Parametric regression models for continuous time removal and recapture studies

We use a class of parametric counting process regression models that are commonly employed in the analysis of failure time data to formulate the subject-specific capture probabilities for removal and recapture studies conducted in continuous time. We estimate the regression parameters by modifying the conventional likelihood score function for left-truncated and right-censored data to accommodate an unknown population size and missing covariates on uncaptured subjects, and we subsequently estimate the population size by a martingale-based estimating function. The resultant estimators for the regression parameters and population size are consistent and asymptotically normal under appropriate regularity conditions. We assess the small sample properties of the proposed estimators through Monte Carlo simulation and we present an application to a bird banding exercise.     

Nonparametric maximum likelihood estimation of population size based on the counting distribution

Summary.  The paper discusses the estimation of an unknown population size n . Suppose that an identification mechanism can identify n _obs cases. The Horvitz–Thompson estimator of n adjusts this number by the inverse of 1− p ₀, where the latter is the probability of not identifying a case. When repeated counts of identifying the same case are available, we can use the counting distribution for estimating p ₀ to solve the problem. Frequently, the Poisson distribution is used and, more recently, mixtures of Poisson distributions. Maximum likelihood estimation is discussed by means of the EM algorithm. For truncated Poisson mixtures, a nested EM algorithm is suggested and illustrated for several application cases. The algorithmic principles are used to show an inequality, stating that the Horvitz–Thompson estimator of n by using the mixed Poisson model is always at least as large as the estimator by using a homogeneous Poisson model. In turn, if the homogeneous Poisson model is misspecified it will, potentially strongly, underestimate the true population size. Examples from various areas illustrate this finding.     

Analysis of band‐recovery data in a multistate capture‐recapture framework

Dead recoveries of marked animals are commonly used to estimate survival probabilities. Band‐recovery models can be parameterized either by r (the probability of recovering a band conditional on death of the animal) or by f (the probability that an animal will be killed, retrieved, and have its band reported). The T parametrization can be implemented in a capture‐recapture framework with two states (alive and newly dead), mortality being the transition probability between the two states. The authors show here that the f parametrization can also be implemented in a multistate framework by imposing simple constraints on some parameters. They illustrate it using data on the mallard and the snow goose. However, they mention that because it does not entirely separate the individual survival and encounter processes, the f parametrization must be used with care on reduced models, or in the presence of estimates at the boundary of the parameter space. As they show, a multistate framework allows the use of powerful software for model fitting or testing the goodness‐of‐fit of models; it also affords the implementation of complex models such as those based on mixture of information or uncertain states     

Effects of omitting a covariate in poisson models when the data are balanced

The authors show that for balanced data, the estimates of effects of interest and of their standard errors are unaffected when a covariate is removed from a multiplicative Poisson model. As they point out, this is not verified in the analogous linear model, nor in the logistic model. In the first case, only the estimated coefficients remain the same, while in the second case, both the estimated effects and their standard errors can change.     

Estimation in capture–recapture models when covariates are subject to measurement errors and missing data

For capture–recapture models when covariates are subject to measurement errors and missing data, a set of estimating equations is constructed to estimate population size and relevant parameters. These estimating equations can be solved by an algorithm similar to the EM algorithm. The proposed method is also applicable to the situation when covariates with no measurement errors have missing data. Simulation studies are used to assess the performance of the proposed estimator. The estimator is also applied to a capture–recapture experiment on the bird species Prinia flaviventris in Hong Kong. The Canadian Journal of Statistics 37: 645–658; 2009 © 2009 Statistical Society of Canada     

Bayesian estimation of the complete sample size from an incomplete poisson sample

For the Poisson a posterior distribution for the complete sample size, N, is derived from an incomplete sample when any specified subset of the classes are missing.Means as well as other posterior characteristics of N are obtained for two examples with various classes removed. For the special case of a truncated ‘missing zero class’ Poisson sample a simulation experiment is performed for the small ‘N=25’ sample situation applying both Bayesian and maximum likelihood methods of estimation.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

Empirical Bayes estimation for the mean of the selected normal population when there are additional data

ABSTRACT

This paper is concerned with the problem of estimation for the mean of the selected population from two normal populations with unknown means and common known variance in a Bayesian framework. The empirical Bayes estimator, when there are available additional observations, is derived and its bias and risk function are computed. The expected bias and risk of the empirical Bayes estimator and the intuitive estimator are compared. It is shown that the empirical Bayes estimator is asymptotically optimal and especially dominates the intuitive estimator in terms of Bayes risk, with respect to any normal prior. Also, the Bayesian correlation between the mean of the selected population (random parameter) and some interested estimators are obtained and compared.     

Sample size re‐estimation incorporating prior information on a nuisance parameter

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter–based sample size re‐estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta‐analytic‐predictive approach. To incorporate external information into the sample size re‐estimation, we propose to update the meta‐analytic‐predictive prior based on the results of the internal pilot study and to re‐estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re‐estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior‐data conflict is present, incorporating external information into the sample size re‐estimation improves the operating characteristics compared to the traditional approach. In the case of a prior‐data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re‐estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re‐estimation, the potential gains should be balanced against the risks.     

Estimating the prevalence of male clients of prostitute women in Vancouver with a simple capture–recapture method

Summary.  Capture–recapture techniques are widely used to estimate the size of difficult-to-count human populations. Applications often focus on the overlap between two or more samples, but another type of data that is encountered in human studies involves only the number of times that particular individuals were encountered in the study period. We present a method for estimating the population size in this situation. This method is simple and technically accessible and allows for entries and exits by individuals and for a difference between probabilities of initial and subsequent contacts. We apply the method to arrest data on male clients of prostitute women in Vancouver.     

A review and re‐interpretation of a group‐sequential approach to sample size re‐estimation in two‐stage trials

In this paper, we review the adaptive design methodology of Li et al. (Biostatistics 3 :277–287) for two‐stage trials with mid‐trial sample size adjustment. We argue that it is closer in principle to a group sequential design, in spite of its obvious adaptive element. Several extensions are proposed that aim to make it even more attractive and transparent alternative to a standard (fixed sample size) trial for funding bodies to consider. These enable a cap to be put on the maximum sample size and for the trial data to be analysed using standard methods at its conclusion. The regulatory view of trials incorporating unblinded sample size re‐estimation is also discussed. © 2014 The Authors. Pharmaceutical Statistics published by John Wiley & Sons, Ltd.     

On optimising the estimation of high quantiles of a probability distribution

One of the major aims of one-dimensional extreme-value theory is to estimate quantiles outside the sample or at the boundary of the sample. The underlying idea of any method to do this is to estimate a quantile well inside the sample but near the boundary and then to shift it somehow to the right place. The choice of this “anchor quantile” plays a major role in the accuracy of the method. We present a bootstrap method to achieve the optimal choice of sample fraction in the estimation of either high quantile or endpoint estimation which extends earlier results by Hall and Weissman (1997) in the case of high quantile estimation. We give detailed results for the estimators used by Dekkers et al. (1989). An alternative way of attacking problems like this one is given in a paper by Drees and Kaufmann (1998).     

On exact computation of minimum sample size for restricted estimation of a binomial parameter

The problem of determining minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level is considered. It is assumed that available auxiliary information allows to restrict the parameter space to some interval whose left boundary is above zero. A range-preserving estimator resulting from the conditional maximization of the likelihood function is considered. A method for exact computation of minimum sample size controlling for the relative error is proposed. Several tables of minimum sample sizes for typical situations are also presented. The range-preserving estimator achieves the same precision and confidence level as the unrestricted maximum likelihood estimator but with a smaller sample.     

Some properties of a nonparametric estimator of the size of an open population

It has been demonstrated in the literature that local polynomial models may be used to estimate the size of an open population using capture–recapture data. However, very little is known about their properties. Here we develop a setting in which the properties of nonparametric estimators of the size of an open population using capture–recapture data can be examined and establish conditions under which expressions for the bias and variance may be determined.     

Unbiased estimation of the parameter of a selected binomial population

The problem is to estimate the parameter of a selected binomial population. The selction rule is to choose the population with the greatest number of successes and, in the case of a tie, to follow one of two schemes: either choose the population with the smallest index or randomize among the tied populations. Since no unbiased estimator exists in the above case, we employ a second stage of sampling and take additional observations on the selected population. We find the uniformly minimum variance unbiased estimator (UMVUE) under the first tie break scheme and we prove that no UMVUE exists under the second. We find an unbiased estimator with desirable properties in the case where no UMVUE exists.     

On the estimation of homogeneous population size from a complex dual-record system

ABSTRACT

A dual-record system (DRS) (equivalently two sample capture–recapture experiments) model, with time and behavioural response variation, has attracted much attention specifically in the domain of official statistics and epidemiology, as the assumption of list independence often fails. The relevant model suffers from parameter identifiability problem, and suitable Bayesian methodologies could be helpful. In this article, we formulate population size estimation in DRS as a missing data problem and two empirical Bayes approaches are proposed along with the discussion of an existing Bayes treatment. Some features and associated posterior convergence for these methods are mentioned. Investigation through an extensive simulation study finds that our proposed approaches compare favourably with the existing Bayes approach for this complex model depending upon the availability of directional nature of underlying behavioural response effect. A real-data example is given to illustrate these methods.     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.     

Adapting by calibration the sample size of a phase III trial on the basis of phase II data

The problem of estimating the sample size for a phase III trial on the basis of existing phase II data is considered, where data from phase II cannot be combined with those of the new phase III trial. Focus is on the test for comparing the means of two independent samples. A launching criterion is adopted in order to evaluate the relevance of phase II results: phase III is run if the effect size estimate is higher than a threshold of clinical importance. The variability in sample size estimation is taken into consideration. Then, the frequentist conservative strategies with a fixed amount of conservativeness and Bayesian strategies are compared. A new conservative strategy is introduced and is based on the calibration of the optimal amount of conservativeness – calibrated optimal strategy (COS). To evaluate the results we compute the Overall Power (OP) of the different strategies, as well as the mean and the MSE of sample size estimators. Bayesian strategies have poor characteristics since they show a very high mean and/or MSE of sample size estimators. COS clearly performs better than the other conservative strategies. Indeed, the OP of COS is, on average, the closest to the desired level; it is also the highest. COS sample size is also the closest to the ideal phase III sample size M_I, showing averages and MSEs lower than those of the other strategies. Costs and experimental times are therefore considerably reduced and standardized. However, if the ideal sample size M_I is to be estimated the phase II sample size n should be around the ideal phase III sample size, i.e. n?2M_I/3. Copyright © 2010 John Wiley & Sons, Ltd.     

A note on the asymptotic optimality of a two population sequential probability ratio test

It is shown that a two population sequential probability ratio test studied by a number of recent authors in the context of sequential medical trials is asymptotically optimal.