Covariate adjustment in randomized trials with binary outcomes: targeted maximum likelihood estimation

Covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment. However, for binary outcomes, investigators generally rely on the unadjusted estimate as the literature indicates that covariate-adjusted estimates based on the logistic regression models are less efficient. The crucial step that has been missing when adjusting for covariates is that one must integrate/average the adjusted estimate over those covariates in order to obtain the marginal effect. We apply the method of targeted maximum likelihood estimation (tMLE) to obtain estimators for the marginal effect using covariate adjustment for binary outcomes. We show that the covariate adjustment in randomized trials using the logistic regression models can be mapped, by averaging over the covariate(s), to obtain a fully robust and efficient estimator of the marginal effect, which equals a targeted maximum likelihood estimator. This tMLE is obtained by simply adding a clever covariate to a fixed initial regression. We present simulation studies that demonstrate that this tMLE increases efficiency and power over the unadjusted method, particularly for smaller sample sizes, even when the regression model is mis-specified.     

Combining biomarkers for classification with covariate adjustment

Combining multiple markers can improve classification accuracy compared with using a single marker. In practice, covariates associated with markers or disease outcome can affect the performance of a biomarker or biomarker combination in the population. The covariate‐adjusted receiver operating characteristic (ROC) curve has been proposed as a tool to tease out the covariate effect in the evaluation of a single marker; this curve characterizes the classification accuracy solely because of the marker of interest. However, research on the effect of covariates on the performance of marker combinations and on how to adjust for the covariate effect when combining markers is still lacking. In this article, we examine the effect of covariates on classification performance of linear marker combinations and propose to adjust for covariates in combining markers by maximizing the nonparametric estimate of the area under the covariate‐adjusted ROC curve. The proposed method provides a way to estimate the best linear biomarker combination that is robust to risk model assumptions underlying alternative regression‐model‐based methods. The proposed estimator is shown to be consistent and asymptotically normally distributed. We conduct simulations to evaluate the performance of our estimator in cohort and case/control designs and compare several different weighting strategies during estimation with respect to efficiency. Our estimator is also compared with alternative regression‐model‐based estimators or estimators that maximize the empirical area under the ROC curve, with respect to bias and efficiency. We apply the proposed method to a biomarker study from an human immunodeficiency virus vaccine trial. Copyright © 2017 John Wiley & Sons, Ltd.     

Induced smoothing for rank‐based regression with recurrent gap time data

Various semiparametric regression models have recently been proposed for the analysis of gap times between consecutive recurrent events. Among them, the semiparametric accelerated failure time (AFT) model is especially appealing owing to its direct interpretation of covariate effects on the gap times. In general, estimation of the semiparametric AFT model is challenging because the rank‐based estimating function is a nonsmooth step function. As a result, solutions to the estimating equations do not necessarily exist. Moreover, the popular resampling‐based variance estimation for the AFT model requires solving rank‐based estimating equations repeatedly and hence can be computationally cumbersome and unstable. In this paper, we extend the induced smoothing approach to the AFT model for recurrent gap time data. Our proposed smooth estimating function permits the application of standard numerical methods for both the regression coefficients estimation and the standard error estimation. Large‐sample properties and an asymptotic variance estimator are provided for the proposed method. Simulation studies show that the proposed method outperforms the existing nonsmooth rank‐based estimating function methods in both point estimation and variance estimation. The proposed method is applied to the data analysis of repeated hospitalizations for patients in the Danish Psychiatric Center Register.     

Time‐to‐event data with time‐varying biomarkers measured only at study entry,with applications to Alzheimer's disease

Relating time‐varying biomarkers of Alzheimer's disease to time‐to‐event using a Cox model is complicated by the fact that Alzheimer's disease biomarkers are sparsely collected, typically only at study entry; this is problematic since Cox regression with time‐varying covariates requires observation of the covariate process at all failure times. The analysis might be simplified by using study entry as the time origin and treating the time‐varying covariate measured at study entry as a fixed baseline covariate. In this paper, we first derive conditions under which using an incorrect time origin of study entry results in consistent estimation of regression parameters when the time‐varying covariate is continuous and fully observed. We then derive conditions under which treating the time‐varying covariate as fixed at study entry results in consistent estimation. We provide methods for estimating the regression parameter when a functional form can be assumed for the time‐varying biomarker, which is measured only at study entry. We demonstrate our analytical results in a simulation study and apply our methods to data from the Rush Religious Orders Study and Memory and Aging Project and data from the Alzheimer's Disease Neuroimaging Initiative.     

A unified approach for synthesizing population-level covariate effect information in semiparametric estimation with survival data

There has been a growing interest in developing methodologies to combine information from public domains to improve efficiency in the analysis of relatively small-scale studies that collect more detailed patient-level information. The auxiliary information is usually given in the form of summary statistics or regression coefficients. Thus, the question arises as to how to incorporate the summary information in the model estimation procedure. In this article, we consider statistical analysis of right-censored survival data when additional information about the covariate effects evaluated in a reduced Cox model is available. Recognizing that such external information can be summarized using population moments, we present a unified framework by employing the generalized method of moments to combine information from different sources for the analysis of survival data. The proposed estimator can be shown to be consistent and asymptotically normal; moreover, it is more efficient than the maximum partial likelihood estimator. We also consider incorporating uncertainty of the external information in the inference procedure. Simulation studies show that, by incorporating the additional summary information, the proposed estimators enjoy a substantial gain in efficiency over the conventional approach. A data analysis of a pancreatic cancer cohort study is presented to illustrate the methods and theory.     

Modeling zero‐inflated count data using a covariate‐dependent random effect model

In various medical related researches, excessive zeros, which make the standard Poisson regression model inadequate, often exist in count data. We proposed a covariate‐dependent random effect model to accommodate the excess zeros and the heterogeneity in the population simultaneously. This work is motivated by a data set from a survey on the dental health status of Hong Kong preschool children where the response variable is the number of decayed, missing, or filled teeth. The random effect has a sound biological interpretation as the overall oral health status or other personal qualities of an individual child that is unobserved and unable to be quantified easily. The overall measure of oral health status, responsible for accommodating the excessive zeros and also the heterogeneity among the children, is covariate dependent. This covariate‐dependent random effect model allows one to distinguish whether a potential covariate has an effect on the conceived overall oral health condition of the children, that is, the random effect, or has a direct effect on the magnitude of the counts, or both. We proposed a multiple imputation approach for estimation of the parameters. We discussed the choice of the imputation size. We evaluated the performance of the proposed estimation method through simulation studies, and we applied the model and method to the dental data. Copyright © 2012 John Wiley & Sons, Ltd.     

Augmented mixed beta regression models for periodontal proportion data

Continuous (clustered) proportion data often arise in various domains of medicine and public health where the response variable of interest is a proportion (or percentage) quantifying disease status for the cluster units, ranging between zero and one. However, because of the presence of relatively disease‐free as well as heavily diseased subjects in any study, the proportion values can lie in the interval [0,1]. While beta regression can be adapted to assess covariate effects in these situations, its versatility is often challenged because of the presence/excess of zeros and ones because the beta support lies in the interval (0,1). To circumvent this, we augment the probabilities of zero and one with the beta density, controlling for the clustering effect. Our approach is Bayesian with the ability to borrow information across various stages of the complex model hierarchy and produces a computationally convenient framework amenable to available freeware. The marginal likelihood is tractable and can be used to develop Bayesian case‐deletion influence diagnostics based on q‐divergence measures. Both simulation studies and application to a real dataset from a clinical periodontology study quantify the gain in model fit and parameter estimation over other ad hoc alternatives and provide quantitative insight into assessing the true covariate effects on the proportion responses. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized monotonic regression using random change points

We introduce a procedure for generalized monotonic curve fitting that is based on a Bayesian analysis of the isotonic regression model. Conventional isotonic regression fits monotonically increasing step functions to data. In our approach we treat the number and location of the steps as random. For each step level we adopt the conjugate prior to the sampling distribution of the data as if the curve was unconstrained. We then propose to use Markov chain Monte Carlo simulation to draw samples from the unconstrained model space and retain only those samples for which the monotonic constraint holds. The proportion of the samples collected for which the constraint holds can be used to provide a value for the weight of evidence in terms of Bayes factors for monotonicity given the data. Using the samples, probability statements can be made about other quantities of interest such as the number of change points in the data and posterior distributions on the location of the change points can be provided. The method is illustrated throughout by a reanalysis of the leukaemia data studied by Schell and Singh.     

Regression analysis using dependent Polya trees

Many commonly used models for linear regression analysis force overly simplistic shape and scale constraints on the residual structure of data. We propose a semiparametric Bayesian model for regression analysis that produces data‐driven inference by using a new type of dependent Polya tree prior to model arbitrary residual distributions that are allowed to evolve across increasing levels of an ordinal covariate (e.g., time, in repeated measurement studies). By modeling residual distributions at consecutive covariate levels or time points using separate, but dependent Polya tree priors, distributional information is pooled while allowing for broad pliability to accommodate many types of changing residual distributions. We can use the proposed dependent residual structure in a wide range of regression settings, including fixed‐effects and mixed‐effects linear and nonlinear models for cross‐sectional, prospective, and repeated measurement data. A simulation study illustrates the flexibility of our novel semiparametric regression model to accurately capture evolving residual distributions. In an application to immune development data on immunoglobulin G antibodies in children, our new model outperforms several contemporary semiparametric regression models based on a predictive model selection criterion. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparison of the variance estimation methods for heteroscedastic nonlinear models

Heteroscedasticity is commonly encountered when fitting nonlinear regression models in practice. We discuss eight different variance estimation methods for nonlinear regression models with heterogeneous response variances, and present a simulation study to compare the performance of the eight methods in terms of estimating the standard errors of the fitted model parameters. The simulation study suggests that when the true variance is a function of the mean model, the power of the mean variance function estimation method and the transform‐both‐sides method are the best choices for estimating the standard errors of the estimated model parameters. In general, the wild bootstrap estimator and two modified versions of the standard sandwich variance estimator are reasonably accurate with relatively small bias, especially when the heterogeneity is nonsystematic across values of the covariate. Furthermore, we note that the two modified sandwich estimators are appealing choices in practice, considering the computational advantage of these two estimation methods relative to the variance function estimation method and the transform‐both‐sides approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Structured fusion lasso penalized multi‐state models

Multi‐state models generalize survival or duration time analysis to the estimation of transition‐specific hazard rate functions for multiple transitions. When each of the transition‐specific risk functions is parametrized with several distinct covariate effect coefficients, this leads to a model of potentially high dimension. To decrease the parameter space dimensionality and to work out a clear image of the underlying multi‐state model structure, one can either aim at setting some coefficients to zero or to make coefficients for the same covariate but two different transitions equal. The first issue can be approached by penalizing the absolute values of the covariate coefficients as in lasso regularization. If, instead, absolute differences between coefficients of the same covariate on different transitions are penalized, this leads to sparse competing risk relations within a multi‐state model, that is, equality of covariate effect coefficients. In this paper, a new estimation approach providing sparse multi‐state modelling by the aforementioned principles is established, based on the estimation of multi‐state models and a simultaneous penalization of the L₁‐norm of covariate coefficients and their differences in a structured way. The new multi‐state modelling approach is illustrated on peritoneal dialysis study data and implemented in the R package penMSM . Copyright © 2016 John Wiley & Sons, Ltd.     

Circular‐circular regression model with a spike at zero

With reference to a real data on cataract surgery, we discuss the problem of zero‐inflated circular‐circular regression when both covariate and response are circular random variables and a large proportion of the responses are zeros. The regression model is proposed, and the estimation procedure for the parameters is discussed. Some relevant test procedures are also suggested. Simulation studies and real data analysis are performed to illustrate the applicability of the model.     

Penalized estimation for proportional hazards models with current status data

We provide a simple and practical, yet flexible, penalized estimation method for a Cox proportional hazards model with current status data. We approximate the baseline cumulative hazard function by monotone B‐splines and use a hybrid approach based on the Fisher‐scoring algorithm and the isotonic regression to compute the penalized estimates. We show that the penalized estimator of the nonparametric component achieves the optimal rate of convergence under some smooth conditions and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a simple variance estimation method is considered for inference on the regression parameters. We perform 2 extensive Monte Carlo studies to evaluate the finite‐sample performance of the penalized approach and compare it with the 3 competing R packages: C1.coxph, intcox, and ICsurv. A goodness‐of‐fit test and model diagnostics are also discussed. The methodology is illustrated with 2 real applications.     

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.     

A semiparametric imputation approach for regression with censored covariate with application to an AMD progression study

This research is motivated by studying the progression of age‐related macular degeneration where both a covariate and the response variable are subject to censoring. We develop a general framework to handle regression with censored covariate where the response can be different types and the censoring can be random or subject to (constant) detection limits. Multiple imputation is a popular technique to handle missing data that requires compatibility between the imputation model and the substantive model to obtain valid estimates. With censored covariate, we propose a novel multiple imputation‐based approach, namely, the semiparametric two‐step importance sampling imputation (STISI) method, to impute the censored covariate. Specifically, STISI imputes the missing covariate from a semiparametric accelerated failure time model conditional on fully observed covariates (Step 1) with the acceptance probability derived from the substantive model (Step 2). The 2‐step procedure automatically ensures compatibility and takes full advantage of the relaxed semiparametric assumption in the imputation. Extensive simulations demonstrate that the STISI method yields valid estimates in all scenarios and outperforms some existing methods that are commonly used in practice. We apply STISI on data from the Age‐related Eye Disease Study, to investigate the association between the progression time of the less severe eye and that of the more severe eye. We also illustrate the method by analyzing the urine arsenic data for patients from National Health and Nutrition Examination Survey (2003‐2004) where the response is binary and 1 covariate is subject to detection limit.     

Assessing age‐at‐onset risk factors with incomplete covariate current status data under proportional odds models

Motivated by an epidemiological survey of fracture in elderly women, we develop a semiparametric regression analysis of current status data with incompletely observed covariate under the proportional odds model. To accommodate both the interval‐censored nature of current status failure time data and the incompletely observed covariate data, we propose an analysis based on the validation likelihood (VL), which is derived from likelihood pertaining to the validation sample, namely the subset of the sample where the data are completely observed. The missing data mechanism is assumed to be missing at random and is explicitly modeled and estimated in the VL approach. We propose implementing the VL method by integrating self‐consistency and Newton–Raphson algorithms. Asymptotic normality and standard error estimation for the proposed estimator of the regression parameter are guaranteed. Simulation results reveal good performance of the VL estimator. The VL method has some gain in efficiency compared with the naive complete case method. But the VL method leads to unbiased estimators, whereas the complete case method does not when missing covariates are not missing completely at random. Application of the VL approach to the fracture data confirms that osteoporosis (low bone density) is a strong risk factor for the age at onset of fracture in elderly women. Copyright © 2012 John Wiley & Sons, Ltd.     

Collaborative targeted learning using regression shrinkage

Causal inference practitioners are routinely presented with the challenge of model selection and, in particular, reducing the size of the covariate set with the goal of improving estimation efficiency. Collaborative targeted minimum loss‐based estimation (CTMLE) is a general framework for constructing doubly robust semiparametric causal estimators that data‐adaptively limit model complexity in the propensity score to optimize a preferred loss function. This stepwise complexity reduction is based on a loss function placed on a strategically updated model for the outcome variable through which the error is assessed using cross‐validation. We demonstrate how the existing stepwise variable selection CTMLE can be generalized using regression shrinkage of the propensity score. We present 2 new algorithms that involve stepwise selection of the penalization parameter(s) in the regression shrinkage. Simulation studies demonstrate that, under a misspecified outcome model, mean squared error and bias can be reduced by a CTMLE procedure that separately penalizes individual covariates in the propensity score. We demonstrate these approaches in an example using electronic medical data with sparse indicator covariates to evaluate the relative safety of 2 similarly indicated asthma therapies for pregnant women with moderate asthma.     

Evaluation of the Effect of a Continuous Treatment: A Machine Learning Approach with an Application to Treatment for Traumatic Brain Injury

For a continuous treatment, the generalised propensity score (GPS) is defined as the conditional density of the treatment, given covariates. GPS adjustment may be implemented by including it as a covariate in an outcome regression. Here, the unbiased estimation of the dose–response function assumes correct specification of both the GPS and the outcome‐treatment relationship. This paper introduces a machine learning method, the ‘Super Learner’, to address model selection in this context. In the two‐stage estimation approach proposed, the Super Learner selects a GPS and then a dose–response function conditional on the GPS, as the convex combination of candidate prediction algorithms. We compare this approach with parametric implementations of the GPS and to regression methods. We contrast the methods in the Risk Adjustment in Neurocritical care cohort study, in which we estimate the marginal effects of increasing transfer time from emergency departments to specialised neuroscience centres, for patients with acute traumatic brain injury. With parametric models for the outcome, we find that dose–response curves differ according to choice of specification. With the Super Learner approach to both regression and the GPS, we find that transfer time does not have a statistically significant marginal effect on the outcomes. © 2015 The Authors. Health Economics Published by John Wiley & Sons Ltd.