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This paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models. We propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable. When the answer is affirmative, we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population. The results of this paper provide guidance for reliable causal inference, based on the recovered covariance matrix obtained from the statistical information with selection bias.

Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population, in this paper, the authors propose an infection age multi-group SEIR epidemic model. The model system also incorporates the feedback variables, where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and for R0 > 1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level. Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable.

The population size has far-reaching effects on the fitness of the population, that, in its turn influences the population extinction or persistence. Understanding the density- and age-dependent factors will facilitate more accurate predictions about the population dynamics and its asymptotic behaviour. In this paper, we develop a rigourous mathematical analysis to study positive and negative effects of increased population density in the classical nonlinear age-structured population model introduced by Gurtin \& MacCamy in the late 1970s. One of our main results expresses the global stability of the system in terms of the newborn function only. We also derive the existence of a threshold population size implying the population extinction, which is well-known in population dynamics as an Allee effect., 20 pages, submitted to J. Math. Sci

SUMMARY An important and often discussed research problem in statistics is how to compare several populations; examples arise in medical science, engineering, finance and other fields. Often population means or medians are compared. However, one population may have a higher mean income, for example, because of a small number of super-rich individuals; the mean therefore may not reflect the wealth of the general population. Instead, an index of the degree of stochastic dominance of one population over another would better reflect their relative wealth. Currently, we can estimate such an index under restrictive conditions, but there is no generic estimator with a known asymptotic distribution. In this paper, we suggest linking the populations via the density ratio model. Under this model, we develop an empirical likelihood estimator and establish its asymptotic normality. In addition, we improve the estimation efficiency by examining the similarities between the populations. Furthermore, we provide a valid bootstrap method for hypothesis testing and the construction of confidence intervals. Simulation experiments show that the proposed estimator substantially improves the estimation efficiency and power of the test, and leads to confidence intervals with satisfactorily precise coverage probabilities. It is also robust with respect to mild model misspecification. Two examples are given to demonstrate the usefulness of both the method and the concept.

There is growing interest in implementation of the mixed model to account for heterogeneity across population observations. However, it has been argued that the assumption of independent and identically distributed (i.i.d) error terms might not be realistic, and for some observations the scale of the error is greater than others. Consequently, that might result in the error terms’ scale to be varied across those observations. As the standard mixed model could not account for the aforementioned attribute of the observations, extended model, allowing for scale heterogeneity, has been proposed to relax the equal error terms across observations. Thus, in this study we extended the mixed model to the model with heterogeneity in scale, or generalized multinomial logit model (GMNL), to see if accounting for the scale heterogeneity, by adding more flexibility to the distribution, would result in an improvement in the model fit. The study used the choice data related to wearing seat belt across front-seat passengers in Wyoming, with all attributes being individual-specific. The results highlighted that although the effect of the scale parameter was significant, the scale effect was trivial, and accounting for the effect at the cost of added parameters would result in a loss of model fit compared with the standard mixed model. Besides considering the standard mixed and the GMNL, the models with correlated random parameters were considered. The results highlighted that despite having significant correlation across the majority of the random parameters, the goodness of fits favors more parsimonious models with no correlation. The results of this study are specific to the dataset used in this study, and due to the possible fact that the heterogeneity in observations related to the front-seat passengers seat belt use might not be extreme, and do not require extra layer to account for the scale heterogeneity, or accounting for the scale heterogeneity at the cost of added parameters might not be required. Extensive discussion has been made in the content of this paper about the model parameters’ estimations and the mathematical formulation of the methods.

There is continued uncertainty in how long it takes a person infected by the COVID-19 virus to become infectious. In this paper, we quantify how this uncertainty affects estimates of the basic replication number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0, and thus estimates of the fraction of the population that would become infected in the absence of effective interventions. The analysis is general, and applies to all SEIR-based models, not only those associated with COVID-19. We find that when modeling a rapidly spreading epidemic, seemingly minor differences in how latency is treated can lead to vastly different estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0. We also derive a simple formula relating the replication number to the fraction of the population that is eventually infected. This formula is robust and applies to all compartmental models whose parameters do not depend on time.

This paper deals with the modeling of wealth distribution considering a society with non-constant population and non-conservative wealth trades. The modeling approach is based on the kinetic theory of active particles, where individuals are distinguished by a scalar variable (the activity) which expresses their social state. A qualitative analysis of the model focusing on asymptotic behaviors and measurement of inequality through the Gini coefficient is presented. Finally, some specific case-studies are proposed in order to carry out numerical experiments to validate our model, characterize societies and investigate emerging behaviors. Fil: Buffa, Bruno Adolfo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina Fil: Knopoff, Damián Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina Fil: Torres, German Ariel. Instituto de Modelado e Innovación Tecnológica (Consejo Nacional de Investigaciones Científicas y Técnicas - Universidad Nacional del Nordeste); Argentina. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura. Departamento de Matemática; Argentina

This paper addresses the problem of extinction in continuous models of population dynamics associated with small numbers of individuals. We begin with an extended discussion of extinction in the particular case of a stochastic logistic model, and how it relates to the corresponding continuous model. Two examples of ‘small number dynamics’ are then considered. The first is what Mollison calls the ‘atto-fox’ problem (in a model of fox rabies), referring to the problematic theoretical occurrence of a predicted rabid fox density of $$10^{-18}$$ 10 - 18 (atto-) per square kilometre. The second is how the production of large numbers of eggs by an individual can reliably lead to the eventual survival of a handful of adults, as it would seem that extinction then becomes a likely possibility. We describe the occurrence of the atto-fox problem in other contexts, such as the microbial ‘yocto-cell’ problem, and we suggest that the modelling resolution is to allow for the existence of a reservoir for the extinctively challenged individuals. This is functionally similar to the concept of a ‘refuge’ in predator–prey systems and represents a state for the individuals in which they are immune from destruction. For what I call the ‘frogspawn’ problem, where only a few individuals survive to adulthood from a large number of eggs, we provide a simple explanation based on a Holling type 3 response and elaborate it by means of a suitable nonlinear age-structured model.

In this paper, the adaptive lasso method is used to screen variables, and different neural network models of seven countries are established by choosing variables. Gross domestic product (GDP) is a function of land area in the country, cultivated land, population, enrollment rate, total capital formation, exports of goods and services, and the general government&rsquo, s final consumption of collateral and broad money. Based on the empirical analysis of the above factors from 1973 to 2016, the results show that the BP neural network model has better performance based on multiple summary statistics, without increasing the number of parameters and better predicting short-term GDP. In addition, the change and the error of the model are small and have a certain reference value.

SUMMARY This paper considers from a Bayesian viewpoint inferences about the size of a closed animal population from data obtained by a multiple-recapture sampling scheme. The method developed enables prior information about the population size and the catch probabilities to be utilized to produce considerable improvements in certain cases on ordinary maximum likelihood methods. Several ways of expressing such prior information are explored and a practical example of the uses of these ways is given. The main result of the paper is an approximation to the posterior distribution of sample size that exhibits the contributions made by the likelihood and the prior ideas. The multiple-recapture sampling scheme involves taking samples from a population of animals, at each stage counting the number of marked animals in the sample, marking the previously unmarked animals and returning the sample to the population. The literature is reviewed by Cormack (1968) and the papers most relevant to our problem are those of Chapman (1952) and Darroch (1958). In these papers, maximum likelihood estimates for the population parameters and variances for these estimates are given. Our approach differs from previous ones by introducing prior information about the population size, and about the propensities of the animals to be captured, called 'catch probabilities' by some previous authors. The incorporation of these propensities is a feature not shared by many previous approaches. Darroch (1958) found their introduction into his model does not change the maximum likelihood estimates of the population size. Our approach will show that strong prior knowledge about the catch probabilities can greatly affect inference about the population size, and it is in this respect that the greatest difference from previous approaches lies. Although it is recognized that models of an open population incorporating death and immigration such as those of Jolly (1965) and Seber (1965) are more practically realistic, we feel that our method, which deals only with a closed population, is worth investigating as a step towards providing a Bayesian treatment of the open population problem.

A nonparametric measure of intraclass correlation based on the probability of certain types of concordances among the observations and which can be estimated from ranked data is proposed. One estimator whose variance can be estimated in an unbiased way is shown to be asymptotically normally distributed. Properties of the measure and its estimator are studied for a normal model; in particular the method is shown to provide a relatively powerful test of the null hypothesis of zero intraclass correlation in a normal population. The intraclass correlation coefficient is often used as a measure of the degree to which individuals from the same family resemble one another in some variable such as height or weight. From a random sample of families and a random selection of individuals from each family, both the between and within family components of variance can be estimated and an estimate of the intraclass correlation obtained. When the observations are ranks this approach is not feasible since direct estimates of the components of variance are not available. An example occurs in a study of aggressiveness in male red grouse (Moss, Watson & Parr, 1974). One measure of aggressiveness of an individual bird is a dominance rank, the position of that bird in the dominance hierarchy of a given group of birds. Comparison of the degree of familial resemblance in different populations is complicated by the fact that the data are ranks and that dominance hierarchies can be variable in size with different family sizes. The development of standard test statistics, such as the Kruskal-Wallis statistic (Kruskal, 1952), as measures of intraclass correlation is unsatisfactory because of the complicated dependence on the structure of the sample; see ? 6. In this paper a measure of intraclass correlation is proposed which overcomes this problem. It is independent of the structure of the sample and it can be estimated in an unbiased way from ranked data. The interclass rank correlation coefficients, Kendall's Xr and Spearman's p (Kendall, 1955), can be interpreted in terms of probabilities of concordances among the observations from a bivariate distribution. In response to Hill (1974), Kerridge (1975) has used this probabilistic interpretation of r in an analysis of Football League results. The measure presented in the present paper is based on concordances among observations from a population in which individuals occur in families. Its properties for a general class of populations are discussed and some particular results for a one-way classification model with normally distributed effects are presented. The power of the associated test is assessed for a normal model for a range of sample structures.

SUMMARY The probability integral for the distribution of range of a sample from a population whose distribution can be represented by the first few terms of an Edgeworth series has been obtained in this paper. The numerical values of the corrective functions arising due to nonnormality are tabulated. The new theoretical results are compared with the earlier results, where available. The distribution of range of samples from nonnormal populations was first studied empirically by Pearson & Adyanthdya (1928) and later, among others, by Pearson (1950), Cox (1954) and David (1954). These studies have been limited mainly to the mean range and to the probability integral in some simple nonnormal cases, and from these likely effects of nonnormality on the distribution of range have been conjectured. Singh (1967) obtained some theoretical results regarding the expectation and the variance of range of samples from a population whose distribution can be represented by the first few terms of an Edgeworth series. These results provided some additional information regarding the effects of parental excess and skewness on the mean and variance of the range. In the present paper the probability integral for the distribution of range of samples from the same type of population has been obtained and evaluated for small samples to examine the effects of parental excess and skewness.

In this paper we consider the following problem. Given a multivariate normal population with known variance matrix, what test is appropriate to determine whether the means are slipped to the right? In the case when the population is univariate normal, this problem can be solved by the ordinary one-sided test using either the normal or the t-distribution functions. It is the purpose of this paper to develop what may be termed a multivariate analogue of the one-sided test of significance.

SUMMARY This paper gives a two-sample procedure for selecting the population with the largest mean from k normal populations with unknown variances. The method is based on a twosample procedure proposed by Stein (1945). Tables necessary for the application of the procedure are given for selected values of k. Comparisons of the minimum values of the expected sample sizes using the proposed procedure are made with the corresponding single-sample sizes for known variances (Bechhofer, 1954). Comparisons are also made of the expected total sample sizes for the single-sample procedure, the two-sample procedure given in this paper and the two-sample procedure proposed by Bechhofer, Dunnett & Sobel (1954) which assumes that the populations have known variance ratios. It is shown that the expected total sample sizes are not much increased by ignorance of the variance ratios.

This paper is concerned with a stochastic logistic model driven by martingales with jumps. In the model, generalized noise and jump noise are taken into account. This model is new and more feasible. The explicit global positive solution of the system is presented, and then sufficient conditions for extinction and persistence are established. The critical value of extinction, nonpersistence in the mean, and weak persistence in the mean are obtained. The path-wise and moment properties are also investigated. Finally, some simulation figures are introduced to illustrate the main results.

SUMMARY A probability sampling plan is suggested in this paper for drawing the sample so that the linear least-squares predictor of the total of a finite population is made asymptotically design-unbiased. Conditions are given for the sampling plan to reduce to the more familiar sample designs discussed in the literature. The linear least-squares predictor of the total of a finite population (Royall, 1970, 1976) derived under a superpopulation model with explanatory variables omitted is still best under the correct model provided that it is unbiased under the correct model (Tam, 1986). The present paper specifies probability sampling schemes which render the linear least-squares predictor asymptoti- cally design-unbiased. Such probability schemes enable some general form of balancing in the sample for the explanatory variables omitted from the assumed superpopulation model, and provide protection of the predictor against this type of model failure. Conditions are also presented in this paper for the specified sample design to become the equal probability selection sample design or the optimal sample design considered by Godambe (1955, 1982).

The hydrological frequency analysis (HFA) is the foundation for the hydraulic engineering design and water resources management. Hydrological extreme observations or samples are the basis for HFA; the representativeness of a sample series to the population distribution is extremely important for the estimation reliability of the hydrological design value or quantile. However, for most of hydrological extreme data obtained in practical application, the size of the samples is usually small, for example, in China about 40~50 years. Generally, samples with small size cannot completely display the statistical properties of the population distribution, thus leading to uncertainties in the estimation of hydrological design values. In this paper, a new method based on bootstrap is put forward to analyze the impact of sampling uncertainty on the design value. By bootstrap resampling technique, a large number of bootstrap samples are constructed from the original flood extreme observations; the corresponding design value or quantile is estimated for each bootstrap sample, so that the sampling distribution of design value is constructed; based on the sampling distribution, the uncertainty of quantile estimation can be quantified. Compared with the conventional approach, this method provides not only the point estimation of a design value but also quantitative evaluation on uncertainties of the estimation.

Climate change impacts population distributions, forcing some species to migrate poleward if they are to survive and keep up with the suitable habitat that is shifting with the temperature isoclines. Previous studies have analysed whether populations have the capacity to keep up with shifting temperature isoclines, and have mathematically determined the combination of growth and dispersal that is needed to achieve this. However, the rate of isocline movement can be highly variable, with much uncertainty associated with yearly shifts. The same is true for population growth rates. Growth rates can be variable and uncertain, even within suitable habitats for growth. In this paper, we reanalyse the question of population persistence in the context of the uncertainty and variability in isocline shifts and rates of growth. Specifically, we employ a stochastic integrodifference equation model on a patch of suitable habitat that shifts poleward at a random rate. We derive a metric describing the asymptotic growth rate of the linearised operator of the stochastic model. This metric yields a threshold criterion for population persistence. We demonstrate that the variability in the yearly shift and in the growth rate has a significant negative effect on the persistence in the sense that it decreases the threshold criterion for population persistence. Mathematically, we show how the persistence metric can be connected to the principal eigenvalue problem for a related integral operator, at least for the case where isocline shifting speed is deterministic. Analysis of dynamics for the case where the dispersal kernel is Gaussian leads to the existence of a critical shifting speed, above which the population will go extinct, and below which the population will persist. This leads to clear bounds on rate of environmental change if the population is to persist. Finally, we illustrate our different results for butterfly population using numerical simulations and demonstrate how increased variances in isocline shifts and growth rates translate into decreased likelihoods of persistence.

The Akaike information criterion for model selection presupposes that the parameter space is not subject to order restrictions or inequality constraints. Anraku (1999) proposed a modified version of this criterion, called the order-restricted information criterion, for model selection in the one-way analysis of variance model when the population means are monotonic. We propose a generalization of this to the case when the population means may be restricted by a mixture of linear equality and inequality constraints. If the model has no inequality constraints, then the generalized order-restricted information criterion coincides with the Akaike information criterion. Thus, the former extends the applicability of the latter to model selection in multi-way analysis of variance models when some models may have inequality constraints while others may not. Simulation shows that the information criterion proposed in this paper performs well in selecting the correct model.

Random imputation methods are often used in practice because they tend to preserve the distribution of the variable being imputed, which is an important property when the goal is to estimate population quantiles. However, this type of imputation method introduces additional variability, the imputation variance, due to the random selection of residuals. In this paper, we propose a class of random balanced imputation methods under which the imputation variance is eliminated while the distribution of the variable being imputed is preserved. The rationale behind balanced imputation is to select residuals at random so that appropriate constraints are satisfied. We describe an algorithm for selecting the random residuals that can be viewed as an adaptation of the cube algorithm proposed in the context of balanced sampling (Deville & Tille, 2004). Results of a simulation study support our findings. Copyright 2011, Oxford University Press.

This paper presents a study of the intertemporal propagation of distributional properties of phenotypes in general polygenic multisex inheritance models with sex- and time-dependent heritabilities. It further analyzes the implications of these models under heavy-tailedness of traits' initial distributions. Our results suggest the optimality of a flexible asexual/binary mating system. Switching between asexual and binary inheritance mechanisms allows the population effectively to achieve a fast suppression of negative traits and a fast dispersion of positive traits, regardless of the distributional properties of the phenotypes in the initial period.

The importance of measurement error in studies of asymmetry has been acknowledged for a long time. It is now common practice to acquire independent repeated measurements of trait values and to estimate the degree of measurement error relative to the amount of asymmetry. Methods also allow obtaining unbiased estimates of asymmetry, both at the population and individual level. One aspect that has been ignored is potential between-individual variation in measurement error. In this paper, I develop a new method to investigate this variation in measurement error and to generate unbiased estimates of individual asymmetries. Simulations show that variation in measurement error can indeed result in biased estimates of individual asymmetry and that the proposed method adequately provides unbiased estimates. The method is applied to two empirical datasets and shows that, at least in some traits, substantial variations in measurement occur. The limitations of the model are discussed.

SUMMARY This paper focuses on the Akaike information criterion, AIC, for linear mixed-effects models in the analysis of clustered data. We make the distinction between questions regarding the population and questions regarding the particular clusters in the data. We show that the AIC in current use is not appropriate for the focus on clusters, and we propose instead the conditional Akaike information and its corresponding criterion, the conditional AIC, CAIC. The penalty term in CAIC is related to the effective degrees of freedom p for a linear mixed model proposed by Hodges & Sargent (2001); p reflects an intermediate level of complexity between a fixed-effects model with no cluster effect and a corresponding model with fixed cluster effects. The CAIC is defined for both maximum likelihood and residual maximum likelihood estimation. A pharmacokinetics data appli cation is used to illuminate the distinction between the two inference settings, and to illustrate the use of the conditional AIC in model selection.

Populations of specific components are often heterogeneous and consist of a small number of different sub-populations. For example there are often two groups: defective components with shorter lifetimes and standard components with longer lifetimes. Another heterogeneous population results when components produced by two different manufacturing lines are combined. In either case a mixture results. The resulting population can be described using the statistical concept of a mixture. It is a well-known result that a mixture of distributions with decreasing failure rates has a decreasing failure rate. However, little is known about the monotonicity of a mixture when the various subpopulations have failure rates which are not necessarily decreasing. In this paper we study and attempt to determine the shape as well as the overall behavior of the failure rate of a mixture from two subpopulations each of which has increasing linear failure rate.

SUMMARY This paper considers estimation of the unknown size N of a population based on multiple capture-recapture samples. We extend the Bayesian multiple recapture model to accommodate possible heterogeneity and dependence among the samples and possible heterogeneity within the samples. In the dependent model, we show that posterior inference for N is independent of almost all the nuisance parameters. We develop a flexible Bayesian model for heterogeneity within samples and demonstrate how Gibbs sampling can be used to calculate the Bayesian estimator for N and other quantities of interest. The performance of the proposed estimators is evaluated by simulation under both correct and incorrect model specifications, and we illustrate our methods in two examples about software review and estimation of a cottontail rabbit population.

When it comes to testing hypotheses regarding two population means, the most commonly used test is the two-sample t-test. There are two versions of this test, one is used when the variances of the two populations are equal (the pooled test) and the other one is used when the variances of the two populations are unequal (the unpooled test). The pooled test seems to have fallen into some disfavor because of its 'claimed' sensitivity to departures from the assumptions of equal population variances. Through a simulation study, this paper demonstrates that although both the pooled and the unpooled test underperform at times in their allocated settings, the overall performance of the pooled t-test is significantly superior to that of the unpooled t-test.

Recently, new concepts have been proposed for assessing bioequivalence of two drug formulations, namely the so-called population and individual bioequivalence. Using moment-based and probability-based measures for evaluating the proposed bioequivalence concepts, criteria have been formulated to decide whether two formulations should be regarded as bioequivalent or not. This decision has of course to be based on an adequate statistical method where the Food and Drug Administration (FDA) guidance (1997) recommends the use of a bootstrap percentile interval. In this paper, we discuss theoretical properties such as consistency and accuracy of the recommended bootstrap intervals. We focus our investigations on the concept of individual bioequivalence and here especially on the scaled versions of the moment-based as well as the probability-based measures as recommended by the FDA. As estimates for the former, we consider those obtained from an according analysis of variance and restricted maximum likelihood estimators under mixed effect models, where an unbiased estimator of the latter can be derived from the corresponding relative frequencies.

This paper addresses the problem of modelling heterogeneous individual characteristics in a population. A flexible unified approach for stochastic parametrization dynamics of the distribution in population data is proposed. To approximate data with multiple observations per individual, models based on Markov processes are constructed. The method can be applied to scalar or multivariate characteristics, and its application to growth and allometry data is considered. Different stochastic versions of known growth and allometry functions are developed, which enable wide applicability. Simple informative growth indices are calculated as the moments of distribution. The three-parameter Gompertz growth model for size-at-age data was reparametrized to a size-increment data model with two parameters.

SUMMARY We consider fitting categorical regression models to data obtained by either stratified or nonstratified case-control, or response selective, sampling from a finite population with known population totals in each response category. With certain models, such as the logistic with appropriate constant terms, a method variously known as conditional maximum likelihood (Breslow & Cain, 1988) or pseudo-conditional likelihood (Wild, 1991), which involves the prospective fitting of a pseudo-model, results in maximum likelihood estimates of case-control data. We extend these results by showing the maximum likelihood estimates for any model can be found by iterating this process with a simple updating of offset parameters. Attention is also paid to estimation of the asymptotic covariance matrix. One benefit of the results of this paper is the ability to obtain maximum likelihood estimates of the parameters of logistic models for stratified case-control studies, compare Breslow & Cain (1988), Scott & Wild (1991), using an ordinary logistic regression program, even when the stratum constants are modelled.

Attributable fractions are commonly used to measure the impact of risk factors on disease incidence in the population. These static measures can be extended to functions of time when the time to disease occurrence or event time is of interest. The present paper deals with nonparametric and semiparametric estimation of attributable fraction functions for cohort studies with potentially censored event time data. The semiparametric models include the familiar proportional hazards model and a broad class of transformation models. The proposed estimators are shown to be consistent, asymptotically normal and asymptotically efficient. Extensive simulation studies demonstrate that the proposed methods perform well in practical situations. A cardiovascular health study is provided. Connections to causal inference are discussed. Copyright 2010, Oxford University Press.

1. Copulas Copulas are functions that join univariate distribution functions to form multivariate distribution functions. They were first introduced in 1959 by A. Sklar [10] to answer some questions posed by M. Frechet concerning the relationship between a multidimensional probability distribution function and its lower dimensional marginals. Over the past 30 years or so, copulas have played an important role in several areas of probability and statistics, including multivariate distribution theory, nonparametric statistics, and Markov processes; but only recently have they come to the attention of the general statistical and mathematical communities. In this paper, we will survey some of the important properties of copulas, and demonstrate ways in which copulas can be employed in probability and mathematical statistics courses to enhance and illuminate the presentation of a number of topics. Specifically, we will show how copulas can be used in statistics i) to characterize some dependence concepts for two random variables; ii) to obtain a geometric interpretation of the population version of a nonparametric correlation coefficient, and to illustrate how that coefficient measures dependence; and iii) to facilitate the generation of counterexamples.

This paper first considers a simple deterministic epidemic in a growing population. It then examines a simple deterministic epidemic in several related growing communities, where infection is transmitted between individuals both within and across the communities. An explicit parametric solution is derived for the number of susceptibles in these communities as a function of time.

alessio.farcomeni@uniroma1.it SUMMARY We introduce a general class of capture‐recapture models in which capture probabilities depend on the previous capture history. We discuss constrained versions of the saturated model based on equality constraints. Despite the seeming complexity, inference can be performed through a simple estimating equation. The approach is illustrated on a data set about the population of Great Copper butterflies in Willamette Valley of O regon. It has long been recognized that in capture‐recapture exper iments the previous capture history may influence capture probabilities, and failure to acknowl edge this fact may lead to biased estimates of the unknown population size. One of the first model s used, denoted by Mb, included behavioural response to capture by using different probabi lities before and after first capture (Otis et al., 1978; Pollock, 1991). Other examples are given by the Markov chain models of Yang & Chao (2005), and the persistence models detailed in Ramsey & Usner (2003) and Ramsey & Severns (2010). In this paper we propose a general class of models which are based on conditional capture probabilities. We restrict the satura ted model through equality constraints, arguing that the resulting class of models is rich enough to s ummarize a plethora of real data situations. The restricted class can be also defined through partitions of the capture histories, where capture probabilities conditional on different hist ories belonging to the same partition are equal. Surprisingly enough, the approach based on maximizing the conditional likelihood leads to a straightforward inference based on a simple estimating equation. An interesting feature of the proposed model class is that the previous capture histor y can be allowed to influence the capture probability in very flexible ways. Different models can be fitted and compared through information criteria or likelihood ratio tests.

SUMMARY We develop several methods for estimating the treatment effect difference defined as the overall log-odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response. A simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions. Multicentre randomized clinical trials are frequently used to test the efficacy of new medical regimens. These trials are attractive due to the ease of getting the desired sample size in a short period of time and the broad coverage of the patient population and medical practitioners. The analysis of data from such trials requires combining information from the centres in a way that properly accounts for the variation due to the centres and the differential efficacy of treatments between the centres. In this paper we investigate several approaches to pooling the data across centres and for handling the effects associated with the centres. Chakravarthi & Grizzle (1975) and Beitler & Landis (1985) argue for treating these effects as random, and Fleiss (1986) argues against because the centres are seldom chosen at random from a well-defined population of clinics and therefore the effects cannot be associated with any population. We prefer the random effect model approach for the following reasons. The goal of a trial is to extend the results of the study to the general population, duly accounting for variation in the patient population and the medical skills of practitioners. Hence, if we believe that the variability between centres as estimated by the data is typical of the variability in the population of centres, then the inference based on the random effects model is more appropriate than the one based on the fixed effects model. Furthermore, the variation among the centres may be more than that predicted from samples from a binomial distribution. Treating the centre effects as random introduces this intra-class correlation into our model assumptions. Let the outcome variable from each centre be a pair of binomial random variables XAj

SUMMARY In finite population inference problems, auxiliary population information is often available. We show in this paper that the empirical likelihood method can be naturally applied to such problems to make effective use of the auxiliary information. We prove that the resulting estimates have smaller asymptotic variances than the usual estimates which do not use auxiliary information. A Bahadur-type representation for empirical likelihood sample quantiles is given. Simulation results show that the empirical likelihood estimates perform well among a number of competitors and are model robust.

SUMMARY Poststratification is often used to increase the precision of survey estimators. When applied to regression estimators, it assumes that the regression relationships in the various strata are different. This gives rise to the use of the separate regression estimator. In this paper, we consider situations where the strata affiliation of the population units is not specified in the frame so that the separate regression estimator is not available, the strata means of the regressor variable and the strata sizes being unknown. We propose a new regression type estimator which accounts for the different regression relationships in the various strata, but no longer depends on the unknown strata means and sizes. We show that this estimator is more precise than estimators that ignore the differences in the regression relationships. The theoretical comparisons are illustrated by simulation results.

A brief review, with references, of the literature on capture-recapture theory is given in Jolly (1963). More recently, Cormack (1964) gives a solution, including asymptotic variances, for a specific situation involving the marking and release of a non-random sample of fulmar petrels. His model is stochastic and will be referred to in ? 4. Seber (1962, 1965) has also produced some interesting solutions, and these together with Darroch (1958, 1959) are most directly relevant to our present problem. Darroch (1958, 1959) shows that in a fully stochastic model with either immigration (often called dilution) or death (or emigration), the population parameters can be easily estimated by maximum likelihood. For the more general case when death and immigration are operating simultaneously he derives estimation equations by equating certain observations to their expectations, but does not give variances or covariances for the estimates. In a later paper, (Darroch, 1961), he considers estimation for a closed population consisting of different strata. Seber (1962) establishes a stochastic model for what he calls the multi-sample single recapture census in which an individual cannot be recaptured more than once. This situation arises, for example, when the recaptures are made in the course of hunting or fishing. He allows for both death and immigration in the population, provides explicit maximumlikelihood estimates of the parameters with variances, and suggests tests for certain of the assumptions. In a second paper, (Seber, 1965), he considers a multiple-recapture model differing only slightly from that of Darroch (1959), with both death and immigration. Again, he provides explicit maximum-likelihood estimates of the parameters with variances. A test is also given for equi-catchability in a closed population of individuals with different capture histories. The first purpose of the present paper is to derive a general probability distribution designed to fit the majority of capture-recapture problems involving a 'single' population. The word 'single' here denotes a population covering an area within whose boundaries the animals (or, in general, individuals or members) are free to move and to mix with others of their kind, but which is regarded as a single area in respect of which parameters are to be estimated. The type of situation which is thus excluded by this definition is one where the population is split into a number of defined areas, and separate population estimates are required for each area as well as for numbers of animals moving from one area to another. The single population, however, need not be homogeneous but may consist of different classes of animals behaving in different ways. The other assumptions underlying the model are stated with the notation in ? 2, and the generalized probability distribution is derived in ?3.

Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probability p, independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.

In a recent paper on inference about the number of defectives in a finite population, Wright (1992) provides a Bayesian view on the sample size required to give high assurance that the probability of unobserved defectives is low. We reconsider Wright's approach and derive alternative formulae appropriate when there is considerable, but not infinite, prior information

Sample survey inference is historically concerned with finite-population parameters, that is, functions (like means and totals) of the observations for the individuals in the population. In scientific applications, however, interest usually focuses on the “superpopulation” parameters associated with a stochastic mechanismhypothesized to generate the observations in the population rather than the finite-population parameters. Two relevant findings discussed in this paper are that (1) with stratified sampling, it is not sufficient to drop finite-population correction factors from standard design-based variance formulas to obtain appropriate variance formulas for superpopulation inference, and (2) with cluster sampling, standard design-based variance formulas can dramatically underestimate superpopulation variability, even with a small sampling fraction of the final units. A literature review of inference for superpopulation parameters is given, with emphasis on why these findings have not been previously appreciated. Examples are provided for estimating superpopulation means, linear regression coefficients and logistic regression coefficients using U.S. data from the 1987 National Health Interview Survey, the third National Health and Nutrition Examination Survey and the 1986 National Hospital Discharge Survey.

Many population models which are far from stationarity can nevertheless be written in autoregressive format, perhaps with random coefficient. It is the thesis of this paper that procedures developed for stationary time series models are a useful guide to inferential results for population processes and may indeed be directly applicable. The illustrations concentrate on estimation of the matrix of mean vital rates in an age-structured population.

SUMMARY Possibilities for moderating effects of a subgrouping variable on strength or direction of an association have been much discussed by social scientists but have not been given satisfactory statistical formulations. The results concern directed measures of associations in linear models containing just three variables. Some key words: Analysis of covariance; Analysis of variance; cG-distribution; Conditional independence; Graphical chain model; Parallel regressions; Yule-Simpson paradox. 1. INTRODUCTION Linear models are commonly used as a framework to estimate and test how a continuous response variable depends on potential influencing variables. This paper is concerned with the situation in which two influences are random variables as well, which may be discrete or continuous. The continuous random variables are to stand for quantitative properties of observational units, while the discrete variables either capture qualitative properties or they represent subgroups of the population to which the observational units belong. To think of all variables as random variables is usually appropriate for data obtained from observational studies, where one cannot control which levels or values of the influencing variables are to be observed. The linear model permits an analysis conditional on fixed levels and values of the influences. But frequently, the association structure among the influences is, in social science applications, of interest itself and, as it will turn out, it is important for a correct interpretation of how the response depends on the influences. Awareness of such problems in interpretation is widespread among social scientists as is documented by extensive discussions of moderating effects; see, for example, Saunders (1956), Zedeck (1971) and Baron & Kenny (1986). There, a variable is called a moderator if its presence changes the strength or direction of an association. However, methods recommended to social scientists for identifying moderating effects given in the above literature or by Cohen & Cohen (1983, pp. 310-4) have been shown to be seriously deficient (Wermuth, 1988). Thus, there is need for clarification. We treat the case of a potential discrete moderator variable and call its categories or levels the subgroups in the linear model. A moderating effect is closely tied to the notion of consistent results on an association (Wermuth, 1987). In the present context results are said to be weakly consistent if the associations within subgroups coincide in direction and strongly consistent if the associ- ations within subgroups coincide in direction and strength. This notion depends typically on the chosen measure of association. In linear models we look at directed measures of

SUMMARY A comparison is made between play-the-winner sampling and vector-at-a-time sampling for selecting the better of two binomial populations when the selection requirement is defined in terms of the ratio of the single trial probabilities of success rather than the difference. Thus the probability of correct selection is to be at least P* when the true ratio of probabilities of success 0 is at least 0*, where P* and 0* are prescribed. Termination rules based on the difference in the number of successes and inverse sampling are considered. It is shown that play-the-winner sampling is uniformly preferable to vector-at-a-time sampling for both termination rules, and that for 0* close to one the success lead termination rule leads to a smaller expected number of trials on the poorer treatment unless 0 is even closer to one. The problem of selecting the better of two binomial populations, i.e. the one with the higher probability of success p on a single trial, has usually been stated using the framework of ranking and selection procedures as follows. If P* and A* are preassigned constants, with < P* < 1 and 0 < A* < 1, the probability of a correct selection is to be at least P* when the true difference in the p-values is at least A*. Sobel & Weiss (1970) consider two sequential procedures for this selection problem in which sampling is terminated when the absolute difference in the number of successes for the two populations first reaches a pre- determined integer. We will refer to this as the success-lead rule for termination. In the first procedure, play-the-winner sampling is used in which sampling continues with the same population after each success and switches to the other population after each failure. In the second procedure, vector-at-a-time sampling is used in which two observations are taken at each stage, one from each population. In many contexts, it is desirable to have a small expected number of trials on the poorer population or treatment but using the success-lead rule it is not true uniformly in the parameter space for the p's that play-the- winner sampling is superior to vector-at-a-time sampling in this respect. In a second paper Sobel & Weiss (1971) consider the use of an inverse sampling rule when sampling is termi- nated when any one population has r successes. They show that in this case play-the-winner sampling is uniformly preferable to vector-at-a-time sampling in the sense that for the same probability requirement, the expected number of trials on the poorer treatment is always smaller. Truncated versions of the inverse sampling rule and the success-lead rule for play- the-winner sampling were proposed by Hoel (1972) and Fushimi (1973). Their stopping rules include the number of failures as well as the number of successes and mean that excessive sample sizes are avoided when the population probabilities of success are small.

Taking account of heterogeneity between the individuals in population based mortality studies is important. A systematic way of describing heterogeneity is by an unobserved quantity called frailty, entering the hazard multiplicatively. Until now most studies have used a gamma distributed frailty, which is mathematically convenient; for example, the distribution among survivors is also gamma. This paper shows that several other distributions have equally simple properties, the main example being the inverse Gaussian distribution. Consequences of the different distributions are examined; the inverse Gaussian makes the population homogeneous with time, whereas for the gamma the relative heterogeneity is constant.

SUMMARY Suppose that we have a sample from a bivariate normal population with variances 0-2, 0-2 and correlation coefficient p, but some observations on one of the two variables are missing. Based on these data, we wish to test the hypothesis of equality of means of the two variables. In this paper, classes of test procedures are given for when the alternative is one-sided and for when it is two-sided. The tests satisfy the property that the probability of type I error does not exceed a preassigned level a when p is negative, and when p is positive but small. The test that is to be recommended in a particular situation depends on what information, if any, is available about the nuisance parameters. Comparisons of the tests with the paired t test and the Mehta & Gurland (1969, 1973) test are also included.

of the results, the long-term effects of these factors were apparent. However the problem was not completely solved, for Moran and Watterson (1959) found that factor (iv) was completely confounded with (v) distribution of the number of offspring per parent and the later evidence suggested that (v) was producing all the effect previously attributed to (iv). Another extension was the discussion by Watterson (1959) of the models when mutation and selection were absent. Two features required further investigation: the behaviour of the models when mutation was absent and selection present, and more importantly, a theory to unify all the known results. The aim of this paper is to settle these topics by showing that the models are approximately one-dimensional diffusion processes. Although it is now clear that all the results could have been derived by diffusion theory, it was not clear at the time that this theory was applicable. However, the agreement between the results stimulated an examination of certain general conditions sufficient for the applicability of diffusion theory to population genetics (Watterson (1962)), and here it is proposed to show how these two models are examples of that theory (?2, ?3) and to thereby derive some further results for them (?4). Some comments will be made concerning the concept of "effective population size".

SUMMARY In an earlier paper Sobel & Weiss (1970) examined two sampling schemes for choosing the binomial population with the larger probability of success. The two sequential allocation procedures were discussed in the light of their ability to reach a decision while using as small a number as possible of tests with the poorer population. It was shown that neither sampling scheme was uniformly better than the other with respect to that criterion. However, one sampling procedure, the Play-the-Winner rule (Robbins, 1956; Zelen, 1969), was superior, provided that the probability of success of the better population was greater than a number roughly equal to A. Here we investigate a two-stage procedure in which the first stage is used to estimate the probability of success of the better population. It is shown that a fairly small estimation stage, approximately five tests with each population, allows one to benefit considerably from the choice between the two sampling methods available in the second stage.

A multiple decision approach to the problem of ranking populations according to their population means has been formulated by Bechhofer (1954). A single-sample solution to this problem was presented by him for the case of normal populations with known variances. In the present paper the case of normal populations with unknown equal variances is considered. A two-sample procedure is proposed as a solution to the latter problem; this procedure is of the type used by Stein (1945) for obtaining a test of Student's hypothesis with power independent of the variance. Tables are available which enable the experimenter to apply this procedure with little computational effort to the ranking of two or three populations with unknown but equal variances. The same tables can be used when the variance ratios are arbitrary and known. Graphic comparisons of the expected sample size are made for the single-sample and two-sample procedures.

The theory of multiplicative (or branching) population processes where the states of each individual in the population range over a fixed finite set has been studied by a number of authors: see for example, Everett and Ulam, Kolmogorov and Sevastyanov, Harris; the continuous parameter case is treated in Arley; further references will be found in the books by Bartlett and Bharucha-Reid. The purpose of the present paper is to study the theory of such processes in the case where the individual state space is arbitrary (i.e., an abstract space).

All the methods which have been presented in recent years for the analysis of capturerecapture data have aimed at estimating total population size, survival rates and dilution rates. These methods, whether deterministic or stochastic, all assume that the animals captured on each occasion are a random sample from the whole population. The proportion of marked animals in each sample then contains information about the size of the population at that instant. Darroch (1958, 1959) has shown for a stochastic model that, in the cases where the population is subject to either death or to immigration but not both, the parameters of the population together with their variances and covariances can be very simply estimated. In the more general cases with both death and immigration, he obtains estimation equations which can readily be solved numerically, and suggests that the variances of the estimates can be obtained by standard asymptotic techniques. For a deterministic survival rate Jolly (1963) obtains explicit formulae for population size and survival and dilution rates, and asymptotic formulae for their variance and covariances. This theory has been derived to meet the case where the animals are sampled completely at random. However, for some species, the markings which are applied to render an animal individually identifiable may be visible without capture. For the past thirteen years zoologists from the University of Aberdeen have been carrying out a study of the fulmar petrel (Fulmarus glacialis) on a small island in Orkney, the details of which have been describedby Carrick & Dunnet (1954) andDunnet, Anderson & Cormack (1963). Throughout this study birds have been caught individually on their nests and have been marked by an identifying set of coloured rings on the leg. These rings are clearly visible in flight, so that in years subsequent to marking the continued presence of a bird in the colony may be established without any capture being effected. New individuals may be marked each year by further capture. The aim of this paper is to examine a model applicable to this biological situation. Clearly no useful inference about the population size is possible, but valid estimates of survival rate can be obtained. In ?? 2 and 3 the notation used and the general formulation of the theory are set out. Maximum-likelihood estimates of survival rates are obtained in ? 4, and their asymptotic variances and covariances in ? 5. These are applied to the over-all survival rate and the life span in ? 6. In ? 7, the theory is illustrated for records from the study of fulmars. 2. NOTATION