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Failure time data subject to various types of censoring commonly arise in epidemiological and biomedical studies. Motivated by an AIDS clinical trial, we consider regression analysis of failure time data that include exact and left-, interval-, and/or right-censored observations, which are often referred to as partly interval-censored failure time data. We study the effects of potentially time-dependent covariates on partly interval-censored failure time via a class of semiparametric transformation models that includes the widely used proportional hazards model and the proportional odds model as special cases. We propose an EM algorithm for the nonparametric maximum likelihood estimation and show that it unifies some existing approaches developed for traditional right-censored data or purely interval-censored data. In particular, the proposed method reduces to the partial likelihood approach in the case of right-censored data under the proportional hazards model. We establish that the resulting estimator is consistent and asymptotically normal. In addition, we investigate the proposed method via simulation studies and apply it to the motivating AIDS clinical trial.

A common scientific problem is to determine a surrogate outcome for a long-term outcome so that future randomized studies can restrict themselves to only collecting the surrogate outcome. We consider the setting that we observe n independent and identically distributed observations of a random variable consisting of baseline covariates, a treatment, a vector of candidate surrogate outcomes at an intermediate time point, and the final outcome of interest at a final time point. We assume the treatment is randomized, conditional on the baseline covariates. The goal is to use these data to learn a most-promising surrogate for use in future trials for inference about a mean contrast treatment effect on the final outcome. We define an optimal surrogate for the current study as the function of the data generating distribution collected by the intermediate time point that satisfies the Prentice definition of a valid surrogate endpoint and that optimally predicts the final outcome: this optimal surrogate is an unknown parameter. We show that this optimal surrogate is a conditional mean and present super-learner and targeted super-learner based estimators, whose predicted outcomes are used as the surrogate in applications. We demonstrate a number of desirable properties of this optimal surrogate and its estimators, and study the methodology in simulations and an application to dengue vaccine efficacy trials.

This article develops hypothesis testing procedures for the stratified mark-specific proportional hazards model with missing covariates where the baseline functions may vary with strata. The mark-specific proportional hazards model has been studied to evaluate mark-specific relative risks where the mark is the genetic distance of an infecting HIV sequence to an HIV sequence represented inside the vaccine. This research is motivated by analyzing the RV144 phase 3 HIV vaccine efficacy trial, to understand associations of immune response biomarkers on the mark-specific hazard of HIV infection, where the biomarkers are sampled via a two-phase sampling nested case-control design. We test whether the mark-specific relative risks are unity and how they change with the mark. The developed procedures enable assessment of whether risk of HIV infection with HIV variants close or far from the vaccine sequence are modified by immune responses induced by the HIV vaccine; this question is interesting because vaccine protection occurs through immune responses directed at specific HIV sequences. The test statistics are constructed based on augmented inverse probability weighted complete-case estimators. The asymptotic properties and finite-sample performances of the testing procedures are investigated, demonstrating double-robustness and effectiveness of the predictive auxiliaries to recover efficiency. The finite-sample performance of the proposed tests are examined through a comprehensive simulation study. The methods are applied to the RV144 trial.

Studying the incidence of rare events is both scientifically important and statistically challenging. When few events are observed, standard survival analysis estimators behave erratically, particularly if covariate adjustment is necessary. In these settings, it is possible to improve upon existing estimators by considering estimation in a bounded statistical model. This bounded model incorporates existing scientific knowledge about the incidence of an event in the population. Estimators that are guaranteed to agree with existing scientific knowledge on event incidence may exhibit superior behavior relative to estimators that ignore this knowledge. Focusing on the setting of competing risks, we propose estimators of cumulative incidence that are guaranteed to respect a bounded model and show that when few events are observed, the proposed estimators offer improvements over existing estimators in bias and variance. We illustrate the proposed estimators using data from a recent preventive HIV vaccine efficacy trial. Copyright © 2017 John Wiley & Sons, Ltd.

Summary This article investigates a generalized semiparametric varying-coefficient model for longitudinal data that can flexibly model three types of covariate effects: time-constant effects, time-varying effects, and covariate-varying effects. Different link functions can be selected to provide a rich family of models for longitudinal data. The model assumes that the time-varying effects are unspecified functions of time and the covariate-varying effects are parametric functions of an exposure variable specified up to a finite number of unknown parameters. The estimation procedure is developed using local linear smoothing and profile weighted least squares estimation techniques. Hypothesis testing procedures are developed to test the parametric functions of the covariate-varying effects. The asymptotic distributions of the proposed estimators are established. A working formula for bandwidth selection is discussed and examined through simulations. Our simulation study shows that the proposed methods have satisfactory finite sample performance. The proposed methods are applied to the ACTG 244 clinical trial of HIV infected patients being treated with Zidovudine to examine the effects of antiretroviral treatment switching before and after HIV develops the T215Y/F drug resistance mutation. Our analysis shows benefits of treatment switching to the combination therapies as compared to continuing with ZDV monotherapy before and after developing the 215-mutation.

Two-phase designs are commonly used to subsample subjects from a cohort in order to study covariates that are too expensive to ascertain for everyone in the cohort. This is particularly true for the study of immune response biomarkers in vaccine immunology, where new, elaborate assays are constantly being developed to improve our understanding of the human immune responses to vaccines and how the immune response may protect humans from virus infection. It has long being recognized that if there exist variables that are correlated with expensive variables and can be measured for every subject in the cohort, they can be leveraged to improve the estimation efficiency for the effects of the expensive variables. In this research article, we developed an improved inverse probability weighted estimation approach for semiparametric transformation models with a two-phase study design. Semiparametric transformation models are a class of models that include the Cox PH and proportional odds models. They provide an attractive way to model the effects of immune response biomarkers as human immune responses generally wane over time. Our approach is based on weights calibration, which has its origin in survey statistics and was used by Breslow et al. to improve inverse probability weighted estimation of the Cox regression model. We develop asymptotic theory for our estimator and examine its performance through simulation studies. We illustrate the proposed method with application to two HIV-1 vaccine efficacy trials.

Principal surrogate endpoints are useful as targets for Phase I and II trials. In many recent trials, multiple post-randomization biomarkers are measured. However, few statistical methods exist for comparison of or combination of biomarkers as principal surrogates and none of these methods to our knowledge utilize time-to-event clinical endpoint information. We propose a Weibull model extension of the semi-parametric estimated maximum likelihood method of Huang and Gilbert [1] that allows for the inclusion of multiple biomarkers in the same risk model as multivariate candidate principal surrogates. We propose several methods for comparing candidate principal surrogates and evaluating multivariate principal surrogates. These include the time-dependent and surrogate-dependent true and false positive fraction, the time-dependent and the integrated standardized total gain and the cumulative distribution function of the risk difference. We illustrate the operating characteristics of our proposed methods in simulations and outline how these statistics can be used to evaluate and compare candidate principal surrogates. We use these methods to investigate candidate surrogates in the Diabetes Control and Complications Trial.

In vaccine research, immune biomarkers that can reliably predict a vaccine's effect on the clinical endpoint (i.e., surrogate markers) are important tools for guiding vaccine development. This article addresses issues on optimizing two-phase sampling study design for evaluating surrogate markers in a principal surrogate framework, motivated by the design of a future HIV vaccine trial. To address the problem of missing potential outcomes in a standard trial design, novel trial designs have been proposed that utilize baseline predictors of the immune response biomarker(s) and/or augment the trial by vaccinating uninfected placebo recipients at the end of the trial and measuring their immune biomarkers. However, inefficient use of the augmented information can lead to counter-intuitive results on the precision of estimation. To remedy this problem, we propose a pseudo-score type estimator suitable for the augmented design and characterize its asymptotic properties. This estimator has superior performance compared with existing estimators and allows calculation of analytical variances useful for guiding study design. Based on the new estimator we investigate in detail the problem of optimizing the sampling scheme of a biomarker in a vaccine efficacy trial for efficiently estimating its surrogate effect, as characterized by the vaccine efficacy curve (a causal effect predictiveness curve) and by the predicted overall vaccine efficacy using the biomarker.

An objective of randomized placebo-controlled preventive HIV vaccine efficacy trials is to assess the relationship between the vaccine effect to prevent infection and the genetic distance of the exposing HIV to the HIV strain represented in the vaccine construct. Motivated by this objective, recently a mark-specific proportional hazards model with a continuum of competing risks has been studied, where the genetic distance of the transmitting strain is the continuous `mark' defined and observable only in failures. A high percentage of genetic marks of interest may be missing for a variety of reasons, predominantly due to rapid evolution of HIV sequences after transmission before a blood sample is drawn from which HIV sequences are measured. This research investigates the stratified mark-specific proportional hazards model with missing marks where the baseline functions may vary with strata. We develop two consistent estimation approaches, the first based on the inverse probability weighted complete-case (IPW) technique, and the second based on augmenting the IPW estimator by incorporating auxiliary information predictive of the mark. We investigate the asymptotic properties and finite-sample performance of the two estimators, and show that the augmented IPW estimator, which satisfies a double robustness property, is more efficient.

Recently a new definition of surrogate endpoint, the "principal surrogate," was proposed based on causal associations between treatment effects on the biomarker and on the clinical endpoint. Despite its appealing interpretation, limited research has been conducted to evaluate principal surrogates, and existing methods focus on risk models that consider a single biomarker. How to compare principal surrogate value of biomarkers or general risk models that consider multiple biomarkers remains an open research question. We propose to characterize a marker or risk model's principal surrogate value based on the distribution of risk difference between interventions. In addition, we propose a novel summary measure (the standardized total gain) that can be used to compare markers and to assess the incremental value of a new marker. We develop a semiparametric estimated-likelihood method to estimate the joint surrogate value of multiple biomarkers. This method accommodates two-phase sampling of biomarkers and is more widely applicable than existing nonparametric methods by incorporating continuous baseline covariates to predict the biomarker(s), and is more robust than existing parametric methods by leaving the error distribution of markers unspecified. The methodology is illustrated using a simulated example set and a real data set in the context of HIV vaccine trials.

After one or more Phase 2 trials show that a candidate preventive vaccine induces immune responses that putatively protect against an infectious disease for which there is no licensed vaccine, the next step is to evaluate the efficacy of the candidate. The trial-designer faces the question of what is the optimal size of the initial efficacy trial? Part of the answer will entail deciding between a large Phase 3 licensure trial or an intermediate-sized Phase 2b screening trial, the latter of which may be designed to directly contribute to the evidence-base for licensing the candidate, or, to test a scientific concept for moving the vaccine field forward, acknowledging that the particular candidate will never be licensable. Using the HIV vaccine field as a case study, we describe distinguishing marks of Phase 2b and Phase 3 prevention efficacy trials, and compare the expected utility of these trial types using Pascal's decision-theoretic framework. By integrating values/utilities on (1) correct or incorrect conclusions resulting from the trial; (2) timeliness of obtaining the trial results; (3) precision for estimating the intervention effect; and (4) resources expended; this decision framework provides a more complete approach to selecting the optimal efficacy trial size than a traditional approach that is based primarily on power calculations. Our objective is to help inform the decision-process for planning an initial efficacy trial design.

Given a randomized treatment Z, a clinical outcome Y, and a biomarker S measured some fixed time after Z is administered, we may be interested in addressing the surrogate endpoint problem by evaluating whether S can be used to reliably predict the effect of Z on Y. Several recent proposals for the statistical evaluation of surrogate value have been based on the framework of principal stratification. In this paper, we consider two principal stratification estimands: joint risks and marginal risks. Joint risks measure causal associations of treatment effects on S and Y, providing insight into the surrogate value of the biomarker, but are not statistically identifiable from vaccine trial data. While marginal risks do not measure causal associations of treatment effects, they nevertheless provide guidance for future research, and we describe a data collection scheme and assumptions under which the marginal risks are statistically identifiable. We show how different sets of assumptions affect the identifiability of these estimands; in particular, we depart from previous work by considering the consequences of relaxing the assumption of no individual treatment effects on Y before S is measured. Based on algebraic relationships between joint and marginal risks, we propose a sensitivity analysis approach for assessment of surrogate value, and show that in many cases the surrogate value of a biomarker may be hard to establish, even when the sample size is large.

In replying to Andrew Dunning’s helpful letter, we endeavor to clarify the value of incorporating baseline participant predictors of the potential surrogate endpoint and of the clinical endpoint into the evaluation of a surrogate endpoint. Such baseline predictors are useful for the evaluation of surrogates defined within each of the statistical and principal surrogate frameworks, which are complementary.

This article considers three nonparametric estimators of the joint distribution function for a survival time and a continuous mark variable when the survival time is interval censored and the mark variable may be missing for interval-censored observations. Finite and large sample properties are described for the nonparametric maximum likelihood estimator (NPMLE) as well as estimators based on midpoint imputation (MIDMLE) and coarsening the mark variable (CMLE). The estimators are compared using data from a simulation study and a recent phase III HIV vaccine efficacy trial where the survival time is the time from enrollment to infection and the mark variable is the genetic distance from the infecting HIV sequence to the HIV sequence in the vaccine. Theoretical and empirical evidence are presented indicating the NPMLE and MIDMLE are inconsistent. Conversely, the CMLE is shown to be consistent in general and thus is preferred.

To support the design of the world's first proof-of-concept (POC) efficacy trial of a cell-mediated immunity-based HIV vaccine, we evaluate eight methods for testing the composite null hypothesis of no-vaccine effect on either the incidence of HIV infection or the viral load set point among those infected, relative to placebo. The first two methods use a single test applied to the actual values or ranks of a burden-of-illness (BOI) outcome that combines the infection and viral load endpoints. The other six methods combine separate tests for the two endpoints using unweighted or weighted versions of the two-part z, Simes', and Fisher's methods. Based on extensive simulations that were used to design the landmark POC trial, the BOI methods are shown to have generally low power for rejecting the composite null hypothesis (and hence advancing the vaccine to a subsequent large-scale efficacy trial). The unweighted Simes' and Fisher's combination methods perform best overall. Importantly, this conclusion holds even after the test for the viral load component is adjusted for bias that can be introduced by conditioning on a postrandomization event (HIV infection). The adjustment is derived using a selection bias model based on the principal stratification framework of causal inference.

Considerons l'etude de deux groupes d'individus infectes par une population mixte de virus genetiquement relies et considerons que de multiples sequences virales ont ete echantillonnees chez chaqueindividu. En se fondant sur la distance genetique entre les paires de sequences virales alignees au sein des individus, nous proposons quatre nouveaux tests pour comparer la diversite de sequence intra-individuelle entre deux groupes. Ce probleme est complique par deux niveaux de dependance dans la structure des donnees : 1) au sein d'un individu, tous les couples de distances qui partage une sequence commune sont positivement correlees et 2) pour chaque couple d'individus qui partage un individu, les deux differences en termes de distances intra-individuelles entre les deux couples d'individus sont positivement correlees. Le premier test que nous proposons repose sur la difference dans les distances moyennes observees entre les paires de sequence au sein d'un individu, poolees sur l'ensemble des individus de chaque groupe et standardises en utilisant une estimation de la variance qui prend en compte la structure de correlation a l'aide de la theorie de la statistique U. La seconde approche est analogue au premier test en prenant une statistique de rang non parametrique et le troisieme test compare l'ensemble des distances moyennes intra-individuelles couplees. Ces tests sont facile a utiliser et resolvent le probleme de la correlation n°1. La quatrieme approche repose sur une combinaison lineaire de toutes les statistiques U calculees sur des sous echantillons de sequences independant et identiquement distribues, sur les deux niveaux de dependances des donnees, i.e. 1 et 2. Cette approche est un peu plus compliquee que les quatre tests proposes mais s'avere generalement plus puissante. Bien que les methodes proposees soient empiriques et n'utilisent pas completement les connaissances de genetique des populations, les tests refletent la biologie a travers les modeles evolutifs utilises pour deriver les distances entre les couples de sequences. Les tests sont evalues theoriquement et par une etude de simulation et sont appliques a un jeu de donnees de 200 sequences VIH echantillonnees chez 21 enfants.

Vaccines with limited ability to prevent HIV infection may positively impact the HIV/AIDS pandemic by preventing secondary transmission and disease in vaccine recipients who become infected. To evaluate the impact of vaccination on secondary transmission and disease, efficacy trials assess vaccine effects on HIV viral load and other surrogate endpoints measured after infection. A standard test that compares the distribution of viral load between the infected subgroups of vaccine and placebo recipients does not assess a causal effect of vaccine, because the comparison groups are selected after randomization. To address this problem, we formulate clinically relevant causal estimands using the principal stratification framework developed by Frangakis and Rubin (2002, Biometrics 58, 21-29), and propose a class of logistic selection bias models whose members identify the estimands. Given a selection model in the class, procedures are developed for testing and estimation of the causal effect of vaccination on viral load in the principal stratum of subjects who would be infected regardless of randomization assignment. We show how the procedures can be used for a sensitivity analysis that quantifies how the causal effect of vaccination varies with the presumed magnitude of selection bias.

From a prospective cohort study of 1948 initially human immunodeficiency virus (HIV) uninfected female commercial sex workers followed between 1985 and 1999 in Dakar, Senegal, the authors compared the male to female per infectious sexual exposure transmission probability of HIV types one (HIV-1) and two (HIV-2). New non-parametric competing risks failure time methods were used, which minimized modelling assumptions and controlled for risk factors for HIV infection. The HIV-1 versus HIV-2 infectivity ratio over time was estimated by the ratio of smoothed non-parametric kernel estimates of the HIV-1 and HIV-2 infection hazard functions in sex workers, adjusted by an estimate of the relative HIV-1 versus HIV-2 prevalence in the partner population. HIV-1 was found to be significantly more infectious than HIV-2 throughout the follow-up period (P < 0.001). The HIV-1/HIV-2 infectivity ratio was inferred to be approximately constant over time, with estimated common value 3.55. The finding of greater HIV-1 infectivity persisted in sensitivity analyses and in covariate-adjusted analyses, with adjusted infectivity ratio estimates ranging between 3.40 and 3.86. Understanding the mechanisms by which HIV-1 infects more efficiently than HIV-2 may be useful in the development of HIV-1 vaccines. Additionally, the methodology developed here may be useful for analysing other data sets. Copyright © 2003 John Wiley & Sons, Ltd.

At the present time, many AIDS clinical trials compare drug therapies by a time-to-event primary endpoint that measures the durability of suppression of HIV replication. For such studies, survival differences tend to occur early and/or late in the follow-up period due to drug differences in initial potency and/or durability of efficacy, and detecting these differences is of primary interest. We propose a weighted log-rank statistic that emphasizes early and/or late survival differences. We also consider some versatile tests that also emphasize these differences but are sensitive to a wider range of alternatives. The performances of the new tests are evaluated in numerical studies. For the alternatives of interest, the new tests show greater power and flexibility than commonly used weighted log-rank tests and related versatile tests. When the main interest is in detecting early and/or late survival differences, these tests may be preferable to the other versatile and weighted log-rank tests that have been studied.

Summary. This article develops omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. Confidence bands for the difference and the ratio of two conditional cumulative incidence functions are also constructed. The omnibus test is formulated in terms of a test process given by a weighted difference of estimates of cumulative cause-specific hazard rates under Cox proportional hazards models. A simulation procedure is devised for sampling from the null distribution of the test process, leading to graphical and numerical techniques for detecting significant differences in the risks. The approach is applied to a cohort study of type-specific HIV infection rates.

From data on HIV-1 characteristics measured on viruses isolated from vaccinated and unvaccinated persons infected while enrolled in preventive HIV-1 vaccine trials, interpretable inferences into strain variations of vaccine efficacy can be made with recently developed sieve analysis models. Four assumptions are needed for the parameters in these models to have meaningful interpretations in terms of vaccine-induced reductions in strain-specific per-contact transmission probabilities: (A1) vaccination impacts each strain-specific transmission probability homogeneously in vaccinated persons (leaky vaccine effect); (A2) for each strain biological susceptibility to infection given exposure is homogeneous among vaccinated trial participants and among unvaccinated trial participants; (A3) the distribution of exposure is equal in vaccinated and unvaccinated trial participants; (A4) the relative prevalence of circulating HIV-1 strains during the trial follow-up period is constant. Through theoretical considerations and simulations of an ongoing phase III HIV-1 vaccine efficacy trial in Bangkok, we evaluate the importance and necessity of these assumptions. We show that the models still provide estimates of biologically interpretable parameters when A1 is violated, but with bias the extent to which vaccine protection is heterogeneous. We also show that the models are highly robust to departures from A4, with implication that the time-independent models are adequate for applications. In addition, we suggest extensions of the sieve analysis models which incorporate random effects that account for unmeasured heterogeneity in infection risk. With these mixed models, usefully interpretable strain-specific vaccine efficacy parameters can be estimated without requiring A2. The conclusion is that A3, which is justified by randomization and blinding, is the essential assumption for the sieve models to provide reliable interpretable inferences into strain variations in vaccine efficacy.

In a preventive vaccine efficacy trial of a vaccine for a genotypically and phenotypically diverse pathogen, it is important to assess if and how vaccine protection against infection or disease varies with characteristics of the exposing pathogen. Gilbert, Self and Ashby developed statistical methods for this problem when the outcome data are counts of the number of vaccinated and unvaccinated trial participants infected by each pathogen strain. However, in many vaccine trials time-to-case information is available, and the extent to which this information improves investigation of differential vaccine protection is unclear. We describe how cause-specific proportional hazards models and other popular competing risks failure time techniques can be applied to this problem. This includes new results on the assumptions required for these methods to give valid inferences about strain-specific vaccine efficacy, and a comparison of theoretical and finite-sample properties between these methods and the time-independent methods. Theoretical considerations, a cholera vaccine trial example, and an extensive simulation study of a human immunodeficiency virus type 1 (HIV-1) vaccine trial show that information about failure times does not appreciably improve estimation or testing unless the pathogen has a high attack rate and the relative prevalence of pathogen strains shifts substantially during the trial follow-up period. An important implication is that practically optimal evaluation of strain-specific vaccine efficacy in HIV-1 vaccine trials will not require knowledge of infection times.

Design of Observational Studies (P. R. Rosenbaum) Peter B. Gilbert Longitudinal Data Analysis (G. Fitzmaurice, M. Davidian, G. Verbeke, and G. Molenberghs, Editors) Joseph W. Hogan Multiple Testing Problems in Pharmaceutical Statistics (A. Dmitrienko, A. C. Tamhane, and F. Bretz, Editors) Jason Stover Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families (A. W. Marshall and I. Olkin) Yosef Rinott