Statistical Efficiency in Distance Sampling

Distance sampling is a technique for estimating the abundance of animals or other objects in a region, allowing for imperfect detection. This paper evaluates the statistical efficiency of the method when its assumptions are met, both theoretically and by simulation. The theoretical component of the paper is a derivation of the asymptotic variance penalty for the distance sampling estimator arising from uncertainty about the unknown detection parameters. This asymptotic penalty factor is tabulated for several detection functions. It is typically at least 2 but can be much higher, particularly for steeply declining detection rates. The asymptotic result relies on a model which makes the strong assumption that objects are uniformly distributed across the region. The simulation study relaxes this assumption by incorporating over-dispersion when generating object locations. Distance sampling and strip transect estimators are calculated for simulated data, for a variety of overdispersion factors, detection functions, sample sizes and strip widths. The simulation results confirm the theoretical asymptotic penalty in the non-overdispersed case. For a more realistic overdispersion factor of 2, distance sampling estimation outperforms strip transect estimation when a half-normal distance function is correctly assumed, confirming previous literature. When the hazard rate model is correctly assumed, strip transect estimators have lower mean squared error than the usual distance sampling estimator when the strip width is close enough to its optimal value (± 75% when there are 100 detections; ± 50% when there are 200 detections). Whether the ecologist can set the strip width sufficiently accurately will depend on the circumstances of each particular study.     

Genetic variability due to geographical inhomogeneity

The frequency of one of two alleles is studied as a function of position and time in a one, two, or three dimensional region. A nonlinear diffusion equation is employed. Each allele is assumed to have a selective advantage in some part of the region. An asymptotic solution is constructed for the case when the selection coefficient is large compared to the diffusion coefficient, i.e. when selection acts more rapidly than diffusion. Then as time increases, the solution tends to a cline, i.e. an equilibrium distribution in which both alleles are present everywhere, each predominating where it has the advantage. In a narrow region around the boundary where the selective advantage switches from one allele to the other, both alleles are present with comparable frequencies. Along a line normal to this boundary, the frequency varies as in a one dimensional habitat with a simple variation in selective advantage. The asymptotic solution is compared with the numerical solution for a special two dimensional case, and the agreement is found to be good.Research supported by the National Science Foundation.     

Structure of the femora and autotomous (postpygal) caudal vertebrae in the three-toed skink Chalcides chalcides (Reptilia: Squamata: Scincidae) and its applicability for age and growth rate determination

The limbs of the three-toed skink Chalcides chalcides are reduced to such a degree that the three digits are too small for skeletochronology. This study, performed on animals collected near Isernia (central Italy), describes the structure of the caudal vertebrae, which are often naturally lost due to autotomy, in order to determine whether they can be used to obtain data on age and growth with skeletochronological techniques. The reliability of the autotomous caudal vertebrae for skeletochronology was verified by performing skeletochronological analyses also on femora. Although the identification of the lines of arrested growth (LAGs) was easier in femora than in autotomous caudal vertebrae, a high correspondence of the LAG count between the two bones was observed. Females were larger and lived longer than males (4 and 3 years, respectively). For both sexes, the snout vent length (SVL) was significantly correlated with age. For both sexes, sexual maturity was attained after two hibernations from birth, beginning at the 20th month of age. At first reproduction, males had an SVL of 91–106 mm and females one of 111–150 mm. Von Bertalanffy growth curves of age versus SVL showed that females had slower growth rates than males for attaining their asymptotic SVL (females: 197 mm; males: 143 mm). The results provide the first data on age and growth of C. chalcide, and show that autotomous caudal vertebrae are reliable alternatives for obtaining such data for limb-reduced reptiles, avoiding the need to sacrifice or disable animals, as occurs when long bones are used.     

Exact tests for one sample correlated binary data

In this paper we developed exact tests for one sample correlated binary data whose cluster sizes are at most two. Although significant progress has been made in the development and implementation of the exact tests for uncorrelated data, exact tests for correlated data are rare. Lack of a tractable likelihood function has made it difficult to develop exact tests for correlated binary data. However, when cluster sizes of binary data are at most two, only three parameters are needed to characterize the problem. One parameter is fixed under the null hypothesis, while the other two parameters can be removed by both conditional and unconditional approaches, respectively, to construct exact tests. We compared the exact and asymptotic p-values in several cases. The proposed method is applied to real-life data.     

Comparison of recent estimators of interclass correlation from familial data

The asymptotic and finite-sample properties of several recent estimators of interclass correlation are compared to more traditional estimators in the case of a variable number of siblings per family. It is shown that Karlin's family-weighted pairwise estimator (Karlin, Cameron, and Williams, 1981, Proceedings of the National Academy of Science 78, 2664-2668) is virtually equivalent to the ensemble estimator (Rosner, Donner, and Hennekens, 1977, Applied Statistics 26, 179-187), thus suggesting an estimator of the former's asymptotic variance. Further, an estimator proposed by Srivastava (1984, Biometrika 71, 177-185) is shown to be identical to the modified sib-mean estimator (Konishi, 1982, Annals of the Institute of Statistical Mathematics 34, 505-515) when the sib-sib correlation is estimated by the method of unweighted group means. Although the estimator due to Srivastava has smaller asymptotic variance than the other two, the gain in efficiency is slight, for familial data, both asymptotically and in finite samples.     

Effect of covariate omission in Weibull accelerated failure time model: A caution

The accelerated failure time model is presented as an alternative to the proportional hazard model in the analysis of survival data. We investigate the effect of covariates omission in the case of applying a Weibull accelerated failure time model. In an uncensored setting, the asymptotic bias of the treatment effect is theoretically zero when important covariates are omitted; however, the asymptotic variance estimator of the treatment effect could be biased and then the size of the Wald test for the treatment effect is likely to exceed the nominal level. In some cases, the test size could be more than twice the nominal level. In a simulation study, in both censored and uncensored settings, Type I error for the test of the treatment effect was likely inflated when the prognostic covariates are omitted. This work remarks the careless use of the accelerated failure time model. We recommend the use of the robust sandwich variance estimator in order to avoid the inflation of the Type I error in the accelerated failure time model, although the robust variance is not commonly used in the survival data analyses.     

Body growth of North Atlantic right whales (Eubalaena glacialis) revisited

Knowing size-at-age is important for determining food requirements and making inferences about the nutritional status of individuals and their populations. Accurate growth curves are also needed to quantify drug dosages to treat wounded or entangled animals. However, body sizes are often based on small numbers of measured animals that must be improved as new data become available. We updated an existing body growth model for North Atlantic right whales (NARWs) using new data from dead animals and from older individuals. Our models indicate that NARWs attain mean lengths and weights of 4.3 m and 1.0 mt at birth, and 13.1 m and 31.7 mt when sexually mature. Calves more than double their length and attain nearly three-quarters of their asymptotic adult size during their first year of life. Overall, our length estimates agreed well with previous estimates, but our mass-at-age values were considerably higher. These differences revealed that necropsy data used alone in allometric models underestimate mass due possibly to several of the stranded animals in the database having been chronically entangled and in poor body condition. Augmenting the database with healthier individuals, such as harvested North Pacific right whales, yielded mass predictions that reflect both healthy and unhealthy individuals.     

The effects of conditioning on probands to correct for multiple ascertainment.

Cannings and Thompson suggested conditioning on the phenotypes of the probands to correct for ascertainment in the analysis of pedigree data. The method assumes single ascertainment and can be expected to yield asymptotically biased parameter estimates except in this specific case. However, because the method is easy to apply, we investigated the degree of bias in the more typical situation of multiple ascertainment, in the hope that the bias might be small and that the method could be applied more generally. To explore the utility of conditioning on probands to correct for multiple ascertainment, we calculated the asymptotic value of the segregation ratio for two versions of the simple Mendelian segregation model on sibship data. For both versions, we found that this asymptotic value decreased approximately linearly as the ascertainment probability increased. When ascertainment was complete, the segregation-ratio estimates were zero, not just asymptotically but for finite sample size as well. In some cases, conditioning on probands actually resulted in greater parameter bias than no ascertainment correction at all. These results hold for a variety of sibship-size distributions, several modes of inheritance, and a wide range of population prevalences of affected individuals.     

Efficiency of age-adjusted tests in animal carcinogenicity experiments

Carcinogenicity experiments may be analysed by a simple comparison of the control and exposed groups with respect to the proportions of observed tumors amongst dead animals, when longevity is identical in control and exposed groups. This simple-proportions test is invalid when longevity varies between groups and age-adjusted methods such as the Hoel-Walburg or logrank test are needed. This paper stresses that there is an advantage to using age-adjusted tests, even when the simple-proportions test is valid. The argument is based on calculations of the asymptotic relative efficiency of the simple-proportions test with respect to both the Hoel-Walburg and the logrank tests.     

A Note on Interval Estimation of the Simple Difference in Data with Correlated Matched Pairs

When we employ cluster sampling to collect data with matched pairs, the assumption of independence between all matched pairs is not likely true. This paper notes that applying interval estimators, that do not account for the intraclass correlation between matched pairs, to estimate the simple difference between two proportions of response can be quite misleading, especially when both the number of matched pairs per cluster and the intraclass correlation between matched pairs within clusters are large. This paper develops two asymptotic interval estimators of the simple difference, that accommodate the data of cluster sampling with correlated matched pairs. This paper further applies Monte Carlo simulation to compare the finite sample performance of these estimators and demonstrates that the interval estimator, derived from a quadratic equation proposed here, can actually perform quite well in a variety of situations.     

Asymptotic distribution of stage-grouped population models

Matrix models are often used to predict the dynamics of size-structured or age-structured populations. The asymptotic behaviour of such models is defined by their malthusian growth rate lambda, and by their stationary distribution w that gives the asymptotic proportion of individuals in each stage. As the coefficients of the transition matrix are estimated from a sample of observations, lambda and w can be considered as random variables whose law depends on the distribution of the observations. The goal of this study is to specify the asymptotic law of lambda and w when using the maximum likelihood estimators of the coefficients of the transition matrix. We prove that lambda and w are asymptotically normal, and the expressions of the asymptotic variance of lambda and of the asymptotic covariance matrix of w are given. The convergence speed of lambda and w towards their asymptotic law is studied using simulations. The results are applied to a real case study that consists of a Usher model for a tropical rain forest in French Guiana. They permit to assess the number of trees to measure to get a given precision on the estimated asymptotic diameter distribution, which is an important information on tropical forest management.     

Behavioral characterization of attractin,a water-borne peptide pheromone in the genus aplysia

Pheromones play a significant role in coordinating reproductive activity in many animals, including opisthobranch molluscs of the genus APLYSIA: Although solitary during most of the year, these simultaneous hermaphrodites gather into breeding aggregations during the reproductive season. The aggregations contain both mating and egg-laying animals and are associated with masses of egg cordons. The egg cordons are a source of pheromones that attract other Aplysia to the area, reduce their latency to mating, and induce egg laying. One of these water-borne egg cordon pheromones ("attractin") has been characterized and shown to be attractive in T-maze assays. Attractin is the first water-borne peptide pheromone characterized in invertebrates. In the current studies, behavioral assays were used to better characterize the attraction, and to examine whether attractin can induce mating. Although the two activities could be related (i.e., attraction occurring because animals were looking for a partner), this was not tested. T-maze assays showed that attractin works as part of a bouquet of odors: the peptide is attractive only when Aplysia brasiliana is part of the stimulus. The animal does not need to be a conspecific, perhaps explaining why multiple species may be associated with one aggregation. Native and recombinant attractin are equally attractive, verifying that N-glycosylation at residue 8 is not required for attraction. Mating studies showed that both native and recombinant attractin reduce the latency to mating. The effects are larger when hermaphroditic mating is considered: in addition to reducing latency, attractin doubles the number of pairs mating as hermaphrodites. The effect may result from attractin stimulating both animals to mate as males and would be consistent with behaviors previously seen in the T-maze. Attractin may thus be contributing to the formation of copulatory chains and rings seen in aggregations in the field. These results may be interpreted in two ways: (1). attractin has multiple activities that contribute to the establishment and maintenance of the aggregation; or (2). the induced desire to mate may make attractin attractive when it is presented in conjunction with an animal. In either case, the results open the door for cellular and molecular studies of mechanism of action.     

Does your species have memory? Analyzing capture–recapture data with memory models

We examine memory models for multisite capture–recapture data. This is an important topic, as animals may exhibit behavior that is more complex than simple first‐order Markov movement between sites, when it is necessary to devise and fit appropriate models to data. We consider the Arnason–Schwarz model for multisite capture–recapture data, which incorporates just first‐order Markov movement, and also two alternative models that allow for memory, the Brownie model and the Pradel model. We use simulation to compare two alternative tests which may be undertaken to determine whether models for multisite capture–recapture data need to incorporate memory. Increasing the complexity of models runs the risk of introducing parameters that cannot be estimated, irrespective of how much data are collected, a feature which is known as parameter redundancy. Rouan et al. (JABES, 2009, pp 338–355) suggest a constraint that may be applied to overcome parameter redundancy when it is present in multisite memory models. For this case, we apply symbolic methods to derive a simpler constraint, which allows more parameters to be estimated, and give general results not limited to a particular configuration. We also consider the effect sparse data can have on parameter redundancy and recommend minimum sample sizes. Memory models for multisite capture–recapture data can be highly complex and difficult to fit to data. We emphasize the importance of a structured approach to modeling such data, by considering a priori which parameters can be estimated, which constraints are needed in order for estimation to take place, and how much data need to be collected. We also give guidance on the amount of data needed to use two alternative families of tests for whether models for multisite capture–recapture data need to incorporate memory.     

The nonparametric Behrens–Fisher problem in partially complete clustered data

In randomized trials or observational studies involving clustered units, the assumption of independence within clusters is not practical. Existing parametric or semiparametric methods assume specific dependence structures within a cluster. Furthermore, parametric model assumptions may not even be realistic when data are measured in a nonmetric scale as commonly happens, for example, in quality‐of‐life outcomes. In this paper, nonparametric effect‐size measures for clustered data that allow meaningful and interpretable probabilistic comparisons of treatments or intervention programs will be introduced. The dependence among observations within a cluster can be arbitrary. Point estimators along with their asymptotic properties for computing confidence intervals and performing hypothesis test will be discussed. Small sample approximations that retain some of the optimal asymptotic behaviors will be presented. In our setup, some clusters may involve observations coming from both intervention groups (referred to as complete clusters), while others may contain observations from one group only (referred to as incomplete clusters). In deriving the asymptotic theories, we do not impose any relation in the rate of divergence of the numbers of complete and incomplete clusters. Simulations show favorable performance of the methods for arbitrary combinations of complete and incomplete clusters. The developed nonparametric methods are illustrated using data from a randomized trial of indoor wood smoke reduction to improve asthma symptoms and a cluster‐randomized trial for smoking cessation.     

Structure of the femora and autotomous (postpygal) caudal vertebrae in the three-toed skink Chalcides chalcides (Reptilia: Squamata: Scincidae) and its applicability for age and growth rate determination

The limbs of the three-toed skink Chalcides chalcides are reduced to such a degree that the three digits are too small for skeletochronology. This study, performed on animals collected near Isernia (central Italy), describes the structure of the caudal vertebrae, which are often naturally lost due to autotomy, in order to determine whether they can be used to obtain data on age and growth with skeletochronological techniques. The reliability of the autotomous caudal vertebrae for skeletochronology was verified by performing skeletochronological analyses also on femora. Although the identification of the lines of arrested growth (LAGs) was easier in femora than in autotomous caudal vertebrae, a high correspondence of the LAG count between the two bones was observed. Females were larger and lived longer than males (4 and 3 years, respectively). For both sexes, the snout vent length (SVL) was significantly correlated with age. For both sexes, sexual maturity was attained after two hibernations from birth, beginning at the 20th month of age. At first reproduction, males had an SVL of 91–106 mm and females one of 111–150 mm. Von Bertalanffy growth curves of age versus SVL showed that females had slower growth rates than males for attaining their asymptotic SVL (females: 197 mm; males: 143 mm). The results provide the first data on age and growth of C. chalcide, and show that autotomous caudal vertebrae are reliable alternatives for obtaining such data for limb-reduced reptiles, avoiding the need to sacrifice or disable animals, as occurs when long bones are used.     

The Semiparametric Case‐Only Estimator

Summary We propose a semiparametric case‐only estimator of multiplicative gene–environment or gene–gene interactions, under the assumption of conditional independence of the two factors given a vector of potential confounding variables. Our estimator yields valid inferences on the interaction function if either but not necessarily both of two unknown baseline functions of the confounders is correctly modeled. Furthermore, when both models are correct, our estimator has the smallest possible asymptotic variance for estimating the interaction parameter in a semiparametric model that assumes that at least one but not necessarily both baseline models are correct.     

Linear Discrete Population Models with Two Time Scales in Fast Changing Environments I: Autonomous Case

In this work we consider a structured population with groups and subgroups of individuals. The intra-group dynamics is assumed to be fast in comparison with the inter-group dynamics. We study linear discrete models where the slow dynamics is represented by a single matrix and the fast dynamics is described by means of the first k terms of a converging sequence of different matrices. The number k can be interpreted as the ratio between the two time scales.The aim of this work is to extend aggregation techniques to the case of fast changing environments. The main idea of aggregation is to build up a new system, with lower dimension, that summarizes the information concerning the fast process. This "aggregated" system provides essential information on the original one. It is shown that the asymptotic behavior of the original system can be approximated by the asymptotic behavior of the aggregated system when the ratio between the two time scales is large enough.We present an example of an age structured population in a patchy environment. The migration process is assumed to be fast in comparison with the demographic process. Numerical simulations illustrate that the asymptotic growth rate and the stable age distribution of the population in the original and the aggregated systems are getting closer as the ratio k increases.     

How to estimate variability in affinity relationships in partially observed groups of domestic herbivores?

Animal sociability measurements based on inter-individual distances or nearest-neighbour distributions can be obtained automatically with telemetry collars. So far, all the indices that have been used require the whole group to be observed. Here, we propose an index of the variability in affinity relationships in groups of domestic herbivores, whose definition does not depend on group size and that can be used even if some data are missing. This index and its estimators are based on a function that measures how frequently an animal is closer than another one from a third animal. When no data are missing, we show that our estimator and the variance of the sociability matrixsensu Sibbald (considered as the reference method) are strongly correlated. We then consider two cases of missing data. In the first case, some animals are randomly missing, that is, to account for random breakdown of telemetry collars. Our estimator is unbiased by such missing data and its variance decreases as the number of observation dates increases. In the second case, the same animals are missing at all observation dates, that is, in large herds where there are more individuals to be observed than available telemetry collars. Our estimator of affinity variance within a group is biased by such missing data. Thus, it requires changing animals equipped with telemetry collars regularly during the experiment. Conversely, the estimator remains unbiased at the population level, that is, if several independent groups are being analysed. We finally illustrate how this estimator can be used by investigating changes in the variability of affinities according to group size in grazing heifers.     

Exact significance testing to establish treatment equivalence with ordered categorical data

This communication concerns the problem of establishing the therapeutic equivalence of two treatments that are being compared on the basis of ordered categorical data. The problem is formulated as a significance test in which the null hypothesis specifies a treatment difference. An efficient numerical algorithm for computing the exact significance level is provided, along with a simple method for obtaining the asymptotic significance level. Both methods are applied to a clinical trial of a new agent versus an active control. Guidelines for when to use the exact procedure and when to rely on asymptotic theory are provided.     

Incomplete generalized U-statistics for food risk assessment

This article proposes statistical tools for quantitative evaluation of the risk due to the presence of some particular contaminants in food. We focus on the estimation of the probability of the exposure to exceed the so-called provisional tolerable weekly intake (PTWI), when both consumption data and contamination data are independently available. A Monte Carlo approximation of the plug-in estimator, which may be seen as an incomplete generalized U-statistic, is investigated. We obtain the asymptotic properties of this estimator and propose several confidence intervals, based on two estimators of the asymptotic variance: (i) a bootstrap type estimator and (ii) an approximate jackknife estimator relying on the Hoeffding decomposition of the original U-statistics. As an illustration, we present an evaluation of the exposure to Ochratoxin A in France.