Modeling the dependence between number of trials and success probability in beta-binomial-Poisson mixture distributions

Beta-binomial models are widely used for overdispersed binomial data, with the binomial success probability modeled as following a beta distribution. The number of binary trials in each binomial is assumed to be nonrandom and unrelated to the success probability. In many behavioral studies, however, binomial observations demonstrate more complex structures. In this article, a general beta-binomial-Poisson mixture model is developed, to allow for a relation between the number of trials and the success probability for overdispersed binomial data. An EM algorithm is implemented to compute both the maximum likelihood estimates of the model parameters and the corresponding standard errors. For illustration, the methodology is applied to study the feeding behavior of green-backed herons in two southeastern Missouri streams.     

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.     

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.     

A runs test based on run lengths

A procedure is proposed for testing the hypothesis that Bernoulli trials (successes and failures) are independent with common probability of success. Equivalently, the procedure may be used to test the hypothesis that the arrangement of a fixed number of successes and failures was determined randomly. The procedure is based on a weighted linear combination of the variances of run lengths of successes and failures. It is shown to have desirable asymptotic properties and to be generally more powerful than the usual runs test, while preserving computational simplicity.     

Modeling the dependence between the number of trials and the success probability in binary trials

A model for binary trials based on a bivariate generalization of the Poisson process for both the number of successes and number of trials with the transition rates dependent on the accumulating numbers of successes and trials is used to reanalyze some recently published data of Zhu, Eickhoff, and Kaiser (2003, Biometrics59, 955-961). This modeling admits alternative distributions for the numbers of trials and the numbers of successes conditional on the number of trials which generalize the Poisson and binomial distributions, without some of the restrictions apparent in the beta-binomial-Poisson mixed modeling of Zhu et al. (2003). Some quite marked differences between the results of this analysis and those described in Zhu et al. (2003) are apparent.     

Tutorial on large deviations for the binomial distribution

We present, in an easy to use form, the large deviation theory of the binomial distribution: how to approximate the probability ofk or more successes inn independent trials, each with success probabilityp, when the specified fraction of successes,a≡k/n, satisfies 0<p<a<1. Supported by NIH grant GM 36230 and NSF grant DMS 8601986. Supported by NIH grant GM 36230 and a grant from the System Development Foundation.     

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).     

Monitoring futility and efficacy in phase II trials with Bayesian posterior distributions—A calibration approach

A multistage single arm phase II trial with binary endpoint is considered. Bayesian posterior probabilities are used to monitor futility in interim analyses and efficacy in the final analysis. For a beta‐binomial model, decision rules based on Bayesian posterior probabilities are converted to “traditional” decision rules in terms of number of responders among patients observed so far. Analytical derivations are given for the probability of stopping for futility and for the probability to declare efficacy. A workflow is presented on how to select the parameters specifying the Bayesian design, and the operating characteristics of the design are investigated. It is outlined how the presented approach can be transferred to statistical models other than the beta‐binomial model.     

Accuracy and selection success in yield trial analyses

Summary Yield trials serve research purposes of estimation and selection. Order statistics are used here to quantify the successes or problems to be expected in selection tasks commonly encountered in breeding and agronomy. Greater accuracy of yield estimates implies greater selection success. A New York soybean yield trial serves as a specific example. The Additive Main effects and Multiplicative Interaction (AMMI) statistical model is used to increase the accuracy of these soybean yield estimates, thereby increasing the probability of successfully selecting, on the basis of the empirical yield data, that genotype which has the maximum true mean. The statistical strategy for increasing accuracy is extremely cost effective relative to the alternative strategy of increasing the number of replications. Better selections increase the speed and effectiveness of breeding programs, and increase the reliability of variety recommendations. Selection tasks are frequently more difficult than may be suspected.This research was supported by the Rhizobotany Project of the USDA-ARS     

The effects of turbidity and an invasive species on foraging success of rosyside dace (Clinostomus funduloides)

1.  Habitat degradation and biological invasions are important threats to fish diversity worldwide. We experimentally examined the effects of turbidity, velocity and intra- and interspecific competition on prey capture location, reactive distance and prey capture success of native rosyside dace ( Clinostomus funduloides ) and invasive yellowfin shiners ( Notropis lutipinnis ) in Coweeta Creek, North Carolina, U.S.A.
2.  Increased turbidity and velocity produced significant decreases in the number of prey captured forward of the fish's location. It is possible that this represents an increase in the amount of energy expended per prey captured.
3.  We used Akaike's Information Criterion (AIC) to evaluate competing explanatory models for reactive distance (10 generalised linear models, GLM) and prey capture success (9 generalised linear mixed models, GLMM).
4.  Reactive distance decreased by 12% with an increase from 2 to 4 conspecifics, whereas a 10 NTU increase in turbidity reduced reactive distance by 9%. Capture success was affected by velocity, dominance and competition, and varied among species. A 6 cm s⁻¹ increase in velocity produced a 28% decline in capture probability; however, dominant fish were 3.2 times more likely to capture a prey item than non-dominant fish. Yellowfin shiners only were 0.62 times as likely to capture a prey item as rosyside dace. Both intra- and interspecific competition reduced capture probability, and fish in high density intraspecific or interspecific trials were 0.46 times and 0.44 times as likely to capture prey, respectively, as fish in two fish intraspecific trials.
5.  These results suggest behavioural variables are as important as physical factors in determining reactive distance and capture probability by these minnows.     

The effect of a gradual response to the environment on species distribution modeling performance

Species distribution models (SDMs) have been widely used in ecology, biogeography, and conservation. Although ecological theory predicts that species occupancy is dynamic, the outputs of SDMs are generally converted into a single occurrence map, and model performance is evaluated in terms of success to predict presences and absences. The aim of this study was to characterize the effects of a gradual response in species occupancy to environmental gradients into the performance of SDMs. First we outline guidelines for the appropriate simulation of artificial species that allows controlling for gradualism and prevalence in the occupancy patterns over an environmental gradient. Second, we derive theoretical expected values for success measures based on presence‐absence predictions (AUC, Kappa, sensitivity and specificity). And finally we used artificial species to exemplify and test the effect of a gradual probabilistic occupancy response to environmental gradients on SDM performance. Our results show that when a species responds gradually to an environmental gradient, conventional measures of SDM predictive success based on presence‐absence cannot be expected to attain currently accepted performance values considered as good, even for a model that recovers perfectly well the true probability of occurrence. A gradual response imposes a theoretical expected value for these measures of performance that can be calculated from the species properties. However, irrespective of the statistical modeling strategy used and of how gradual the species response is, one can recover the true probability of occurrence as a function of environmental variables provided that species and sample prevalence are similar. Therefore, model performance based on presence‐absence should be judged against the theoretical expected value rather than to absolute values currently in use such as AUC > 0.8. Overall, we advocate for a wider use of the probability of occurrence and emphasize the need for further technical developments in this sense.     

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.     

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.     

Using probability of persistence to identify important areas for biodiversity conservation

Most attempts to identify important areas for biodiversity have sought to represent valued features from what is known of their current distribution, and have treated all included records as equivalent. We develop the idea that a more direct way of planning for conservation success is to consider the probability of persistence for the valued features. Probabilities also provide a consistent basis for integrating the many pattern and process factors affecting conservation success. To apply the approach, we describe a method for seeking networks of conservation areas that maximize probabilities of persistence across species. With data for European trees, this method requires less than half as many areas as an earlier method to represent all species with a probability of at least 0.95 (where possible). Alternatively, for trials choosing any number of areas between one and 50, the method increases the mean probability among species by more than 10%. This improvement benefits the least-widespread species the most and results in greater connectivity among selected areas. The proposed method can accommodate local differences in viability, vulnerability, threats, costs, or other social and political constraints, and is applicable in principle to any surrogate measure for biodiversity value.     

Site occupancy models with heterogeneous detection probabilities

Models for estimating the probability of occurrence of a species in the presence of imperfect detection are important in many ecological disciplines. In these "site occupancy" models, the possibility of heterogeneity in detection probabilities among sites must be considered because variation in abundance (and other factors) among sampled sites induces variation in detection probability (p). In this article, I develop occurrence probability models that allow for heterogeneous detection probabilities by considering several common classes of mixture distributions for p. For any mixing distribution, the likelihood has the general form of a zero-inflated binomial mixture for which inference based upon integrated likelihood is straightforward. A recent paper by Link demonstrates that in closed population models used for estimating population size, different classes of mixture distributions are indistinguishable from data, yet can produce very different inferences about population size. I demonstrate that this problem can also arise in models for estimating site occupancy in the presence of heterogeneous detection probabilities. The implications of this are discussed in the context of an application to avian survey data and the development of animal monitoring programs.     

我国林火发生预测模型研究进展

通过文献回顾,总结了国内林火发生预测模型的研究现状,并从林火发生驱动因子、林火发生概率预测模型、林火发生频次预测模型和模型检验方法等方面进行归纳分析。得出以下结论: 1)气象、地形、植被、可燃物、人类活动等因素是影响林火发生及模型预测精度的主要驱动因子;2)林火发生概率模型中,地理加权逻辑斯蒂回归模型考虑了变量之间的空间相关性,Gompit回归模型适宜非对称结构的林火数据,随机森林模型不需要多重共线性检验,在避免过度拟合的同时提高了预测精度,是林火发生概率预测模型的优选方法之一;3)林火发生频次模型中,负二项回归模型更适合对过度离散数据进行模拟,零膨胀模型和栅栏模型可以处理林火数据中包含大量零值的问题;4)ROC检验、AIC检验、似然比检验和Wald检验方法是林火概率和频次模型的常用检验方法。林火发生预测模型研究仍是我国当前林火管理工作的重点,预测模型的选择需要依据不同地区林火数据特点。此外,构建林火预测模型时需要考虑更多的影响因素,以提高模型预测精度;未来,需要进一步探索其他数学模型在林火发生预测中的应用,不断提高林火发生预测模型的准确度。     

Beals smoothing revisited

Beals smoothing is a multivariate transformation specially designed for species presence/absence community data containing noise and/or a lot of zeros. This transformation replaces the observed values of the target species by predictions of occurrence on the basis of its co-occurrences with the remaining species. In many applications, the transformed values are used as input for multivariate analyses. As Beals smoothing values provide a sense of “probability of occurrence”, they have also been used for inference. However, this transformation can produce spurious results, and it must be used with caution. Here we study the statistical and ecological bases underlying the Beals smoothing function, and the factors that may affect the reliability of transformed values are explored using simulated data sets. Our simulations demonstrate that Beals predictions are unreliable for target species that are not related to the overall ecological structure. Furthermore, the presence of these “random” species may diminish the quality of Beals smoothing values for the remaining species. A statistical test is proposed to determine when observed values can be replaced with Beals smoothing predictions. Two real-data example applications are presented to illustrate the potentially false predictions of Beals smoothing and the necessary checking step performed by the new test. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

A frequency matrix for probabilistic identification of some bacilli

A matrix comprising frequencies for positive results for 44 Bacillus taxa for 30 characters has been constructed. The 44 taxa include most of the common species and several clusters of environmental isolates including those described as B. firmus-B. lentus intermediates. The tests, which were chosen for their high diagnostic value, included some of the traditional tests used for identification of bacilli supplemented with a range of sugar fermentations and other characterization tests. The matrix was evaluated by identifying hypothetical median organisms, cluster representatives and a panel of 23 reference strains. All reference strains achieved Willcox probabilities above 0.995. Fifty-eight environmental isolates were also subjected to the 30 tests and identification was attempted. Forty-one strains (70%) achieved a Willcox probability greater than 0.95, which was considered an acceptable identification, and were assigned to 12 taxa. If the SE of taxonomic distance was also considered in the identification score (an acceptable value being less than 7.0), the number of acceptable identifications was reduced to 34 (59%). It was encouraging that bacteria from garden soils identified to the common species such as B. subtilis, B. cereus and B. licheniformis whereas some of the bacteria from an estuarine habitat were identified as species such as B. firmus which are normally identified with that habitat.     

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.     

A model for estimating the minimum number of offspring to sample in studies of reproductive success

Molecular parentage permits studies of selection and evolution in fecund species with cryptic mating systems, such as fish, amphibians, and insects. However, there exists no method for estimating the number of offspring that must be assigned parentage to achieve robust estimates of reproductive success when only a fraction of offspring can be sampled. We constructed a 2-stage model that first estimated the mean (μ) and variance (v) in reproductive success from published studies on salmonid fishes and then sampled offspring from reproductive success distributions simulated from the μ and v estimates. Results provided strong support for modeling salmonid reproductive success via the negative binomial distribution and suggested that few offspring samples are needed to reject the null hypothesis of uniform offspring production. However, the sampled reproductive success distributions deviated significantly (χ(2) goodness-of-fit test p value < 0.05) from the known simulated reproductive success distribution at rates often >0.05 and as high as 0.24, even when hundreds of offspring were assigned parentage. In general, reproductive success patterns were less accurate when offspring were sampled from cohorts with larger numbers of parents and greater variance in reproductive success. Our model can be reparameterized with data from other species and will aid researchers in planning reproductive success studies by providing explicit sampling targets required to accurately assess reproductive success.