The use of auxiliary variables in capture-recapture modelling: An overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

The use of auxiliary variables in capture-recapture modelling: an overview

I review the use of auxiliary variables in capture-recapture models for estimation of demographic parameters (e.g. capture probability, population size, survival probability, and recruitment, emigration and immigration numbers). I focus on what has been done in current research and what still needs to be done. Typically in the literature, covariate modelling has made capture and survival probabilities functions of covariates, but there are good reasons also to make other parameters functions of covariates as well. The types of covariates considered include environmental covariates that may vary by occasion but are constant over animals, and individual animal covariates that are usually assumed constant over time. I also discuss the difficulties of using time-dependent individual animal covariates and some possible solutions. Covariates are usually assumed to be measured without error, and that may not be realistic. For closed populations, one approach to modelling heterogeneity in capture probabilities uses observable individual covariates and is thus related to the primary purpose of this paper. The now standard Huggins-Alho approach conditions on the captured animals and then uses a generalized Horvitz-Thompson estimator to estimate population size. This approach has the advantage of simplicity in that one does not have to specify a distribution for the covariates, and the disadvantage is that it does not use the full likelihood to estimate population size. Alternately one could specify a distribution for the covariates and implement a full likelihood approach to inference to estimate the capture function, the covariate probability distribution, and the population size. The general Jolly-Seber open model enables one to estimate capture probability, population sizes, survival rates, and birth numbers. Much of the focus on modelling covariates in program MARK has been for survival and capture probability in the Cormack-Jolly-Seber model and its generalizations (including tag-return models). These models condition on the number of animals marked and released. A related, but distinct, topic is radio telemetry survival modelling that typically uses a modified Kaplan-Meier method and Cox proportional hazards model for auxiliary variables. Recently there has been an emphasis on integration of recruitment in the likelihood, and research on how to implement covariate modelling for recruitment and perhaps population size is needed. The combined open and closed 'robust' design model can also benefit from covariate modelling and some important options have already been implemented into MARK. Many models are usually fitted to one data set. This has necessitated development of model selection criteria based on the AIC (Akaike Information Criteria) and the alternative of averaging over reasonable models. The special problems of estimating over-dispersion when covariates are included in the model and then adjusting for over-dispersion in model selection could benefit from further research.     

复杂抽样推断方法体系的比较研究

对复杂样本进行推断通常有两种体系,一种是传统的基于随机化理论的统计推断,另一种是基于模型的统计推断。传统的抽样理论以随机化理论为基础,将总体取值视为固定,随机性仅体现在样本的选取上,对总体的推断依赖于抽样设计。该方法在大样本情况下具有稳健估计量,但在小样本、数据缺失等情况下失效。基于模型的抽样推断认为总体是超总体模型中抽取的一个随机样本,对总体的推断取决于模型的建立,但在不可忽略抽样设计下估计量是有偏估计。在对这两类推断方法分析的基础上,提出抽样设计辅助的模型推断,并指出该方法在复杂抽样中具有重要的应用价值。     

Environmental risk assessment model for low dose extrapolation ∗

Use of experimental data from animal studies to estimate human risk due to long-term exposure to very low doses of chemicals in the environment poses a number of biological and statistical problems. One of the statistical problems is to extrapolate the animal dose-response relation from the high dose levels where data are available to low dose, which humans might encounter. Here, a quantal dose-response model is developed based on a multi-hit theory of toxic response. The development of the model utilizes a weighted Lagrange-Poisson distribution for the number of hits. When spontaneous background toxic response is included, the model involves three unknown parameters. The maximum likelihood estimators for these parameters are given as the solution of a nonlinear iterative algorithm. The use of this model for low-dose extrapolation is indicated. The results are applied to nine sets of toxic response data.     

Performance of distance sampling estimators: a simulation study for designs based on footpaths

ABSTRACT

Recently, distance sampling emerged as an advantageous technique to estimate the abundance of many animal populations, including ungulates. Its basic design involves the random selection of several samplers (transects or points) within the population range, and a Horvitz–Thompson-like estimator is then applied to estimate the population abundance while correcting for animal detectability. Ensuring even coverage probability is essential for subsequent inference on the population size, but it may not be achievable because of limited access to parts of the population range. Moreover, in several environmental conditions, a random selection of samplers may induce very high survey costs because it does not minimize the displacement time of the observer(s) between successive samplers. We thus tested whether two-stage designs – based on the random selection of points and then of nearby samplers – could be more cost-effective, for a given population size and when even area coverage cannot be guaranteed. Here, we further extend our analyses to assess the performance of two-stage designs under varying animal densities.     

Statistical inference in dynamic panel data models

Anderson and his collaborators have made seminal contributions to inference with instrumental variables and to dynamic panel data models. We review these contributions and the extensive economic and statistical literature that these contributions spawned. We describe our recent work in these two areas, presenting new approaches to (a) making valid inferences in the presence of weak instruments and (b) instrument and model selection for dynamic panel data models. Both approaches use empirical likelihood and resampling. For inference in the presence of weak instruments, our approach uses model averaging to achieve asymptotic efficiency with strong instruments but maintain valid inferences with weak instruments. For instrument and model selection, our approach aims at choosing valid instruments that are strong enough to be useful.     

Hierarchical spatially varying coefficient and temporal dynamic process models using spTDyn

Bayesian hierarchical spatio-temporal models are becoming increasingly important due to the increasing availability of space-time data in various domains. In this paper we develop a user friendly R package, spTDyn, for spatio-temporal modelling. It can be used to fit models with spatially varying and temporally dynamic coefficients. The former is used for modelling the spatially varying impact of explanatory variables on the response caused by spatial misalignment. This issue can arise when the covariates only vary over time, or when they are measured over a grid and hence do not match the locations of the response point-level data. The latter is to examine the temporally varying impact of explanatory variables in space-time data due, for example, to seasonality or other time-varying effects. The spTDyn package uses Markov chain Monte Carlo sampling written in C, which makes computations highly efficient, and the interface is written in R making these sophisticated modelling techniques easily accessible to statistical analysts. The models and software, and their advantages, are illustrated using temperature and ozone space-time data.     

Analysis of two-phase sampling data with semiparametric additive hazards models

Under the case-cohort design introduced by Prentice (Biometrica 73:1–11, 1986), the covariate histories are ascertained only for the subjects who experience the event of interest (i.e., the cases) during the follow-up period and for a relatively small random sample from the original cohort (i.e., the subcohort). The case-cohort design has been widely used in clinical and epidemiological studies to assess the effects of covariates on failure times. Most statistical methods developed for the case-cohort design use the proportional hazards model, and few methods allow for time-varying regression coefficients. In addition, most methods disregard data from subjects outside of the subcohort, which can result in inefficient inference. Addressing these issues, this paper proposes an estimation procedure for the semiparametric additive hazards model with case-cohort/two-phase sampling data, allowing the covariates of interest to be missing for cases as well as for non-cases. A more flexible form of the additive model is considered that allows the effects of some covariates to be time varying while specifying the effects of others to be constant. An augmented inverse probability weighted estimation procedure is proposed. The proposed method allows utilizing the auxiliary information that correlates with the phase-two covariates to improve efficiency. The asymptotic properties of the proposed estimators are established. An extensive simulation study shows that the augmented inverse probability weighted estimation is more efficient than the widely adopted inverse probability weighted complete-case estimation method. The method is applied to analyze data from a preventive HIV vaccine efficacy trial.     

Ancestral Inference in Population Genetics Models with Selection (with Discussion)

A new algorithm is presented for exact simulation from the conditional distribution of the genealogical history of a sample, given the composition of the sample, for population genetics models with general diploid selection. The method applies to the usual diffusion approximation of evolution at a single locus, in a randomly mating population of constant size, for mutation models in which the distribution of the type of a mutant does not depend on the type of the progenitor allele; this includes any model with only two alleles. The new method is applied to ancestral inference for the two‐allele case, both with genic selection and heterozygote advantage and disadvantage, where one of the alleles is assumed to have resulted from a unique mutation event. The paper describes how the method could be used for inference when data are also available at neutral markers linked to the locus under selection. It also informally describes and constructs the non‐neutral Fleming–Viot measure‐valued diffusion.     

Bayesian diffusion process models with time-varying parameters

The diffusion process is a widely used statistical model for many natural dynamic phenomena but its inference is very complicated because complete data describing the diffusion sample path is not necessarily available. In addition, data is often collected with substantial uncertainty and it is not uncommon to have missing observations. Thus, the observed process will be discrete over a finite time period and the marginal likelihood given by this discrete data is not always available. In this paper, we consider a class of nonstationary diffusion process models with not only the measurement error but also discretely time-varying parameters which are modeled via a state space model. Hierarchical Bayesian inference for such a diffusion process model with time-varying parameters is applied to financial data.     

Proportional hazards models based on biased samples and estimated selection probabilities

In non‐randomized biomedical studies using the proportional hazards model, the data often constitute an unrepresentative sample of the underlying target population, which results in biased regression coefficients. The bias can be avoided by weighting included subjects by the inverse of their respective selection probabilities, as proposed by Horvitz & Thompson (1952) and extended to the proportional hazards setting for use in surveys by Binder (1992) and Lin (2000). In practice, the weights are often estimated and must be treated as such in order for the resulting inference to be accurate. The authors propose a two‐stage weighted proportional hazards model in which, at the first stage, weights are estimated through a logistic regression model fitted to a representative sample from the target population. At the second stage, a weighted Cox model is fitted to the biased sample. The authors propose estimators for the regression parameter and cumulative baseline hazard. They derive the asymptotic properties of the parameter estimators, accounting for the difference in the variance introduced by the randomness of the weights. They evaluate the accuracy of the asymptotic approximations in finite samples through simulation. They illustrate their approach in an analysis of renal transplant patients using data obtained from the Scientific Registry of Transplant Recipients     

Statistical inference for restricted partially linear varying coefficient errors-in-variables models

As a useful extension of partially linear models and varying coefficient models, the partially linear varying coefficient model is useful in statistical modelling. This paper considers statistical inference for the semiparametric model when the covariates in the linear part are measured with additive error and some additional linear restrictions on the parametric component are available. We propose a restricted modified profile least-squares estimator for the parametric component, and prove the asymptotic normality of the proposed estimator. To test hypotheses on the parametric component, we propose a test statistic based on the difference between the corrected residual sums of squares under the null and alterative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotically standard chi-squared distribution. Some simulation studies are conducted to illustrate our approaches.     

Approximating bayesian inference by weighted likelihood

The author proposes to use weighted likelihood to approximate Bayesian inference when no external or prior information is available. He proposes a weighted likelihood estimator that minimizes the empirical Bayes risk under relative entropy loss. He discusses connections among the weighted likelihood, empirical Bayes and James‐Stein estimators. Both simulated and real data sets are used for illustration purposes.     

Nonparametric adaptive likelihood weights

The weighted likelihood can be used to make inference about one population when data from similar populations are available. The author shows heuristically that the weighted likelihood can be seen as a special case of the entropy maximization principle. This leads him to propose the minimum averaged mean squared error (MAMSE) weights. He describes an algorithm for calculating these weights and shows its convergence using the Kuhn‐Tucker conditions. He explores the performance and properties of the weighted likelihood based on MAMSE weights through simulations.     

A robust approach to longitudinal data analysis

The author introduces robust techniques for estimation, inference and variable selection in the analysis of longitudinal data. She first addresses the problem of the robust estimation of the regression and nuisance parameters, for which she derives the asymptotic distribution. She uses weighted estimating equations to build robust quasi‐likelihood functions. These functions are then used to construct a class of test statistics for variable selection. She derives the limiting distribution of these tests and shows its robustness properties in terms of stability of the asymptotic level and power under contamination. An application to a real data set allows her to illustrate the benefits of a robust analysis.     

Random coefficients regression models. a review 1

A review is given of random regression coefficients models. The emphasis is put on the problem of estimating the mean regression coefficients and the covariance matrix of the coefficients. Prediction of the individual random coefficients is not discussed. The main purpose of the review is to point to the practical aspects of the models and the problem of statistical inference in finite samples. Some problems for future research are indicated.     

Efficient Estimation in Heteroscedastic Partially Linear Varying Coefficient Models

This article considers statistical inference for the heteroscedastic partially linear varying coefficient models. We construct an efficient estimator for the parametric component by applying the weighted profile least-squares approach, and show that it is semiparametrically efficient in the sense that the inverse of the asymptotic variance of the estimator reaches the semiparametric efficiency bound. Simulation studies are conducted to illustrate the performance of the proposed method.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

On estimation of linear transformation models with nested case–control sampling

Nested case–control (NCC) sampling is widely used in large epidemiological cohort studies for its cost effectiveness, but its data analysis primarily relies on the Cox proportional hazards model. In this paper, we consider a family of linear transformation models for analyzing NCC data and propose an inverse selection probability weighted estimating equation method for inference. Consistency and asymptotic normality of our estimators for regression coefficients are established. We show that the asymptotic variance has a closed analytic form and can be easily estimated. Numerical studies are conducted to support the theory and an application to the Wilms’ Tumor Study is also given to illustrate the methodology.     

On Semiparametric EV Models with Serially Correlated Errors in Both Regression Models and Mismeasured Covariates

Abstract.  We consider inference for a semiparametric regression model where some covariates are measured with errors, and the errors in both the regression model and the mismeasured covariates are serially correlated. We propose a weighted estimating equations-based estimator (WEEBE) for the regression coefficients. We show that the WEEBE is asymptotically more efficient than the estimators that neglect the serial correlations. This is an interesting new finding since earlier results in the statistical literature have shown that the weighted estimation is not as efficient as the unweighted estimation when the measurement errors and serially correlated errors of the regression models exist simultaneously (Biometrics, 49, 1993, 1262; Technometrics, 42, 2000, 137). The proposed WEEBE does not require undersmoothing the regressor functions in order to make it attain the root- n consistency. Simulation studies show that the proposed estimator has nice finite sample properties. A real data set is used to illustrate the proposed method.