Volume 8, Issue 1

The determination of a cut-off value for a continuous prognostic test is an important problem, which is statistically challenging and practically important for risk assessment. We propose in this paper a method to estimate the optimal cut-off from this type of longitudinal data with censored failure times. The principle is to combine the prognostic error rates of false positives and false negatives with a cost function, which has the advantages to be statistically convenient and to be directly associated with the decision-making. Simulations were performed and the results demonstrate the interest of our approach compared to a reference method. The method is also illustrated by predicting the long-term survival of kidney transplant recipients from the 1-year creatinine clearance.

The instrumental variables framework is commonly used for the estimation of causal effects from cohort samples. However, the combination of instrumental variables with more efficient designs such as case-control sampling requires new methodological consideration. For example, as the use of Mendelian randomization studies is increasing and the cost of genotyping and gene expression data can be high, the analysis of data gathered from more cost-effective sampling designs is of prime interest. We show that the standard instrumental variables analysis does not appropriately estimate the causal effects of interest when the instrumental variables design is combined with the case-control design. We also propose a method that can estimate the causal effects in such combined designs. We illustrate the method with a study in oncology.

In many causal inference problems, one is interested in the direct causal effect of an exposure on an outcome of interest that is not mediated by certain intermediate variables. Robins and Greenland (1992) and Pearl (2001) formalized the definition of two types of direct effects (natural and controlled) under the counterfactual framework. The efficient scores (under a nonparametric model) for the various natural effect parameters and their general robustness conditions, as well as an estimating equation based estimator using the efficient score, are provided in Tchetgen Tchetgen and Shpitser (2011b). In this article, we apply the targeted maximum likelihood framework of van der Laan and Rubin (2006) and van der Laan and Rose (2011) to construct a semiparametric efficient, multiply robust, substitution estimator for the natural direct effect which satisfies the efficient score equation derived in Tchetgen Tchetgen and Shpitser (2011b). We note that the robustness conditions in Tchetgen Tchetgen and Shpitser (2011b) may be weakened, thereby placing less reliance on the estimation of the mediator density. More precisely, the proposed estimator is asymptotically unbiased if either one of the following holds: i) the conditional mean outcome given exposure, mediator, and confounders, and the mediated mean outcome difference are consistently estimated; (ii) the exposure mechanism given confounders, and the conditional mean outcome are consistently estimated; or (iii) the exposure mechanism and the mediator density, or the exposure mechanism and the conditional distribution of the exposure given confounders and mediator, are consistently estimated. If all three conditions hold, then the effect estimate is asymptotically efficient. Extensions to the natural indirect effect are also discussed.

In many large clinical trials there are delays between the time at which events occur and the time at which they are reported. Estimators of the mean frequency function for recurrent events that are currently used are inconsistent in these circumstances. We propose two new estimators to be used when events are reported with delay. One method is a basic inverse probability of censoring weighting approach, while the other explicitly estimates the distribution of the reporting delays. The asymptotic properties of these estimators are discussed and variance estimators are given. We examine the results of simulations comparing the new estimators to each other and to existing estimators that do not properly account for the delays. We also calculate some of these quantities using data from TNT, a clinical trial in which there were delays and events of interest were recurrent.

Most of the methodological literature on evaluating an additional marker for risk prediction involves purely statistical measures of classification performance. A disadvantage of a purely statistical measure is the difficulty in deciding the improvement in the measure that would make inclusion of the additional marker worthwhile. In contrast, a medical decision making approach can weigh the cost or harm of ascertaining an additional marker against the benefit of a higher true positive rate for a given false positive rate that may be associated with risk prediction involving the additional marker. An appealing form of the medical decision making approach involves the risk threshold, which is the risk at which the expected utility of treatment and no treatment is the same. In this framework, a readily interpretable evaluation of the net benefit of an additional marker is the test tradeoff corresponding to the risk threshold. The test tradeoff is the minimum number of tests for a new marker that need to be traded for a true positive to yield an increase in the net benefit of risk prediction with the additional marker. For a sensitivity analysis the test tradeoff is computed over multiple risk thresholds. This article updates the theory and estimation of the test tradeoff. An example is provided.

This paper extends the previous probability model for the distribution of lead time in periodic cancer screening exams, namely, in that the lifetime T is treated as a random variable, instead of a fixed value. Hence the number of screens for a given individual is a random variable as well. We use the actuarial life table from the Social Security Administration to obtain the lifetime distribution, and then use this information to project the lead time distribution for someone with a future screening schedule. Simulation studies using the HIP study group data provide estimates of the lead time under different screening frequencies. The projected lead time has two components: a point mass at zero (corresponding to interval cases detected between screening exams) and a continuous probability density. We present estimates of the projected lead time for participants in a breast cancer screening program. The model is more realistic and can inform optimal screening frequency. This study focuses on breast cancer screening, but is applicable to other kinds of cancer screening also.

We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are in fact log-concave. In addition, we empirically show that our proposed estimator exhibits an efficiency gain for finite sample sizes with respect to the standard empirical estimate in various scenarios and that it is only slightly less efficient, if at all, compared to the fully parametric binormal estimate in case the underlying distributions are normal. The estimator is also quite robust against modest deviations from the log-concavity assumption. We show that bootstrap confidence intervals for the value of the ROC curve at a fixed false positive fraction based on the new estimate are on average shorter compared to the approach by Zhou and Qin (2005), while maintaining coverage probability. Computation of our proposed estimate uses the R package logcondens that implements univariate log-concave density estimation and can be done very efficiently using only one line of code. These obtained results lead us to advocate our estimate for a wide range of scenarios.

We consider estimation of the effect of a multiple time point intervention on an outcome of interest, where the intervention nodes are subject to time-dependent confounding by intermediate covariates.In previous work van der Laan (2010) and Stitelman and van der Laan (2011a) developed and implemented a closed form targeted maximum likelihood estimator (TMLE) relying on the log-likelihood loss function, and demonstrated important gains relative to inverse probability of treatment weighted estimators and estimating equation based estimators. This TMLE relies on an initial estimator of the entire probability distribution of the longitudinal data structure. To enhance the finite sample performance of the TMLE of the target parameter it is of interest to select the smallest possible relevant part of the data generating distribution, which is estimated and updated by TMLE. Inspired by this goal, we develop a new closed form TMLE of an intervention specific mean outcome based on general longitudinal data structures. The target parameter is represented as an iterative sequence of conditional expectations of the outcome of interest. This collection of conditional means represents the relevant part, which is estimated and updated using the general TMLE algorithm. We also develop this new TMLE for other causal parameters, such as parameters defined by working marginal structural models. The theoretical properties of the TMLE are also practically demonstrated with a small scale simulation study.The proposed TMLE is building upon a previously proposed estimator Bang and Robins (2005) by integrating some of its key and innovative ideas into the TMLE framework.

Modern data-rich analyses may call for fitting a large number of nonparametric quantile regressions. For example, growth charts may be constructed for each of a collection of variables, to identify those for which individuals with a disorder tend to fall in the tails of their age-specific distribution; such variables might serve as developmental biomarkers. When such a large set of analyses are carried out by penalized spline smoothing, reliable automatic selection of the smoothing parameter is particularly important. We show that two popular methods for smoothness selection may tend to overfit when estimating extreme quantiles as a smooth function of a predictor such as age; and that improved results can be obtained by multifold cross-validation or by a novel likelihood approach. A simulation study, and an application to a functional magnetic resonance imaging data set, demonstrate the favorable performance of our methods.

Targeted minimum loss based estimation (TMLE) provides a template for the construction of semiparametric locally efficient double robust substitution estimators of the target parameter of the data generating distribution in a semiparametric censored data or causal inference model (van der Laan and Rubin (2006), van der Laan (2008), van der Laan and Rose (2011)). In this article we demonstrate how to construct a TMLE that also satisfies the property that it is at least as efficient as a user supplied asymptotically linear estimator. In particular it is shown that this type of TMLE can incorporate empirical efficiency maximization as in Rubin and van der Laan (2008), Tan (2008, 2010), Rotnitzky et al. (2012), and retain double robustness. For the sake of illustration we focus on estimation of the additive average causal effect of a point treatment on an outcome, adjusting for baseline covariates.

Traditional resampling-based tests for homogeneity in covariance matrices across multiple groups resample residuals, that is, data centered by group means. These residuals do not share the same second moments when the null hypothesis is false, which makes them difficult to use in the setting of multiple testing. An alternative approach is to resample standardized residuals, data centered by group sample means and standardized by group sample covariance matrices. This approach, however, has been observed to inflate type I error when sample size is small or data are generated from heavy-tailed distributions. We propose to improve this approach by using robust estimation for the first and second moments. We discuss two statistics: the Bartlett statistic and a statistic based on eigen-decomposition of sample covariance matrices. Both statistics can be expressed in terms of standardized errors under the null hypothesis. These methods are extended to test homogeneity in correlation matrices. Using simulation studies, we demonstrate that the robust resampling approach provides comparable or superior performance, relative to traditional approaches, for single testing and reasonable performance for multiple testing. The proposed methods are applied to data collected in an HIV vaccine trial to investigate possible determinants, including vaccine status, vaccine-induced immune response level and viral genotype, of unusual correlation pattern between HIV viral load and CD4 count in newly infected patients.

Sequential Randomized Controlled Trials (SRCTs) are rapidly becoming essential tools in the search for optimized treatment regimes in ongoing treatment settings. Analyzing data for multiple time-point treatments with a view toward optimal treatment regimes is of interest in many types of afflictions: HIV infection, Attention Deficit Hyperactivity Disorder in children, leukemia, prostate cancer, renal failure, and many others. Methods for analyzing data from SRCTs exist but they are either inefficient or suffer from the drawbacks of estimating equation methodology. We describe an estimation procedure, targeted maximum likelihood estimation (TMLE), which has been fully developed and implemented in point treatment settings, including time to event outcomes, binary outcomes and continuous outcomes. Here we develop and implement TMLE in the SRCT setting. As in the former settings, the TMLE procedure is targeted toward a pre-specified parameter of the distribution of the observed data, and thereby achieves important bias reduction in estimation of that parameter. As with the so-called Augmented Inverse Probability of Censoring Weight (A-IPCW) estimator, TMLE is double-robust and locally efficient. We report simulation results corresponding to two data-generating distributions from a longitudinal data structure.

Semi-Markov and modulated renewal processes provide a large class of multi-state models which can be used for analysis of longitudinal failure time data. In biomedical applications, models of this kind are often used to describe evolution of a disease and assume that patient may move among a finite number of states representing different phases in the disease progression. Several authors proposed extensions of the proportional hazard model for regression analysis of these processes. In this paper, we consider a general class of censored semi-Markov and modulated renewal processes and propose use of transformation models for their analysis. Special cases include modulated renewal processes with interarrival times specified using transformation models, and semi-Markov processes with with one-step transition probabilities defined using copula-transformation models. We discuss estimation of finite and infinite dimensional parameters and develop an extension of the Gaussian multiplier method for setting confidence bands for transition probabilities and related parameters. A transplant outcome data set from the Center for International Blood and Marrow Transplant Research is used for illustrative purposes.

In the statistical literature, the conditional density model specification is commonly used to study regression effects. One attractive model is the semiparametric density ratio model, under which the conditional density function is the product of an unknown baseline density function and a known parametric function containing the covariate information. This model has a natural connection with generalized linear models and is closely related to biased sampling problems. Despite the attractive features and importance of this model, most existing methods are too restrictive since they are based on multi-sample data or conditional likelihood functions. The conditional likelihood approach can eliminate the unknown baseline density but cannot estimate it. We propose efficient estimation procedures based on the nonparametric likelihood. The nonparametric likelihood approach allows for general forms of covariates and estimates the regression parameters and the baseline density simultaneously. Therefore, the nonparametric likelihood approach is more versatile than the conditional likelihood approach especially when estimation of the conditional mean or other quantities of the outcome is of interest. We show that the nonparametric maximum likelihood estimators are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in practical settings. A real example is used for illustration.

Detection and estimation of measures of familial aggregation is considered the first step to establish whether a certain disease has genetic component. Such measures are usually estimated from observational studies on siblings, parent-offspring, extended pedigrees or twins. When the trait of interest is quantitative (e.g. Blood pressures, body mass index, blood glucose levels, etc.) efficient likelihood estimation of such measures is feasible under the assumption of multivariate normality of the distributions of the traits. In this case the intra-class and inter-class correlations are used to assess the similarities among family members. When the trail is measured on the binary scale, we establish a full likelihood inference on such measures among siblings, parents, and parent-offspring. We illustrate the methodology on nuclear family data where the trait is the presence or absence of hypertension.

This paper presents a targeted minimum loss based estimator (TMLE) that incorporates known conditional bounds on a continuous outcome. Subject matter knowledge regarding the bounds of a continuous outcome within strata defined by a subset of covariates, X, translates into statistical knowledge that constrains the model space of the true joint distribution of the data. In settings where there is low Fisher Information in the data for estimating the desired parameter, as is common when X is high dimensional relative to sample size, incorporating this domain knowledge can improve the fit of the targeted outcome regression, thereby improving bias and variance of the parameter estimate. We show that TMLE, a substitution estimator defined as a mapping from a density to a (possibly d-dimensional) real number, readily incorporates this global knowledge, resulting in improved finite sample performance.

In a prevalent cohort study with follow-up subjects identified as prevalent cases are followed until failure (defined suitably) or censoring. When the dates of the initiating events of these prevalent cases are ascertainable, each observed datum point consists of a backward recurrence time and a possibly censored forward recurrence time. Their sum is well known to be the left truncated lifetime. It is common to term these left truncated lifetimes "length biased" if the initiating event times of all the incident cases (including those not observed through the prevalent sampling scheme) follow a stationary Poisson process. Statistical inference is then said to be carried out under stationarity. Whether or not stationarity holds, a further assumption needed for estimation of the incident survivor function is the independence of the lifetimes and their accompanying truncation times. That is, it must be assumed that survival does not depend on the calendar date of the initiating event. We show how this assumption may be checked under stationarity, even though only the backward recurrence times and their associated (possibly censored) forward recurrence times are observed. We prove that independence of the lifetimes and truncation times is equivalent to equality in distribution of the backward and forward recurrence times, and exploit this equivalence as a means of testing the former hypothesis. A simulation study is conducted to investigate the power and Type 1 error rate of our proposed tests, which include a bootstrap procedure that takes into account the pairwise dependence between the forward and backward recurrence times, as well as the potential censoring of only one of the members of each pair. We illustrate our methods using data from the Canadian Study of Health and Aging. We also point out an equivalence of the problem presented here to a non-standard changepoint problem.

In this paper we discuss estimation of transition probabilities for semi–Markov multi–state models. Non–parametric and semi–parametric estimators of the transition probabilities for a large class of models (forward going models) are proposed. Large sample theory is derived using the functional delta method and the use of resampling is proposed to derive confidence bands for the transition probabilities. The last part of the paper concerns the presentation of the main ideas of the R implementation of the proposed estimators, and data from a renal replacement study are used to illustrate the behavior of the estimators proposed.

Noncompliance with assigned treatment is an important problem of randomized clinical trials. In this situation, the structural mean model (SMM) approach focuses on the average treatment effect among patients actually treated (ATT). In contrast, the principal stratification (PS) approach addresses the effect on a certain subgroup defined by latent compliance behavior. While these approaches target different causal effects, the estimators have the same form as the classical instrumental variable estimator, under the assumption of no effect modification (NEM) and monotonic selection. In this article, we clarify the relation between SMM and PS under the monotonic selection assumption. Specifically, we translate the NEM assumption for the SMM estimator into the words of the PS approach. Then, we propose a new bound for the ATT by making a possibly more plausible assumption than the NEM assumption based on the PS approach. Furthermore, we extend these results to the average treatment effect for the entire population. The proposed bounds are illustrated with applications to a real clinical trial data. Although our assumption cannot be empirically verified, the proposed bounds can be considerably tighter than those previously proposed.

Propensity score (Pscore) matching and inverse probability of treatment weighting (IPTW) can remove bias due to observed confounders, if the Pscore is correctly specified. Genetic Matching (GenMatch) matches on the Pscore and individual covariates using an automated search algorithm to balance covariates. This paper compares common ways of implementing Pscore matching and IPTW, with Genmatch for balancing time-constant baseline covariates}. The methods are considered when estimates of treatment effectiveness are required for patient subgroups, and the treatment allocation process differs by subgroup. We apply these methods in a prospective cohort study that estimates the effectiveness of Drotrecogin alfa activated, for subgroups of patients with severe sepsis. In a simulation study we compare the methods when the Pscore is correctly specified, and then misspecified by ignoring the subgroup-specific treatment allocation. The simulations also consider poor overlap in baseline covariates, and different sample sizes. In the case study, GenMatch reports better covariate balance than IPTW or Pscore matching. In the simulations with correctly specified Pscores, good overlap and reasonable sample sizes, all methods report minimal bias. When the Pscore is misspecified, GenMatch reports the least imbalance and bias. With small sample sizes, IPTW is the most efficient approach, but all methods report relatively high bias of treatment effects. This study shows that overall GenMatch achieves the best covariate balance for each subgroup, and is more robust to Pscore misspecification than common alternative Pscore approaches.

In many randomized controlled trials the outcome of interest is a time to event, and one measures on each subject baseline covariates and time-dependent covariates until the subject either drops-out, the time to event is observed, or the end of study is reached. The goal of such a study is to assess the causal effect of the treatment on the survival curve. We present a targeted maximum likelihood estimator of the causal effect of treatment on survival fully utilizing all the available covariate information, resulting in a double robust locally efficient substitution estimator that will be consistent and asymptotically linear if either the censoring mechanism is consistently estimated, or if the maximum likelihood based estimator is already consistent. In particular, under the independent censoring assumption assumed by current methods, this TMLE is always consistent and asymptotically linear so that it provides valid confidence intervals and tests. Furthermore, we show that when both the censoring mechanism and the initial maximum likelihood based estimator are mis-specified, and thus inconsistent, the TMLE exhibits stability when inverse probability weighted estimators and double robust estimating equation based methods break down The TMLE is used to analyze the Tshepo study, a study designed to evaluate the efficacy, tolerability, and development of drug resistance of six different first-line antiretroviral therapies. Most importantly this paper presents a general algorithm that may be used to create targeted maximum likelihood estimators of a large class of parameters of interest for general longitudinal data structures.

Censored data are characteristics of many bioassays in HIV/AIDS studies where assays may not be sensitive enough to determine gradations in viral load determination among those below a detectable threshold. Not accounting for such left-censoring appropriately can lead to biased parameter estimates in most data analysis. To properly adjust for left-censoring, this paper presents an extension of the Tobit model for fitting nonlinear dynamic mixed-effects models with skew distributions. Such extensions allow one to specify the conditional distributions for viral load response to account for left-censoring, skewness and heaviness in the tails of the distributions of the response variable. A Bayesian modeling approach via Markov Chain Monte Carlo (MCMC) algorithm is used to estimate model parameters. The proposed methods are illustrated using real data from an HIV/AIDS study.

A semi-parametric spatial model for spatial dependence is proposed in Poisson regressions to study the effects of risk factors on incidence outcomes. The spatial model is constructed through an application of reproducing kernels. A Bayesian framework is proposed to infer the unknown parameters. Simulations are performed to compare the reproducing kernel-based method with several commonly used approaches in spatial modeling, including independent Gaussian and CAR models. Compared with these models, the reproducing kernel-based method is easy to implement and more flexible in terms of the ability to model various spatial dependence patterns. To further demonstrate the proposed method, two real data applications are discussed: Scottish lip cancer data and Florida smoke-related cancer data.

The Kaplan-Meier and closely related Lynden-Bell estimators are used to provide nonparametric estimation of the distribution of a left-truncated random variable. These estimators assume that the left-truncation variable is independent of the time-to-event. This paper proposes a semiparametric method for estimating the marginal distribution of the time-to-event that does not require independence. It models the conditional distribution of the time-to-event given the truncation variable using Cox's model for left truncated data, and uses inverse probability weighting. We report the results of simulations and illustrate the method using a survival study.

Estimators of the conditional expectation, i.e., prediction, function involve a global bias-variance trade off. In some cases, an estimator that yields unbiased estimates of the conditional expectation for a particular partitioning of the data may be desirable. Such estimators are calibrated with respect to the partitioning. We identify the conditional expectation given a particular partitioning as a smooth parameter of the distribution of the data, where the partitioning may be defined on the covariate space or on the prediction space of the estimator. We propose a targeted maximum likelihood estimation (TMLE) procedure that updates an initial prediction estimator such that the updated estimator yields an unbiased and efficient estimator of this smooth parameter in the nonparametric statistical model. When the partitioning is defined on the prediction space of the estimator, our TMLE involves enforcing an implicit constraint on the estimator itself. We show that our resulting estimator of the smooth parameter is equal to the empirical estimator, which is also known to be unbiased and efficient in the nonparametric statistical model. We derive the TMLE for single time-point prediction and also time-dependent prediction in a counting process framework.

There is quite an extensive literature on the deleterious impact of exposure misclassification when inferring exposure-disease associations, and on statistical methods to mitigate this impact. Virtually all of this work, however, presumes a common number of states for the true exposure status and the classified exposure status. In the simplest situation, for instance, both the true status and the classified status are binary. The present work diverges from the norm, in considering classification into three states when the actual exposure status is simply binary. Intuitively, the classification states might be labeled as `unlikely exposed,' `maybe exposed,' and `likely exposed.' While this situation has been discussed informally in the epidemiological literature, we provide some theory concerning what can be learned about the exposure-disease relationship, under various assumptions about the classification scheme. We focus on the challenging situation whereby no validation data is available from which to infer classification probabilities, but some prior assertions about these probabilities might be justified.

The attributable fraction of a disease due to an exposure is the fraction of disease cases in a population that can be attributed to that exposure. We consider the attributable fraction for a semi-continuous exposure, that is an exposure for which a clump of people have zero exposure and the rest of the people have a continuously distributed positive exposure. Estimation of the attributable fraction involves estimating the conditional probability of having the disease given the exposure. Three main approaches to estimating the probability function are (1) a classical method based on sample averages; (2) parametric regression methods such as logistic regression models and power models; and (3) nonparametric regression methods including local linear smoothing and isotonic regression. We compare performance of these methods in estimating the attributable fraction for a semi-continuous exposure in a simulation study and in an example.

Age-period-cohort models provide a useful method for modeling incidence and mortality rates. There is great interest in estimating the rates of disease at given future time-points in order that plans can be made for the provision of the required future services. In the setting of using age-period-cohort models incorporating restricted cubic splines, a new technique for projecting incidence is proposed. The new technique projects the period and cohort terms linearly from 10 years within the range of the available data in order to give projections that are based on recent trends. The method is validated via a comparison with existing methods in the setting of Finnish cancer registry data. The reasons for the improvements seen for the newly proposed method are twofold. Firstly, improvements are seen due to the finer splitting of the timescale to give a more continuous estimate of the incidence rate. Secondly, the new method uses more recent trends to dictate the future projections than previously proposed methods.

Longitudinal data arise frequently in medical studies and it is a common practice to analyze such complex data with nonlinear mixed-effects (NLME) models, which enable us to account for between-subject and within-subject variations. To partially explain the variations, time-dependent covariates are usually introduced to these models. Some covariates, however, may be often measured with substantial errors and missing observations. It is often the case that model random error is assumed to be distributed normally, but the normality assumption may not always give robust and reliable results, particularly if the data exhibit skewness. In the literature, there has been considerable interest in accommodating either skewed response or covariate measured with error and missing data in such models, but there has been relatively little study concerning all these features simultaneously. This article is to address simultaneous impact of skewness in response and measurement error and missing data in covariate by jointly modeling the response and covariate processes under a framework of Bayesian semiparametric nonlinear mixed-effects models. In particular, we aim at exploring how mixed-effects joint models based on one-compartment model with one phase time-varying decay rate and two-compartment model with two phase time-varying decay rates contribute to modeling results and inference. The method is illustrated by an AIDS data example to compare potential models with different distributional specifications and various scenarios. The findings from this study suggest that the one-compartment model with a skew-normal distribution may provide more reasonable results if the data exhibit skewness in response and/or have measurement error and missing observations in covariates.

Pearl (2011) invites researchers to contribute to a discussion on the logic and utility of principal stratification in causal inference, raising some thought-provoking questions. In our commentary, we discuss the role of principal stratification in causal inference, describing why we view the principal stratification framework as useful for addressing causal inference problems where causal estimands are defined in terms of intermediate outcomes. We focus on mediation analysis and principal stratification analysis, showing that they generally involve different causal estimands and answer different questions. We argue that even when principal stratification may not answer the causal questions of primary interest, it can be a preliminary analysis of the data to assess the plausibility of identifying assumptions. We also discuss the use of principal stratification to address issues of surrogate outcomes. Our discussion stresses that a principal stratification analysis should account for all the principal strata and evaluate the distributions of potential outcomes in each of the principal strata. To this end, we view a Bayesian analysis particularly suited for drawing inference on principal strata membership and principal strata effects.

We extend Pearl's criticisms of principal stratification analysis as a method for interpreting and adjusting for intermediate variables in a causal analysis. We argue that this can be meaningful only in those rare cases that involve strong functional dependence, and even then may not be appropriate.