Two-stage instrumental variable methods for estimating the causal odds ratio: analysis of bias

We present closed-form expressions of asymptotic bias for the causal odds ratio from two estimation approaches of instrumental variable logistic regression: (i) the two-stage predictor substitution (2SPS) method and (ii) the two-stage residual inclusion (2SRI) approach. Under the 2SPS approach, the first stage model yields the predicted value of treatment as a function of an instrument and covariates, and in the second stage model for the outcome, this predicted value replaces the observed value of treatment as a covariate. Under the 2SRI approach, the first stage is the same, but the residual term of the first stage regression is included in the second stage regression, retaining the observed treatment as a covariate. Our bias assessment is for a different context from that of Terza (J. Health Econ. 2008; 27(3):531-543), who focused on the causal odds ratio conditional on the unmeasured confounder, whereas we focus on the causal odds ratio among compliers under the principal stratification framework. Our closed-form bias results show that the 2SPS logistic regression generates asymptotically biased estimates of this causal odds ratio when there is no unmeasured confounding and that this bias increases with increasing unmeasured confounding. The 2SRI logistic regression is asymptotically unbiased when there is no unmeasured confounding, but when there is unmeasured confounding, there is bias and it increases with increasing unmeasured confounding. The closed-form bias results provide guidance for using these IV logistic regression methods. Our simulation results are consistent with our closed-form analytic results under different combinations of parameter settings.     

A general approach to evaluating the bias of 2‐stage instrumental variable estimators

Unmeasured confounding is a common concern when researchers attempt to estimate a treatment effect using observational data or randomized studies with nonperfect compliance. To address this concern, instrumental variable methods, such as 2‐stage predictor substitution (2SPS) and 2‐stage residual inclusion (2SRI), have been widely adopted. In many clinical studies of binary and survival outcomes, 2SRI has been accepted as the method of choice over 2SPS, but a compelling theoretical rationale has not been postulated. We evaluate the bias and consistency in estimating the conditional treatment effect for both 2SPS and 2SRI when the outcome is binary, count, or time to event. We demonstrate analytically that the bias in 2SPS and 2SRI estimators can be reframed to mirror the problem of omitted variables in nonlinear models and that there is a direct relationship with the collapsibility of effect measures. In contrast to conclusions made by previous studies (Terza et al, 2008), we demonstrate that the consistency of 2SRI estimators only holds under the following conditions: (1) when the null hypothesis is true; (2) when the outcome model is collapsible; or (3) when estimating the nonnull causal effect from Cox or logistic regression models, the strong and unrealistic assumption that the effect of the unmeasured covariates on the treatment is proportional to their effect on the outcome needs to hold. We propose a novel dissimilarity metric to provide an intuitive explanation of the bias of 2SRI estimators in noncollapsible models and demonstrate that with increasing dissimilarity between the effects of the unmeasured covariates on the treatment versus outcome, the bias of 2SRI increases in magnitude.     

Bias testing,bias correction,and confounder selection using an instrumental variable model

Instrumental variable (IV) analysis can be used to address bias due to unobserved confounding when estimating the causal effect of a treatment on an outcome of interest. However, if a proposed IV is correlated with unmeasured confounders and/or weakly correlated with the treatment, the standard IV estimator may be more biased than an ordinary least squares (OLS) estimator. Several methods have been proposed that compare the bias of the IV and OLS estimators relying on the belief that measured covariates can be used as proxies for the unmeasured confounder. Despite these developments, there is lack of discussion about approaches that can be used to formally test whether the IV estimator may be less biased than the OLS estimator. Thus, we have developed a testing framework to compare the bias and a criterion to select informative measured covariates for bias comparison and regression adjustment. We also have developed a bias-correction method, which allows one to use an invalid IV to correct the bias of the OLS or IV estimator. Numerical studies demonstrate that the proposed methods perform well with realistic sample sizes.     

Bias due to two‐stage residual‐outcome regression analysis in genetic association studies

Association studies of risk factors and complex diseases require careful assessment of potential confounding factors. Two‐stage regression analysis, sometimes referred to as residual‐ or adjusted‐outcome analysis, has been increasingly used in association studies of single nucleotide polymorphisms (SNPs) and quantitative traits. In this analysis, first, a residual‐outcome is calculated from a regression of the outcome variable on covariates and then the relationship between the adjusted‐outcome and the SNP is evaluated by a simple linear regression of the adjusted‐outcome on the SNP. In this article, we examine the performance of this two‐stage analysis as compared with multiple linear regression (MLR) analysis. Our findings show that when a SNP and a covariate are correlated, the two‐stage approach results in biased genotypic effect and loss of power. Bias is always toward the null and increases with the squared‐correlation between the SNP and the covariate (). For example, for , 0.1, and 0.5, two‐stage analysis results in, respectively, 0, 10, and 50% attenuation in the SNP effect. As expected, MLR was always unbiased. Since individual SNPs often show little or no correlation with covariates, a two‐stage analysis is expected to perform as well as MLR in many genetic studies; however, it produces considerably different results from MLR and may lead to incorrect conclusions when independent variables are highly correlated. While a useful alternative to MLR under , the two ‐stage approach has serious limitations. Its use as a simple substitute for MLR should be avoided. Genet. Epidemiol. 2011. © 2011 Wiley Periodicals, Inc. 35: 592‐596, 2011     

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.     

On a preference‐based instrumental variable approach in reducing unmeasured confounding‐by‐indication

Treatment preferences of groups (e.g., clinical centers) have often been proposed as instruments to control for unmeasured confounding‐by‐indication in instrumental variable (IV) analyses. However, formal evaluations of these group‐preference‐based instruments are lacking. Unique challenges include the following: (i) correlations between outcomes within groups; (ii) the multi‐value nature of the instruments; (iii) unmeasured confounding occurring between and within groups. We introduce the framework of between‐group and within‐group confounding to assess assumptions required for the group‐preference‐based IV analyses. Our work illustrates that, when unmeasured confounding effects exist only within groups but not between groups, preference‐based IVs can satisfy assumptions required for valid instruments. We then derive a closed‐form expression of asymptotic bias of the two‐stage generalized ordinary least squares estimator when the IVs are valid. Simulations demonstrate that the asymptotic bias formula approximates bias in finite samples quite well, particularly when the number of groups is moderate to large. The bias formula shows that when the cluster size is finite, the IV estimator is asymptotically biased; only when both the number of groups and cluster size go to infinity, the bias disappears. However, the IV estimator remains advantageous in reducing bias from confounding‐by‐indication. The bias assessment provides practical guidance for preference‐based IV analyses. To increase their performance, one should adjust for as many measured confounders as possible, consider groups that have the most random variation in treatment assignment and increase cluster size. To minimize the likelihood for these IVs to be invalid, one should minimize unmeasured between‐group confounding. Copyright © 2014 John Wiley & Sons, Ltd.     

Identifying the odds ratio estimated by a two‐stage instrumental variable analysis with a logistic regression model

An adjustment for an uncorrelated covariate in a logistic regression changes the true value of an odds ratio for a unit increase in a risk factor. Even when there is no variation due to covariates, the odds ratio for a unit increase in a risk factor also depends on the distribution of the risk factor. We can use an instrumental variable to consistently estimate a causal effect in the presence of arbitrary confounding. With a logistic outcome model, we show that the simple ratio or two‐stage instrumental variable estimate is consistent for the odds ratio of an increase in the population distribution of the risk factor equal to the change due to a unit increase in the instrument divided by the average change in the risk factor due to the increase in the instrument. This odds ratio is conditional within the strata of the instrumental variable, but marginal across all other covariates, and is averaged across the population distribution of the risk factor. Where the proportion of variance in the risk factor explained by the instrument is small, this is similar to the odds ratio from a RCT without adjustment for any covariates, where the intervention corresponds to the effect of a change in the population distribution of the risk factor. This implies that the ratio or two‐stage instrumental variable method is not biased, as has been suggested, but estimates a different quantity to the conditional odds ratio from an adjusted multiple regression, a quantity that has arguably more relevance to an epidemiologist or a policy maker, especially in the context of Mendelian randomization. Copyright © 2013 John Wiley & Sons, Ltd.     

Methods of designing two‐stage winner trials with survival outcomes

In drug development, especially for oncology studies, a recent proposal is to combine a costly phase II dose selection study with a subsequent phase III study into a single trial that compares the selected (winning) dose from the first stage with the control group. This design may also be used in phase III trials, in which the winning active treatment regimen, selected at the first stage, is compared with the control group at the second stage. This design is known as a two‐stage winner design, as proposed by Shun et al. (2008) for continuous outcomes. Time‐to‐event data are often analyzed in oncology trials. In order to derive the critical value and power of this design, per Shun et al. (2008), it is essential to calculate the asymptotic covariance and correlation of the log‐rank statistics for survival outcomes between the two stages. In this paper, we derive the asymptotic covariance and correlation, and provide additional approximate design parameters. Examples are given to illustrate the method, and simulations are performed to evaluate the veracity of these approximate design parameters. Copyright © 2013 John Wiley & Sons, Ltd.     

One‐stage and two‐stage designs for phase II clinical trials with survival endpoints

This work is motivated by trials in rapidly lethal cancers or cancers for which measuring shrinkage of tumours is infeasible. In either case, traditional phase II designs focussing on tumour response are unsuitable. Usually, tumour response is considered as a substitute for the more relevant but longer‐term endpoint of death. In rapidly lethal cancers such as pancreatic cancer, there is no need to use a surrogate, as the definitive endpoint is (sadly) available so soon. In uveal cancer, there is no counterpart to tumour response, and so, mortality is the only realistic response available. Cytostatic cancer treatments do not seek to kill tumours, but to mitigate their effects. Trials of such therapy might also be based on survival times to death or progression, rather than on tumour shrinkage. Phase II oncology trials are often conducted with all study patients receiving the experimental therapy, and this approach is considered here. Simple extensions of one‐stage and two‐stage designs based on binary responses are presented. Outcomes based on survival past a small number of landmark times are considered: here, the case of three such times is explored in examples. This approach allows exact calculations to be made for both design and analysis purposes. Simulations presented here show that calculations based on normal approximations can lead to loss of power when sample sizes are small. Two‐stage versions of the procedure are also suggested. Copyright © 2014 John Wiley & Sons, Ltd.     

Meta‐analysis using individual participant data: one‐stage and two‐stage approaches,and why they may differ

Meta‐analysis using individual participant data (IPD) obtains and synthesises the raw, participant‐level data from a set of relevant studies. The IPD approach is becoming an increasingly popular tool as an alternative to traditional aggregate data meta‐analysis, especially as it avoids reliance on published results and provides an opportunity to investigate individual‐level interactions, such as treatment‐effect modifiers. There are two statistical approaches for conducting an IPD meta‐analysis: one‐stage and two‐stage. The one‐stage approach analyses the IPD from all studies simultaneously, for example, in a hierarchical regression model with random effects. The two‐stage approach derives aggregate data (such as effect estimates) in each study separately and then combines these in a traditional meta‐analysis model. There have been numerous comparisons of the one‐stage and two‐stage approaches via theoretical consideration, simulation and empirical examples, yet there remains confusion regarding when each approach should be adopted, and indeed why they may differ. In this tutorial paper, we outline the key statistical methods for one‐stage and two‐stage IPD meta‐analyses, and provide 10 key reasons why they may produce different summary results. We explain that most differences arise because of different modelling assumptions, rather than the choice of one‐stage or two‐stage itself. We illustrate the concepts with recently published IPD meta‐analyses, summarise key statistical software and provide recommendations for future IPD meta‐analyses. © 2016 The Authors. Statistics in Medicine published by 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.     

Sample size for two‐stage studies with maintenance therapy

An adaptive treatment strategy (ATS) is defined as a sequence of treatments and intermediate responses. ATS' arise when chronic diseases such as cancer and depression are treated over time with various treatment alternatives depending on intermediate responses to earlier treatments. Clinical trials are often designed to compare ATSs based on appropriate designs such as sequential randomization designs. Although recent literature provides statistical methods for analyzing data from such trials, very few articles have focused on statistical power and sample size issues. This paper presents a sample size formula for comparing the survival probabilities under two treatment strategies sharing same initial, but different maintenance treatment. The formula is based on the large sample properties of inverse‐probability‐weighted estimator. Simulation study shows strong evidence that the proposed sample size formula guarantees desired power, regardless of the true distributions of survival times. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian hierarchical modeling of longitudinal glaucomatous visual fields using a two‐stage approach

The Bayesian approach has become increasingly popular because it allows to fit quite complex models to data via Markov chain Monte Carlo sampling. However, it is also recognized nowadays that Markov chain Monte Carlo sampling can become computationally prohibitive when applied to a large data set. We encountered serious computational difficulties when fitting an hierarchical model to longitudinal glaucoma data of patients who participate in an ongoing Dutch study. To overcome this problem, we applied and extended a recently proposed two‐stage approach to model these data. Glaucoma is one of the leading causes of blindness in the world. In order to detect deterioration at an early stage, a model for predicting visual fields (VFs) in time is needed. Hence, the true underlying VF progression can be determined, and treatment strategies can then be optimized to prevent further VF loss. Because we were unable to fit these data with the classical one‐stage approach upon which the current popular Bayesian software is based, we made use of the two‐stage Bayesian approach. The considered hierarchical longitudinal model involves estimating a large number of random effects and deals with censoring and high measurement variability. In addition, we extended the approach with tools for model evaluation. Copyright © 2017 John Wiley & Sons, Ltd.     

Controlling the type I error rate in two‐stage sequential adaptive designs when testing for average bioequivalence

In a 2×2 crossover trial for establishing average bioequivalence (ABE) of a generic agent and a currently marketed drug, the recommended approach to hypothesis testing is the two one‐sided test (TOST) procedure, which depends, among other things, on the estimated within‐subject variability. The power of this procedure, and therefore the sample size required to achieve a minimum power, depends on having a good estimate of this variability. When there is uncertainty, it is advisable to plan the design in two stages, with an interim sample size reestimation after the first stage, using an interim estimate of the within‐subject variability. One method and 3 variations of doing this were proposed by Potvin et al. Using simulation, the operating characteristics, including the empirical type I error rate, of the 4 variations (called Methods A, B, C, and D) were assessed by Potvin et al and Methods B and C were recommended. However, none of these 4 variations formally controls the type I error rate of falsely claiming ABE, even though the amount of inflation produced by Method C was considered acceptable. A major disadvantage of assessing type I error rate inflation using simulation is that unless all possible scenarios for the intended design and analysis are investigated, it is impossible to be sure that the type I error rate is controlled. Here, we propose an alternative, principled method of sample size reestimation that is guaranteed to control the type I error rate at any given significance level. This method uses a new version of the inverse‐normal combination of p‐values test, in conjunction with standard group sequential techniques, that is more robust to large deviations in initial assumptions regarding the variability of the pharmacokinetic endpoints. The sample size reestimation step is based on significance levels and power requirements that are conditional on the first‐stage results. This necessitates a discussion and exploitation of the peculiar properties of the power curve of the TOST testing procedure. We illustrate our approach with an example based on a real ABE study and compare the operating characteristics of our proposed method with those of Method B of Povin et al.     

One‐ and two‐stage design proposals for a phase II trial comparing three active treatments with control using an ordered categorical endpoint

Phase II clinical trials are performed to investigate whether a novel treatment shows sufficient promise of efficacy to justify its evaluation in a subsequent definitive phase III trial, and they are often also used to select the dose to take forward. In this paper we discuss different design proposals for a phase II trial in which three active treatment doses and a placebo control are to be compared in terms of a single‐ordered categorical endpoint. The sample size requirements for one‐stage and two‐stage designs are derived, based on an approach similar to that of Dunnett. Detailed computations are prepared for an illustrative example concerning a study in stroke. Allowance for early stopping for futility is made. Simulations are used to verify that the specified type I error and power requirements are valid, despite certain approximations used in the derivation of sample size. The advantages and disadvantages of the different designs are discussed, and the scope for extending the approach to different forms of endpoint is considered. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimation of infection prevalence and sensitivity in a stratified two‐stage sampling design employing highly specific diagnostic tests when there is no gold standard

In this work, we describe a two‐stage sampling design to estimate the infection prevalence in a population. In the first stage, an imperfect diagnostic test was performed on a random sample of the population. In the second stage, a different imperfect test was performed in a stratified random sample of the first sample. To estimate infection prevalence, we assumed conditional independence between the diagnostic tests and develop method of moments estimators based on expectations of the proportions of people with positive and negative results on both tests that are functions of the tests' sensitivity, specificity, and the infection prevalence. A closed‐form solution of the estimating equations was obtained assuming a specificity of 100% for both tests. We applied our method to estimate the infection prevalence of visceral leishmaniasis according to two quantitative polymerase chain reaction tests performed on blood samples taken from 4756 patients in northern Ethiopia. The sensitivities of the tests were also estimated, as well as the standard errors of all estimates, using a parametric bootstrap. We also examined the impact of departures from our assumptions of 100% specificity and conditional independence on the estimated prevalence. Copyright © 2015 John Wiley & Sons, Ltd.     

A sequential Cox approach for estimating the causal effect of treatment in the presence of time‐dependent confounding applied to data from the Swiss HIV Cohort Study

When estimating the effect of treatment on HIV using data from observational studies, standard methods may produce biased estimates due to the presence of time‐dependent confounders. Such confounding can be present when a covariate, affected by past exposure, is both a predictor of the future exposure and the outcome. One example is the CD4 cell count, being a marker for disease progression for HIV patients, but also a marker for treatment initiation and influenced by treatment. Fitting a marginal structural model (MSM) using inverse probability weights is one way to give appropriate adjustment for this type of confounding. In this paper we study a simple and intuitive approach to estimate similar treatment effects, using observational data to mimic several randomized controlled trials. Each ‘trial’ is constructed based on individuals starting treatment in a certain time interval. An overall effect estimate for all such trials is found using composite likelihood inference. The method offers an alternative to the use of inverse probability of treatment weights, which is unstable in certain situations. The estimated parameter is not identical to the one of an MSM, it is conditioned on covariate values at the start of each mimicked trial. This allows the study of questions that are not that easily addressed fitting an MSM. The analysis can be performed as a stratified weighted Cox analysis on the joint data set of all the constructed trials, where each trial is one stratum. The model is applied to data from the Swiss HIV cohort study. Copyright © 2010 John Wiley & Sons, Ltd.     

Improve efficiency and reduce bias of Cox regression models for two‐stage randomization designs using auxiliary covariates

Two‐stage randomization designs are broadly accepted and becoming increasingly popular in clinical trials for cancer and other chronic diseases to assess and compare the effects of different treatment policies. In this paper, we propose an inferential method to estimate the treatment effects in two‐stage randomization designs, which can improve the efficiency and reduce bias in the presence of chance imbalance of a robust covariate‐adjustment without additional assumptions required by Lokhnygina and Helterbrand (Biometrics, 63:422‐428)'s inverse probability weighting (IPW) method. The proposed method is evaluated and compared with the IPW method using simulations and an application to data from an oncology clinical trial. Given the predictive power of baseline covariates collected in this real data, our proposed method obtains 17–38% gains in efficiency compared with the IPW method in terms of overall survival outcome. Copyright © 2017 John Wiley & Sons, Ltd.     

Extensions to the two‐stage randomized trial design for testing treatment,self‐selection,and treatment preference effects to binary outcomes

While traditional clinical trials seek to determine treatment efficacy within a specified population, they often ignore the role of a patient's treatment preference on his or her treatment response. The two‐stage (doubly) randomized preference trial design provides one approach for researchers seeking to disentangle preference effects from treatment effects. Currently, this two‐stage design is limited to the design and analysis of continuous outcome variables; in this presentation, we extend this current design to include binary variables. We present test statistics for testing preference, selection, and treatment effects in a two‐stage randomized design with a binary outcome measure, with and without stratification. We also derive closed‐form sample size formulas to indicate the number of patients needed to detect each effect. A series of simulation studies explore the properties and efficiency of both the unstratified and stratified two‐stage randomized trial designs. Finally, we demonstrate the applicability of these methods using an example of a trial of Hepatitis C treatment.     

Dose–response modeling in mental health using stein‐like estimators with instrumental variables

A mental health trial is analyzed using a dose–response model, in which the number of sessions attended by the patients is deemed indicative of the dose of psychotherapeutic treatment. Here, the parameter of interest is the difference in causal treatment effects between the subpopulations that take part in different numbers of therapy sessions. For this data set, interactions between random treatment allocation and prognostic baseline variables provide the requisite instrumental variables. While the corresponding two‐stage least squares (TSLS) estimator tends to have smaller bias than the ordinary least squares (OLS) estimator; the TSLS suffers from larger variance. It is therefore appealing to combine the desirable properties of the OLS and TSLS estimators. Such a trade‐off is achieved through an affine combination of these two estimators, using mean squared error as a criterion. This produces the semi‐parametric Stein‐like (SPSL) estimator as introduced by Judge and Mittelhammer (2004). The SPSL estimator is used in conjunction with multiple imputation with chained equations, to provide an estimator that can exploit all available information. Simulated data are also generated to illustrate the superiority of the SPSL estimator over its OLS and TSLS counterparts. A package entitled SteinIV implementing these methods has been made available through the R platform. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.