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Negative binomial related distributions have been widely used in practice. The calculation of the corresponding Fisher information matrices involves the expectation of trigamma function values which can only be calculated numerically and approximately. In this paper, we propose a trigamma-free approach to approximate the expectations involving the trigamma function, along with theoretical upper bounds for approximation errors. We show by numerical studies that our approach is highly efficient and much more accurate than previous methods. We also apply our approach to compute the Fisher information matrices of zero-inflated negative binomial (ZINB) and beta negative binomial (ZIBNB) probabilistic models, as well as ZIBNB regression models.

The estimation of abundance from presence–absence data is an intriguing problem in applied statistics. The classical Poisson model makes strong independence and homogeneity assumptions and in practice generally underestimates the true abundance. A controversial ad hoc method based on negative-binomial counts (Am. Nat.) has been empirically successful but lacks theoretical justification. We first present an alternative estimator of abundance based on a paired negative binomial model that is consistent and asymptotically normally distributed. A quadruple negative binomial extension is also developed, which yields the previous ad hoc approach and resolves the controversy in the literature. We examine the performance of the estimators in a simulation study and estimate the abundance of 44 tree species in a permanent forest plot.

Permutation tests are increasingly being used as a reliable method for inference in neuroimaging analysis. However, they are computationally intensive. For small, non-imaging datasets, recomputing a model thousands of times is seldom a problem, but for large, complex models this can be prohibitively slow, even with the availability of inexpensive computing power. Here we exploit properties of statistics used with the general linear model (GLM) and their distributions to obtain accelerations irrespective of generic software or hardware improvements. We compare the following approaches: (i) performing a small number of permutations; (ii) estimating the p-value as a parameter of a negative binomial distribution; (iii) fitting a generalised Pareto distribution to the tail of the permutation distribution; (iv) computing p-values based on the expected moments of the permutation distribution, approximated from a gamma distribution; (v) direct fitting of a gamma distribution to the empirical permutation distribution; and (vi) permuting a reduced number of voxels, with completion of the remainder using low rank matrix theory. Using synthetic data we assessed the different methods in terms of their error rates, power, agreement with a reference result, and the risk of taking a different decision regarding the rejection of the null hypotheses (known as the resampling risk). We also conducted a re-analysis of a voxel-based morphometry study as a real-data example. All methods yielded exact error rates. Likewise, power was similar across methods. Resampling risk was higher for methods (i), (iii) and (v). For comparable resampling risks, the method in which no permutations are done (iv) was the absolute fastest. All methods produced visually similar maps for the real data, with stronger effects being detected in the family-wise error rate corrected maps by (iii) and (v), and generally similar to the results seen in the reference set. Overall, for uncorrected p-values, method (iv) was found the best as long as symmetric errors can be assumed. In all other settings, including for familywise error corrected p-values, we recommend the tail approximation (iii). The methods considered are freely available in the tool PALM — Permutation Analysis of Linear Models. •Permutation methods can be accelerated through additional statistical approaches.•Six approaches are described and assessed.•Methods can be 100 times faster than in the non-accelerated case.•Recommendations are provided for various common scenarios.

In this paper, a new mixed negative binomial (NB) distribution named as negative binomial-weighted Garima (NB-WG) distribution has been introduced for modeling count data. Two special cases of the formulation distribution including negative binomial- Garima (NB-G) and negative binomial-size biased Garima (NB-SBG) are obtained by setting the specified parameter. Some statistical properties such as the factorial moments, the first four moments, variance and skewness have also been derived. Parameter estimation is implemented using maximum likelihood estimation (MLE) and real data sets are discussed to demonstrate the usefulness and applicability of the proposed distribution.

A Poisson process is a commonly used starting point for modeling stochastic variation of ecological count data around a theoretical expectation. However, data typically show more variation than implied by the Poisson distribution. Such overdispersion is often accounted for by using models with different assumptions about how the variance changes with the expectation. The choice of these assumptions can naturally have apparent consequences for statistical inference. We propose a parameterization of the negative binomial distribution, where two overdispersion parameters are introduced to allow for various quadratic mean-–variance relationships, including the ones assumed in the most commonly used approaches. Using bird migration as an example, we present hypothetical scenarios on how overdispersion can arise due to sampling, flocking behavior or aggregation, environmental variability, or combinations of these factors. For all considered scenarios, mean-–variance relationships can be appropriately described by the negative binomial distribution with two overdispersion parameters. To illustrate, we apply the model to empirical migration data with a high level of overdispersion, gaining clearly different model fits with different assumptions about mean-–variance relationships. The proposed framework can be a useful approximation for modeling marginal distributions of independent count data in likelihood-based analyses.

The time taken to detect a species during site occupancy surveys contains information about the observation process. Accounting for the observation process leads to better inference about site occupancy. We explore the gain in efficiency that can be obtained from time-to-detection (TTD) data and show that this model type has a significant benefit for estimating the parameters related to detection intensity. However, for estimating occupancy probability parameters, the efficiency improvement is generally very minor. To explore whether TTD data could add valuable information when detection intensities vary between sites and surveys, we developed a mixed exponential TTD occupancy model. This new model can simultaneously estimate the detection intensity and aggregation parameters when the number of detectable individuals at the site follows a negative binomial distribution. We found that this model provided a much better description of the occupancy patterns than conventional detection/nondetection methods among 63 bird species data from the Karoo region of South Africa. Ignoring the heterogeneity of detection intensity in the TTD model generally yielded a negative bias in the estimated occupancy probability. Using simulations, we briefly explore study design trade offs between numbers of sites and surveys for different occupancy modeling strategies.

Abstract The crayfishes of Chile are endemic species and have been poorly studied with respect to their biology. The available literature is restricted to species descriptions, whereas there is no information about other biological topics such as ectocommensals and parasites, which can affect crayfishes. The aim of the present study was to do a literature review and update the information about ectocommensals and parasites that have been reported from Chilean crayfishes. The literature mentioned the presence of Temnocephala chilensis (Platyhelminthes), Stratiodrilus pugnaxi (Polychaeta) and Protozoa such as species of the genera Operculigera and Lagenophrys as ectocommensals for Parastacus pugnax and Samastacus spinifrons. As an endoparasite, Psorospermium haeckelii was reported, a unicellular eukaryotic organism that was found in P. pugnax ovaria. An additional analysis of collected specimens of P. pugnax and S. spinifrons revealed the presence of T. chilensis, albeit at a low incidence. Ecological, biogeographical and evolutionary topics are discussed considering the case of a marked microendemism of an organism associated with Chilean crayfishes.

Abstract The intertidal rocky shores in continental Chile have high species diversity mainly in northern Chile (18-27° S), and one of the most widespread species is the gastropod Echinolittorina peruviana (Lamarck, 1822). The aim of the present study is do a first characterization of spatial distribution of E. peruviana in along rocky shore in Antofagasta town in northern Chile. Individuals were counted in nine different sites that also were determined their spectral properties using remote sensing techniques (LANDSAT ETM+). The results revealed that sites without marked human intervention have more abundant in comparison to sites located in the town, also in all studied sites was found an aggregated pattern, and in six of these sites were found a negative binomial distribution. The low density related to sites with human intervention is supported when spectral properties for sites were included. These results would agree with other similar results for rocky shore in northern and southern Chile. Resumo As costas rochosas entremarés no Chile continental apresentam alta diversidade de espécies, principalmente no norte do país (18-27 ° S), e uma das espécies mais difundidas é o gastrópode Echinolittorina peruviana (Lamarck, 1822). O objetivo do presente estudo é fazer uma primeira caracterização da distribuição espacial de E. peruviana no costão rochoso da cidade de Antofagasta no norte do Chile. Os indivíduos foram contados em nove locais diferentes onde também foram determinadas suas propriedades espectrais usando técnicas de sensoriamento remoto (LANDSAT ETM +). Os resultados revelaram que os locais sem intervenção humana marcada apresentam maior abundância em comparação aos locais localizados no município. Também em todos os locais estudados foi encontrado um padrão agregado, sendo que em seis desses locais foi encontrada uma distribuição binomial negativa. A baixa densidade relacionada a sites com intervenção humana é suportada quando as propriedades espectrais para sites foram incluídas. Esses resultados concordariam com outros resultados semelhantes para costões rochosos no norte e no sul do Chile.

The spatial distribution of species is not random; instead, individuals tend to gather, resulting in a non-random pattern. Previous studies used the independent negative binomial distribution (NBD) to model the distributional aggregation of a single species, in which the independence of the distribution of individuals of a species in different quadrats had been assumed. This way of analyzing aggregation will result in the scale-dependent estimation of the aggregation or shape parameter. However, because non-random (and therefore non-independent) distribution of individuals of a species in a finite area can be caused by either correlated or clumped distribution of individuals of a species between neighboring sites, an alternative model would assume that the distribution of individuals of a species over different sampling areas is multinomial. Here, we showed that, by assuming that regional species abundance followed a NBD while using a multinomial distribution to assign individuals of species in different non-overlapped sampling quadrats that are from a partition of the entire region (quantifying positive correlation or synchrony), the estimation of the shape parameter in this probabilistic model, which is the negative multinomial distribution (NMD), was scale-invariant (i.e., the estimated shape parameter is identical across different partitions of the study region). Accordingly, the estimation of the shape parameter was related to regional species distribution alone. This implied that, the shape parameter at the community level, using the NMD model, reflected the evenness of interspecific abundance. As a comparison, if the distribution of individuals of a single species followed independent NBDs as studied previously, the shape parameter would measure the evenness of intraspecific abundance (quantifying single-species’ distributional aggregation). Moreover, our study highlighted the necessity for adjusting the model for the effects of unsampled species when studying community-level distributional patterns. Collectively, as long as a target area is partitioned into non-overlapping quadrats (no matter how their sizes vary), the proposed NMD model in this study, along with the independent NBDs model, can be jointly formulated as a framework to reconcile the scale-dependent debate on the shape parameter, unifying the relationship between inter- or intraspecific abundance and distributional patterns.

A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y, as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton-Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.