Logistic regression in capture-recapture models

The effect of population heterogeneity in capture-recapture, or dual registration, models is discussed. An estimator of the unknown population size based on a logistic regression model is introduced. The model allows different capture probabilities across individuals and across capture times. The probabilities are estimated from the observed data using conditional maximum likelihood. The resulting population estimator is shown to be consistent and asymptotically normal. A variance estimator under population heterogeneity is derived. The finite-sample properties of the estimators are studied via simulation. An application to Finnish occupational disease registration data is presented.     

Capture-recapture when time and behavioral response affect capture probabilities

We consider a capture-recapture model in which capture probabilities vary with time and with behavioral response. Two inference procedures are developed under the assumption that recapture probabilities bear a constant relationship to initial capture probabilities. These two procedures are the maximum likelihood method (both unconditional and conditional types are discussed) and an approach based on optimal estimating functions. The population size estimators derived from the two procedures are shown to be asymptotically equivalent when population size is large enough. The performance and relative merits of various population size estimators for finite cases are discussed. The bootstrap method is suggested for constructing a variance estimator and confidence interval. An example of the deer mouse analyzed in Otis et al. (1978, Wildlife Monographs 62, 93) is given for illustration.     

Bayes and Empirical Bayes Estimators of Abundance and Density from Spatial Capture-Recapture Data

In capture-recapture and mark-resight surveys, movements of individuals both within and between sampling periods can alter the susceptibility of individuals to detection over the region of sampling. In these circumstances spatially explicit capture-recapture (SECR) models, which incorporate the observed locations of individuals, allow population density and abundance to be estimated while accounting for differences in detectability of individuals. In this paper I propose two Bayesian SECR models, one for the analysis of recaptures observed in trapping arrays and another for the analysis of recaptures observed in area searches. In formulating these models I used distinct submodels to specify the distribution of individual home-range centers and the observable recaptures associated with these individuals. This separation of ecological and observational processes allowed me to derive a formal connection between Bayes and empirical Bayes estimators of population abundance that has not been established previously. I showed that this connection applies to every Poisson point-process model of SECR data and provides theoretical support for a previously proposed estimator of abundance based on recaptures in trapping arrays. To illustrate results of both classical and Bayesian methods of analysis, I compared Bayes and empirical Bayes esimates of abundance and density using recaptures from simulated and real populations of animals. Real populations included two iconic datasets: recaptures of tigers detected in camera-trap surveys and recaptures of lizards detected in area-search surveys. In the datasets I analyzed, classical and Bayesian methods provided similar – and often identical – inferences, which is not surprising given the sample sizes and the noninformative priors used in the analyses.     

Small Sample Bias and Variance of Lincoln-Petersen and Bailey Estimators

The Lincoln-Petersen and Bailey estimators of an unknown population size were compared in a computer simulation of capture-recapture sampling with replacement from small populations. The Bailey estimator was negatively biased and had smaller variance than the Petersen estimator. The Petersen estimator tended to be positively biased. The Lincoln-Petersen variance estimator tended to be positively biased while the Bailey variance estimator was negatively biased.     

Small‐Sample Estimation of Species Richness Applied to Forest Communities

Summary Many well‐known methods are available for estimating the number of species in a forest community. However, most existing methods result in considerable negative bias in applications, where field surveys typically represent only a small fraction of sampled communities. This article develops a new method based on sampling with replacement to estimate species richness via the generalized jackknife procedure. The proposed estimator yields small bias and reasonably accurate interval estimation even with small samples. The performance of the proposed estimator is compared with several typical estimators via simulation study using two complete census datasets from Panama and Malaysia.     

The relationship between species detection probability and local extinction probability

In community-level ecological studies, generally not all species present in sampled areas are detected. Many authors have proposed the use of estimation methods that allow detection probabilities that are <1 and that are heterogeneous among species. These methods can also be used to estimate community-dynamic parameters such as species local extinction probability and turnover rates (Nichols et al. Ecol Appl 8:1213–1225; Conserv Biol 12:1390–1398). Here, we present an ad hoc approach to estimating community-level vital rates in the presence of joint heterogeneity of detection probabilities and vital rates. The method consists of partitioning the number of species into two groups using the detection frequencies and then estimating vital rates (e.g., local extinction probabilities) for each group. Estimators from each group are combined in a weighted estimator of vital rates that accounts for the effect of heterogeneity. Using data from the North American Breeding Bird Survey, we computed such estimates and tested the hypothesis that detection probabilities and local extinction probabilities were negatively related. Our analyses support the hypothesis that species detection probability covaries negatively with local probability of extinction and turnover rates. A simulation study was conducted to assess the performance of vital parameter estimators as well as other estimators relevant to questions about heterogeneity, such as coefficient of variation of detection probabilities and proportion of species in each group. Both the weighted estimator suggested in this paper and the original unweighted estimator for local extinction probability performed fairly well and provided no basis for preferring one to the other.     

Robust design and the use of auxiliary variables in estimating population size for an open population

We consider the problem of estimating the population size for an open population where the data are collected over secondary periods within primary periods according to a robust design suggested by Pollock (1982, Journal of Wildlife Management 46, 757-760). A conditional likelihood is used to estimate the parameters associated with a generalized linear model in which the capture probability is assumed to have a logistic form depending on individual covariates. A Horvitz-Thompson-type estimator is used to estimate the population size for each primary period and the survival probabilities between primary periods. The asymptotic properties of the proposed estimators are investigated through simulation and are found to perform well. A data set for such a robust design of a small-mammal capture-recapture study conducted at Dummy Bottom within Browns Park National Wildlife Refuge is analyzed.     

Robust estimation of population size in closed animal populations from capture-recapture experiments

In this paper the problem of finding robust estimators of population size in closed K-sample capture-recapture experiments is considered. Particular attention is paid to models where heterogeneity of capture probabilities is allowed. First, a general estimation procedure is given which does not depend on any assumptions about the form of the distribution of capture probabilities. This is followed by a detailed discussion of the usefulness of the generalized jackknife technique to reduce bias. Numerical comparisons of the bias and variance of various estimators are given. Finally, a general discussion is given with several recommendations on estimators to be used in practice.     

Estimation of the size of an open population from capture-recapture data using weighted martingale methods

A weighted martingale method, akin to a moving average, is proposed to allow the use of modified closed-population methods in the estimation of the size of a smoothly changing open population when there are frequent capture occasions. We concentrate here on modifications to martingale estimating functions for model Mt, but a wide range of closed-population estimators may be modified in this fashion. The method is motivated by and applied to weekly capture-recapture data from the Mai Po bird sanctuary in Hong Kong. Simulations show that the weighted martingale estimator compared well with the Jolly-Seber estimator when the conditions for the latter to be valid are met, and it performed far better when individuals were allowed to leave and reenter the population. Expressions are derived for the asymptotic bias and variance of the estimator in an appendix.     

Random-Effects,Fixed-Effects and the within-between Specification for Clustered Data in Observational Health Studies: A Simulation Study

Background

When unaccounted-for group-level characteristics affect an outcome variable, traditional linear regression is inefficient and can be biased. The random- and fixed-effects estimators (RE and FE, respectively) are two competing methods that address these problems. While each estimator controls for otherwise unaccounted-for effects, the two estimators require different assumptions. Health researchers tend to favor RE estimation, while researchers from some other disciplines tend to favor FE estimation. In addition to RE and FE, an alternative method called within-between (WB) was suggested by Mundlak in 1978, although is utilized infrequently.

Methods

We conduct a simulation study to compare RE, FE, and WB estimation across 16,200 scenarios. The scenarios vary in the number of groups, the size of the groups, within-group variation, goodness-of-fit of the model, and the degree to which the model is correctly specified. Estimator preference is determined by lowest mean squared error of the estimated marginal effect and root mean squared error of fitted values.

Results

Although there are scenarios when each estimator is most appropriate, the cases in which traditional RE estimation is preferred are less common. In finite samples, the WB approach outperforms both traditional estimators. The Hausman test guides the practitioner to the estimator with the smallest absolute error only 61% of the time, and in many sample sizes simply applying the WB approach produces smaller absolute errors than following the suggestion of the test.

Conclusions

Specification and estimation should be carefully considered and ultimately guided by the objective of the analysis and characteristics of the data. The WB approach has been underutilized, particularly for inference on marginal effects in small samples. Blindly applying any estimator can lead to bias, inefficiency, and flawed inference.     

Comparison of One‐Step and Two‐Step Meta‐Analysis Models Using Individual Patient Data

The problem of combining information from separate trials is a key consideration when performing a meta‐analysis or planning a multicentre trial. Although there is a considerable journal literature on meta‐analysis based on individual patient data (IPD), i.e. a one‐step IPD meta‐analysis, versus analysis based on summary data, i.e. a two‐step IPD meta‐analysis, recent articles in the medical literature indicate that there is still confusion and uncertainty as to the validity of an analysis based on aggregate data. In this study, we address one of the central statistical issues by considering the estimation of a linear function of the mean, based on linear models for summary data and for IPD. The summary data from a trial is assumed to comprise the best linear unbiased estimator, or maximum likelihood estimator of the parameter, along with its covariance matrix. The setup, which allows for the presence of random effects and covariates in the model, is quite general and includes many of the commonly employed models, for example, linear models with fixed treatment effects and fixed or random trial effects. For this general model, we derive a condition under which the one‐step and two‐step IPD meta‐analysis estimators coincide, extending earlier work considerably. The implications of this result for the specific models mentioned above are illustrated in detail, both theoretically and in terms of two real data sets, and the roles of balance and heterogeneity are highlighted. Our analysis also shows that when covariates are present, which is typically the case, the two estimators coincide only under extra simplifying assumptions, which are somewhat unrealistic in practice.     

Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996     

Estimating survival and association in a semicompeting risks model

Summary .   In many follow-up studies, patients are subject to concurrent events. In this article, we consider semicompeting risks data as defined by Fine, Jiang, and Chappell (2001, Biometrika 88 , 907–919) where one event is censored by the other but not vice versa. The proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency. This article suggests a general method for estimating the dependence parameter when the dependency is modeled with an Archimedean copula. It uses the copula-graphic estimator of Zheng and Klein (1995, Biometrika 82 , 127–138) for estimating the survival function of the nonterminal event, subject to dependent censoring. Asymptotic properties of these estimators are derived. Simulations show that the new methods work well with finite samples. The copula-graphic estimator is shown to be more accurate than the estimator proposed by Fine et al. (2001) ; its performances are similar to those of the self-consistent estimator of Jiang, Fine, Kosorok, and Chappell (2005, Scandinavian Journal of Statistics 33, 1–20). The analysis of a data set, emphasizing the estimation of characteristics of the observable region, is presented as an illustration.     

Shrinkage Estimators for Covariance Matrices

Estimation of covariance matrices in small samples has been studied by many authors. Standard estimators, like the unstructured maximum likelihood estimator (ML) or restricted maximum likelihood (REML) estimator, can be very unstable with the smallest estimated eigenvalues being too small and the largest too big. A standard approach to more stably estimating the matrix in small samples is to compute the ML or REML estimator under some simple structure that involves estimation of fewer parameters, such as compound symmetry or independence. However, these estimators will not be consistent unless the hypothesized structure is correct. If interest focuses on estimation of regression coefficients with correlated (or longitudinal) data, a sandwich estimator of the covariance matrix may be used to provide standard errors for the estimated coefficients that are robust in the sense that they remain consistent under misspecification of the covariance structure. With large matrices, however, the inefficiency of the sandwich estimator becomes worrisome. We consider here two general shrinkage approaches to estimating the covariance matrix and regression coefficients. The first involves shrinking the eigenvalues of the unstructured ML or REML estimator. The second involves shrinking an unstructured estimator toward a structured estimator. For both cases, the data determine the amount of shrinkage. These estimators are consistent and give consistent and asymptotically efficient estimates for regression coefficients. Simulations show the improved operating characteristics of the shrinkage estimators of the covariance matrix and the regression coefficients in finite samples. The final estimator chosen includes a combination of both shrinkage approaches, i.e., shrinking the eigenvalues and then shrinking toward structure. We illustrate our approach on a sleep EEG study that requires estimation of a 24 x 24 covariance matrix and for which inferences on mean parameters critically depend on the covariance estimator chosen. We recommend making inference using a particular shrinkage estimator that provides a reasonable compromise between structured and unstructured estimators.     

Estimating population size for capture-recapture data when capture probabilities vary by time and individual animal.

There have been no estimators of population size associated with the capture-recapture model when the capture probabilities vary by time and individual animal. This work proposes a nonparametric estimation technique that is appropriate for such a model using the idea of sample coverage, which is defined as the proportion of the total individual capture probabilities of the captured animals. A simulation study was carried out to show the performance of the proposed estimation procedure. Numerical results indicate that it generally works satisfactorily when the coefficient of variation of the individual capture probabilities is relatively large. An example is also given for illustration.     

Estimation of population size based on additive hazards models for continuous-time recapture experiments

We use the semiparametric additive hazards model to formulate the effects of individual covariates on the capture rates in the continuous-time capture-recapture experiment, and then construct a Horvitz-Thompson-type estimator for the unknown population size. The resulting estimator is consistent and asymptotically normal with an easily estimated variance. Simulation studies show that the asymptotic approximations are adequate for practical use when the average capture probabilities exceed .5. Ignoring covariates would underestimate the population size and the coverage probability is poor. A wildlife example is provided.     

The one-sample problem from eradication studies of chronic viral infections

In studies of chronic viral infections, the objective is to estimate probabilities of developing viral eradication and resistance. Complications arise as the laboratory methods used to assess eradication status result in unusual types of censored observations. This paper proposes nonparametric methods for the one-sample analysis of viral eradication/resistance data. We show that the unconstrained nonparametric maximum likelihood estimator of the subdistributions of eradication and resistance are obtainable in closed form. In small samples, these estimators may be inadmissible; thus, we also present an algorithm for obtaining the constrained MLEs based on an isotonic regression of the unconstrained MLEs. Estimators of several functionals of the eradication and resistance subdistributions are also developed and discussed. The methods are illustrated with results from recent hepatitis C clinical trials.     

A Jackknife Variance Estimator for Unistage Stratified Samples with Unequal Probabilities

Existing jackknife variance estimators used with sample surveyscan seriously overestimate the true variance under unistagestratified sampling without replacement with unequal probabilities.A novel jackknife variance estimator is proposed which is asnumerically simple as existing jackknife variance estimators.Under certain regularity conditions, the proposed variance estimatoris consistent under stratified sampling without replacementwith unequal probabilities. The high entropy regularity conditionnecessary for consistency is shown to hold for the Rao–Sampforddesign. An empirical study of three unequal probability samplingdesigns supports our findings.     

On Estimation of Allele Frequencies via Next-Generation DNA Resequencing with Barcoding

Next Generation Sequencing (NGS) has revolutionized biomedical research in recent years. It is now commonly used to identify rare variants through resequencing individual genomes. Due to the cost of NGS, researchers have considered pooling samples as a cost-effective alternative to individual sequencing. In this article, we consider the estimation of allele frequencies of rare variants through the NGS technologies with pooled DNA samples with or without barcodes. We consider three methods for estimating allele frequencies from such data, including raw sequencing counts, inferred genotypes, and expected minor allele counts, and compare their performance. Our simulation results suggest that the estimator based on inferred genotypes overall performs better than or as well as the other two estimators. When the sequencing coverage is low, biases and MSEs can be sensitive to the choice of the prior probabilities of genotypes for the estimators based on inferred genotypes and expected minor allele counts so that more accurate specification of prior probabilities is critical to lower biases and MSEs. Our study shows that the optimal number of barcodes in a pool is relatively robust to the frequencies of rare variants at a specific coverage depth. We provide general guidelines on using DNA pooling with barcoding for the estimation of allele frequencies of rare variants.     

Estimating the proportion of uncatchable animals in a population by double-sampling

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).