Should a propensity score model be super? The utility of ensemble procedures for causal adjustment

In investigations of the effect of treatment on outcome, the propensity score is a tool to eliminate imbalance in the distribution of confounding variables between treatment groups. Recent work has suggested that Super Learner, an ensemble method, outperforms logistic regression in nonlinear settings; however, experience with real-data analyses tends to show overfitting of the propensity score model using this approach. We investigated a wide range of simulated settings of varying complexities including simulations based on real data to compare the performances of logistic regression, generalized boosted models, and Super Learner in providing balance and for estimating the average treatment effect via propensity score regression, propensity score matching, and inverse probability of treatment weighting. We found that Super Learner and logistic regression are comparable in terms of covariate balance, bias, and mean squared error (MSE); however, Super Learner is computationally very expensive thus leaving no clear advantage to the more complex approach. Propensity scores estimated by generalized boosted models were inferior to the other two estimation approaches. We also found that propensity score regression adjustment was superior to either matching or inverse weighting when the form of the dependence on the treatment on the outcome is correctly specified.     

Estimating inverse probability weights using super learner when weight‐model specification is unknown in a marginal structural Cox model context

Correct specification of the inverse probability weighting (IPW) model is necessary for consistent inference from a marginal structural Cox model (MSCM). In practical applications, researchers are typically unaware of the true specification of the weight model. Nonetheless, IPWs are commonly estimated using parametric models, such as the main‐effects logistic regression model. In practice, assumptions underlying such models may not hold and data‐adaptive statistical learning methods may provide an alternative. Many candidate statistical learning approaches are available in the literature. However, the optimal approach for a given dataset is impossible to predict. Super learner (SL) has been proposed as a tool for selecting an optimal learner from a set of candidates using cross‐validation. In this study, we evaluate the usefulness of a SL in estimating IPW in four different MSCM simulation scenarios, in which we varied the specification of the true weight model specification (linear and/or additive). Our simulations show that, in the presence of weight model misspecification, with a rich and diverse set of candidate algorithms, SL can generally offer a better alternative to the commonly used statistical learning approaches in terms of MSE as well as the coverage probabilities of the estimated effect in an MSCM. The findings from the simulation studies guided the application of the MSCM in a multiple sclerosis cohort from British Columbia, Canada (1995–2008), to estimate the impact of beta‐interferon treatment in delaying disability progression. Copyright © 2017 John Wiley & Sons, Ltd.     

Bayesian dose‐finding designs for combination of molecularly targeted agents assuming partial stochastic ordering

Molecularly targeted agent (MTA) combination therapy is in the early stages of development. When using a fixed dose of one agent in combinations of MTAs, toxicity and efficacy do not necessarily increase with an increasing dose of the other agent. Thus, in dose‐finding trials for combinations of MTAs, interest may lie in identifying the optimal biological dose combinations (OBDCs), defined as the lowest dose combinations (in a certain sense) that are safe and have the highest efficacy level meeting a prespecified target. The limited existing designs for these trials use parametric dose–efficacy and dose–toxicity models. Motivated by a phase I/II clinical trial of a combination of two MTAs in patients with pancreatic, endometrial, or colorectal cancer, we propose Bayesian dose‐finding designs to identify the OBDCs without parametric model assumptions. The proposed approach is based only on partial stochastic ordering assumptions for the effects of the combined MTAs and uses isotonic regression to estimate partially stochastically ordered marginal posterior distributions of the efficacy and toxicity probabilities. We demonstrate that our proposed method appropriately accounts for the partial ordering constraints, including potential plateaus on the dose–response surfaces, and is computationally efficient. We develop a dose‐combination‐finding algorithm to identify the OBDCs. We use simulations to compare the proposed designs with an alternative design based on Bayesian isotonic regression transformation and a design based on parametric change‐point dose–toxicity and dose–efficacy models and demonstrate desirable operating characteristics of the proposed designs. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

The impact of unmeasured baseline effect modification on estimates from an inverse probability of treatment weighted logistic model

We present the results of a Monte Carlo simulation study in which we demonstrate how strong baseline interactions between a confounding variable and a treatment can create an important difference between the marginal effect of exposure on outcome (as estimated by an inverse probability of treatment weighted logistic model) and the conditional effect (as estimated by an adjusted logistic regression model). The scenarios that we explored included one with a rare outcome and a strong and prevalent effect measure modifier where, across 1,000 simulated data sets, the estimates from an adjusted logistic regression model (mean β = 0.475) and an inverse probability of treatment weighted logistic model (mean β = 2.144) do not coincide with the known true effect (β = 0.68925) when the effect measure modifier is not accounted for. When the marginal and conditional estimates do not coincide despite a rare outcome this may suggest that there is heterogeneity in the effect of treatment between individuals. Failure to specify effect measure modification in the statistical model appears to results in systematic differences between the conditional and marginal estimates. When these differences in estimates are observed, testing for or including interactions or non-linear modeling terms may be advised.     

Simulation from a known Cox MSM using standard parametric models for the g‐formula

It is routinely argued that, unlike standard regression‐based estimates, inverse probability weighted (IPW) estimates of the parameters of a correctly specified Cox marginal structural model (MSM) may remain unbiased in the presence of a time‐varying confounder affected by prior treatment. Previously proposed methods for simulating from a known Cox MSM lack knowledge of the law of the observed outcome conditional on the measured past. Although unbiased IPW estimation does not require this knowledge, standard regression‐based estimates rely on correct specification of this law. Thus, in typical high‐dimensional settings, such simulation methods cannot isolate bias due to complex time‐varying confounding as it may be conflated with bias due to misspecification of the outcome regression model. In this paper, we describe an approach to Cox MSM data generation that allows for a comparison of the bias of IPW estimates versus that of standard regression‐based estimates in the complete absence of model misspecification. This approach involves simulating data from a standard parametrization of the likelihood and solving for the underlying Cox MSM. We prove that solutions exist and computations are tractable under many data‐generating mechanisms. We show analytically and confirm in simulations that, in the absence of model misspecification, the bias of standard regression‐based estimates for the parameters of a Cox MSM is indeed a function of the coefficients in observed data models quantifying the presence of a time‐varying confounder affected by prior treatment. We discuss limitations of this approach including that implied by the ‘g‐null paradox’. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian adjustment for covariate measurement errors: A flexible parametric approach

In most epidemiological investigations, the study units are people, the outcome variable (or the response) is a health‐related event, and the explanatory variables are usually environmental and/or socio‐demographic factors. The fundamental task in such investigations is to quantify the association between the explanatory variables (covariates/exposures) and the outcome variable through a suitable regression model. The accuracy of such quantification depends on how precisely the relevant covariates are measured. In many instances, we cannot measure some of the covariates accurately. Rather, we can measure noisy (mismeasured) versions of them. In statistical terminology, mismeasurement in continuous covariates is known as measurement errors or errors‐in‐variables. Regression analyses based on mismeasured covariates lead to biased inference about the true underlying response–covariate associations. In this paper, we suggest a flexible parametric approach for avoiding this bias when estimating the response–covariate relationship through a logistic regression model. More specifically, we consider the flexible generalized skew‐normal and the flexible generalized skew‐t distributions for modeling the unobserved true exposure. For inference and computational purposes, we use Bayesian Markov chain Monte Carlo techniques. We investigate the performance of the proposed flexible parametric approach in comparison with a common flexible parametric approach through extensive simulation studies. We also compare the proposed method with the competing flexible parametric method on a real‐life data set. Though emphasis is put on the logistic regression model, the proposed method is unified and is applicable to the other generalized linear models, and to other types of non‐linear regression models as well. Copyright © 2009 John Wiley & Sons, Ltd.     

An alternative parameterization of the general linear mixture model for longitudinal data with non-ignorable drop-outs

This paper considers the mixture model methodology for handling non-ignorable drop-outs in longitudinal studies with continuous outcomes. Recently, Hogan and Laird have developed a mixture model for non-ignorable drop-outs which is a standard linear mixed effects model except that the parameters which characterize change over time depend also upon time of drop-out. That is, the mean response is linear in time, other covariates and drop-out time, and their interactions. One of the key attractions of the mixture modelling approach to drop-outs is that it is relatively easy to explore the sensitivity of results to model specification. However, the main drawback of mixture models is that the parameters that are ordinarily of interest are not immediately available, but require marginalization of the distribution of outcome over drop-out times. Furthermore, although a linear model is assumed for the conditional mean of the outcome vector given time of drop out, after marginalization, the unconditional mean of the outcome vector is not, in general, linear in the regression parameters. As a result, it is not possible to parsimoniously describe the effects of covariates on the marginal distribution of the outcome in terms of regression coefficients. The need to explicitly average over the distribution of the drop-out times and the absence of regression coefficients that describe the effects of covariates on the outcome are two unappealing features of the mixture modelling approach. In this paper we describe a particular parameterization of the general linear mixture model that circumvents both of these problems.     

Causal inference for Mann–Whitney–Wilcoxon rank sum and other nonparametric statistics

The nonparametric Mann–Whitney–Wilcoxon (MWW) rank sum test is widely used to test treatment effect by comparing the outcome distributions between two groups, especially when there are outliers in the data. However, such statistics generally yield invalid conclusions when applied to nonrandomized studies, particularly those in epidemiologic research. Although one may control for selection bias by using available approaches of covariates adjustment such as matching, regression analysis, propensity score matching, and marginal structural models, such analyses yield results that are not only subjective based on how the outliers are handled but also often difficult to interpret. A popular alternative is a conditional permutation test based on randomization inference [Rosenbaum PR. Covariance adjustment in randomized experiments and observational studies. Statistical Science 2002; 17 (3):286–327]. Because it requires strong and implausible assumptions that may not be met in most applications, this approach has limited applications in practice. In this paper, we address this gap in the literature by extending MWW and other nonparametric statistics to provide causal inference for nonrandomized study data by integrating the potential outcome paradigm with the functional response models (FRM). FRM is uniquely positioned to model dynamic relationships between subjects, rather than attributes of a single subject as in most regression models, such as the MWW test within our context. The proposed approach is illustrated with data from both real and simulated studies. Copyright © 2013 John Wiley & Sons, Ltd.     

Regression modeling of longitudinal data with outcome‐dependent observation times: extensions and comparative evaluation

Conventional longitudinal data analysis methods assume that outcomes are independent of the data‐collection schedule. However, the independence assumption may be violated, for example, when a specific treatment necessitates a different follow‐up schedule than the control arm or when adverse events trigger additional physician visits in between prescheduled follow‐ups. Dependence between outcomes and observation times may introduce bias when estimating the marginal association of covariates on outcomes using a standard longitudinal regression model. We formulate a framework of outcome‐observation dependence mechanisms to describe conditional independence given observed observation‐time process covariates or shared latent variables. We compare four recently developed semi‐parametric methods that accommodate one of these mechanisms. To allow greater flexibility, we extend these methods to accommodate a combination of mechanisms. In simulation studies, we show how incorrectly specifying the outcome‐observation dependence may yield biased estimates of covariate‐outcome associations and how our proposed extensions can accommodate a greater number of dependence mechanisms. We illustrate the implications of different modeling strategies in an application to bladder cancer data. In longitudinal studies with potentially outcome‐dependent observation times, we recommend that analysts carefully explore the conditional independence mechanism between the outcome and observation‐time processes to ensure valid inference regarding covariate‐outcome associations. Copyright © 2014 John Wiley & Sons, Ltd.     

An information criterion for marginal structural models

Marginal structural models were developed as a semiparametric alternative to the G‐computation formula to estimate causal effects of exposures. In practice, these models are often specified using parametric regression models. As such, the usual conventions regarding regression model specification apply. This paper outlines strategies for marginal structural model specification and considerations for the functional form of the exposure metric in the final structural model. We propose a quasi‐likelihood information criterion adapted from use in generalized estimating equations. We evaluate the properties of our proposed information criterion using a limited simulation study. We illustrate our approach using two empirical examples. In the first example, we use data from a randomized breastfeeding promotion trial to estimate the effect of breastfeeding duration on infant weight at 1 year. In the second example, we use data from two prospective cohorts studies to estimate the effect of highly active antiretroviral therapy on CD4 count in an observational cohort of HIV‐infected men and women. The marginal structural model specified should reflect the scientific question being addressed but can also assist in exploration of other plausible and closely related questions. In marginal structural models, as in any regression setting, correct inference depends on correct model specification. Our proposed information criterion provides a formal method for comparing model fit for different specifications. Copyright © 2012 John Wiley & Sons, Ltd.     

A mixed effects model for the analysis of ordinal longitudinal pain data subject to informative drop-out

We extend the model of Pulkstenis et al. that models binary longitudinal data, subject to informative drop-out through remedication, to the ordinal response case. We present a selection model shared-parameter approach that specifies mixed models for both ordinal response and discrete survival time to remedication. In this fashion, the random parameter present in both models completely characterizes the relationship between response and time to remedication inducing their conditional independence. With a log-log link function for both response and study 'survival', as well as specification of a log-gamma distribution for the random effect, we obtain a closed-form expression for the marginal log-likelihood of response and time to remedication that does not require approximation or numerical integration techniques. A data analysis is performed and simulation results presented which support the consistency of parameter and standard error estimates.     

SOME COMMENTS ON MISSPECIFICATION OF PRIORS IN BAYESIAN MODELLING OF MEASUREMENT ERROR PROBLEMS

In this paper we discuss some aspects of misspecification of prior distributions in the context of Bayesian modelling of measurement error problems. A Bayesian approach to the treatment of common measurement error situations encountered in epidemiology has been recently proposed. Its implementation involves, first, the structural specification, through conditional independence relationships, of three submodels – a measurement model, an exposure model and a disease model – and secondly, the choice of functional forms for the distributions involved in the submodels. We present some results indicating how the estimation of the regression parameters of interest, which is carried out using Gibbs sampling, can be influenced by a misspecification of the parametric shape of the prior distribution of exposure. © 1997 by John Wiley & Sons, Ltd.     

Estimation of Causal Effects using Propensity Score Weighting: An Application to Data on Right Heart Catheterization

We consider methods for estimating causal effects of treatments when treatment assignment is unconfounded with outcomes conditional on a possibly large set of covariates. Robins and Rotnitzky (1995) suggested combining regression adjustment with weighting based on the propensity score (Rosenbaum and Rubin, 1983). We adopt this approach, allowing for a flexible specification of both the propensity score and the regression function. We apply these methods to data on the effects of right heart catheterization (RHC) studied in Connors et al (1996), and we find that our estimator gives stable estimates over a wide range of values for the two parameters governing the selection of variables.     

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.     

Power and sample size calculations for discrete bounded outcome scores

We consider power and sample size calculations for randomized trials with a bounded outcome score (BOS) as primary response adjusted for a priori chosen covariates. We define BOS to be a random variable restricted to a finite interval. Typically, a BOS has a J- or U-shaped distribution hindering traditional parametric methods of analysis. When no adjustment for covariates is needed, a non-parametric test could be chosen. However, there is still a problem with calculating the power since the common location-shift alternative does not hold in general for a BOS. In this paper, we consider a parametric approach and assume that the observed BOS is a coarsened version of a true BOS, which has a logit-normal distribution in each treatment group allowing correction for covariates. A two-step procedure is used to calculate the power. Firstly, the power function is defined conditionally on the covariate values. Secondly, the marginal power is obtained by averaging the conditional power with respect to an assumed distribution for the covariates using Monte Carlo integration. A simulation study evaluates the performance of our method which is also applied to the ECASS-1 stroke study.     

Estimation of adjusted rate differences using additive negative binomial regression

Rate differences are an important effect measure in biostatistics and provide an alternative perspective to rate ratios. When the data are event counts observed during an exposure period, adjusted rate differences may be estimated using an identity‐link Poisson generalised linear model, also known as additive Poisson regression. A problem with this approach is that the assumption of equality of mean and variance rarely holds in real data, which often show overdispersion. An additive negative binomial model is the natural alternative to account for this; however, standard model‐fitting methods are often unable to cope with the constrained parameter space arising from the non‐negativity restrictions of the additive model. In this paper, we propose a novel solution to this problem using a variant of the expectation–conditional maximisation–either algorithm. Our method provides a reliable way to fit an additive negative binomial regression model and also permits flexible generalisations using semi‐parametric regression functions. We illustrate the method using a placebo‐controlled clinical trial of fenofibrate treatment in patients with type II diabetes, where the outcome is the number of laser therapy courses administered to treat diabetic retinopathy. An R package is available that implements the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

A weighting approach to causal effects and additive interaction in case-control studies: marginal structural linear odds models

Estimates of additive interaction from case-control data are often obtained by logistic regression; such models can also be used to adjust for covariates. This approach to estimating additive interaction has come under some criticism because of possible misspecification of the logistic model: If the underlying model is linear, the logistic model will be misspecified. The authors propose an inverse probability of treatment weighting approach to causal effects and additive interaction in case-control studies. Under the assumption of no unmeasured confounding, the approach amounts to fitting a marginal structural linear odds model. The approach allows for the estimation of measures of additive interaction between dichotomous exposures, such as the relative excess risk due to interaction, using case-control data without having to rely on modeling assumptions for the outcome conditional on the exposures and covariates. Rather than using conditional models for the outcome, models are instead specified for the exposures conditional on the covariates. The approach is illustrated by assessing additive interaction between genetic and environmental factors using data from a case-control study.     

Mixed‐effects Gaussian process functional regression models with application to dose–response curve prediction

We propose a new semiparametric model for functional regression analysis, combining a parametric mixed‐effects model with a nonparametric Gaussian process regression model, namely a mixed‐effects Gaussian process functional regression model. The parametric component can provide explanatory information between the response and the covariates, whereas the nonparametric component can add nonlinearity. We can model the mean and covariance structures simultaneously, combining the information borrowed from other subjects with the information collected from each individual subject. We apply the model to dose–response curves that describe changes in the responses of subjects for differing levels of the dose of a drug or agent and have a wide application in many areas. We illustrate the method for the management of renal anaemia. An individual dose–response curve is improved when more information is included by this mechanism from the subject/patient over time, enabling a patient‐specific treatment regime. Copyright © 2012 John Wiley & Sons, Ltd.     

Mixed binary-continuous copula regression models with application to adverse birth outcomes

Bivariate copula regression allows for the flexible combination of two arbitrary, continuous marginal distributions with regression effects being placed on potentially all parameters of the resulting bivariate joint response distribution. Motivated by the risk factors for adverse birth outcomes, many of which are dichotomous, we consider mixed binary-continuous responses that extend the bivariate continuous framework to the situation where one response variable is discrete (more precisely, binary) whereas the other response remains continuous. Utilizing the latent continuous representation of binary regression models, we implement a penalized likelihood–based approach for the resulting class of copula regression models and employ it in the context of modeling gestational age and the presence/absence of low birth weight. The analysis demonstrates the advantage of the flexible specification of regression impacts including nonlinear effects of continuous covariates and spatial effects. Our results imply that racial and spatial inequalities in the risk factors for infant mortality are even greater than previously suggested.     

Spline‐based procedures for dose‐finding studies with active control

In a dose‐finding study with an active control, several doses of a new drug are compared with an established drug (the so‐called active control). One goal of such studies is to characterize the dose–response relationship and to find the smallest target dose concentration d^*, which leads to the same efficacy as the active control. For this purpose, the intersection point of the mean dose–response function with the expected efficacy of the active control has to be estimated. The focus of this paper is a cubic spline‐based method for deriving an estimator of the target dose without assuming a specific dose–response function. Furthermore, the construction of a spline‐based bootstrap CI is described. Estimator and CI are compared with other flexible and parametric methods such as linear spline interpolation as well as maximum likelihood regression in simulation studies motivated by a real clinical trial. Also, design considerations for the cubic spline approach with focus on bias minimization are presented. Although the spline‐based point estimator can be biased, designs can be chosen to minimize and reasonably limit the maximum absolute bias. Furthermore, the coverage probability of the cubic spline approach is satisfactory, especially for bias minimal designs. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.