A Mixture Point Process for Repeated Failure Times,with an Application to a Recurrent Disease

We present a model that describes the distribution of recurring times of a disease in presence of covariate effects. After a first occurrence of the disease in an individual, the time intervals between successive cases are supposed to be independent and to be a mixture of two distributions according to the issue of the previous treatment. Both sub‐distributions of the model and the mixture proportion are allowed to involve covariates. Parametric inference is considered and we illustrate the methods with data of a recurrent disease and with simulations, using piecewise constant baseline hazard functions.     

Distribution fields for aquatic ecosystem components: method of identification of correlation zones

1. Data from the two multidisciplinary surveys of Lake Kinneret (Israel), including acoustic surveys of fish concentrations and concurrent sampling of plankton at stations, were used to reconstruct fish and plankton distributions. No significant lake-wide correlations for the distribution fields were found. With respect to patchiness of the fish and plankton distributions, we hypothesized that they might be correlated in localized zones. 2. A method is suggested for the identification of areas where there are strong correlations of two distribution fields. The method is based on outlining zones where the gradients of the two fields of interest are in the same direction (or are opposite). Only areas larger than the autocorrelation circles (or ellipses) for the fields are considered. The correlation of the fields is calculated for each of the zones selected. 3. The method was used in analysing data from the multidisciplinary surveys. We were able to detect areas of the lake where there were correlations for fish and plankton distributions. Analysis of specific conditions inside the correlation zones made it possible to construct hypotheses concerning the causes for the observed patterns of fish and plankton distributions.     

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.     

Quantitative classification of life-style types in predaceous phytoseiid mites

Classification of species into different functional groups based on biological criteria has been a difficult problem in ecology. The difficulty mainly arises because natural classification patterns are not necessarily mutually exclusive. The more group characteristics overlap, the more difficult it is to identify the membership of a species in the overlapping portions of any two groups. In this paper, we present an application of discriminant analysis by creating classification models from life history and morphological data for two specialist and two generalist life-styles type of predaceous phytoseiid mites. Two stages can be distinguished in our method: life-style group membership assignment and trait variable evaluation. We use a Bayesian framework to create a classifier system to locate or assign species within a mixture of trait distributions. The method assumes that a mixture of trait distributions can represent the multiple dimensions of biological data. The mixture is most evident near the boundaries between groups. Because of the complexity of analytical solution, an iterative method is used to estimate the unknown means, variances, and mixing proportion between groups. We also developed a criterion based on information theory to evaluate model performance with different combinations of input variables and different hypotheses. We present a working example of our proposed methods. We apply these methods to the problem of selecting key species for inoculative release and for classical introductions of biological pest control agents.     

Estimation of european corn borer egg masses density by sampling of runs

Spatial distributions of individuals may be considered in two ways: Firstly, individuals are situated on a continuum. Secondly, individuals are situated on discrete units, for example, insect eggs on plants. The aim of this paper is to show that sometimes, in the latter case, sampling methods based on infestation runs can be of interest in estimating population density. Analytical results are obtained under complete spatial randomness hypothesis and alternative hypotheses. Sampling procedures with limited cost are discussed and the European corn borer (ECB) case is mainly considered.     

A Bagging‐Based Correction for the Mixture Model Estimator of Population Size

Estimation of a population size by means of capture‐recapture techniques is an important problem occurring in many areas of life and social sciences. We consider the frequencies of frequencies situation, where a count variable is used to summarize how often a unit has been identified in the target population of interest. The distribution of this count variable is zero‐truncated since zero identifications do not occur in the sample. As an application we consider the surveillance of scrapie in Great Britain. In this case study holdings with scrapie that are not identified (zero counts) do not enter the surveillance database. The count variable of interest is the number of scrapie cases per holding. For count distributions a common model is the Poisson distribution and, to adjust for potential heterogeneity, a discrete mixture of Poisson distributions is used. Mixtures of Poissons usually provide an excellent fit as will be demonstrated in the application of interest. However, as it has been recently demonstrated, mixtures also suffer under the so‐called boundary problem, resulting in overestimation of population size. It is suggested here to select the mixture model on the basis of the Bayesian Information Criterion. This strategy is further refined by employing a bagging procedure leading to a series of estimates of population size. Using the median of this series, highly influential size estimates are avoided. In limited simulation studies it is shown that the procedure leads to estimates with remarkable small bias. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exploring Extra-Binomial Variation in Teratology Data Using Continuous Mixtures

Discrete data from animal teratology experiments are known to exhibit extra-binomial variation. For example, we discuss a dominant lethal assay experiment in which male mice are exposed to various levels of radiation and are then mated to females. The response of interest is the number of resorptions out of the number of implantations. Most statistical work on analyzing such data has focused on modeling response rates as a function of dose of a suspected teratogen (radiation in this case) while accounting for the extra-binomial variability when calculating standard errors of the regression coefficients. Sometimes, however, when an unobserved genetic or exposure variable is suspected, the shape of the mixing distribution is of interest. We propose a mixture of beta-binomials (MBB) family of distributions that includes the non-parametric mixture of binomials model of Laird (1978) as a special case. The MBB family can accommodate a mixing distribution with one or more modes, and we develop a bootstrap test for multimodality. We apply the method to data from a dominant lethal teratology experiment.     

On Identifiability in Capture–Recapture Models

Summary .   We study the issue of identifiability of mixture models in the context of capture–recapture abundance estimation for closed populations. Such models are used to take account of individual heterogeneity in capture probabilities, but their validity was recently questioned by Link (2003, Biometrics 59, 1123–1130) on the basis of their nonidentifiability. We give a general criterion for identifiability of the mixing distribution, and apply it to establish identifiability within families of mixing distributions that are commonly used in this context, including finite and beta mixtures. Our analysis covers binomial and geometrically distributed outcomes. In an example we highlight the difference between the identifiability issue considered here and that in classical binomial mixture models.     

Pairwise rank-based likelihood for estimation and inference on the mixture proportion

We consider the problem of estimation and inference on the mixture parameter in the two-sample problem when sample data from the two distributions as well as from a third population consisting of a mixture of the two are used. Under a general nonparametric model, where the relationship between the two populations is unspecified, we develop a pairwise rank-based likelihood. Simultaneous inference on the mixture proportion and a parameter representing the probability an observation from one population is greater than an observation from the other population is based on this likelihood. Under some regularity conditions, it is shown that the maximum pairwise rank likelihood estimator is consistent and has an asymptotic normal distribution. Simulation results indicate that the performance of this statistic is satisfactory. The methodology is demonstrated on a data set in prostate cancer.     

Bayesian covariance selection in generalized linear mixed models

The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identifying the subset of predictors that have random effects, random effects selection can be challenging, particularly when outcome distributions are nonnormal. This article proposes a fully Bayesian approach to the problem of simultaneous selection of fixed and random effects in GLMMs. Integrating out the random effects induces a covariance structure on the multivariate outcome data, and an important problem that we also consider is that of covariance selection. Our approach relies on variable selection-type mixture priors for the components in a special Cholesky decomposition of the random effects covariance. A stochastic search MCMC algorithm is developed, which relies on Gibbs sampling, with Taylor series expansions used to approximate intractable integrals. Simulated data examples are presented for different exponential family distributions, and the approach is applied to discrete survival data from a time-to-pregnancy study.     

Sequential sampling: a novel method in farm animal welfare assessment

Lameness in dairy cows is an important welfare issue. As part of a welfare assessment, herd level lameness prevalence can be estimated from scoring a sample of animals, where higher levels of accuracy are associated with larger sample sizes. As the financial cost is related to the number of cows sampled, smaller samples are preferred. Sequential sampling schemes have been used for informing decision making in clinical trials. Sequential sampling involves taking samples in stages, where sampling can stop early depending on the estimated lameness prevalence. When welfare assessment is used for a pass/fail decision, a similar approach could be applied to reduce the overall sample size. The sampling schemes proposed here apply the principles of sequential sampling within a diagnostic testing framework. This study develops three sequential sampling schemes of increasing complexity to classify 80 fully assessed UK dairy farms, each with known lameness prevalence. Using the Welfare Quality herd-size-based sampling scheme, the first ‘basic’ scheme involves two sampling events. At the first sampling event half the Welfare Quality sample size is drawn, and then depending on the outcome, sampling either stops or is continued and the same number of animals is sampled again. In the second ‘cautious’ scheme, an adaptation is made to ensure that correctly classifying a farm as ‘bad’ is done with greater certainty. The third scheme is the only scheme to go beyond lameness as a binary measure and investigates the potential for increasing accuracy by incorporating the number of severely lame cows into the decision. The three schemes are evaluated with respect to accuracy and average sample size by running 100 000 simulations for each scheme, and a comparison is made with the fixed size Welfare Quality herd-size-based sampling scheme. All three schemes performed almost as well as the fixed size scheme but with much smaller average sample sizes. For the third scheme, an overall association between lameness prevalence and the proportion of lame cows that were severely lame on a farm was found. However, as this association was found to not be consistent across all farms, the sampling scheme did not prove to be as useful as expected. The preferred scheme was therefore the ‘cautious’ scheme for which a sampling protocol has also been developed.     

The metapopulation dynamics of an infectious disease: tuberculosis in possums

An SEI metapopulation model is developed for the spread of an infectious agent by migration. The model portrays two age classes on a number of patches connected by migration routes which are used as host animals mature. A feature of this model is that the basic reproduction ratio may be computed directly, using a scheme that separates topography, demography, and epidemiology. We also provide formulas for individual patch basic reproduction numbers and discuss their connection with the basic reproduction ratio for the system. The model is applied to the problem of spatial spread of bovine tuberculosis in a possum population. The temporal dynamics of infection are investigated for some generic networks of migration links, and the basic reproduction ratio is computed-its value is not greatly different from that for a homogeneous model. Three scenarios are considered for the control of bovine tuberculosis in possums where the spatial aspect is shown to be crucial for the design of disease management operations.     

Fitting mixture models to grouped and truncated data via the EM algorithm

The fitting of finite mixture models via the EM algorithm is considered for data which are available only in grouped form and which may also be truncated. A practical example is presented where a mixture of two doubly truncated log-normal distributions is adopted to model the distribution of the volume of red blood cells in cows during recovery from anemia.     

Stage duration distributions in matrix population models

Matrix population models are a standard tool for studying stage‐structured populations, but they are not flexible in describing stage duration distributions. This study describes a method for modeling various such distributions in matrix models. The method uses a mixture of two negative binomial distributions (parametrized using a maximum likelihood method) to approximate a target (true) distribution. To examine the performance of the method, populations consisting of two life stages (juvenile and adult) were considered. The juvenile duration distribution followed a gamma distribution, lognormal distribution, or zero‐truncated (over‐dispersed) Poisson distribution, each of which represents a target distribution to be approximated by a mixture distribution. The true population growth rate based on a target distribution was obtained using an individual‐based model, and the extent to which matrix models can approximate the target dynamics was examined. The results show that the method generally works well for the examined target distributions, but is prone to biased predictions under some conditions. In addition, the method works uniformly better than an existing method whose performance was also examined for comparison. Other details regarding parameter estimation and model development are also discussed.     

A transformation‐based approach to Gaussian mixture density estimation for bounded data

Finite mixture of Gaussian distributions provide a flexible semiparametric methodology for density estimation when the continuous variables under investigation have no boundaries. However, in practical applications, variables may be partially bounded (e.g., taking nonnegative values) or completely bounded (e.g., taking values in the unit interval). In this case, the standard Gaussian finite mixture model assigns nonzero densities to any possible values, even to those outside the ranges where the variables are defined, hence resulting in potentially severe bias. In this paper, we propose a transformation‐based approach for Gaussian mixture modeling in case of bounded variables. The basic idea is to carry out density estimation not on the original data but on appropriately transformed data. Then, the density for the original data can be obtained by a change of variables. Both the transformation parameters and the parameters of the Gaussian mixture are jointly estimated by the expectation‐maximization (EM) algorithm. The methodology for partially and completely bounded data is illustrated using both simulated data and real data applications.     

Bayesian inference for disease prevalence using negative binomial group testing

Group testing, also known as pooled testing, and inverse sampling are both widely used methods of data collection when the goal is to estimate a small proportion. Taking a Bayesian approach, we consider the new problem of estimating disease prevalence from group testing when inverse (negative binomial) sampling is used. Using different distributions to incorporate prior knowledge of disease incidence and different loss functions, we derive closed form expressions for posterior distributions and resulting point and credible interval estimators. We then evaluate our new estimators, on Bayesian and classical grounds, and apply our methods to a West Nile Virus data set.     

Allele frequency estimation from ambiguous data: using resampling schema in validating frequency estimates and in selective neutrality testing

The development of molecular typing techniques applied to the study of population genetic diversity originates data with increasing precision but at the cost of some ambiguities. As distinct techniques may produce distinct kinds of ambiguities, a crucial issue is to assess the differences between frequency distributions estimated from data produced by alternative techniques for the same sample. To that aim, we developed a resampling scheme that allows evaluating, by statistical means, the significance of the difference between two frequency distributions. The same approach is then shown to be applicable to test selective neutrality when only sample frequencies are known. The use of these original methods is presented here through an application to the genetic study of a Munda human population sample, where three different HLA loci were typed using two different molecular methods (reverse PCR-SSO typing on microbeads arrays based on Luminex technology and PCR-SSP typing), as described in details in the companion article by Riccio et al. [The Austroasiatic Munda population from India and its enigmatic origin: An HLA diversity study. Hum. Biol. 38:405-435 (2011)]. The differences between the frequency estimates of the two typing techniques were found to be smaller than those resulting from sampling. Overall, we show that using a resampling scheme in validating frequency estimates is effective when alternative frequency estimates are available. Moreover, resampling appears to be the unique way to test selective neutrality when only frequency data are available to describe the genetic structure of populations.     

Two-phase importance sampling for inference about transmission trees

There has been growing interest in the statistics community to develop methods for inferring transmission pathways of infectious pathogens from molecular sequence data. For many datasets, the computational challenge lies in the huge dimension of the missing data. Here, we introduce an importance sampling scheme in which the transmission trees and phylogenies of pathogens are both sampled from reasonable importance distributions, alleviating the inference. Using this approach, arbitrary models of transmission could be considered, contrary to many earlier proposed methods. We illustrate the scheme by analysing transmissions of Streptococcus pneumoniae from household to household within a refugee camp, using data in which only a fraction of hosts is observed, but which is still rich enough to unravel the within-household transmission dynamics and pairs of households between whom transmission is plausible. We observe that while probability of direct transmission is low even for the most prominent cases of transmission, still those pairs of households are geographically much closer to each other than expected under random proximity.     

Performance of individual vs. group sampling for inferring dispersal under isolation‐by‐distance

Models of isolation‐by‐distance formalize the effects of genetic drift and gene flow in a spatial context where gene dispersal is spatially limited. These models have been used to show that, at an appropriate spatial scale, dispersal parameters can be inferred from the regression of genetic differentiation against geographic distance between sampling locations. This approach is compelling because it is relatively simple and robust and has rather low sampling requirements. In continuous populations, dispersal can be inferred from isolation‐by‐distance patterns using either individuals or groups as sampling units. Intrigued by empirical findings where individual samples seemed to provide more power, we used simulations to compare the performances of the two methods in a range of situations with different dispersal distributions. We found that sampling individuals provide more power in a range of dispersal conditions that is narrow but fits many realistic situations. These situations were characterized not only by the general steepness of isolation‐by‐distance but also by the intrinsic shape of the dispersal kernel. The performances of the two approaches are otherwise similar, suggesting that the choice of a sampling unit is globally less important than other settings such as a study's spatial scale.     

Equivalence of truncated count mixture distributions and mixtures of truncated count distributions

This article is about modeling count data with zero truncation. A parametric count density family is considered. The truncated mixture of densities from this family is different from the mixture of truncated densities from the same family. Whereas the former model is more natural to formulate and to interpret, the latter model is theoretically easier to treat. It is shown that for any mixing distribution leading to a truncated mixture, a (usually different) mixing distribution can be found so that the associated mixture of truncated densities equals the truncated mixture, and vice versa. This implies that the likelihood surfaces for both situations agree, and in this sense both models are equivalent. Zero-truncated count data models are used frequently in the capture-recapture setting to estimate population size, and it can be shown that the two Horvitz-Thompson estimators, associated with the two models, agree. In particular, it is possible to achieve strong results for mixtures of truncated Poisson densities, including reliable, global construction of the unique NPMLE (nonparametric maximum likelihood estimator) of the mixing distribution, implying a unique estimator for the population size. The benefit of these results lies in the fact that it is valid to work with the mixture of truncated count densities, which is less appealing for the practitioner but theoretically easier. Mixtures of truncated count densities form a convex linear model, for which a developed theory exists, including global maximum likelihood theory as well as algorithmic approaches. Once the problem has been solved in this class, it might readily be transformed back to the original problem by means of an explicitly given mapping. Applications of these ideas are given, particularly in the case of the truncated Poisson family.