Volume 7, Issue 1

Principal stratification has recently become a popular tool to address certain causal inference questions, particularly in dealing with post-randomization factors in randomized trials. Here, we analyze the conceptual basis for this framework and invite response to clarify the value of principal stratification in estimating causal effects of interest.

Technological advances facilitating the acquisition of large arrays of biomarker data have led to new opportunities to understand and characterize disease progression over time. This creates an analytical challenge, however, due to the large numbers of potentially informative markers, the high degrees of correlation among them, and the time-dependent trajectories of association. We propose a mixed ridge estimator, which integrates ridge regression into the mixed effects modeling framework in order to account for both the correlation induced by repeatedly measuring an outcome on each individual over time, as well as the potentially high degree of correlation among possible predictor variables. An expectation-maximization algorithm is described to account for unknown variance and covariance parameters. Model performance is demonstrated through a simulation study and an application of the mixed ridge approach to data arising from a study of cardiometabolic biomarker responses to evoked inflammation induced by experimental low-dose endotoxemia.

In this paper, we present a histogram-like estimator of a conditional density that uses cross-validation to estimate the histogram probabilities, as well as the optimal number and position of the bins. This estimator is an alternative to kernel density estimators when the dimension of the covariate vector is large. We demonstrate its applicability to estimation of Marginal Structural Model (MSM) parameters in which an initial estimator of the exposure mechanism is needed. MSM estimation based on the proposed density estimator results in less biased estimates, when compared to estimates based on a misspecified parametric model.

It is common for novel dose-finding designs to be presented without a study of their convergence properties. In this article we suggest that examination of convergence is a necessary quality check for dose-finding designs. We present a new convergence proof for a nonparametric family of methods called “interval designs,” under certain conditions on the toxicity-frequency function F. We compare these conditions with the convergence conditions for the popular CRM one-parameter Phase I cancer design, via an innovative numerical sensitivity study generating a diverse sample of dose-toxicity scenarios. Only a small fraction of scenarios meet the Shen-O'Quigley convergence conditions for CRM. Conditions for “interval design” convergence are met more often, but still less than half the time. In the discussion, we illustrate how convergence properties and limitations help provide insight about small-sample behavior.

There has been great public health interest in estimating usual, i.e., long-term average, intake of episodically consumed dietary components that are not consumed daily by everyone, e.g., fish, red meat and whole grains. Short-term measurements of episodically consumed dietary components have zero-inflated skewed distributions. So-called two-part models have been developed for such data in order to correct for measurement error due to within-person variation and to estimate the distribution of usual intake of the dietary component in the univariate case. However, there is arguably much greater public health interest in the usual intake of an episodically consumed dietary component adjusted for energy (caloric) intake, e.g., ounces of whole grains per 1000 kilo-calories, which reflects usual dietary composition and adjusts for different total amounts of caloric intake. Because of this public health interest, it is important to have models to fit such data, and it is important that the model-fitting methods can be applied to all episodically consumed dietary components. We have recently developed a nonlinear mixed effects model (Kipnis, et al., 2010), and have fit it by maximum likelihood using nonlinear mixed effects programs and methodology (the SAS NLMIXED procedure). Maximum likelihood fitting of such a nonlinear mixed model is generally slow because of 3-dimensional adaptive Gaussian quadrature, and there are times when the programs either fail to converge or converge to models with a singular covariance matrix. For these reasons, we develop a Monte-Carlo (MCMC) computation of fitting this model, which allows for both frequentist and Bayesian inference. There are technical challenges to developing this solution because one of the covariance matrices in the model is patterned. Our main application is to the National Institutes of Health (NIH)-AARP Diet and Health Study, where we illustrate our methods for modeling the energy-adjusted usual intake of fish and whole grains. We demonstrate numerically that our methods lead to increased speed of computation, converge to reasonable solutions, and have the flexibility to be used in either a frequentist or a Bayesian manner.

The reproducibility of a biomarker plays a paramount role in determining whether it provides an accurate indication of the true underlying disease or risk status of an individual. When biomarker measurement involves obtaining samples of tissue at random from the organ of interest, sampling variability based on spatial effects can affect this reproducibility. This situation arises when a target organ, such as the prostate or esophagus, is evaluated by multiple random needle biopsies or when an excised organ is randomly sampled. We present a general approach toward estimating reproducibility in the presence of different variance-covariance structures needed to account for possible spatial or temporal variation and correlation. Specifically, we extend the work of previous authors involving applications of the concordance correlation coefficient (CCC) by allowing for different variance-covariance structures of the data. A general concordance correlation matrix representing pairwise concordance correlation coefficients is presented along with an overall concordance correlation coefficient both of which may be obtained from models assuming different variance-covariance structures. The overall concordance correlation coefficient provides a measure of the overall reproducibility and its validity relative to various assumed covariance structures can be assessed by examining commonly employed goodness-of-fit measures. We illustrate these methods to minichromosome maintenance protein 2 (MCM2) data coming from the prostate glands of seven subjects having prostate biopsies between 2002 and 2003.

This paper describes a method for comparisons of several treatments with a control, simultaneously for multiple endpoints. These endpoints are assumed to be normally distributed with different scales and variances. An approximate multivariate t-distribution is used to obtain quantiles for test decisions, multiplicity-adjusted p-values, and simultaneous confidence intervals. Simulation results show that this approach controls the family-wise error type I over both the comparisons and the endpoints in an admissible range. The approach will be applied to a randomized clinical trial comparing two new sets of extracorporeal circulations with a standard for three primary endpoints. A related R package is available.

The problem of covariate measurement error with heteroscedastic measurement error variance is considered. Standard regression calibration assumes that the measurement error has a homoscedastic measurement error variance. An estimator is proposed to correct regression coefficients for covariate measurement error with heteroscedastic variance. Point and interval estimates are derived. Validation data containing the gold standard must be available. This estimator is a closed-form correction of the uncorrected primary regression coefficients, which may be of logistic or Cox proportional hazards model form, and is closely related to the version of regression calibration developed by Rosner et al. (1990). The primary regression model can include multiple covariates measured without error. The use of these estimators is illustrated in two data sets, one taken from occupational epidemiology (the ACE study) and one taken from nutritional epidemiology (the Nurses' Health Study). In both cases, although there was evidence of moderate heteroscedasticity, there was little difference in estimation or inference using this new procedure compared to standard regression calibration. It is shown theoretically that unless the relative risk is large or measurement error severe, standard regression calibration approximations will typically be adequate, even with moderate heteroscedasticity in the measurement error model variance. In a detailed simulation study, standard regression calibration performed either as well as or better than the new estimator. When the disease is rare and the errors normally distributed, or when measurement error is moderate, standard regression calibration remains the method of choice.

The relative risk is a clinically important measure of the effect of treatment on binary outcomes in randomized controlled trials (RCTs). An adjusted relative risk can be estimated using log binomial regression; however, convergence problems are common with this model. While alternative methods have been proposed for estimating relative risks, comparisons between methods have been limited, particularly in the context of RCTs. We compare ten different methods for estimating relative risks under a variety of scenarios relevant to RCTs with independent observations. Results of a large simulation study show that some methods may fail to overcome the convergence problems of log binomial regression, while others may substantially overestimate the treatment effect or produce inaccurate confidence intervals. Further, conclusions about the effectiveness of treatment may differ depending on the method used. We give recommendations for choosing a method for estimating relative risks in the context of RCTs with independent observations.

This paper deals with the problem of discrimination between two sets of complex signals generated by stationary processes with both random effects and mixed spectral distributions. The presence of outlier signals and their influence on the classification process is also considered. As an initial input, a feature vector obtained from estimations of the spectral distribution is proposed and used with two different learning machines, namely a single artificial neural network and the LogitBoost classifier. Performance of both methods is evaluated on five simulation studies as well as on a set of actual data of electroencephalogram (EEG) records obtained from both normal subjects and others having experienced epileptic seizures. Of the different classification methods, Logitboost is shown to be more robust to the presence of outlier signals.

In recent years, various mixed-effects models have been suggested for estimating viral decay rates in HIV dynamic models for complex longitudinal data. Among those models are linear mixed-effects (LME), nonlinear mixed-effects (NLME), and semiparametric nonlinear mixed-effects (SNLME) models. However, a critical question is whether these models produce coherent estimates of viral decay rates, and if not, which model is appropriate and should be used in practice. In addition, one often assumes that a model random error is normally distributed, but the normality assumption may be unrealistic, particularly if the data exhibit skewness. Moreover, some covariates such as CD4 cell count may be often measured with substantial errors. This paper addresses these issues simultaneously by jointly modeling the response variable with skewness and a covariate process with measurement errors using a Bayesian approach to investigate how estimated parameters are changed or different under these three models. A real data set from an AIDS clinical trial study was used to illustrate the proposed models and methods. It was found that there was a significant incongruity in the estimated decay rates in viral loads based on the three mixed-effects models, suggesting that the decay rates estimated by using Bayesian LME or NLME joint models should be interpreted differently from those estimated by using Bayesian SNLME joint models. The findings also suggest that the Bayesian SNLME joint model is preferred to other models because an arbitrary data truncation is not necessary; and it is also shown that the models with a skew-normal distribution and/or measurement errors in covariate may achieve reliable results when the data exhibit skewness.

It is often necessary to compare two measurement methods in medicine and other experimental sciences. This problem covers a broad range of data with applications arising from many different fields. The Bland-Altman method has been a favorite method for concordance assessment. However, the Bland-Altman approach creates a problem of interpretation for many applications when a mixture of fixed bias, proportional bias and/or proportional error occurs. In this paper, an improved Bland-Altman method is proposed to handle more complicated scenarios in practice. This new approach includes Bland-Altman's approach as its special case. We evaluate concordance by defining an agreement interval for each individual paired observation and assessing the overall concordance. The proposed interval approach is very informative and offers many advantages over existing approaches. Data sets are used to demonstrate the advantages of the new method.

This article is devoted to the asymptotic study of adaptive group sequential designs in the case of randomized clinical trials (RCTs) with binary treatment, binary outcome and no covariate. By adaptive design, we mean in this setting a RCT design that allows the investigator to dynamically modify its course through data-driven adjustment of the randomization probability based on data accrued so far, without negatively impacting on the statistical integrity of the trial. By adaptive group sequential design, we refer to the fact that group sequential testing methods can be equally well applied on top of adaptive designs. We obtain that, theoretically, the adaptive design converges almost surely to the targeted unknown randomization scheme. In the estimation framework, we obtain that our maximum likelihood estimator of the parameter of interest is a strongly consistent estimator, and it satisfies a central limit theorem. We can estimate its asymptotic variance, which is the same as that it would feature had we known in advance the targeted randomization scheme and independently sampled from it. Consequently, inference can be carried out as if we had resorted to independent and identically distributed (iid) sampling. In the testing framework, we obtain that the multidimensional t-statistic that we would use under iid sampling still converges to the same canonical distribution under adaptive sampling. Consequently, the same group sequential testing can be carried out as if we had resorted to iid sampling. Furthermore, a comprehensive simulation study that we undertake in a companion article validates the theory. A three-sentence take-home message is “Adaptive designs do learn the targeted optimal design and inference, and testing can be carried out under adaptive sampling as they would under the targeted optimal randomization probability iid sampling. In particular, adaptive designs achieve the same efficiency as the fixed oracle design. This is confirmed by a simulation study, at least for moderate or large sample sizes, across a large collection of targeted randomization probabilities.'”

We undertake here a comprehensive simulation study of the theoretical properties that we derive in a companion article devoted to the asymptotic study of adaptive group sequential designs in the case of randomized clinical trials (RCTs) with binary treatment, binary outcome and no covariate. By adaptive design, we mean in this setting a RCT design that allows the investigator to dynamically modify its course through data-driven adjustment of the randomization probability based on data accrued so far without negatively impacting on the statistical integrity of the trial. By adaptive group sequential design, we refer to the fact that group sequential testing methods can be equally well applied on top of adaptive designs. The simulation study validates the theory. It notably shows in the estimation framework that the confidence intervals we obtain achieve the desired coverage even for moderate sample sizes. In addition, it shows in the testing framework that type I error control at the prescribed level is guaranteed and that all sampling procedures only suffer from a very slight increase of the type II error. A three-sentence take-home message is “Adaptive designs do learn the targeted optimal design and inference and testing can be carried out under adaptive sampling as they would under the targeted optimal randomization probability iid sampling. In particular, adaptive designs achieve the same efficiency as the fixed oracle design. This is confirmed by a simulation study, at least for moderate or large sample sizes, across a large collection of targeted randomization probabilities.”

Consider the problem of testing s null hypotheses simultaneously. In order to deal with the multiplicity problem, the classical approach is to restrict attention to multiple testing procedures that control the familywise error rate (FWE). The closure method of Marcus et al. (1976) reduces the problem of constructing such procedures to one of constructing single tests that control the usual probability of a Type 1 error. It was shown by Sonnemann (1982, 2008) that any coherent multiple testing procedure can be constructed using the closure method. Moreover, it was shown by Sonnemann and Finner (1988) that any incoherent multiple testing procedure can be replaced by a coherent multiple testing procedure which is at least as good. In this paper, we first show an analogous result for dissonant and consonant multiple testing procedures. We show further that, in many cases, the improvement of the consonant multiple testing procedure over the dissonant multiple testing procedure may in fact be strict in the sense that it has strictly greater probability of detecting a false null hypothesis while still maintaining control of the FWE. Finally, we show how consonance can be used in the construction of some optimal maximin multiple testing procedures. This last result is especially of interest because there are very few results on optimality in the multiple testing literature.

In disease classification, a traditional technique is the receiver operative characteristic (ROC) curve and the area under the curve (AUC). With high-dimensional data, the ROC techniques are needed to conduct classification and variable selection. The current ROC methods do not explicitly incorporate unequal misclassification costs or do not have a theoretical grounding for optimizing the AUC. Empirical studies in the literature have demonstrated that optimizing the hinge loss can maximize the AUC approximately. In theory, minimizing the hinge rank loss is equivalent to minimizing the AUC in the asymptotic limit. In this article, we propose a novel nonparametric method HingeBoost to optimize a weighted hinge loss incorporating misclassification costs. HingeBoost can be used to construct linear and nonlinear classifiers. The estimation and variable selection for the hinge loss are addressed by a new boosting algorithm. Furthermore, the proposed twin HingeBoost can select more sparse predictors. Some properties of HingeBoost are studied as well. To compare HingeBoost with existing classification methods, we present empirical study results using data from simulations and a prostate cancer study with mass spectrometry-based proteomics.

When cost data are collected in a clinical study, interest centers on the between-treatment difference in mean cost. When censoring is present, the resulting loss of information can be limited by collecting cost data for several pre-specified time intervals, leading to censored longitudinal cost data. Most models for marginal costs stratify by time interval. However, in few other areas of biostatistics would we stratify by default. We argue that there are benefits to considering more general models: for example, in some settings, pooling regression coefficients across intervals can improve the precision of the estimated between-treatment difference in mean cost. Previous work has used inverse-probability-weighted GEEs coupled with an independent working variance to estimate parameters from these more general models. We show that the greatest precision benefits of non-stratified models are achieved by using more sophisticated working variance matrices.

With a binary response Y, the dose-response model under consideration is logistic in flavor with pr(Y=1 | D) = R (1+R)-1, R = λ0 + EAR D, where λ0 is the baseline incidence rate and EAR is the excess absolute risk per gray. The calculated thyroid dose of a person i is expressed as Dimes = fiQimes/Mimes. Here, Qimes is the measured content of radioiodine in the thyroid gland of person i at time tmes, Mimes is the estimate of the thyroid mass, and fi is the normalizing multiplier. The Qi and Mi are measured with multiplicative errors ViQ and ViM, so that Qimes = QitrViQ (this is classical measurement error model) and Mitr = MimesViM (this is Berkson measurement error model). Here, Qitr is the true content of radioactivity in the thyroid gland, and Mitr is the true value of the thyroid mass. The error in fi is much smaller than the errors in (Qimes, Mimes) and ignored in the analysis. By means of Parametric Full Maximum Likelihood and Regression Calibration (under the assumption that the data set of true doses has lognormal distribution), Nonparametric Full Maximum Likelihood, Nonparametric Regression Calibration, and by properly tuned SIMEX method we study the influence of measurement errors in thyroid dose on the estimates of λ0 and EAR. The simulation study is presented based on a real sample from the epidemiological studies. The doses were reconstructed in the framework of the Ukrainian-American project on the investigation of Post-Chernobyl thyroid cancers in Ukraine, and the underlying subpolulation was artificially enlarged in order to increase the statistical power. The true risk parameters were given by the values to earlier epidemiological studies, and then the binary response was simulated according to the dose-response model.

Various assumptions have been used in the literature to identify natural direct and indirect effects in mediation analysis. These effects are of interest because they allow for effect decomposition of a total effect into a direct and indirect effect even in the presence of interactions or non-linear models. In this paper, we consider the relation and interpretation of various identification assumptions in terms of causal diagrams interpreted as a set of non-parametric structural equations. We show that for such causal diagrams, two sets of assumptions for identification that have been described in the literature are in fact equivalent in the sense that if either set of assumptions holds for all models inducing a particular causal diagram, then the other set of assumptions will also hold for all models inducing that diagram. We moreover build on prior work concerning a complete graphical identification criterion for covariate adjustment for total effects to provide a complete graphical criterion for using covariate adjustment to identify natural direct and indirect effects. Finally, we show that this criterion is equivalent to the two sets of independence assumptions used previously for mediation analysis.

We consider two-stage sampling designs, including so-called nested case control studies, where one takes a random sample from a target population and completes measurements on each subject in the first stage. The second stage involves drawing a subsample from the original sample, collecting additional data on the subsample. This data structure can be viewed as a missing data structure on the full-data structure collected in the second-stage of the study. Methods for analyzing two-stage designs include parametric maximum likelihood estimation and estimating equation methodology. We propose an inverse probability of censoring weighted targeted maximum likelihood estimator (IPCW-TMLE) in two-stage sampling designs and present simulation studies featuring this estimator.

A meta-analysis that uses individual-level data instead of study-level data is widely considered to be a gold standard approach, in part because it allows a time-to-event analysis. Unfortunately, with the common practice of presenting Kaplan-Meier survival curves after pooling subjects across randomized trials, using individual-level data can actually be a step backwards; a Simpson's paradox can occur in which pooling incorrectly reverses the direction of an association. We introduce a nonparametric procedure for synthesizing survival curves across studies that is designed to avoid this difficulty and preserve the integrity of randomization. The technique is based on a counterfactual formulation in which we ask what pooled survival curves would look like if all subjects in all studies had been assigned treatment, or if all subjects had been assigned to control arms. The method is related to a Kaplan-Meier adjustment proposed in 2005 by Xie and Liu to correct for confounding in nonrandomized studies, but is formulated for the meta-analysis setting. The procedure is discussed in the context of examining rosiglitazone and cardiovascular adverse events.

The Cox proportional hazards model or its discrete time analogue, the logistic failure time model, posit highly restrictive parametric models and attempt to estimate parameters which are specific to the model proposed. These methods are typically implemented when assessing effect modification in survival analyses despite their flaws. The targeted maximum likelihood estimation (TMLE) methodology is more robust than the methods typically implemented and allows practitioners to estimate parameters that directly answer the question of interest. TMLE will be used in this paper to estimate two newly proposed parameters of interest that quantify effect modification in the time to event setting. These methods are then applied to the Tshepo study to assess if either gender or baseline CD4 level modify the effect of two cART therapies of interest, efavirenz (EFV) and nevirapine (NVP), on the progression of HIV. The results show that women tend to have more favorable outcomes using EFV while males tend to have more favorable outcomes with NVP. Furthermore, EFV tends to be favorable compared to NVP for individuals at high CD4 levels.

Variable selection for high dimensional data has recently received a great deal of attention. However, due to the complex structure of the likelihood, only limited developments have been made for time-to-event data where censoring is present. In this paper, we propose a Bayesian variable selection scheme for a Bayesian semiparametric survival model for right censored survival data sets. A special shrinkage prior on the coefficients corresponding to the predictor variables is used to handle cases when the explanatory variables are of very high-dimension. The shrinkage prior is obtained through a scale mixture representation of Normal and Gamma distributions. Our proposed variable selection prior corresponds to the well known lasso penalty. The likelihood function is based on the Cox proportional hazards model framework, where the cumulative baseline hazard function is modeled a priori by a gamma process. We assign a prior on the tuning parameter of the shrinkage prior and adaptively control the sparsity of our model. The primary use of the proposed model is to identify the important covariates relating to the survival curves. To implement our methodology, we have developed a fast Markov chain Monte Carlo algorithm with an adaptive jumping rule. We have successfully applied our method on simulated data sets under two different settings and real microarray data sets which contain right censored survival time. The performance of our Bayesian variable selection model compared with other competing methods is also provided to demonstrate the superiority of our method. A short description of the biological relevance of the selected genes in the real data sets is provided, further strengthening our claims.

This work considers the estimation of the size N of a closed population using incomplete lists of its members. Capture histories are constructed by establishing the presence or the absence of each individual in all the lists available. Models for data featuring a heterogeneous catchability and list dependencies are considered. A log-linear model leading to a lower bound for the population size is derived for a known set of list dependencies and a latent catchability variable with an arbitrary distribution. This generalizes Chao’s lower bound to models with interactions. The proposed model can be used to carry out a search for important list interactions. It also provides diagnostic information about the nature of the underlying heterogeneity. Indeed, it is shown that the Poisson maximum likelihood estimator of N under a dichotomous latent class model does not exist for a particular set of LB models. Several distributions for the heterogeneous catchability are considered; they allow to investigate the sensitivity of the population size estimate to the model for the heterogeneous catchability.

We investigate a competing risks model, using the specification of the Gompertz distribution for failure times from competing causes and the inverse Gaussian distribution for failure times from the cause of interest. The expectation-maximization algorithm is used for parameter estimation and the model is applied to real data on breast cancer and melanoma. In these applications, our models compare favourably with existing techniques. The proposed method provides a useful technique that may be more broadly applicable than existing alternatives.

Relative risks have become a popular measure of treatment effect for binary outcomes in randomized controlled trials (RCTs). Relative risks can be estimated directly using log binomial regression but the model may fail to converge. Alternative methods are available for estimating relative risks but these have generally only been evaluated for independent data. As some of these methods are now being applied in cluster RCTs, investigation of their performance in this context is needed. We compare log binomial regression and three alternative methods (expanded logistic regression, log Poisson regression and log normal regression) for estimating relative risks in cluster RCTs. Clustering is taken into account using generalized estimating equations (GEEs) with an independence or exchangeable working correlation structure. The results of our large simulation study show that the log binomial GEE generally performs well for clustered data but suffers from convergence problems, as expected. Both the log Poisson GEE and log normal GEE have advantages in certain settings in terms of type I error, bias and coverage. The expanded logistic GEE can perform poorly and is sensitive to the chosen working correlation structure. Conclusions about the effectiveness of treatment often differ depending on the method used, highlighting the need to pre-specify an analysis approach. We recommend pre-specifying that either the log Poisson GEE or log normal GEE will be used in the event that the log binomial GEE fails to converge.

The assumptions that anchor large clinical trials are rooted in smaller, Phase II studies. In addition to specifying the target population, intervention delivery, and patient follow-up duration, physician-scientists who design these Phase II studies must select the appropriate response variables (endpoints). However, endpoint measures can be problematic. If the endpoint assesses the change in a continuous measure over time, then the occurrence of an intervening significant clinical event (SCE), such as death, can preclude the follow-up measurement. Finally, the ideal continuous endpoint measurement may be contraindicated in a fraction of the study patients, a change that requires a less precise substitution in this subset of participants. A score function that is based on the U-statistic can address these issues of 1) intercurrent SCE’s and 2) response variable ascertainments that use different measurements of different precision. The scoring statistic is easy to apply, clinically relevant, and provides flexibility for the investigators’ prospective design decisions. Sample size and power formulations for this statistic are provided as functions of clinical event rates and effect size estimates that are easy for investigators to identify and discuss. Examples are provided from current cardiovascular cell therapy research.

There is an active debate in the literature on censored data about the relative performance of model based maximum likelihood estimators, IPCW-estimators, and a variety of double robust semiparametric efficient estimators. Kang and Schafer (2007) demonstrate the fragility of double robust and IPCW-estimators in a simulation study with positivity violations. They focus on a simple missing data problem with covariates where one desires to estimate the mean of an outcome that is subject to missingness. Responses by Robins, et al. (2007), Tsiatis and Davidian (2007), Tan (2007) and Ridgeway and McCaffrey (2007) further explore the challenges faced by double robust estimators and offer suggestions for improving their stability. In this article, we join the debate by presenting targeted maximum likelihood estimators (TMLEs). We demonstrate that TMLEs that guarantee that the parametric submodel employed by the TMLE procedure respects the global bounds on the continuous outcomes, are especially suitable for dealing with positivity violations because in addition to being double robust and semiparametric efficient, they are substitution estimators. We demonstrate the practical performance of TMLEs relative to other estimators in the simulations designed by Kang and Schafer (2007) and in modified simulations with even greater estimation challenges.

We present a model for longitudinal measures of fetal weight as a function of gestational age. We use a linear mixed model, with a Box-Cox transformation of fetal weight values, and restricted cubic splines, in order to flexibly but parsimoniously model median fetal weight. We systematically compare our model to other proposed approaches. All proposed methods are shown to yield similar median estimates, as evidenced by overlapping pointwise confidence bands, except after 40 completed weeks, where our method seems to produce estimates more consistent with observed data. Sex-based stratification affects the estimates of the random effects variance-covariance structure, without significantly changing sex-specific fitted median values. We illustrate the benefits of including sex-gestational age interaction terms in the model over stratification. The comparison leads to the conclusion that the selection of a model for fetal weight for gestational age can be based on the specific goals and configuration of a given study without affecting the precision or value of median estimates for most gestational ages of interest.

Suppose that having established a marginal total effect of a point exposure on a time-to-event outcome, an investigator wishes to decompose this effect into its direct and indirect pathways, also known as natural direct and indirect effects, mediated by a variable known to occur after the exposure and prior to the outcome. This paper proposes a theory of estimation of natural direct and indirect effects in two important semiparametric models for a failure time outcome. The underlying survival model for the marginal total effect and thus for the direct and indirect effects, can either be a marginal structural Cox proportional hazards model, or a marginal structural additive hazards model. The proposed theory delivers new estimators for mediation analysis in each of these models, with appealing robustness properties. Specifically, in order to guarantee ignorability with respect to the exposure and mediator variables, the approach, which is multiply robust, allows the investigator to use several flexible working models to adjust for confounding by a large number of pre-exposure variables. Multiple robustness is appealing because it only requires a subset of working models to be correct for consistency; furthermore, the analyst need not know which subset of working models is in fact correct to report valid inferences. Finally, a novel semiparametric sensitivity analysis technique is developed for each of these models, to assess the impact on inference, of a violation of the assumption of ignorability of the mediator.

Background: The evidence for the effectiveness of antihypertensive medication use for slowing decline in kidney function in older persons is sparse. We addressed this research question by the application of novel methods in a marginal structural model. Methods: Change in kidney function was measured by two or more measures of cystatin C in 1,576 hypertensive participants in the Cardiovascular Health Study over 7 years of follow-up (1989-1997 in four U.S. communities). The exposure of interest was antihypertensive medication use. We used a novel estimator in a marginal structural model to account for bias due to confounding and informative censoring. Results: The mean annual decline in eGFR was 2.41 ± 4.91 mL/min/1.73 m2. In unadjusted analysis, antihypertensive medication use was not associated with annual change in kidney function. Traditional multivariable regression did not substantially change these estimates. Based on a marginal structural analysis, persons on antihypertensives had slower declines in kidney function; participants had an estimated 0.88 (0.13, 1.63) ml/min/1.73 m2 per year slower decline in eGFR compared with persons on no treatment. In a model that also accounted for bias due to informative censoring, the estimate for the treatment effect was 2.23 (-0.13, 4.59) ml/min/1.73 m2 per year slower decline in eGFR.Conclusion: In summary, estimates from a marginal structural model suggested that antihypertensive therapy was associated with preserved kidney function in hypertensive elderly adults. Confirmatory studies may provide power to determine the strength and validity of the findings.

Pearl’s article provides a useful springboard for discussing further the benefits and drawbacks of principal stratification and the associated discomfort with attributing effects to post-treatment variables. The basic insights of the approach are important: pay close attention to modification of treatment effects by variables not observable before treatment decisions are made, and be careful in attributing effects to variables when counterfactuals are ill-defined. These insights have often been taken too far in many areas of application of the approach, including instrumental variables, censoring by death, and surrogate outcomes. A novel finding is that the usual principal stratification estimand in the setting of censoring by death is by itself of little practical value in estimating intervention effects.

This commentary takes up Pearl's welcome challenge to clearly articulate the scientific value of principal stratification estimands that we and colleagues have investigated, in the area of randomized placebo-controlled preventive vaccine efficacy trials, especially trials of HIV vaccines. After briefly arguing that certain principal stratification estimands for studying vaccine effects on post-infection outcomes are of genuine scientific interest, the bulk of our commentary argues that the “causal effect predictiveness” (CEP) principal stratification estimand for evaluating immune biomarkers as surrogate endpoints is not of ultimate scientific interest, because it evaluates surrogacy restricted to the setting of a particular vaccine efficacy trial, but is nevertheless useful for guiding the selection of primary immune biomarker endpoints in Phase I/II vaccine trials and for facilitating assessment of transportability/bridging surrogacy.

This Reader’s Reaction contains some brief remarks regarding Pearl’s concerns regarding the value of principal stratification.

Dr. Pearl invites researchers to justify their use of principal stratification. This comment explains how the use of principal stratification simplified a complex mediational problem encountered when evaluating a smoking cessation intervention’s effect on reducing smoking withdrawal symptoms.

The paired availability design for historical controls postulated four classes corresponding to the treatment (old or new) a participant would receive if arrival occurred during either of two time periods associated with different availabilities of treatment. These classes were later extended to other settings and called principal strata. Judea Pearl asks if principal stratification is a goal or a tool and lists four interpretations of principal stratification. In the case of the paired availability design, principal stratification is a tool that falls squarely into Pearl’s interpretation of principal stratification as “an approximation to research questions concerning population averages.” We describe the paired availability design and the important role played by principal stratification in estimating the effect of receipt of treatment in a population using data on changes in availability of treatment. We discuss the assumptions and their plausibility. We also introduce the extrapolated estimate to make the generalizability assumption more plausible. By showing why the assumptions are plausible we show why the paired availability design, which includes principal stratification as a key component, is useful for estimating the effect of receipt of treatment in a population. Thus, for our application, we answer Pearl’s challenge to clearly demonstrate the value of principal stratification.

Pearl (2011) asked for the causal inference community to clarify the role of the principal stratification framework in the analysis of causal effects. Here, I argue that the notion of principal stratification has shed light on problems of non-compliance, censoring-by-death, and the analysis of post-infection outcomes; that it may be of use in considering problems of surrogacy but further development is needed; that it is of some use in assessing “direct effects”; but that it is not the appropriate tool for assessing “mediation.” There is nothing within the principal stratification framework that corresponds to a measure of an “indirect” or “mediated” effect.

Pearl (2011) posed the question of whether confinement of clinical trial analyses involving post-randomization variables to the principal stratum “framework” of Frangakis and Rubin (2002) unduly restricts the scientific questions that can be asked. Frangakis and Rubin illustrated their proposal through examples involving compliance, mediation, and surrogacy. Here the utility of the principal stratum framework, and the potential outcomes formulation from which it derives, are considered for these topics in the specific setting of the Women’s Health Initiative randomized, placebo controlled trials of postmenopausal hormone therapy. It is argued that the essential issues related to study reliability and causal interpretation involve the avoidance of context-specific biases that are typically not closely related to whether or not treatment effects have a representation in terms of potential outcomes contrasts. Also, while the questions posed within principal strata may be of interest, some key questions in the hormone therapy setting would not be addressed if restricted to contrasts within principal strata.

In randomized controlled trials (RCTs), treatment assignment is unconfounded with baseline covariates, allowing outcomes to be directly compared between treatment arms. When outcomes are binary, the effect of treatment can be summarized using relative risks, absolute risk reductions and the number needed to treat (NNT). When outcomes are time-to-event in nature, the effect of treatment on the absolute reduction of the risk of an event occurring within a specified duration of follow-up and the associated NNT can be estimated. In observational studies of the effect of treatments on health outcomes, treatment is frequently confounded with baseline covariates. Regression adjustment is commonly used to estimate the adjusted effect of treatment on outcomes. We highlight several limitations of measures of treatment effect that are directly obtained from regression models. We illustrate how both regression-based approaches and propensity-score based approaches allow one to estimate the same measures of treatment effect as those that are commonly reported in RCTs. The CONSORT statement recommends that both relative and absolute measures of treatment effects be reported for RCTs with dichotomous outcomes. The methods described in this paper will allow for similar reporting in observational studies.