are independent (e.g., gene-environment independence), even under unmeasured confounding, if the estimate of the interaction is nonzero, then either there is a true interaction between the two factors or there is an interaction between one of the factors and the unmeasured confounder; an interaction must be present in either scenario. We apply the results to two examples drawn from the literature.     

Assessing the impact of confounding (measured and unmeasured) in a case-control study to examine the increased risk of pandemic A/H1N1 associated with receipt of the 2008-9 seasonal influenza vaccine

Background

This study examines the role of measured and unmeasured confounding in the relationship between the 2008-9 seasonal influenza vaccine and pandemic H1N1 (pH1N1) influenza virus.

Methods

Data were taken from a test-negative case-control study of 462 lab confirmed pandemic A/H1N1 (pH1N1) cases and 484 test-negative controls. Adjusted odds ratios (OR) and 95% confidence intervals (CI) were derived using multivariate logistic regression. The analysis was repeated using propensity matching. A sensitivity analysis was conducted to quantify the impact of a hypothetical unmeasured confounder.

Results

Cases were more likely to have received the seasonal influenza vaccine after adjusting for multiple confounders using multivariate regression (OR 1.82, 95% CI: 1.25-2.65), using propensity matching (OR 1.86, 95% CI: 1.19-2.92) and in subsequent sensitivity analyses. An unmeasured confounder would need a prevalence of 20%, an odds ratio with the vaccine and pH1N1 of ≥3.5 and ≥3.0 (respectively) to result in a non-significant association. Using a prevalence of 40% the respective associations were 3.0 and 2.5.

Conclusion

A significant positive association between the seasonal influenza vaccine and lab confirmed pH1N1 was observed after considering multiple confounders and using different methods for confounder adjustment. This was not likely explained by an unmeasured confounder given the prevalence and strength of association needed to result in a non-significant association.     

A flexible,interpretable framework for assessing sensitivity to unmeasured confounding

When estimating causal effects, unmeasured confounding and model misspecification are both potential sources of bias. We propose a method to simultaneously address both issues in the form of a semi‐parametric sensitivity analysis. In particular, our approach incorporates Bayesian Additive Regression Trees into a two‐parameter sensitivity analysis strategy that assesses sensitivity of posterior distributions of treatment effects to choices of sensitivity parameters. This results in an easily interpretable framework for testing for the impact of an unmeasured confounder that also limits the number of modeling assumptions. We evaluate our approach in a large‐scale simulation setting and with high blood pressure data taken from the Third National Health and Nutrition Examination Survey. The model is implemented as open‐source software, integrated into the treatSens package for the R statistical programming language. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

A comparison of sensitivity analyses of the effect of wound duration on wound healing

The effect of an unmeasured exposure on the association between an exposure and outcome can never be fully known in an observational study. The purpose of the current study was to evaluate the sensitivity to unmeasured confounding of the previously published association between wound duration and wound healing in a chronic venous leg ulcer. This was estimated using two methods of sensitivity analysis, called Rosenbaum's method and Greenland's external adjustment. The association between wound duration and wound healing was insensitive to confounding unless the odds ratio for the association between the confounder and a younger wound was almost nine (Rosenbaum). In fact, the association between this hypothetical confounder and healing must have an odds ratio of 100 to remove the effect of duration fully (Greenland). Both of these techniques provide useful and complementary information when assessing the effects of a potential unmeasured confounder.     

Reported associations between asthma and acute lymphoblastic leukemia: insights from a hybrid simulation study

Numerous studies have reported a protective association between asthma and acute lymphoblastic leukemia (ALL), but the causal structure of this association remains unclear. We present a hybrid simulation to examine the compatibility of this association with uncontrolled confounding by infection or another unmeasured factor. We generated a synthetic cohort using inputs on the interrelations of asthma, ALL, infections, and other suggested risk factors from the literature and the Danish National Birth Cohort. We computed odds ratios (ORs) between asthma and ALL in the synthetic cohort with and without adjustment for infections and other (including unmeasured) confounders. Only if infection was an extremely strong risk factor for asthma (OR of 10) and an extremely strong protective factor against ALL (OR of 0.1) was the asthma-ALL association compatible with the literature (OR of 0.78). Similarly, strong uncontrolled confounding by an unmeasured factor could downwardly bias the asthma-ALL association, but not enough to replicate findings in the literature. This investigation illustrates that the reported protective association between asthma and ALL is unlikely to be entirely due to uncontrolled confounding by infections or an unmeasured confounder alone. Simulation can be used to advance our understanding of risk factors for rare outcomes as demonstrated by this study.     

“探针变量”：观察性研究中混杂因素识别的工具

观察性研究中往往存在未知或未测量的混杂因素，是流行病学因果关联研究中的重大挑战。本文介绍一种可以应用在观察性研究中的一种对未知/未测量混杂因素进行识别和效应评估的工具——“探针变量”。其主要可以分为暴露探针变量、结局探针变量以及中介探针3种形式，前2种不仅可以对未知/未测量混杂因素进行识别，也可以对其效应量进行估计，从而揭示真实的暴露与结局之间的关联。而中介探针则是针对“中介因子”进行控制，从而识别暴露和结局之间是否存在未测量混杂因素。该理论实践过程中最大的困难在于“探针变量”的选择和确定，不恰当的“探针变量”可能引入新的混杂，导致未测量混杂因素识别不准确。“探针变量”可以推荐作为观察性研究报告中的一项敏感性分析内容，有助于读者真实理解暴露与结局之间的关联，增加观察性流行病学研究中的证据力度。     

Indirect adjustment of relative risks of an exposure with multiple categories for an unmeasured confounder

Purpose

With observational epidemiologic studies, there is often concern that an unmeasured variable might confound an observed association. Investigators can assess the impact from such unmeasured variables on an observed relative risk (RR) by utilizing externally sourced information and applying an indirect adjustment procedure, for example, the “Axelson adjustment.” Although simple and easy to use, this approach applies to exposure and confounder variables that are binary. Other approaches eschew specific values and provide only bounds on the potential bias.

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.     

Hierarchical priors for bias parameters in Bayesian sensitivity analysis for unmeasured confounding

Recent years have witnessed new innovation in Bayesian techniques to adjust for unmeasured confounding. A challenge with existing methods is that the user is often required to elicit prior distributions for high-dimensional parameters that model competing bias scenarios. This can render the methods unwieldy. In this paper, we propose a novel methodology to adjust for unmeasured confounding that derives default priors for bias parameters for observational studies with binary covariates. The confounding effects of measured and unmeasured variables are treated as exchangeable within a Bayesian framework. We model the joint distribution of covariates by using a log-linear model with pairwise interaction terms. Hierarchical priors constrain the magnitude and direction of bias parameters. An appealing property of the method is that the conditional distribution of the unmeasured confounder follows a logistic model, giving a simple equivalence with previously proposed methods. We apply the method in a data example from pharmacoepidemiology and explore the impact of different priors for bias parameters on the analysis results.     

Use of sensitivity analysis to assess the effects on anti-hepatitis A virus antibodies of access to household water supply

The objective of this study was to assess the impact of a possible unmeasured confounding variable in a previously published association between the effects of household water supply and positive results for hepatitis A serology. This was estimated using a path of integration between two methods of sensitivity analysis, called Rosenbaum's method and Greenland's external adjustment. The association between household water supply and positive results for hepatitis A (outcome) serology was insensitive to confounding unless the odds ratio for the association between the confounder and the outcome was > or = 4. The integration of the two sensitivity analysis methods presented proved useful when assessing the effects of a potential unmeasured confounder.     

Estimating the causal effect of zidovudine on CD4 count with a marginal structural model for repeated measures

Even in the absence of unmeasured confounding factors or model misspecification, standard methods for estimating the causal effect of a time-varying treatment on the mean of a repeated measures outcome (for example, GEE regression) may be biased when there are time-dependent variables that are simultaneously confounders of the effect of interest and are predicted by previous treatment. In contrast, the recently developed marginal structural models (MSMs) can provide consistent estimates of causal effects when unmeasured confounding and model misspecification are absent. We describe an MSM for repeated measures that parameterizes the marginal means of counterfactual outcomes corresponding to prespecified treatment regimes. The parameters of MSMs are estimated using a new class of estimators - inverse-probability of treatment weighted estimators. We used an MSM to estimate the effect of zidovudine therapy on mean CD4 count among HIV-infected men in the Multicenter AIDS Cohort Study. We estimated a potential expected increase of 5.4 (95 per cent confidence interval -1.8,12.7) CD4 lymphocytes/l per additional study visit while on zidovudine therapy. We also explain the theory and implementation of MSMs for repeated measures data and draw upon a simple example to illustrate the basic ideas.     

Testing causal effects in observational survival data using propensity score matching design

Time‐to‐event data are very common in observational studies. Unlike randomized experiments, observational studies suffer from both observed and unobserved confounding biases. To adjust for observed confounding in survival analysis, the commonly used methods are the Cox proportional hazards (PH) model, the weighted logrank test, and the inverse probability of treatment weighted Cox PH model. These methods do not rely on fully parametric models, but their practical performances are highly influenced by the validity of the PH assumption. Also, there are few methods addressing the hidden bias in causal survival analysis. We propose a strategy to test for survival function differences based on the matching design and explore sensitivity of the P‐values to assumptions about unmeasured confounding. Specifically, we apply the paired Prentice‐Wilcoxon (PPW) test or the modified PPW test to the propensity score matched data. Simulation studies show that the PPW‐type test has higher power in situations when the PH assumption fails. For potential hidden bias, we develop a sensitivity analysis based on the matched pairs to assess the robustness of our finding, following Rosenbaum's idea for nonsurvival data. For a real data illustration, we apply our method to an observational cohort of chronic liver disease patients from a Mayo Clinic study. The PPW test based on observed data initially shows evidence of a significant treatment effect. But this finding is not robust, as the sensitivity analysis reveals that the P‐value becomes nonsignificant if there exists an unmeasured confounder with a small impact.     

Model specification and unmeasured confounders in partially ecologic analyses based on group proportions of exposed

OBJECTIVES: The aim of this study was to quantify bias from a partially ecologic analysis due to (i) model misspecification and (ii) an unmeasured confounder, considering various scenarios that may occur in occupational and environmental epidemiology. A study with an aggregate exposure variable, PE, but with outcome, group membership, and covariates assessed individually is partially ecologic. In this paper, PE is the proportion exposed; PE can vary across geographic areas or occupational groups. METHODS: Several hypothetical scenarios were considered, varying the baseline risk, the exposure effect, the exposure distribution across groups, the impact of the (unmeasured) confounder, and the confounder distribution across groups. First, confounding within groups was introduced. Thereafter, confounding between groups was introduced by co-varying PE and the confounder prevalence across the groups. The expected odds ratio (exposed versus unexposed) was calculated in two alternative models, the logistic regression and linear odds models, both with PE as the independent variable. Moreover, empirical data on noise exposure and sleeping disturbances were analyzed. RESULTS: Compared with the logistic regression model, the linear odds model yielded a markedly less biased odds ratio (OR) when the outcome was rare (< or = 5% baseline risk). Confounding within groups resulted in marginal bias, whereas confounding between groups resulted in more pronounced bias. CONCLUSIONS: A logistic regression analysis, with PE as an independent variable, can yield substantial model misspecification bias. By contrast, the linear odds model is valid when the outcome is rare. Confounding between groups should be of more concern than confounding within groups in partially ecologic analyses.     

Robust versus consistent variance estimators in marginal structural Cox models

In survival analyses, inverse‐probability‐of‐treatment (IPT) and inverse‐probability‐of‐censoring (IPC) weighted estimators of parameters in marginal structural Cox models are often used to estimate treatment effects in the presence of time‐dependent confounding and censoring. In most applications, a robust variance estimator of the IPT and IPC weighted estimator is calculated leading to conservative confidence intervals. This estimator assumes that the weights are known rather than estimated from the data. Although a consistent estimator of the asymptotic variance of the IPT and IPC weighted estimator is generally available, applications and thus information on the performance of the consistent estimator are lacking. Reasons might be a cumbersome implementation in statistical software, which is further complicated by missing details on the variance formula. In this paper, we therefore provide a detailed derivation of the variance of the asymptotic distribution of the IPT and IPC weighted estimator and explicitly state the necessary terms to calculate a consistent estimator of this variance. We compare the performance of the robust and consistent variance estimators in an application based on routine health care data and in a simulation study. The simulation reveals no substantial differences between the 2 estimators in medium and large data sets with no unmeasured confounding, but the consistent variance estimator performs poorly in small samples or under unmeasured confounding, if the number of confounders is large. We thus conclude that the robust estimator is more appropriate for all practical purposes.