Analysis of dose-response in flexible dose titration clinical studies

Assessing dose-response from flexible-dose clinical trials (e.g., titration or dose escalation studies) is challenging and often problematic due to the selection bias caused by 'titration-to-response'. We investigate the performance of a dynamic linear mixed-effects (DLME) model and marginal structural model (MSM) in evaluating dose-response from flexible-dose titration clinical trials via simulations. The simulation results demonstrated that DLME models with previous exposure as a time-varying covariate may provide an unbiased and efficient estimator to recover exposure-response relationship from flexible-dose clinical trials. Although the MSM models with independent and exchangeable working correlations appeared to be able to recover the right direction of the dose-response relationship, it tended to over-correct selection bias and overestimated the underlying true dose-response. The MSM estimators were also associated with large variability in the parameter estimates. Therefore, DLME may be an appropriate modeling option in identifying dose-response when data from fixed-dose studies are absent or a fixed-dose design is unethical to be implemented.     

Double robust estimation in longitudinal marginal structural models

In this article we consider estimation of causal parameters in a marginal structural model for the discrete intensity of the treatment specific counting process (e.g. hazard of a treatment specific survival time) based on longitudinal observational data on treatment, covariates and survival. We define three estimators: the inverse probability of treatment weighted (IPTW) estimator, the maximum likelihood estimator (MLE), and a double robust (DR) estimator. The DR estimator is obtained by following a general methodology for constructing double robust estimating functions in censored data models as described in van der Laan and Robins (Unified Methods for Censored Longitudinal Data and Causality, 2002). The double-robust estimator is consistent and asymptotically linear when either the treatment mechanism or the partial likelihood of the observed data is consistently estimated. We illustrate the superiority of the DR estimator relative to the IPTW and ML estimators in a simulation study. The proposed methodology is also applied to estimate the causal effect of exercise on physical functioning in a longitudinal study of seniors in Sonoma County.     

Comparison of causal effect estimators under exposure misclassification

Over the past decades, various principles for causal effect estimation have been proposed, all differing in terms of how they adjust for measured confounders: either via traditional regression adjustment, by adjusting for the expected exposure given those confounders (e.g., the propensity score), or by inversely weighting each subject's data by the likelihood of the observed exposure, given those confounders. When the exposure is measured with error, this raises the question whether these different estimation strategies might be differently affected and whether one of them is to be preferred for that reason. In this article, we investigate this by comparing inverse probability of treatment weighted (IPTW) estimators and doubly robust estimators for the exposure effect in linear marginal structural mean models (MSM) with G-estimators, propensity score (PS) adjusted estimators and ordinary least squares (OLS) estimators for the exposure effect in linear regression models. We find analytically that these estimators are equally affected when exposure misclassification is independent of the confounders, but not otherwise. Simulation studies reveal similar results for time-varying exposures and when the model of interest includes a logistic link.     

Survival analysis using inverse probability of treatment weighted methods based on the generalized propensity score

In survival analysis, treatment effects are commonly evaluated based on survival curves and hazard ratios as causal treatment effects. In observational studies, these estimates may be biased due to confounding factors. The inverse probability of treatment weighted (IPTW) method based on the propensity score is one of the approaches utilized to adjust for confounding factors between binary treatment groups. As a generalization of this methodology, we developed an exact formula for an IPTW log‐rank test based on the generalized propensity score for survival data. This makes it possible to compare the group differences of IPTW Kaplan–Meier estimators of survival curves using an IPTW log‐rank test for multi‐valued treatments. As causal treatment effects, the hazard ratio can be estimated using the IPTW approach. If the treatments correspond to ordered levels of a treatment, the proposed method can be easily extended to the analysis of treatment effect patterns with contrast statistics. In this paper, the proposed method is illustrated with data from the Kyushu Lipid Intervention Study (KLIS), which investigated the primary preventive effects of pravastatin on coronary heart disease (CHD). The results of the proposed method suggested that pravastatin treatment reduces the risk of CHD and that compliance to pravastatin treatment is important for the prevention of CHD. Copyright © 2009 John Wiley & Sons, Ltd.     

Using marginal structural models to analyze the impact of subsequent therapy on the treatment effect in survival data: Simulations and clinical trial examples

We explore the impact of time-varying subsequent therapy on the statistical power and treatment effects in survival analysis. The marginal structural model (MSM) with stabilized inverse probability treatment weights (sIPTW) was used to account for the effects due to the subsequent therapy. Simulations were performed to compare the MSM-sIPTW method with the conventional method without accounting for the time-varying covariate such as subsequent therapy that is dependent on the initial response of the treatment effect. The results of the simulations indicated that the statistical power, thereby the Type I error, of the trials to detect the frontline treatment effect could be inflated if no appropriate adjustment was made for the impact due to the add-on effects of the subsequent therapy. Correspondingly, the hazard ratio between the treatment groups may be overestimated by the conventional analysis methods. In contrast, MSM-sIPTW can maintain the Type I error rate and gave unbiased estimates of the hazard ratio for the treatment. Two real examples were used to discuss the potential clinical implications. The study demonstrated the importance of accounting for time-varying subsequent therapy for obtaining unbiased interpretation of data.     

Landmark estimation of survival and treatment effects in observational studies

Clinical studies aimed at identifying effective treatments to reduce the risk of disease or death often require long term follow-up of participants in order to observe a sufficient number of events to precisely estimate the treatment effect. In such studies, observing the outcome of interest during follow-up may be difficult and high rates of censoring may be observed which often leads to reduced power when applying straightforward statistical methods developed for time-to-event data. Alternative methods have been proposed to take advantage of auxiliary information that may potentially improve efficiency when estimating marginal survival and improve power when testing for a treatment effect. Recently, Parast et al. (J Am Stat Assoc 109(505):384–394, 2014) proposed a landmark estimation procedure for the estimation of survival and treatment effects in a randomized clinical trial setting and demonstrated that significant gains in efficiency and power could be obtained by incorporating intermediate event information as well as baseline covariates. However, the procedure requires the assumption that the potential outcomes for each individual under treatment and control are independent of treatment group assignment which is unlikely to hold in an observational study setting. In this paper we develop the landmark estimation procedure for use in an observational setting. In particular, we incorporate inverse probability of treatment weights (IPTW) in the landmark estimation procedure to account for selection bias on observed baseline (pretreatment) covariates. We demonstrate that consistent estimates of survival and treatment effects can be obtained by using IPTW and that there is improved efficiency by using auxiliary intermediate event and baseline information. We compare our proposed estimates to those obtained using the Kaplan–Meier estimator, the original landmark estimation procedure, and the IPTW Kaplan–Meier estimator. We illustrate our resulting reduction in bias and gains in efficiency through a simulation study and apply our procedure to an AIDS dataset to examine the effect of previous antiretroviral therapy on survival.     

Ensemble Approaches to Estimating the Population Mean with Missing Response

We propose new ensemble approaches to estimate the population mean for missing response data with fully observed auxiliary variables. We first compress the working models according to their categories through a weighted average, where the weights are proportional to the square of the least‐squares coefficients of model refitting. Based on the compressed values, we develop two ensemble frameworks, under which one is to adjust weights in the inverse probability weighting procedure and the other is built upon an additive structure by reformulating the augmented inverse probability weighting function. The asymptotic normality property is established for the proposed estimators through the theory of estimating functions with plugged‐in nuisance parameter estimates. Simulation studies show that the new proposals have substantial advantages over existing ones for small sample sizes, and an acquired immune deficiency syndrome data example is used for illustration.     

Assessing the treatment effect in a randomized controlled trial with extensive non‐adherence: the EVOLVE trial

Intention‐to‐treat (ITT) analysis is widely used to establish efficacy in randomized clinical trials. However, in a long‐term outcomes study where non‐adherence to study drug is substantial, the on‐treatment effect of the study drug may be underestimated using the ITT analysis. The analyses presented herein are from the EVOLVE trial, a double‐blind, placebo‐controlled, event‐driven cardiovascular outcomes study conducted to assess whether a treatment regimen including cinacalcet compared with placebo in addition to other conventional therapies reduces the risk of mortality and major cardiovascular events in patients receiving hemodialysis with secondary hyperparathyroidism. Pre‐specified sensitivity analyses were performed to assess the impact of non‐adherence on the estimated effect of cinacalcet. These analyses included lag‐censoring, inverse probability of censoring weights (IPCW), rank preserving structural failure time model (RPSFTM) and iterative parameter estimation (IPE). The relative hazard (cinacalcet versus placebo) of mortality and major cardiovascular events was 0.93 (95% confidence interval 0.85, 1.02) using the ITT analysis; 0.85 (0.76, 0.95) using lag‐censoring analysis; 0.81 (0.70, 0.92) using IPCW; 0.85 (0.66, 1.04) using RPSFTM and 0.85 (0.75, 0.96) using IPE. These analyses, while not providing definitive evidence, suggest that the intervention may have an effect while subjects are receiving treatment. The ITT method remains the established method to evaluate efficacy of a new treatment; however, additional analyses should be considered to assess the on‐treatment effect when substantial non‐adherence to study drug is expected or observed. Copyright © 2015 John Wiley & Sons, Ltd.     

Using number of failed contact attempts to adjust for non-ignorable non-response

Summary.  We present a general method of adjustment for non-ignorable non-response in studies where one or more further attempts are made to contact initial non-responders. A logistic regression model relates the probability of response at each contact attempt to covariates and outcomes of interest. We assume that the effect of these covariates and outcomes on the probability of response is the same at all contact attempts. Knowledge of the number of contact attempts enables estimation of the model by using only information from the respondents and the number of non-responders. Three approaches for fitting the response models and estimating parameters of substantive interest and their standard errors are compared: a modified conditional likelihood method in which the fitted inverse probabilities of response are used in weighted analyses for the outcomes of interest, an EM procedure with the Louis formula and a Bayesian approach using Markov chain Monte Carlo methods. We further propose the creation of several sets of weights to incorporate uncertainty in the probability weights in subsequent analyses. Our methods are applied as a sensitivity analysis to a postal survey of symptoms in Persian Gulf War veterans and other servicemen.     

Sensitivity analysis for unmeasured confounding in a marginal structural Cox proportional hazards model

Sensitivity analysis for unmeasured confounding should be reported more often, especially in observational studies. In the standard Cox proportional hazards model, this requires substantial assumptions and can be computationally difficult. The marginal structural Cox proportional hazards model (Cox proportional hazards MSM) with inverse probability weighting has several advantages compared to the standard Cox model, including situations with only one assessment of exposure (point exposure) and time-independent confounders. We describe how simple computations provide sensitivity for unmeasured confounding in a Cox proportional hazards MSM with point exposure. This is achieved by translating the general framework for sensitivity analysis for MSMs by Robins and colleagues to survival time data. Instead of bias-corrected observations, we correct the hazard rate to adjust for a specified amount of unmeasured confounding. As an additional bonus, the Cox proportional hazards MSM is robust against bias from differential loss to follow-up. As an illustration, the Cox proportional hazards MSM was applied in a reanalysis of the association between smoking and depression in a population-based cohort of Norwegian adults. The association was moderately sensitive for unmeasured confounding.     

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

The stratified Cox model is commonly used for stratified clinical trials with time‐to‐event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum‐specific Cox model estimates using inverse‐variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50‐200 subjects per treatment), we propose an alternative approach in which stratum‐specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug‐in formula for the variance of RGLR‐based estimated log hazard ratios. We demonstrate using simulations that our proposed two‐step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment‐by‐stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.     

Empirical likelihood weighted composite quantile regression with partially missing covariates

This paper develops a novel weighted composite quantile regression (CQR) method for estimation of a linear model when some covariates are missing at random and the probability for missingness mechanism can be modelled parametrically. By incorporating the unbiased estimating equations of incomplete data into empirical likelihood (EL), we obtain the EL-based weights, and then re-adjust the inverse probability weighted CQR for estimating the vector of regression coefficients. Theoretical results show that the proposed method can achieve semiparametric efficiency if the selection probability function is correctly specified, therefore the EL weighted CQR is more efficient than the inverse probability weighted CQR. Besides, our algorithm is computationally simple and easy to implement. Simulation studies are conducted to examine the finite sample performance of the proposed procedures. Finally, we apply the new method to analyse the US news College data.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

Adaptive truncated weighting for improving marginal structural model estimation of treatment effects informally censored by subsequent therapy

Randomized clinical trials are designed to estimate the direct effect of a treatment by randomly assigning patients to receive either treatment or control. However, in some trials, patients who discontinued their initial randomized treatment are allowed to switch to another treatment. Therefore, the direct treatment effect of interest may be confounded by subsequent treatment. Moreover, the decision on whether to initiate a second‐line treatment is typically made based on time‐dependent factors that may be affected by prior treatment history. Due to these time‐dependent confounders, traditional time‐dependent Cox models may produce biased estimators of the direct treatment effect. Marginal structural models (MSMs) have been applied to estimate causal treatment effects even in the presence of time‐dependent confounders. However, the occurrence of extremely large weights can inflate the variance of the MSM estimators. In this article, we proposed a new method for estimating weights in MSMs by adaptively truncating the longitudinal inverse probabilities. This method provides balance in the bias variance trade‐off when large weights are inevitable, without the ad hoc removal of selected observations. We conducted simulation studies to explore the performance of different methods by comparing bias, standard deviation, confidence interval coverage rates, and mean square error under various scenarios. We also applied these methods to a randomized, open‐label, phase III study of patients with nonsquamous non‐small cell lung cancer. Copyright © 2015 John Wiley & Sons, Ltd.     

Longitudinal data analysis for generalized linear models with follow‐up dependent on outcome‐related variables

In longitudinal studies, observation times are often irregular and subject‐specific. Frequently they are related to the outcome measure or other variables that are associated with the outcome measure but undesirable to condition upon in the model for outcome. Regression analyses that are unadjusted for outcome‐dependent follow‐up then yield biased estimates. The authors propose a class of inverse‐intensity rate‐ratio weighted estimators in generalized linear models that adjust for outcome‐dependent follow‐up. The estimators, based on estimating equations, are very simple and easily computed; they can be used under mixtures of continuous and discrete observation times. The predictors of observation times can be past observed outcomes, cumulative values of outcome‐model covariates and other factors associated with the outcome. The authors validate their approach through simulations and they illustrate it using data from a supported housing program from the US federal government.     

Adjusting for time-varying confounding in the subdistribution analysis of a competing risk

Despite decades of research in the medical literature, assessment of the attributable mortality due to nosocomial infections in the intensive care unit (ICU) remains controversial, with different studies describing effect estimates ranging from being neutral to extremely risk increasing. Interpretation of study results is further hindered by inappropriate adjustment (a) for censoring of the survival time by discharge from the ICU, and (b) for time-dependent confounders on the causal path from infection to mortality. In previous work (Vansteelandt et al. Biostatistics 10:46–59), we have accommodated this through inverse probability of treatment and censoring weighting. Because censoring due to discharge from the ICU is so intimately connected with a patient’s health condition, the ensuing inverse weighting analyses suffer from influential weights and rely heavily on the assumption that one has measured all common risk factors of ICU discharge and mortality. In this paper, we consider ICU discharge as a competing risk in the sense that we aim to infer the risk of ‘ICU mortality’ over time that would be observed if nosocomial infections could be prevented for the entire study population. For this purpose we develop marginal structural subdistribution hazard models with accompanying estimation methods. In contrast to subdistribution hazard models with time-varying covariates, the proposed approach (a) can accommodate high-dimensional confounders, (b) avoids regression adjustment for post-infection measurements and thereby so-called collider-stratification bias, and (c) results in a well-defined model for the cumulative incidence function. The methods are used to quantify the causal effect of nosocomial pneumonia on ICU mortality using data from the National Surveillance Study of Nosocomial Infections in ICU’s (Belgium).     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.     

Estimation of treatment effects for heterogeneous matched‐pairs data with probit models

Estimating the effect of medical treatments on subject responses is one of the crucial problems in medical research. Matched‐pairs designs are commonly implemented in the field of medical research to eliminate confounding and improve efficiency. In this article, new estimators of treatment effects for heterogeneous matched‐pairs data are proposed. Asymptotic properties of the proposed estimators are derived. Simulation studies show that the proposed estimators have some advantages over the famous Heckman's estimator, the conditional maximum likelihood estimator, and the inverse probability weighted estimator. We apply the proposed methodology to a data set from a study of low‐birth‐weight infants.     

Focused information criterion and model averaging based on weighted composite quantile regression

We study the focused information criterion and frequentist model averaging and their application to post‐model‐selection inference for weighted composite quantile regression (WCQR) in the context of the additive partial linear models. With the non‐parametric functions approximated by polynomial splines, we show that, under certain conditions, the asymptotic distribution of the frequentist model averaging WCQR‐estimator of a focused parameter is a non‐linear mixture of normal distributions. This asymptotic distribution is used to construct confidence intervals that achieve the nominal coverage probability. With properly chosen weights, the focused information criterion based WCQR estimators are not only robust to outliers and non‐normal residuals but also can achieve efficiency close to the maximum likelihood estimator, without assuming the true error distribution. Simulation studies and a real data analysis are used to illustrate the effectiveness of the proposed procedure.