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In 2007, there were 33.2 million people around the world living with HIV/AIDS (UNAIDS/WHO, 2007). In May 2003, the U.S. President announced a global program, known as the President's Emergency Plan for AIDS Relief (PEPFAR), to address this epidemic. We seek to estimate patient mortality in PEPFAR in an effort to monitor and evaluate this program. This effort, however, is hampered by loss to follow-up that occurs at very high rates. As a consequence, standard survival data and analysis on observed nondropout data are generally biased, and provide no objective evidence to correct the potential bias. In this article, we apply double-sampling designs and methodology to PEPFAR data, and we obtain substantially different and more plausible estimates compared with standard methods (1-year mortality estimate of 9.6% compared to 1.7%). The results indicate that a double-sampling design is critical in providing objective evidence of possible nonignorable dropout and, thus, in obtaining accurate data in PEPFAR. Moreover, we show the need for appropriate analysis methods coupled with double-sampling designs.

We thank the editorial board for the opportunity to have discussion on the issue of study design, which is often more important than analysis for obtaining reliable information, especially in problems with missing data. Double sampling designs to address dropout require allocating resources for recovering data for a subgroup of dropouts, but there are often positive trade-offs when doing so. Although such ideas have long been used for surveys with onetime enrollment, we have little evidence that they are being used systematically in studies in public health, where enrollment is often longitudinal. The conclusion is that, in these longitudinal settings, either double sampling is not often used or, as we suspect based on communications, it is employed implicitly, and the data are being analyzed with unknown methods. Our goals therefore were to

Design In various types of biomedical studies, subjects drop out for reasons related to failure. For example, less healthy patients may seek treatment elsewhere or healthier subjects may be discharged from hospital prior to the end of the study. Typically, inference requires strong assumptions about the dropout mechanism (Baker, Wax, and Patterson, 1993) (BWP). The primary motivation of double sampling is to provide information to relax the distributional assumptions. There are two basic designs. Dropout double sampling. At the time of dropout, a subject is randomly assigned to follow-up or no follow-up. This is the design in FRangakis and Rubin (FR), where dropout was loss to follow-up, failure time was time from surgery to failure of prosthesis, and administrative censoring varied, depending on time of entry. Initial double sampling. At the start of the study, such as time of surgery or randomization to treatment assignment, subjects are randomly assigned to either a partial follow-up (PF) or full follow-up (FF) group. If a subject in the FF group drops out of the study, he is followed until failure or administrative censoring. If a subject in the partial followup group drops out, he is not followed after dropout. This is the design in BWP, where dropout was discharge from the hospital, failure time was time of infection following surgery, and administrative censoring was fixed at 30 days. Initial double sampling is a special case of dropout double sampling. Imagine that, as a result of initial double sampling, each subject is randomly assigned a label, either PF or FF, and when dropout occurs, subjects are randomly assigned to follow-up or not based on the PF and FF labels. The only difference with dropout double sampling is the additional random assignment to PF and FF among subjects who did not drop out. This extra information is not used in either the FR or BWP analysis.

We investigate the use of follow-up samples of individuals to estimate survival curves from studies that are subject to right censoring from two sources: (i) early termination of the study, namely, administrative censoring, or (ii) censoring due to lost data prior to administrative censoring, so-called dropout. We assume that, for the full cohort of individuals, administrative censoring times are independent of the subjects' inherent characteristics, including survival time. To address the loss to censoring due to dropout, which we allow to be possibly selective, we consider an intensive second phase of the study where a representative sample of the originally lost subjects is subsequently followed and their data recorded. As with double-sampling designs in survey methodology, the objective is to provide data on a representative subset of the dropouts. Despite assumed full response from the follow-up sample, we show that, in general in our setting, administrative censoring times are not independent of survival times within the two subgroups, nondropouts and sampled dropouts. As a result, the stratified Kaplan-Meier estimator is not appropriate for the cohort survival curve. Moreover, using the concept of potential outcomes, as opposed to observed outcomes, and thereby explicitly formulating the problem as a missing data problem, reveals and addresses these complications. We present an estimation method based on the likelihood of an easily observed subset of the data and study its properties analytically for large samples. We evaluate our method in a realistic situation by simulating data that match published margins on survival and dropout from an actual hip-replacement study. Limitations and extensions of our design and analytic method are discussed.

Although ethnic population counts measured by the United States Census are based on self-identification, the same is not necessarily true of cases reported to cancer registries. The use of different ethnic classification methods for numerators and denominators may therefore lead to biased estimates of cancer incidence rates. The extent of such misclassification may be assessed by conducting an ethnicity survey of cancer patients and estimating the proportion misclassified using double sampling models that account for sample stratification. For two ethnic categories, logistic regression may be used to model self-identified ethnicity as a function of demographic variables and the fallible classification method. Incidence rates then may be adjusted for misclassification using regression results to estimate the number of cancer cases of a given age, sex, and site in each self-identified ethnic group. An example is given using this method to estimate ethnic misclassification of San Francisco Bay area Hispanic cancer patients diagnosed in 1990. Results suggest that the number of cancer cases reported as Hispanic is an underestimate of the number of cases self-identified as Hispanic, resulting in an underestimate of Hispanic cancer rates.

Since 1970, epidemiological surveillance of congenital malformations has been carried out in Norway on the basis of data in the nationwide Medical Birth Registry. A separate local registry of cases occurring in the county of Hordaland was established in 1985 to clarify problems inherent in the reporting. This paper deals with double sampling estimators of the prevalence at birth of various malformations, utilizing joint data from two such registries. Particular examples include Down's syndrome, malformations of the central nervous system (CNS), cleft lip or cleft palate, and hypospadias. The methods applied also allow estimation of the probabilities of ascertainment of the different malformations in each registry. Expressions are derived for the variances of the asymptotic distributions of the double sampling estimators, and the possibility of using these values to detect true changes in disease prevalence when ascertainment is uncertain is discussed. The merits of double sampling are assessed comparing the crude estimator based on the Medical Birth Registry alone with the double sampling estimators. Finally, some fully parameterized alternatives to the double sampling models are introduced to evaluate some of the assumptions involved.

To analyze grouped survival data subject to informative censoring, we propose the following two-part model: a logistic regression model for the hazard for failure, given covariates and a logistic regression model for the hazard for informative censoring, given time of failure and covariates. We fit the model to survival data arising from a double sampling design: In a full follow-up (FF) sample subjects are followed after censoring, and in a partial follow-up (PF) sample subjects are not followed after censoring. We illustrate the methodology using data from a study of wound infection in which patients in the PF sample are censored by release from the hospital, whereas patients in the FF sample are followed after hospital release. We discuss identifiability when there is only a PF sample. We also consider how the allocation between the PF and FF samples affects the precision of estimates.

Data from the Garki Project are analyzed to assess how misdiagnosis affects the estimated prevalence of Plasmodium falciparum. Three double-sampling models that account for the fallibility of the expert are derived and applied. The models incorporate information about the density of parasites in the blood to varying degrees. The error in the estimation of prevalence is quantified; and its dependence on calendar time, age, prevalence, and density is investigated. Prevalence and average density are discovered to be good predictors of the error, with the latter being better. Implications of the double-sampling models for the design of epidemiological surveys similar to the one in Garki are investigated.

Estimates of population size obtained by capture-recapture methods refer solely to the catchable portion of a population. Given a population containing marked animals, two closed-form maximum likelihood estimators of the proportion of uncatchable animals are presented. They are based on twice sampling the proportion of marked animals in the population: the first sample is drawn from catchable animals only, the second from mixed catchable and uncatchable animals. If the individuals in the first sample are not available to the second sample, both samples must be taken from a representative subpopulation of known size. The quantities required may be obtained during a standard capture-recapture session, provided the sampling methods meet the relevant assumptions; the ensuing estimate of population size can then be corrected for uncatchability. The technique is illustrated for eider ducks, using data from Coulson (1984, Ibis 126, 525-543).

Tenenbein [1970] presented a double sampling scheme for estimating from binomial data with misclassifications. At the first stage a sample of n units was classified by both an expensive true measuring device which made no errors and a relatively inexpensive fallible device which was subject to misclassification errors. The estimation of p was discussed, and the optimum sample sizes were derived which minimize the variance of estimation, subject to fixed measurement costs, and the cost, subject to a fixed variance of estimation. The optimum sample sizes depend upon the unknown probabilities of misclassification. In this paper we present practical methods of determining n and N by taking a preliminary sample of m true-fallible data pairs order to estimate the unknown parameters, and thus to estimate n and N. The resulting 3-stage scheme is discussed and recommendations for determining the value of m are made.