Nicholas P. Jewell

Nonparametric Estimation of the Case Fatality Ratio with Competing Risks Data: An Application to Severe Acute Respiratory Syndome (SARS)

Nicholas P. Jewell et al. — Fri, 12 Feb 2010 11:31:45 PST

For diseases with some level of associated mortality, the case fatality ratio measures the proportion of diseased individuals who die from the disease. In principle, it is straightforward to estimate this quantity from individual follow-up data that provides times from onset to death or recovery. In particular, in a competing risks context, the case fatality ratio is defined by the limiting value of the sub-distribution function, associated with death, at infinity. When censoring is present, however, estimation of this quantity is complicated by the possibility of little information in the right tail of of the sub-distribution function, requiring use of estimators evaluated at large or the largest observed death times. With right censoring, the variability of such estimators is large in the tail, suggesting the possibility of using estimators evaluated at smaller death times where bias may be increased but overall mean squared error be smaller. These issues are investigated here for nonparametric estimators of the sub-distribution functions for both death and recovery. The ideas are illustrated on case fatality data for individuals infected with severe acute respiratory syndrome (SARS) in Hong Kong in 2003.

Analyzing Direct Effects in Randomized Trials with Secondary Interventions

Michael Rosenblum et al. — Fri, 12 Feb 2010 11:31:44 PST

The Methods for Improving Reproductive Health in Africa (MIRA) trial is a recently completed randomized trial that investigated the effect of diaphragm and lubricant gel use in reducing HIV infection among susceptible women. 5,045 women were randomly assigned to either the active treatment arm or not. Additionally, all subjects in both arms received intensive condom counselling and provision, the "gold standard" HIV prevention barrier method. There was much lower reported condom use in the intervention arm than in the control arm, making it difficult to answer important public health questions based solely on the intention-to-treat analysis. We adapt an analysis technique from causal inference to estimate the "direct effects" of assignment to the diaphragm arm, adjusting for condom use in an appropriate sense. Issues raised in the MIRA trial apply to other trials of HIV prevention methods, some of which are currently being conducted or designed.

Detailed Version: Analyzing Direct Effects in Randomized Trials with Secondary Interventions: An Application to HIV Prevention Trials

Michael A. Rosenblum et al. — Fri, 12 Feb 2010 11:31:43 PST

This is the detailed technical report that accompanies the paper “Analyzing Direct Effects in Randomized Trials with Secondary Interventions: An Application to HIV Prevention Trials” (an unpublished, technical report version of which is available online at http://www.bepress.com/ucbbiostat/paper223).

The version here gives full details of the models for the time-dependent analysis, and presents further results in the data analysis section. The Methods for Improving Reproductive Health in Africa (MIRA) trial is a recently completed randomized trial that investigated the effect of diaphragm and lubricant gel use in reducing HIV infection among susceptible women. 5,045 women were randomly assigned to either the active treatment arm or not. Additionally, all subjects in both arms received intensive condom counselling and provision, the "gold standard" HIV prevention barrier method.

There was much lower reported condom use in the intervention arm than in the control arm, making it difficult to answer important public health questions based solely on the intention-to-treat analysis.

We adapt an analysis technique from causal inference to estimate the "direct effects" of assignment to the diaphragm arm, adjusting for condom use in an appropriate sense.

Issues raised in the MIRA trial apply to other trials of HIV prevention methods, some of which are currently being conducted or designed.

MAXIMUM LIKELIHOOD ESTIMATION OF ORDERED MULTINOMIAL PARAMETERS

Nicholas P. Jewell et al. — Fri, 12 Feb 2010 11:31:43 PST

The pool-adjacent violator-algorithm (Ayer et al., 1955) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see, Barlow et al., 1972). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times. This paper considers an extended problem of maximum likelihood estimation of a series of ‘ordered’ multinomial parameters pi = (p1i, p2i, . . . , pmi) for 1 < = I < = k, where ordered means that pj1 < = pj2 < = .. . < = pjk for each j with 1 < = j < = m-1. The data consist of k independent observations X1, . . . ,Xk where Xi has a multinomial distribution with probability parameter pi and known index ni > = 1. By making use of variants of the pool adjacent violator algorithm, we obtain a simple algorithm to compute the maximum likelihood estimator of p1, . . . , pk, and demonstrate its convergence. The results are applied to nonparametric maximum likelihood estimation of the sub-distribution functions associated with a survival time random variable with competing risks when only current status data are available. (Jewell et al., 2003)

Regression Analysis of a Disease Onset Distribution Using Diagnosis Data

Jessica G. Young et al. — Fri, 12 Feb 2010 11:31:42 PST

We consider methods for estimating the effect of a covariate on a disease onset distribution when the observed data structure consists of right-censored data on diagnosis times and current status data on onset times amongst individuals who have not yet been diagnosed. Dunson and Baird (2001) approached this problem using maximum likelihood, under the assumption that the ratio of the diagnosis and onset distributions is monotonic non-decreasing. As an alternative, we propose a two-step estimator, an extension of the approach of van der Laan, Jewell and Petersen (1997) in the single sample setting, that is computationally much simpler and requires no assumptions on this ratio. A simulation study is performed comparing estimates obtained from these two approaches, as well as that from a standard current status analysis that ignores diagnosis data. Results indicate that the Dunson and Baird estimator outperforms the two-step estimator when the monotonicity assumption holds, but the reverse is true when the assumption fails. The simple current status estimator loses only a small amount of precision in comparison to the two-step procedure but requires monitoring time information for all individuals. In the data that motivated this work, a study of uterine fibroids and chemical exposure to dioxin, the monotonicity assumption is seen to fail. Here, the two-step and current status estimators both show no significant association between the level of dioxin exposure and the hazard for onset of uterine fibroids; the two-step estimator of the relative hazard associated with increasing levels of exposure has the least estimated variance amongst the three estimators considered.

The Impact of Secondary Condom Interventions on the Interpretation of Results from HIV Prevention Trials

Stephen Shiboski et al. — Fri, 12 Feb 2010 11:31:41 PST

Given the recent failure of a number of randomized trials to demonstrate effectiveness of proposed methods for prevention of sexual transmission of HIV, novel approaches to study design and analysis that address adherence and other post-randomization behaviors are of increasing interest. The inclusion of a mandatory condom use intervention in all randomized groups in such trials can significantly impact interpretation of study results, especially when levels of use observed in the study may differ from real world levels.

We use quantitative examples and simulations to investigate this issue, focusing on effectiveness estimated by the standard intention to treat analysis approach. We also assess the application of recently developed methods for estimating the causal effect of treatment assignment, accounting for observed levels of condom use.

Results show that observed levels of condom use may have substantial impacts on the conclusions drawn from standard analyses of prevention trials, with the most serious effects observed in studies with unblinded control groups. Causal estimation methods accounting for post-randomization behaviors can help clarify these impacts by focusing attention on effectiveness for controlled levels of condom use.

Supplemental causal analyses that account for post-randomization condom use may provide useful information about possible efficacy that complement standard analyses. However, interpretation of results may be limited by the quality of available data on adherence behavior, and limited statistical power.

A Machine-Learning Algorithm for Estimating and Ranking the Impact of Environmental Risk Factors in Exploratory Epidemiological Studies

Jessica G. Young et al. — Fri, 12 Feb 2010 11:31:40 PST

Statistics for Epidemiology

Nicholas P. Jewell — Fri, 25 Aug 2006 13:31:53 PDT

Variances for Maximum Penalized Likelihood Estimates Obtained via the EM Algorithm

Mark R. Segal et al. — Tue, 22 Aug 2006 12:13:22 PDT

We address the problem of providing variances for parameter estimates obtained under a penalized likelihood formulation through use of the EM algorithm. The proposed solution represents a synthesis of two existent techniques. Firstly, we exploit the supplemented EM algorithm developed in Meng and Rubin (1991) that provides variance estimates for maximum likelihood estimates obtained via the EM algorithm. Their procedure relies on evaluating the Jacobian of the mapping induced by the EM algorithm. Secondly, we utilize a result from Green (1990) that provides an expression for the Jacobian of the mapping induced by the EM algorithm applied to a penalized likelihood. The resultant procedure requires no additional code to that needed for the penalized EM algorithm itself.

Uncertainty About the Incubation Period of AIDS and its Impact on Backcalculation

Peter R. Bacchetti et al. — Tue, 22 Aug 2006 12:13:18 PDT

We analyze three sets of doubly-censored cohort data on incubation times, estimating incubation distributions using semi-parametric methods and assessing the comparability of the estimates. Weibull models appear to be inappropriate for at least one of the cohorts, and the estimates for the different cohorts are substantially different. We use these estimates as inputs for backcalculation, using a nonparametric method based on maximum penalized likelihood. The different incubations all produce fits to the reported AIDS counts that are as good as the fit from a nonstationary incubation distribution that models treatment effects, but the estimated infection curves are very different. We also develop a method for estimating nonstationarity as part of the backcalculation procedure and find that such estimates also depend very heavily on the assumed incubation distribution. We conclude that incubation distributions are so uncertain that meaningful error bounds are difficult to place on backcalculated estimates and that backcalculation may be too unreliable to be used without being supplemented by other sources of information in HIV prevalence and incidence.

The NPMLE in the Uniform Doubly Censored Current Status Data Model

Mark J. van der Laan et al. — Tue, 22 Aug 2006 12:13:13 PDT

In biostatistical applications interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well understood singly censored current status model, also known as interval censored data, case I. Jewell, Malani and Vittinghoff (1994) extended this current status model by allowing the initial time to be unobserved, but with its distribution over an observed interval [A,B] known to be uniformly distributed; the data is referred to as doubly censored current status data. These authors used this model to handle applications in AIDS partner studies focusing on the nonparametirc maximum likelihood estimate (NPMLE) of the distribution function, G, of T. The model is a submodel of the current status model, but G is essentially the derivative of the distribution function of interest, F, in the current status model. In this paper we establish that the NPMLE of G is uniformly consistent and that the resulting estimators for square root n estimable parameters are efficient. We propose an iterative weighted Pool-Adjacent-Violator-Algorithm to compute the NPMLE of G. The rate of convergence of the NPMLE of F is also established.

The Impact of Uncertainty in the AIDS Incubation Period on Reconstructions of the HIV Epidemic

Peter Bacchetti et al. — Tue, 22 Aug 2006 12:13:09 PDT

Backcalculation is the primary method used to reconstruct past human immunodeficiency virus (HIV) infection rates, to estimate current prevalence of HIV infection, and to project future incidence of acquired immunodeficiency syndrome (AIDS). The method is very sensitive to uncertainty about the incubation period. We estimate incubation distributions from three sets of cohort data and find that the estimates for the cohorts are substantially different. Backcalculations employing the different estimates produce equally good fits to reported AIDS counts but quite different estimates of cumulative infections. These results suggest that the incubation distribution is likely to differ for different populations and that the differences are large enough to have a big impact on the resulting estimates of HIV infection rates. This seriously limits the usefulness of backcalculation for populations (such as intravenous drug users, heterosexuals, and women) that lack precise information on incubation times.

The Effect of Retrospective Sampling on Binary Regression Models for Clustered Data

John M. Neuhaus et al. — Tue, 22 Aug 2006 12:13:05 PDT

Recently a great deal of attention has been given to binary regression models for clustered or correlated observations. The data of interest are of the form of a binary dependent or response variable, together with independent variables, where sets of observations are grouped together into clusters. A number of models and methods of analysis have been suggested to study such data. Many of these are extensions in some way of the familiar logistic regression model for binary data which are not grouped (i.e., each cluster is of size one). In general, the analyses of these clustered data models proceed by assuming that the observed clusters are a simple random sample of clusters selected from a population of clusters. In this paper, we consider the application of these procedures to the case where the clusters are selected randomly in a manner which depends on the pattern of responses in the cluster. For example, we show that ignoring the retrospective nature of the sample design, by fitting standard logistic regression models for clustered binary data, may result in misleading estimates of the effects of covariates and the precision of estimated regression coefficients.

The Design and Analysis of Partner Studies of HIV Transmission

Nicholas P. Jewell et al. — Tue, 22 Aug 2006 12:13:00 PDT

Common goals in epidemiologic studies of infectious diseases include identification of the infectious agent, description of the modes of transmission and characterization of factors that influence the probability of transmission from infected to uninfected individuals. In the case of AIDS, the agent has been identified as the Human Immunodeficiency Virus (HIV), and transmission is known to occur through a variety of contact mechanisms including unprotected sexual intercourse, transfusion of infected blood products and sharing of needles in intravenous drug use. Relatively little is known about the probability of IV transmission associated with the various modes of contact, or the role that other cofactors play in promoting or suppressing transmission. Here, transmission probability refers to the probability that the virus is transmitted to a susceptible individual following exposure consisting of a series of potentially infectious contacts. The infectivity of HIV for a given route of transmission is defined to be the per contact probability of infection. Knowledge of infectivity and its relationship to other factors is important in understanding the dynamics of the AIDS epidemic and in suggesting appropriate measures to control its spread.

The primary source of empirical data about infectivity comes from sexual partners of infected individuals. Partner studies consist of a series of such partnerships, usually heterosexual and monogamous, each composed of an initially infected "index case" and a partner who may or may not be infected by the time of data collection. However, because the infection times of both partners may be unknown and the history of contacts uncertain, any quantitative characterization of infectivity is extremely difficult. Thus, most statistical analyses of partner study data involve the simplifying assumption that infectivity is a constant common to all partnerships.

The major objectives of this work are to describe and discuss the design and analysis of partner studies, providing a general statistical framework for investigations of infectivity and risk factors for HIV transmission. The development is largely based on three papers: Jewell and Shiboski (1990), Kim and Lagakos (1990), and Shiboski and Jewell (1992).

Temporal Stability and Geographic Variation in Cumulative Case Fatality Rates and Average Doubling Times of SARS Epidemics

Alison P. Galvani et al. — Tue, 22 Aug 2006 12:12:55 PDT

We analyze temporal stability and geographic trends in cumulative case fatality rates and average doubling times of severe acute respiratory syndrome (SARS). In part, we account for correlations between case fatality rates and doubling times through differences in control measures. We discuss factors that may alter future estimates of case fatality rates. We also discuss reasons for heterogeneity in doubling times among countries and the implications for the control of SARS in different countries and parameterization of epidemic models.

Statistical Analysis of the Time Dependence of HIV Infectivity Based on Partner Study Data

Stephen C. Shiboski et al. — Tue, 22 Aug 2006 12:12:51 PDT

Statistical analyses of data form studies of Human Immunodeficiency Virus (HIV) transmission in partners of infected individuals often focus on estimation of the per contact probability of virus transmission, or infectivity. Of particular interest is evaluating whether the infectivity changes during the course of a partnership and in identifying factors that influence the infectiousness of the initially infected partner (called the index case) and the susceptibility of the uninfected partner. Estimation and inference are complicated by limitations in partner study data, which may include unknown time of infection for either or both partners, inaccurate or incomplete information on the number and frequency of contacts and uncertain disease status of the index case. Jewell and Shiboski (1990a) developed statistical methods for partner studies in which data is retrospectively ascertained using techniques that rely on knowledge of the numbers of contacts for each partnership. The infectivity was treated as a function of the number of contacts but was assumed not to depend on the length of time of exposure. Here we consider various generalizations of these ideas. In particular, discussion is focused on analysis of data where the (chronological) time of exposure is observed in addition to or rather than the number of contacts and using models that allow variation in the infectivity according to time since infection of the index case. Where possible, methods are illustrated on data sets on heterosexual transmission.

Statistical Analysis of HIV Infectivity Based on Partner Studies

Nicholas P. Jewell et al. — Tue, 22 Aug 2006 12:12:46 PDT

Partner studies produce data on the infection status of partners of individuals known or assumed to be infected with the human immunodeficiency virus (HIV) after a known or estimated number of contacts. Previous studies have assumed a constant probability of transmission (infectivity) of the virus at each contact. Recently, interest has focused on the possibility of heterogeneity of infectivity across partnerships. This paper develops parametric and nonparametric procedures based on partner data in order to examine the risk of infection after a given number of contacts. Graphical methods and inference techniques are presented that allow the investigator to evaluate the constant infectivity model and consider the impact of heterogeneity of infectivity, error in measurement of the number of contacts, and regression effects of other covariates. The majority of the methods can be computationally implemented easily with use of software to fit generalized linear models. The concepts and techniques are closely related to ideas from discrete survival analysis. A data set on heterosexual transmisison is used to illustrate the methods.

Some Variants of the Backcalculation Method for Estimation of Disease Incidence: An Application to Multiple Sclerosis Data from the Faroe Islands

Nicholas P. Jewell et al. — Tue, 22 Aug 2006 12:12:43 PDT

Backcalculation is a technique that was originally developed for the study of HIV incidence. Here we introduce some variants of the estimation technique that allow for (i) correlation of the unobserved disease incidence counts, and (ii) the use of a smoothing step as part of the maximizing step in the EM algorithm to reduce instability due to small diagnosis counts. Both of these issues can be important in the analysis of small “epidemics”. In addition, identification of correlation between diagnosis counts provides indirect evidence of correlation among unobserved incidence counts, hinting at the possibility of an infectious agent. We illustrate the ideas by reconstructing an incidence intensity function for the onset of multiple sclerosis, using data from the Faroe Islands. Previously, this data had been examined statistically, by Joseph, Wolfson & Wolfson (1990), to address the issue of infectiousness of multiple sclerosis. We argue that the incidence function cannot directly shed light on the enigmatic origin of multiple sclerosis in the Faroe Islands during World War II, and, in particular, cannot discriminate between hypotheses of an infectious or environmental agent.

Some Surprising Results About Covariate Adjustment in Logistic Regression Models

Laurence D. Robinson et al. — Tue, 22 Aug 2006 12:12:41 PDT

Results from classic linear regression regarding the effect of adjusting for covariates upon the precision of an estimator of exposure effect are often assumed to apply more generally to other types of regression models. In this paper we show that such an assumption is not justified in the case of logistic regression, where the effect of adjusting for covariates upon precision is quite different. For example, in classic linear regression the adjustment for a non-confounding predictive covariate results in improved precision, whereas such adjustment in logistic regression results in a loss of precision. However, when testing for a treatment effect in randomized studies, it is always more efficient to adjust for predictive covariates when logistic models are used, and thus in this regard the behavior of logistic regression is the same as that of classic linear regression.

Some Statistical Issues in Studies of the Epidemiology of AIDS

Nicholas P. Jewell — Tue, 22 Aug 2006 12:12:37 PDT

Analyses of studies of the epidemiology and natural history of infection with the Human Immunodeficiency Virus and subsequent onset of AIDS are complicated by many statistical issues. Several such problems are associated with the nature of data collection which is often incomplete. Here we briefly survey some of the statistical methods that have been developed to meet the needs of analysis of AIDS data. In particular, we consider projection of the number of future cases, and estimation and identification of two key epidemiological unknowns, namely the properties of the incubation distribution and those of the infectivity associated with transmission.