Statistical considerations when using a composite endpoint for comparing treatment groups.

When comparing two treatment groups in a time-to-event analysis, it is common to use a composite event consisting of two or more distinct outcomes. The goal of this paper is to develop a statistical methodology to derive efficiency guidelines for deciding whether to expand a study primary endpoint from E1 (for example, non-fatal myocardial infarction and cardiovascular death) to the composite of E1 and E2 (for example, non-fatal myocardial infarction, cardiovascular death or revascularisation). We investigate this problem by considering the asymptotic relative efficiency of a log-rank test for comparing treatment groups with respect to a primary relevant endpoint E1 versus the composite primary endpoint, say E, of E1 and E2, where E2 is some additional endpoint.

A likelihood estimation of HIV incidence incorporating information on past prevalence.

SUMMARY: The prevalence and incidence of an epidemic are basic characteristics that are essential for monitoring its impact, determining public health priorities, assessing the effect of interventions, and for planning purposes. A direct approach for estimating incidence is to undertake a longitudinal cohort study where a representative sample of disease free individuals are followed for a specified period of time and new cases of infection are observed and recorded. This approach is expensive, time consuming and prone to bias due to loss-to-follow-up. An alternative approach is to estimate incidence from cross sectional surveys using biomarkers to identify persons recently infected as in (Brookmeyer and Quinn, 1995; Janssen et al., 1998). This paper builds on the work of Janssen et al. (1998) and extends the theoretical framework proposed by Balasubramanian and Lagakos (2010) by incorporating information on past prevalence and deriving maximum likelihood estimators of incidence. The performance of the proposed method is evaluated through a simulation study, and its use is illustrated using data from the Botswana AIDS Impact (BAIS) III survey of 2008.

Testing and interval estimation for two-sample survival comparisons with small sample sizes and unequal censoring.

While the commonly used log-rank test for survival times between 2 groups enjoys many desirable properties, sometimes the log-rank test and its related linear rank tests perform poorly when sample sizes are small. Similar concerns apply to interval estimates for treatment differences in this setting, though their properties are less well known. Standard permutation tests are one option, but these are not in general valid when the underlying censoring distributions in the comparison groups are unequal. We develop 2 methods for testing and interval estimation, for use with small samples and possibly unequal censoring, based on first imputing survival and censoring times and then applying permutation methods. One provides a heuristic justification for the approach proposed recently by Heinze and others (2003, Exact log-rank tests for unequal follow-up. Biometrics 59, 1151-1157). Simulation studies show that the proposed methods have good Type I error and power properties. For accelerated failure time models, compared to the asymptotic methods of Jin and others (2003, Rank-based inference for the accelerated failure time model. Biometrika 90, 341-353), the proposed methods yield confidence intervals with better coverage probabilities in small-sample settings and similar efficiency when sample sizes are large. The proposed methods are illustrated with data from a cancer study and an AIDS clinical trial.

Statistical monitoring of clinical trials with multivariate response and/or multiple arms: a flexible approach.

Randomized clinical trials with a multivariate response and/or multiple treatment arms are increasingly common, in part because of their efficiency and a greater concern about balancing risks with benefits. In some trials, the specific types and magnitudes of treatment group differences that would warrant early termination cannot easily be specified prior to the onset of the trial and/or could change as the trial progresses. This underscores the need for more flexible monitoring methods than traditional approaches. This paper extends the repeated confidence bands approach for interim monitoring to more general settings where there can be a multivariate response and/or multiple treatment arms and where the metrics for comparing treatment groups can change during the conduct of the trial. We illustrate the approach using the results of a recent AIDS clinical trial and examine its efficiency and robustness via simulation.

Augmented cross-sectional prevalence testing for estimating HIV incidence.

Estimation of an HIV incidence rate based on a cross-sectional sample of individuals evaluated with both a sensitive and less-sensitive diagnostic test offers important advantages to incidence estimation based on a longitudinal cohort study. However, the reliability of the cross-sectional approach has been called into question because of two major concerns. One is the difficulty in obtaining a reliable external approximation for the mean "window period" between detectability of HIV infection with the sensitive and less-sensitive test, which is used in the cross-sectional estimation procedure. The other is how to handle false negative results with the less-sensitive diagnostic test; that is, subjects who may test negative-implying a recent infection-long after they are infected. We propose and investigate an augmented design for cross-sectional incidence estimation studies in which subjects found in the recent infection state are followed for transition to the nonrecent infection state. Inference is based on likelihood methods that account for the length-biased nature of the window periods of subjects found in the recent infection state, and relate the distribution of their forward recurrence times to the population distribution of the window period. The approach performs well in simulation studies and eliminates the need for external approximations of the mean window period and, where applicable, the false negative rate.

Estimating HIV incidence based on combined prevalence testing.

Knowledge of incidence rates of HIV and other infectious diseases is important in evaluating the state of an epidemic as well as for designing interventional studies. Estimation of disease incidence from longitudinal studies can be expensive and time consuming. Alternatively, Janssen et al. (1998, Journal of the American Medical Association 280, 42-48) proposed the estimation of HIV incidence at a single point in time based on the combined use of a standard and "detuned" antibody assay. This article frames the problem from a longitudinal perspective, from which the maximum likelihood estimator of incidence is determined and compared with the Janssen estimator. The formulation also allows estimation for general situations, including different batteries of tests among subjects, inclusion of covariates, and a comparative evaluation of different test batteries to help guide study design. The methods are illustrated with data from an HIV interventional trial and a seroprevalence survey recently conducted in Botswana.

Nonparametric inference and uniqueness for periodically observed progressive disease models.

In many studies examining the progression of HIV and other chronic diseases, subjects are periodically monitored to assess their progression through disease states. This gives rise to a specific type of panel data which have been termed "chain-of-events data"; e.g. data that result from periodic observation of a progressive disease process whose states occur in a prescribed order and where state transitions are not observable. Using a discrete time semi-Markov model, we develop an algorithm for nonparametric estimation of the distribution functions of sojourn times in a J state progressive disease model. Issues of uniqueness for chain-of-events data are not well-understood. Thus, a main goal of this paper is to determine the uniqueness of the nonparametric estimators of the distribution functions of sojourn times within states. We develop sufficient conditions for uniqueness of the nonparametric maximum likelihood estimator, including situations where some but not all of its components are unique. We illustrate the methods with three examples.

GENERALIZED LEAST SQUARES ESTIMATION OF THE MEAN FUNCTION OF A COUNTING PROCESS BASED ON PANEL COUNTS.

This paper considers nonparametric estimation of the mean function of a counting process based on periodic observations, i.e., panel counts. We present estimators derived through minimizing a class of generalized sums of squares subject to a monotonicity constraint. We establish consistency of the estimators and provide procedures to implement them with various weight functions. For specific weight functions, they reduce to the estimator given in Sun and Kalbfleisch (1995), and are closely related to the nonparametric maximum likelihood estimator studied in Wellner and Zhang (2000). With other weight functions, the proposed estimators provide alternatives that can have better efficiency in non-Poisson situations than previous approaches. Simulations are used to examine the finite-sample performance of the proposed estimators.

Inference after variable selection using restricted permutation methods.

When confronted with multiple covariates and a response variable, analysts sometimes apply a variable-selection algorithm to the covariate-response data to identify a subset of covariates potentially associated with the response, and then wish to make inferences about parameters in a model for the marginal association between the selected covariates and the response. If an independent data set were available, the parameters of interest could be estimated by using standard inference methods to fit the postulated marginal model to the independent data set. However, when applied to the same data set used by the variable selector, standard ("naive") methods can lead to distorted inferences. The authors develop testing and interval estimation methods for parameters reflecting the marginal association between the selected covariates and response variable, based on the same data set used for variable selection. They provide theoretical justification for the proposed methods, present results to guide their implementation, and use simulations to assess and compare their performance to a sample-splitting approach. The methods are illustrated with data from a recent AIDS study.

Evaluating the role of human papillomavirus vaccine in cervical cancer prevention.

Persistent genital infection with human papillomavirus (HPV) is a natural candidate as a surrogate marker for cervical cancer because of the strong epidemiologic and molecular evidence that HPV infection is the causative agent for almost all cervical cancers. However, while infection with high-risk types of HPV appears to be necessary for the development of cervical cancer, most infections are controlled by host immune response and do not lead to cancer in the vast majority of infected women. Because diagnostic tests cannot distinguish a persistent infection in the pathogenesis of cervical cancer from a transient infection, it is difficult to describe the disease mechanism as a progressive process based on observations. Therefore, the disease pathogenesis pathway does not fit into the usual surrogate marker framework, raising practical concerns about using HPV infection as a surrogate for a clinical endpoint in vaccine trials. In this paper, we describe the challenges in defining HPV infection as a surrogate endpoint in a HPV vaccine trial that is aimed at reducing cervical cancer rates and examine potential effects of the vaccine. We then outline some issues in the design and analysis of HPV vaccine trials, including the use of operationally defined HPV infection events meant to capture persistent infections. We conclude with a recommendation for a multistate model that uses HPV infection to help explain the mechanisms of vaccine action rather than validate it as an endpoint substitute.

On the use of adjusted cross-sectional estimators of HIV incidence.

OBJECTIVE: To elucidate when and how cross-sectional estimators of HIV incidence rates based on a sensitive and less sensitive diagnostic test should be adjusted. METHODS: Evaluate the statistical properties of unadjusted and adjusted cross-sectional estimators of HIV incidence, including the adjusted estimators considered by McDougal et al, for the 2 settings where (a) all infected subjects eventually become reactive to the less sensitive test, and (b) a subset of infected subjects indefinitely remain nonreactive to the less sensitive test. Derive the maximum likelihood estimator of incidence for the latter setting and use analytical results and simulation studies to compare the performance of the various estimators. RESULTS: When every infected subject would eventually become reactive to the less sensitive test, the McDougal adjusted estimator is uniformly less precise than the unadjusted estimator and more susceptible to bias. When a subset of the infected population would indefinitely remain nonreactive to the less sensitive test, the McDougal adjusted estimator is less precise than the maximum likelihood estimator, which coincides with an estimator developed by McWalter and Welte using a mathematical modeling approach. When the assumed model is incorrect, the unadjusted estimator overestimates incidence, whereas the maximum likelihood estimator can be biased in either direction. CONCLUSIONS: The standard unadjusted cross-sectional estimator of HIV incidence should be used when all infected individuals would eventually become reactive to the less sensitive test. When a subset of individuals would indefinitely remain nonreactive to the less sensitive test, the maximum likelihood estimator for this setting should be used. Characterizing the proportion of individuals who would indefinitely remain nonreactive is crucial for accurate estimation of HIV incidence.

Marginal analysis of panel counts through estimating functions.

We develop nonparametric estimation procedures for the marginal mean function of a counting process based on periodic observations, using two types of self-consistent estimating equations. The first is derived from the likelihood studied in Wellner & Zhang (2000), assuming a Poisson counting process, and gives a nondecreasing estimator, which is the same as the nonparametric maximum likelihood estimator of Wellner & Zhang and thus is consistent without the Poisson assumption. Motivated by the construction of parametric generalized estimating equations, the second type is a set of data-adaptive quasi-score functions, which are likelihood estimating functions under a mixed-Poisson assumption. We evaluate the procedures via simulation, and illustrate them with the data from a bladder cancer study.

Analysis of time to a silent event whose occurrence is monitored with error, with application to mother-to-child HIV transmission.

Inferences about the distribution of time to HIV infection in infants are complicated because infection is a silent event and imperfect diagnostic tests are used to detect its occurrence, leading to false-positive and false-negative results. Nonparametric likelihood approaches are computationally hampered by a large number of parameters and a possibly nonconcave likelihood function. To overcome these difficulties, we develop one-sample and regression methods based on profile likelihood and Markov chain Monte Carlo techniques. The methods also provide a useful diagnostic for assessing the infection status of individual subjects, and are illustrated using results from a recent clinical trial for the prevention of mother-to-child HIV transmission.

Statistical methods for panel data from a semi-Markov process, with application to HPV.

Continuous-time, multistate processes can be used to represent a variety of biological processes in the public health sciences; yet the analysis of such processes is complex when they are observed only at a limited number of time points. Inference methods for such panel data have been developed for time homogeneous Markov models, but there has been little research done for other classes of processes. We develop likelihood-based methods for panel data from a semi-Markov process, where transition intensities depend on the duration of time in the current state. The proposed methods account for possible misclassification of states. To illustrate the methods, we investigate a three- and a four-state models in detail and apply the results to model the natural history of oncogenic genital human papillomavirus infections in women.

Nonparametric estimation of the mean function of a stochastic process with missing observations.

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right-censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.