Binomial transformation applied to presence-absence community data

Community data is often transformed or standardized to meet the requirements and assumptions of multivariate analysis. While these methods are usually appropriate for abundance data, they are seldom applied to presence-absence data. Here, a method of transforming a binary matrix using the binomial probability is described. Number of trials (n), number of successes (x) and probability of success (p) are necessary to compute the binomial probability. Successes were defined as the number of sites where the species occurrence can be considered; trials were equal and greater than the number of successes. The actual occurrence of each species along the gradient was considered the probability of success. The Mantel statistic associated with the binomially transformed distance matrix and the distance matrix based on binary data were used to choose an appropriate binomial transformation. The chosen binomial transformation gave greater value to species indicating habitat typologies. Binomially transformed data rendered results closer to expectations.     

The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

On models for binomial data with random numbers of trials

A binomial outcome is a count s of the number of successes out of the total number of independent trials n=s+f, where f is a count of the failures. The n are random variables not fixed by design in many studies. Joint modeling of (s, f) can provide additional insight into the science and into the probability pi of success that cannot be directly incorporated by the logistic regression model. Observations where n= 0 are excluded from the binomial analysis yet may be important to understanding how pi is influenced by covariates. Correlation between s and f may exist and be of direct interest. We propose Bayesian multivariate Poisson models for the bivariate response (s, f), correlated through random effects. We extend our models to the analysis of longitudinal and multivariate longitudinal binomial outcomes. Our methodology was motivated by two disparate examples, one from teratology and one from an HIV tertiary intervention study.     

Biological dosimetry of chernobyl cleanup workers: inclusion of data on age and smoking provides improved radiation dose estimates

We report the results of a study of chromosome translocations in 126 Russian subjects who participated in the cleanup activities at Chernobyl and another 53 subjects, from other places in Russia, who were not exposed at Chernobyl. In agreement with our earlier study, we find increased translocation frequencies among the exposed compared to Russian controls. We describe statistical methods for estimating the dose of ionizing radiation determined by scoring chromosome translocations found in circulating lymphocytes sampled several years after exposure. Two statistical models were fitted to the data. One model assumed that translocation frequencies followed an overdispersed Poisson distribution. The second model assumed that translocation frequencies followed a negative binomial distribution. In addition, the effects of radiation exposure were modeled as additive or as multiplicative to the effects of age and smoking history. We found that the negative binomial model fit the data better than the overdispersed Poisson model. We could not distinguish between the additive and the multiplicative model with our data. Individual dose estimates ranged from 0 (for 43 subjects) to 0.56 Gy (mean 0.14 Gy) under the multiplicative model and from 0 to 0.95 Gy (mean 0.15 Gy) under the additive model. Dose estimates were similar under the two models when the number of translocations was less than 4 per 100 cells. The additive model tended to estimate larger doses when the number of translocations was greater than 4 per 100 cells. We also describe a method for estimating upper 95% tolerance bounds for numbers of translocations in unexposed individuals. We found that inclusion of data on age and smoking history was important for dose estimation. Ignoring these factors could result in gross overestimation of exposures, particularly in older subjects who smoke.     

Point and Interval Estimation of the Population Size Using a Zero‐Truncated Negative Binomial Regression Model

This paper presents the zero‐truncated negative binomial regression model to estimate the population size in the presence of a single registration file. The model is an alternative to the zero‐truncated Poisson regression model and it may be useful if the data are overdispersed due to unobserved heterogeneity. Horvitz–Thompson point and interval estimates for the population size are derived, and the performance of these estimators is evaluated in a simulation study. To illustrate the model, the size of the population of opiate users in the city of Rotterdam is estimated. In comparison to the Poisson model, the zero‐truncated negative binomial regression model fits these data better and yields a substantially higher population size estimate. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Confidence limits of the probability of success in animal experiments and clinical studies: A Bayesian approach

PurposeTo determine from the number of trials, n, and the number of observed successes, k the most probable value, the variance and the confidence limits of the probability of success, p, in animal experiments and clinical studies subject to binomial statistics.MethodIn such experiments the probability of success is an unknown parameter. The Bayesian approach to the problem is advocated, based on constructed distribution of the probability of success.ResultsA simple Matlab code for the calculation of the confidence limits according to the proposed method is provided. The most probable, the mean, the variance and the confidence limits are calculated applying the usual definitions of these characteristics.ConclusionThe proposed method works for any number of trials – large and small and all possible values of the number of successes, including k = 0 and k = n, providing exact formulae for the calculation of the confidence limits in all cases.     

When no catches matter: Coping with zeros in environmental assessments

The environmental legislation of many countries increasingly requires the continuous monitoring of fish assemblages to evaluate the success of river and stream restorations. Predicting species–environment relationships on the basis of monitoring data is central in the evaluation of ecological integrity and planning of rehabilitation strategies. Monitoring data are, however, often plagued by a substantial proportion of zeros (no catch at single sampling points) which are caused by relevant ecological processes, but complicate the use of commonly used statistical methods. This study compares mere count regression models, mixture and hurdle models based on Poisson and negative binomial distribution and logistic regressions with respect to their ability to cope with large zero-inflated data sets obtained by point abundance sampling of young-of-the-year fish from three large German rivers. Only mixture and hurdle models based on negative binomial distribution could satisfactorily be fitted to the zero-inflated and overdispersed count data. The logistic regression models applied to transliterated catch data simplified the computational procedure and yielded qualitative similar results to the count regression models indicating that the use of more complex count data did not generally provide better predictions. Therefore, presence/absence sampling may be a suitable and less costly alternative to abundance surveys for identifying environmental factors which affect the spatial distribution of fish populations at least if information on subtly abundance fluctuations is not needed. Mixture or hurdle models are particularly worth the additional effort if it is reasonable to distinguish between those environmental factors influencing the occurrence probability and others affecting the abundance. All models showed low sensitivity to rare guilds pointing to the need for a further development of statistical models for rare species whose management is a matter of growing environmental concern.     

Estimating divergence times in the presence of an overdispersed molecular clock

Molecular loci that fail relative-rate tests are said to be "overdispersed." Traditional molecular-clock approaches to estimating divergence times do not take this into account. In this study, a method was developed to estimate divergence times using loci that may be overdispersed. The approach was to replace the traditional Poisson process assumption with a more general stationary process assumption. A probability model was developed, and an accompanying computer program was written to find maximum-likelihood estimates of divergence times under both the Poisson process and the stationary process assumptions. In simulation, it was shown that confidence intervals under the traditional Poisson assumptions often vastly underestimate the true confidence limits for overdispersed loci. Both models were applied to two data sets: one from land plants, the other from the higher metazoans. In both cases, the traditional Poisson process model could be rejected with high confidence. Maximum-likelihood analysis of the metazoan data set under the more general stationary process suggested that their radiation occurred well over a billion years ago, but confidence intervals were extremely wide. It was also shown that a model consistent with a Cambrian (or nearly Cambrian) origination of the animal phyla, although significantly less likely than a much older divergence, fitted the data well. It is argued that without an a priori understanding of the variance in the time between substitutions, molecular data sets may be incapable of ever establishing the age of the metazoan radiation.     

Overdispersed fungus germination data: statistical analysis using R

ABSTRACT

Proportion data from dose-response experiments are often overdispersed, characterised by a larger variance than assumed by the standard binomial model. Here, we present different models proposed in the literature that incorporate overdispersion. We also discuss how to select the best model to describe the data and present, using R software, specific code used to fit and interpret binomial, quasi-binomial, beta-binomial, and binomial-normal models, as well as to assess goodness-of-fit. We illustrate applications of these generalized linear models and generalized linear mixed models with a case study from a biological control experiment, where different isolates of Isaria fumosorosea (Hypocreales: Cordycipitaceae) were used to assess which ones presented higher resistance to UV-B radiation. We show how to test for differences between isolates and also how to statistically group isolates presenting a similar behaviour.     

On a Model for Estimating the Parameter of the Human Reproductive Process

Attempt has been made in this paper to estimate certain parameters (data pertaining to which are either not available or easily reportable) of the human reproductive process as the period of postpartum ammenorrhoea (P.P.A.), number of foetal wastages in between live births etc., using a truncated negative binomial probability model. In view of the hypothesis that the probability of foetal wastages varies from mother to mother, the truncated negative binomial distribution has been compounded by weighing with the best prior Beta distribution of the parameter. Estimation has been made by successive approximation using the method of moments.     

A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives

Count data often show a higher incidence of zero counts than would be expected if the data were Poisson distributed. Zero-inflated Poisson regression models are a useful class of models for such data, but parameter estimates may be seriously biased if the nonzero counts are overdispersed in relation to the Poisson distribution. We therefore provide a score test for testing zero-inflated Poisson regression models against zero-inflated negative binomial alternatives.     

Simultaneous confidence intervals for a success probability and intraclass correlation,with an application to screening mammography

This paper provides asymptotic simultaneous confidence intervals for a success probability and intraclass correlation of the beta‐binomial model, based on the maximum likelihood estimator approach. The coverage probabilities of those intervals are evaluated. An application to screening mammography is presented as an example. The individual and simultaneous confidence intervals for sensitivity and specificity and the corresponding intraclass correlations are investigated. Two additional examples using influenza data and sex ratio data among sibships are also considered, where the individual and simultaneous confidence intervals are provided.     

THE DIRICHLET-MULTINOMIAL MODEL: ACCOUNTING FOR INTER-TRIAL VARIATION IN REPLICATED RATINGS

Differences in sensory acuity and hedonic reactions to products lead to latent groups in pooled ratings data. Manufacturing locations and time differences also are sources of rating heterogeneity. Intensity and hedonic ratings are ordered categorical data. Categorical responses follow a multinomial distribution and this distribution can be applied to pooled data over trials if the multinomial probabilities are constant from trial to trial. The common test statistic used for comparing vectors of proportions or frequencies is the Pearson chi-square statistic. When ratings data are obtained from repeated ratings experiments or from a cluster sampling procedure, the covariance matrix for the vector of category proportions can differ dramatically from the one assumed for the multinomial model because of inter-trial. This effect is referred to as overdispersion. The standard multinomial model does not fit overdispersed multinomial data. The practical implication of this is that an inflated Type I error can result in a seriously erroneous conclusion. Another implication is that overdispersion is a measurable quantity that may be of interest because it can be used to signal the presence of latent segments. The Dirichlet-Multinomial (DM) model is introduced in this paper to fit overdispersed intensity and hedonic ratings data. Methods for estimating the parameters of the DM model and the test statistics based on them to test against a specified vector or compare vectors of proportions are given. A novel theoretical contribution of this paper is a method for calculating the power of the tests. This method is useful both in evaluating the tests and determining sample size and the number of trials. A test for goodness of fit of the multinomial model against the DM model is also given. The DM model can be extended further to the Generalized Dirichlet-Multinomial (GDM) model, in which multiple sources of variation are considered. The GDM model and its applications are discussed in this paper. Applications of the DM and GDM models in sensory and consumer research are illustrated using numerical examples.     

Bacterial Density in Water Determined by Poisson or Negative Binomial Distributions

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.     

Efficient Simulation of Multivariate Binomial and Poisson Distributions

Power investigations, for example, in statistical procedures for the assessment of agreement among multiple raters often require the simultaneous simulation of several dependent binomial or Poisson distributions to appropriately model the stochastical dependencies between the raters' results. Regarding the rather large dimensions of the random vectors to be generated and the even larger number of interactions to be introduced into the simulation scenarios to determine all necessary information on their distributions' dependence stucture, one needs efficient and fast algorithms for the simulation of multivariate Poisson and binomial distributions. Therefore two equivalent models for the multivariate Poisson distribution are combined to obtain an algorithm for the quick implementation of its multivariate dependence structure. Simulation of the multivariate Poisson distribution then becomes feasible by first generating and then convoluting independent univariate Poisson variates with appropriate expectations. The latter can be computed via linear recursion formulae. Similar means for simulation are also considered for the binomial setting. In this scenario it turns out, however, that exact computation of the probability function is even easier to perform; therefore corresponding linear recursion formulae for the point probabilities of multivariate binomial distributions are presented, which only require information about the index parameter and the (simultaneous) success probabilities, that is the multivariate dependence structure among the binomial marginals.     

Calculating census efficiency for river birds: a case study with the White-throated Dipper Cinclus cinclus in the Pyrénées

Using the binomial law we modelled field data to estimate the probability ( ̂ ) of detecting pairs of breeding White-throated Dippers, and the population size ( ̂ ± confidence limits). The model was divided into two parts according to whether the actual size of the population under study was known or not; in the latter case the truncated binomial model was used. Dipper abundance data were collected from three 4-km-long river tracts in the Pyrénées (France) during the breeding seasons of different years. Goodness-of-fit tests indicated that the binomial model fitted the data well. For a given visit during the survey, the estimated probability of detecting any pair of Dippers if they were present was always high (0.63–0.94) and constant from year to year but not between sites. Estimations ( ̂ ) of the size of the population provided by the binomial model were very close to that derived from mapping techniques. This study provides the first ever quantification of the number of visits required to detect birds on linear territories: three visits were necessary to detect the whole breeding population.     

Functional heterogeneity at the frog sensory-motor synapse

This article compares four models of amplitude fluctuations in postsynaptic potentials. The convolution of two binomial distributions and the beta model proved the best fit with experimentally obtained data (as compared with the binomial model). The beta model is based on the assumption that the probability of quantal transmitter release is a random variable with a beta distribution. Numbers of quantal generators as estimated by the beta model were found to resemble numbers of morphological identifiable synaptic boutons. Findings from research using this model showed that the binomial parameter n may be interpreted as the number of transmitter release sites functioning with a probability in excess of 0.2. The findings obtained confirm the postulated functional diversity of release sites at interneuronal synapses.I. M. Sechenov Institute of Evolutionary Physiology and Biochemistry, Academy of Sciences of the USSR, Leningrad. Translated from Neirofiziologiya, Vol. 21, No. 6, pp. 780–788, November–December, 1989.     

Mate detection success of maleClethrionomys rufocanus in relation to the spatial distribution of sexually receptive females

Summary Mate detection success of male grey-sided voles,Clethrionomys rufocanus, in relation to the spatial distribution of sexually receptive females was studied in an experimental island population. The spatiotemporal distribution of receptive females was controlled by containing females in small, mobile wire-mesh cages, whereas the response by free-ranging males was monitored by means of radiotelemetry. Males were on average more successful in finding oestrous females when females were spatially clumped than when females were spatially overdispersed. In addition, the variance (CV) in male mate detecting success was highest when females had an overdispersed spatial distribution. These results are consistent with predictions from a theoretical model (Ims, 1988b) analysing the effect of mate distribution on male mating success, and with empirical results on prey detection success of predators searching for prey.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

Neurotransmitter release statistics: Moment estimates for inhomogeneous Bernoulli trials

Summary Evoked release of quanta of neurotransmitter is generally treated as a set of homogeneous, stationary Bernoulli trials, hence governed by the binomial distribution. Relaxing the assumptions of uniformity and stationarity leads to a more realistic physiological model of transmitter release but also introduces systematic biases in the moment estimates of the binomial parameters. We derive probability generating functions for quantal release and expressions for the moment estimates of ¯n and ¯p for a generalized model that incorporates temporal variation and nonuniformity in individual release probabilities and in numbers of release sites.