Multivariate Polya and Inverse Polya Distributions of Order k

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU , ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU , ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU , TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..     

On Mohan's Property of the Poisson Distribution

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to Moran (1952) and Patil & Seshadri (1964), respectively. Morans result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X = Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables. Motivated by Morans result Patil & Seshadri obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by Moran, Y | X is negative hypergeometric if and only if Y, Z are negative binomials. In this paper we examine the results of Moran and Patil & Seshadri in the case where the conditional distribution of Y | X is truncated at an arbitrary point k – 1 (k = 1, 2, …). In fact we attempt to answer the question as to whether Morans property of the Poisson distribution, and subsequently Patil & Seshadris property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k – 1 and negative hypergeometric truncated at k – 1 respectively.     

Characterizations of Continuous Binomial and Negative Binomial Mixtures

Characterizations are given for binomial (n, ?) and negative binomial (k, ?) mixtures in terms of the regression function of the mixing random variable Y on the mixture variable X. In both cases Y is the parameter ? and the mixing distribution is a continuous distribution on (0, 1) and on (0, ∞) respectively. Biological applications are indicated.     

Random effects modeling of multiple binomial responses using the multivariate binomial logit-normal distribution

The multivariate binomial logit-normal distribution is a mixture distribution for which, (i) conditional on a set of success probabilities and sample size indices, a vector of counts is independent binomial variates, and (ii) the vector of logits of the parameters has a multivariate normal distribution. We use this distribution to model multivariate binomial-type responses using a vector of random effects. The vector of logits of parameters has a mean that is a linear function of explanatory variables and has an unspecified or partly specified covariance matrix. The model generalizes and provides greater flexibility than the univariate model that uses a normal random effect to account for positive correlations in clustered data. The multivariate model is useful when different elements of the response vector refer to different characteristics, each of which may naturally have its own random effect. It is also useful for repeated binary measurement of a single response when there is a nonexchangeable association structure, such as one often expects with longitudinal data or when negative association exists for at least one pair of responses. We apply the model to an influenza study with repeated responses in which some pairs are negatively associated and to a developmental toxicity study with continuation-ratio logits applied to an ordinal response with clustered observations.     

Test Based on Independent,but Non-identically Distributed Random Variables

The binomial test is applied for the problem of testing a hypothesis based on a sample of independent, but non-identically distributed random variables. The used basic idea is that each random variable indicates the presence of the hypothesis. Hence each random variable is transformed such that the binomial test can be used as a simple procedure.     

Sampling for the sugarcane borer (Lepidoptera: Crambidae) on sugarcane in Louisiana.

A 3-yr study was conducted in 0.6- to 2.0-ha sugarcane fields throughout south Louisiana under varying sugarcane borer, Diatraea saccharalis (F.), density levels to determine the spatial dispersion of infestations and to develop a sequential sampling plan. Infestations of D. saccharalis were randomly dispersed. Infestation levels (percentage of stalks infested) ranged from 0.6 to 33.3%. Frequency distributions of the number of infested stalks indicated that the Poisson distribution best fit the data Tests of other distributions (negative binomial [aggregated], binomial [uniform], geometric, and hypergeometric) resulted in poorer fits. The sequential sampling plan devised, with lower and upper D. saccharalis infestation limits of 2 and 5% and 5 and 10%, required maximum average sample numbers of 7.1 and 5.5 (20-stalk samples), respectively, to make terminating management decisions. It is our assessment that implementation of these plans would decrease sampling effort by 50-60% when compared with sampling programs currently in use for D. saccharalis management decisions in Louisiana.     

Four Applications of a Bivariate Pareto Distribution

The following model, of “latent structure” type, is considered: in each subpopulation, X and Y are random variables drawn independently from the same exponential distribution, and the parameter of the exponential distribution varies between subpopulations with a Gamma density. Over the whole population, X and Y are then positively correlated, and jointly have a bivariate PARETO distribution. Four examples show how this distribution is useful in analysing ordered contingency tables in which the two dimensions can be regarded as alternative measures of the same thing: the injuries to the two drivers in a road accident, or the severity of a lesion present in a patient as assessed by two physicians, for instance. Two extensions are considered: (a) allowing X and Y to have Gamma distributions, with each subpopulation having the same shape parameter but different scale parameters; (b) allowing the scale parameter for Y to be correlated with the scale parameter for X, rather than being identical to it. A new bivariate distribution with three shape parameters is derived, expressed in terms of a generalised hypergeometric function.     

A dispersal-limited sampling theory for species and alleles

The importance of dispersal for biodiversity has long been recognized. However, it was never advertised as vigorously as Stephen Hubbell did in the context of his neutral community theory. After his book appeared in 2001, several scientists have sought and found analytical expressions for the effect of dispersal limitation on community composition, still in the neutral context. This has been done along two relatively independent lines of research that have a different mathematical approach and focus on different, yet related, types of results. Here, we study both types in a new framework that makes use of the sampling nature of the theory. We present sampling distributions that contain binomial or hypergeometric sampling on the one hand, and dispersal limitation on the other, and thus views dispersal limitation as ubiquitous as sampling effects. Further, we express the results of one line of research in terms of the other and vice versa, using the concept of subsamples. A consequence of our findings is that metacommunity size does not independently affect the outcome of neutral models in contrast to a previous assertion (Ecol. Lett., 7, 2004, p. 904) based on an incorrect formula (Phys. Rev. E, 68, 2003, p. 061902, eqns 11-14). Our framework provides the basis for development of a dispersal-limited non-neutral community theory and applies in population genetics as well, where alleles and mutation play the roles of species and speciation respectively.     

The Hyper-Negative Binomial Distribution

A generalized family of the negative binomial distribution is introduced in a paper by Srivastava, Yousry and Ahmed (1986). It is a solution of the difference equation and is called the hyper-negative binomial distribution. Certain properties including the moments of the distribution are presented. Moment estimators of the parameters are obtained and goodness of fit is illustrated for chromatid aberration data.     

Symmetric Bernoulli Distributions and Generalized Binomial Distributions

The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.     

Characterization of the Negative Binomial and Gamma Distributions

Characterization of the negative binomial and gamma distributions by a conditional distribution and a linear regression, and the gamma distribution by the negative binomial distribution are given. An application to a random shock model is discussed.     

The generalized linear model for spatial data: assessing the effects of environmental covariates on population density in the field

With interest in spatial ecology growing, correlational field studies are likely to become increasingly important. Unfortunately, ecological field data often do not follow the assumptions of classical statistics, so techniques like the popular and powerful multiple linear regression and its variants are often unreliable, and results can be misleading. The generalized linear model (GLM) is a flexible extension of linear regression that has proved especially useful for discrete data. In this paper, the technique is adapted to accommodate spatially correlated, discrete data. Specifically, to demonstrate the approach, Japanese beetle grub [Popillia japonica Newman (Coleoptera, Scarabaeidae)] population density in the field is modeled as a function of soil organic matter content. The response variable (grub counts in small soil samples) was a spatially autocorrelated, discrete random variable. Three classes of GLMs of the association between soil organic matter content and grub density were compared: (i) regression (assuming normally distributed response variable), (ii) GLM assuming negative binomial counts, and (iii) GLM based on the assumption that the counts conformed to Taylor's power law (TPL). Because the grubs were distributed in patches rather than at random, models that explicitly accounted for the spatial autocorrelation of grub counts were constructed, and compared with models that assumed independent observations. The fitted values for the discrete GLMs [viz., (ii) and (iii)] differed noticeably from the fitted values from multiple regression; but fitted values among the negative binomial and TPL GLMs were virtually identical, regardless of whether the spatial covariance was incorporated into a model, whether a spherical or exponential variogram model was used, or whether variance function parameters were estimated over a large or small scale. However, P‐values for the overall significance of the models depended heavily on whether the GLM assumed a discrete or continuous response variable, and whether or not spatial autocorrelation in the response variable was accounted for. On average, P‐values were 45‐fold higher in the spatial GLMs than in the non‐spatial and 23‐fold higher in the discrete GLMs than in the continuous.     

A hierarchical Bayesian model for combining multiple 2 x 2 tables using conditional likelihoods

This paper introduces a hierarchical Bayesian model for combining multiple 2 x 2 tables that allows the flexibility of different odds ratio estimates for different tables and at the same time allows the tables to borrow information from each other. The proposed model, however, is different from a full Bayesian model in that the nuisance parameters are eliminated by conditioning instead of integration. The motivation is a more robust model and a faster and more stable Gibbs algorithm. We work out a Gibbs scheme using the adaptive rejection sampling for log concave density and an algorithm for the mean and variance of the noncentral hypergeometric distribution. The model is applied to a multicenter ulcer clinical trial.     

Correlation between the numbers of two types of children when the family size distribution is zero-truncated negative binomial.

Let family size N be a random variable formed from two types of children B and C (B + C = N). The correlation coefficient between B and C has been recently studied by Rao et al. [1973]. We consider this problem when the frequency of fertile childless families (N = 0) is not available or cannot be estimated. Assuming the distribution of N is zero-truncated negative binomial, a formula for the correlation coefficient is derived. Applying this formula to the data of Reed and Reed [1965] gives a correlation coefficient of 0.27. The corresponding empirical coefficient for non-zero size families calculated directly from the data is 0.24.     

The Non-Central Negative Binomial Distribution

The non-central negative binomial distribution is derived by mixing the POISSON distribution with a certain BESSEL function distribution, or the negative binomial distribution with the POISSON distribution. Some properties of the distribution are discussed, including a characterization. The fitting of the distribution to some data are given together with the efficiency for various procedures of estimation.     

Spatial distribution of nests of Acromyrmex crassispinus (Forel) (Hymenoptera: Formicidae) in Pinus taeda plantations

The spatial distribution of insects is essential to perform control strategies, to improve sample techniques and to estimate economic losses. We aimed to determine the spatial distribution of nests of Acromyrmex crassispinus (Forel) in Pinus taeda plantations. The experiments were carried out in P. taeda plantations with different ages (treatments: recently-planted, three and six-year old plants). The study took place in Rio Negrinho and in Três Barras, SC. Three plots of one hectare were delimited in each treatment, and plots were divided in 64 sample units. The analysis of the dispersion index [variance/mean relationship (I), index of Morisita (Iδ) and k exponent of negative binomial distribution] showed that the majority of the samplings presented random distribution. Among the three distributions of probabilities studied: Poisson, positive binomial and negative binomial, the Poisson distribution was the best model to fit the spatial distribution of A. crassispinus nests in all samplings. The result was a random distribution in the plantings of different ages.     

吉林蛟河阔叶红松林物种多度分布模型研究

以吉林蛟河阔叶红松林的木本植物为研究对象,将30hm²的样地面积划分为5m×5m,10m×10m,20m×20m,25m×25m的连续取样单元,在4个不同尺度下分别统计各物种在每个取样单元中的有无,得到每个物种在不同尺度下的取样单元数。利用随机分布模型和负二项分布模型分析物种的多度分布,对比预测多度与观测多度讨论两个模型的科学性与实用性。结果表明：对于阔叶红松林而言,负二项分布模型在所有研究尺度上的预测精度都要优于随机分布模型。随机分布和负二项分布的模型预测误差随着研究尺度的增大而增大,因此选取较小的取样单元可以切实提高物种多度的预测精度。利用随机分布和负二项分布模型对多度较小的物种进行预测的效果要优于多度较大的物种。负二项分布模型适合用来模拟阔叶红松林的物种多度分布格局,并且模型的拟合效果受取样单元大小影响。     

The number distribution for involved lymph nodes in cancer

The number of involved lymph nodes exhibits considerable heterogeneity within populations. Here, the implications of population heterogeneity are explored with respect to the kinematics of nodal metastases. Data from the National Cancer Institute's Surveillance, Epidemiology, and End Results program for 224656 breast, 12404 gastric, 18015 rectal, 4117 cervical and 2443 laryngeal cancers as well as 9118 melanomas were used to construct frequency distributions for the number of involved nodes which were then fitted to the negative binomial distribution. The negative binomial distribution described the heterogeneity in nodal involvement well. The patterns of nodal involvement can be explained by either of two models: one where involved nodes could seed further nodal metastases, the other where the number of nodal metastases in any individual was randomly distributed, with the deviations between patients accounted for by population heterogeneity. Since the number of sampled nodes similarly approximated a negative binomial distribution, random involvement with superimposed population heterogeneity would more credibly explain both sets of observations.     

A new approach for assessment of mental architecture: repeated tagging

A new approach to the study of a relatively neglected property of mental architecture-whether and when the already-processed elements are separated from the to-be-processed elements-is proposed. The process of numerical proportion discrimination between two sets of elements defined either by color or by orientation can be described as sampling with or without replacement (characterized by binomial or hypergeometric probability distributions respectively) depending on the possibility to tag an element once or repeatedly. All empirical psychometric functions were approximated by a theoretical model showing that the ability to keep track of the already tagged elements is not an inflexible part of the mental architecture but rather an individually variable strategy which also depends on conspicuity of perceptual attributes. Strong evidence is provided that in a considerable number of trials, observers tagged the same element repeatedly which can only be done serially at two separate time moments.     

An efficient binomial model-based measure for sequence comparison and its application

Sequence comparison is one of the major tasks in bioinformatics, which could serve as evidence of structural and functional conservation, as well as of evolutionary relations. There are several similarity/dissimilarity measures for sequence comparison, but challenges remains. This paper presented a binomial model-based measure to analyze biological sequences. With help of a random indicator, the occurrence of a word at any position of sequence can be regarded as a random Bernoulli variable, and the distribution of a sum of the word occurrence is well known to be a binomial one. By using a recursive formula, we computed the binomial probability of the word count and proposed a binomial model-based measure based on the relative entropy. The proposed measure was tested by extensive experiments including classification of HEV genotypes and phylogenetic analysis, and further compared with alignment-based and alignment-free measures. The results demonstrate that the proposed measure based on binomial model is more efficient.