Poisson Lognormal Research Articles

A comparative analysis of alternative approaches to fitting species-abundance models

Fits of species-abundance distributions to empirical data are increasingly used to evaluate models of diversity maintenance and community structure and to infer properties of communities, such as species richness. Two distributions predicted by several models are the Poisson lognormal (PLN) and the negative binomial (NB) distribution; however, at least three different ways to parameterize the PLN have been proposed, which differ in whether unobserved species contribute to the likelihood and in whether the likelihood is conditional upon the total number of individuals in the sample. Each of these has an analogue for the NB. Here, we propose a new formulation of the PLN and NB that includes the number of unobserved species as one of the estimated parameters. We investigate the performance of parameter estimates obtained from this reformulation, as well as the existing alternatives, for drawing inferences about the shape of species abundance distributions and estimation of species richness. We simulate the random sampling of a fixed number of individuals from lognormal and gamma community relative abundance distributions, using a previously developed 'individual-based' bootstrap algorithm. We use a range of sample sizes, community species richness levels and shape parameters for the species abundance distributions that span much of the realistic range for empirical data, generating 1 000 simulated data sets for each parameter combination. We then fit each of the alternative likelihoods to each of the simulated data sets, and we assess the bias, sampling variance and estimation error for each method. Parameter estimates behave reasonably well for most parameter values, exhibiting modest levels of median error. However, for the NB, median error becomes extremely large as the NB approaches either of two limiting cases. For both the NB and PLN, >90% of the variation in the error in model parameters across parameter sets is explained by three quantities that corresponded to the proportion of species not observed in the sample, the expected number of species observed in the sample and the discrepancy between the true NB or PLN distribution and a Poisson distribution with the same mean. There are relatively few systematic differences between the four alternative likelihoods. In particular, failing to condition the likelihood on the total sample sizes does not appear to systematically increase the bias in parameter estimates. Indeed, overall, the classical likelihood performs slightly better than the alternatives. However, our reparameterized likelihood, for which species richness is a fitted parameter, has important advantages over existing approaches for estimating species richness from fitted species-abundance models.

Measuring safety treatment effects using full Bayes non-linear safety performance intervention functions

Full Bayes linear intervention models have been recently proposed to conduct before–after safety studies. These models assume linear slopes to represent the time and treatment effects across the treated and comparison sites. However, the linear slope assumption can only furnish some restricted treatment profiles. To overcome this problem, a first-order autoregressive (AR1) safety performance function (SPF) that has a dynamic regression equation (known as the Koyck model) is proposed. The non-linear ‘Koyck’ model is compared to the linear intervention model in terms of inference, goodness-of-fit, and application. Both models were used in association with the Poisson-lognormal (PLN) hierarchy to evaluate the safety performance of a sample of intersections that have been improved in the Greater Vancouver area. The two models were extended by incorporating random parameters to account for the correlation between sites within comparison–treatment pairs. Another objective of the paper is to compute basic components related to the novelty effects, direct treatment effects, and indirect treatment effects and to provide simple expressions for the computation of these components in terms of the model parameters. The Koyck model is shown to furnish a wider variety of treatment profiles than those of the linear intervention model. The analysis revealed that incorporating random parameters among matched comparison–treatment pairs in the specification of SPFs can significantly improve the fit, while reducing the estimates of the extra-Poisson variation. Also, the proposed PLN Koyck model fitted the data much better than the Poisson-lognormal linear intervention (PLNI) model. The novelty effects were short lived, the indirect (through traffic volumes) treatment effects were approximately within ±10%, whereas the direct treatment effects indicated a non-significant 6.5% reduction during the after period under PLNI compared to a significant 12.3% reduction in predicted collision counts under the PLN Koyck model.

Operating characteristic curves for single, double and multiple fraction defective sampling plans developed for Cronobacter in powder infant formula

The microbiological criteria established in the EC 2073/2005 for Cronobacter in powder infant formula (PIF) are based on two-class attribute sampling plans, where the sample results are qualitative (sample indicates presence or absence) and the lot is rejected if any sample is positive. The performance of a sampling plan is revealed by its Operating Characteristic (OC) curve which plots the probability of acceptance against possible values of proportion defective. The objective of this study is to generate several OC curves for single, double and multiple sampling plans assuming different statistical distributions of Cronobacter in PIF in order to determine and compare the probabilities of rejecting/accepting the lot and the respective level of contamination. In this study, the microbial distribution of Cronobacter in the PIF is described by assuming to be Poisson-lognormal (PLN), Poisson-gamma (PG), Zero- inflated Poisson-lognormal (ZIPLN) and Zero-inflated Poisson-gamma (ZIPG). For each distribution the proportion defective of the population is estimated in order to determine the probability of acceptance for single, double and multiple sampling plans. Furthermore, the effect of clustering of the bacteria on the probability of acceptance of the lot was assessed through a Poisson-logarithmic (PLOG) and a PLogn distribution. Probabilities of accepting the lot at a given level of contamination change drastically according to the statistical distribution assumed and by changing the values of its own parameters. In addition, a procedure of multiple sampling plan based on resampling reveals that the producer's risk (α risk) can be reduced without affecting the consumer's risk (β risk).

The Discrete Weibull Distribution: An Alternative for Correlated Counts with Confirmation for Microbial Counts in Water

Distributions of pathogen counts in treated water over time are highly skewed, power-law-like, and discrete. Over long periods of record, a long tail is observed, which can strongly determine the long-term mean pathogen count and associated health effects. Such distributions have been modeled with the Poisson lognormal (PLN) computed (not closed-form) distribution, and a new discrete growth distribution (DGD), also computed, recently proposed and demonstrated for microbial counts in water (Risk Analysis 29, 841-856). In this article, an error in the original theoretical development of the DGD is pointed out, and the approach is shown to support the closed-form discrete Weibull (DW). Furthermore, an information-theoretic derivation of the DGD is presented, explaining the fit shown for it to the original nine empirical and three simulated (n = 1,000) long-term waterborne microbial count data sets. Both developments result from a theory of multiplicative growth of outcome size from correlated, entropy-forced cause magnitudes. The predicted DW and DGD are first borne out in simulations of continuous and discrete correlated growth processes, respectively. Then the DW and DGD are each demonstrated to fit 10 of the original 12 data sets, passing the chi-square goodness-of-fit test (α= 0.05, overall p = 0.1184). The PLN was not demonstrated, fitting only 4 of 12 data sets (p = 1.6 × 10(-8) ), explained by cause magnitude correlation. Results bear out predictions of monotonically decreasing distributions, and suggest use of the DW for inhomogeneous counts correlated in time or space. A formula for computing the DW mean is presented.

A comparative Full Bayesian before-and-after analysis and application to urban road safety countermeasures in New Jersey

This paper develops a step-by-step methodology for the application of Full Bayes (FB) approach for before-and-after analysis of road safety countermeasures. As part of this methodology, it studies the posterior prediction capability of Bayesian approaches and their use in crash reduction factor (CRF) estimation. A collection of candidate models are developed to investigate the impacts of different countermeasures on road safety when limited data are available. The candidate models include traditional, random effects, non-hierarchical and hierarchical Poisson-Gamma and Poisson-Lognormal (P-LN) distributions. The use of random effects and hierarchical model structures allows treatment of the data in a time-series cross-section panel, and deal with the spatial and temporal effects in the data. Next, the proposed FB estimation methodology is applied to urban roads in New Jersey to investigate the impacts of different treatment measures on the safety of “urban collectors and arterial roads” with speed limits less than 45 mph. The treatment types include (1) increase in lane width, (2) installation of median barriers, (3) vertical and horizontal improvements in the road alignment; and (4) installation of guide rails. The safety performance functions developed via different model structures show that random effects hierarchical P-LN models with informative hyper-priors perform better compared with other model structures for each treatment type. The individual CRF values are also found to be consistent across the road sections, with all showing a decrease in crash rates after the specific treatment except guide rail installation treatment. The highest decrease in the crash rate is observed after the improvement in vertical and horizontal alignment followed by increase in lane width and installation of median barriers. Overall statistical analyses of the results obtained from different candidate models show that when limited data are available, P-LN model structure combined with higher levels of hierarchy and informative priors may reduce the biases in model parameters resulting in more robust estimates.

A method to account for outliers in the development of safety performance functions

Accident data sets can include some unusual data points that are not typical of the rest of the data. The presence of these data points (usually termed outliers) can have a significant impact on the estimates of the parameters of safety performance functions (SPFs). Few studies have considered outliers analysis in the development of SPFs. In these studies, the practice has been to identify and then exclude outliers from further analysis. This paper introduces alternative mixture models based on the multivariate Poisson lognormal (MVPLN) regression. The proposed approach presents outlier resistance modeling techniques that provide robust safety inferences by down-weighting the outlying observations rather than rejecting them. The first proposed model is a scale-mixture model that is obtained by replacing the normal distribution in the Poisson-lognormal hierarchy by the Student t distribution, which has heavier tails. The second model is a two-component mixture (contaminated normal model) where it is assumed that most of the observations come from a basic distribution, whereas the remaining few outliers arise from an alternative distribution that has a larger variance. The results indicate that the estimates of the extra-Poisson variation parameters were considerably smaller under the mixture models leading to higher precision. Also, both mixture models have identified the same set of outliers. In terms of goodness-of-fit, both mixture models have outperformed the MVPLN. The outlier rejecting MVPLN model provided a superior fit in terms of a much smaller DIC and standard deviations for the parameter estimates. However, this approach tends to underestimate uncertainty by producing too small standard deviations for the parameter estimates, which may lead to incorrect conclusions. It is recommended that the proposed outlier resistance modeling techniques be used unless the exclusion of the outlying observations can be justified because of data related reasons (e.g., data collection errors).

Testing species abundance models: a new bootstrap approach applied to Indo‐Pacific coral reefs

Patterns in the commonness and rarity of species are a fundamental characteristic of ecological assemblages; however, testing between alternative models for such patterns remains an important challenge. Conventional approaches to fitting or testing species abundance models often assume that species, not individuals, are the units that are sampled and that species' abundances are independent of one another. Here we test three different models (the Poisson lognormal, the negative binomial, and the neutral, "zero-sum multinomial" [ZSM]) against species abundance distributions of Indo-Pacific corals and reef fishes. We derive and apply several alternative bootstrap analyses of model fit, each of which makes different assumptions about how species abundance data are sampled, and we assess the extent to which tests of model fit are sensitive to such assumptions. For all models, goodness of fit is remarkably consistent, regardless of whether one assumes that species or individuals are the units that are sampled or whether or not one assumes that species' abundances are statistically independent of one another. However, goodness-of-fit estimates are approximately twice as precise and detect lack of model fit more frequently, when based on sampling of individuals, rather than species. Bootstrap analyses indicate that the Poisson lognormal distribution exhibits substantially better fit to species abundance patterns, consistent with model selection analyses. In particular, heterogeneity in species abundances (many rare and few highly abundant species) is too great to be captured by the ZSM model or the negative binomial model and is best explained by models that predict species abundance patterns that are much closer, but not identical, to the lognormal distribution. More broadly, our bootstrap analyses suggest that estimates of model fit are likely to be robust to assumptions about the statistical interdependence of species abundances, but that tests of model fit are more powerful when they assume sampling of individuals, rather than species. Such individual-based tests therefore may be able to identify lack of model fit where previous tests have been inconclusive.

Comparison of Hierarchical Bayesian Models for Overdispersed Count Data using DIC and Bayes' Factors

When replicate count data are overdispersed, it is common practice to incorporate this extra-Poisson variability by including latent parameters at the observation level. For example, the negative binomial and Poisson-lognormal (PLN) models are obtained by using gamma and lognormal latent parameters, respectively. Several recent publications have employed the deviance information criterion (DIC) to choose between these two models, with the deviance defined using the Poisson likelihood that is obtained from conditioning on these latent parameters. The results herein show that this use of DIC is inappropriate. Instead, DIC was seen to perform well if calculated using likelihood that was marginalized at the group level by integrating out the observation-level latent parameters. This group-level marginalization is explicit in the case of the negative binomial, but requires numerical integration for the PLN model. Similarly, DIC performed well to judge whether zero inflation was required when calculated using the group-marginalized form of the zero-inflated likelihood. In the context of comparing multilevel hierarchical models, the top-level DIC was obtained using likelihood that was further marginalized by additional integration over the group-level latent parameters, and the marginal densities of the models were calculated for the purpose of providing Bayes' factors. The computational viability and interpretability of these different measures is considered.

Accident prediction models with random corridor parameters

Recent research advocates the use of count models with random parameters as an alternative method for analyzing accident frequencies. In this paper a dataset composed of urban arterials in Vancouver, British Columbia, is considered where the 392 segments were clustered into 58 corridors. The main objective is to assess the corridor effects with alternate specifications. The proposed models were estimated in a Full Bayes context via Markov Chain Monte Carlo (MCMC) simulation and were compared in terms of their goodness of fit and inference. A variety of covariates were found to significantly influence accident frequencies. However, these covariates resulted in random parameters and thereby their effects on accident frequency were found to vary significantly across corridors. Further, a Poisson-lognormal (PLN) model with random parameters for each corridor provided the best fit. Apart from the improvement in goodness of fit, such an approach is useful in gaining new insights into how accident frequencies are influenced by the covariates, and in accounting for heterogeneity due to unobserved road geometrics, traffic characteristics, environmental factors and driver behavior. The inclusion of corridor effects in the mean function could also explain enough variation that some of the model covariates would be rendered non-significant and thereby affecting model inference.

A New Theoretical Discrete Growth Distribution with Verification for Microbial Counts in Water

Living microbes are discrete, not homogeneously distributed in environmental media, and the form of the distribution of their counts in drinking water has not been well established. However, this count may "scale" or range over orders of magnitude over time, in which case data representing the tail of the distribution, and governing the mean, would be represented only in impractically long data records. In the absence of such data, knowledge of the general form of the full distribution could be used to estimate the true mean accounting for low-probability, high-consequence count events and provide a basis for a general environmental dose-response function. In this article, a new theoretical discrete growth distribution (DGD) is proposed for discrete counts in environmental media and other discrete growth systems. The term growth refers not to microbial growth but to a general abiotic first-order growth/decay of outcome sizes in many complex systems. The emergence and stability of the DGD in such systems, defined in simultaneous work, are also described. The DGD is then initially verified versus 12 of 12 simulated long-term drinking water and short-term treated and untreated water microbial count data sets. The alternative Poisson lognormal (PLN) distribution was rejected for 2 (17%) of the 12 data sets with 95% confidence and, like other competitive distributions, was not found stable (in simultaneous work). Sample averages are compared with means assessed from the fitted DGD, with varying results. Broader validation of the DGD for discrete counts arising as outcomes of mathematical growth systems is suggested.

Collision prediction models using multivariate Poisson-lognormal regression

This paper advocates the use of multivariate Poisson-lognormal (MVPLN) regression to develop models for collision count data. The MVPLN approach presents an opportunity to incorporate the correlations across collision severity levels and their influence on safety analyses. The paper introduces a new multivariate hazardous location identification technique, which generalizes the univariate posterior probability of excess that has been commonly proposed and applied in the literature. In addition, the paper presents an alternative approach for quantifying the effect of the multivariate structure on the precision of expected collision frequency. The MVPLN approach is compared with the independent (separate) univariate Poisson-lognormal (PLN) models with respect to model inference, goodness-of-fit, identification of hot spots and precision of expected collision frequency. The MVPLN is modeled using the WinBUGS platform which facilitates computation of posterior distributions as well as providing a goodness-of-fit measure for model comparisons. The results indicate that the estimates of the extra Poisson variation parameters were considerably smaller under MVPLN leading to higher precision. The improvement in precision is due mainly to the fact that MVPLN accounts for the correlation between the latent variables representing property damage only ( PDO) and injuries plus fatalities ( I + F). This correlation was estimated at 0.758, which is highly significant, suggesting that higher PDO rates are associated with higher I + F rates, as the collision likelihood for both types is likely to rise due to similar deficiencies in roadway design and/or other unobserved factors. In terms of goodness-of-fit, the MVPLN model provided a superior fit than the independent univariate models. The multivariate hazardous location identification results demonstrated that some hazardous locations could be overlooked if the analysis was restricted to the univariate models.

The distribution of new enhancing lesion counts in multiple sclerosis: further explorations

Background A statistical distribution describing the number of new enhancing lesions seen on MRI in patients with MS is of great importance for improving the statistical methodology of clinical trials using new enhancing lesions as outcome measure. We examined whether there are superior alternatives for the currently proposed negative binomial (NB) distribution. Objective To determine the optimal statistical distribution describing new enhancing lesion counts from a selection of six conceivable models, and to assess the effect on the distribution of a treatment effect, varying follow-up duration and selection for activity at baseline. Methods The statistical NB, Poisson-Inverse Gaussian (P-IG), Poisson- Lognormal (P-LN), Neyman type A (NtA), Pólya-Aeppli (PA) and Zero Inflated Poisson (ZIP) distribution were fitted on new enhancing lesion data derived from one treated and two untreated cohorts of RRMS and relapsing SPMS patients and on subgroups of varying follow-up duration and selection for baseline activity. Measure of comparison was Akaike's information criterion (AIC). Results Both the subgroup analyses as well as a treatment had a noticeable effect on the distributional characteristics of new enhancing lesion counts. The NB distribution generally provided the most optimal fit, closely followed by the P-IG distribution and the P-LN distribution. Fits of the PA and NtA distribution were suboptimal, while the ZIP distribution was the least adequate for modelling new enhancing lesion counts. Conclusion The NB distribution is the optimal distribution for modelling new enhancing lesion counts, irrespective of the effect of treatment, follow-up duration or a baseline activity selection criterion.

Joint modelling of the total amount and the number of claims by conditionals

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple ( S , N ) , where N is a count variable and S = X 1 + ⋯ + X N is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of S | N and N | S belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson–Gamma and the Poisson–Binomial conditionals models. Other conditionals models are proposed, including the Poisson–Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.

Lognormal distribution of cellular uptake of radioactivity: statistical analysis of alpha-particle track autoradiography.

Recently, the distribution of radioactivity among a population of cells labeled with 210Po was shown to be well described by a lognormal (LN) distribution function (J Nucl Med. 2006;47:1049-1058) with the aid of autoradiography. To ascertain the influence of Poisson statistics on the interpretation of the autoradiographic data, the present work reports on a detailed statistical analysis of these earlier data. The measured distributions of alpha-particle tracks per cell were subjected to statistical tests with Poisson, LN, and Poisson-lognormal (P-LN) models. The LN distribution function best describes the distribution of radioactivity among cell populations exposed to 0.52 and 3.8 kBq/mL of 210Po-citrate. When cells were exposed to 67 kBq/mL, the P-LN distribution function gave a better fit; however, the underlying activity distribution remained lognormal. The present analysis generally provides further support for the use of LN distributions to describe the cellular uptake of radioactivity. Care should be exercised when analyzing autoradiographic data on activity distributions to ensure that Poisson processes do not distort the underlying LN distribution.

Species Abundance in a Forest Community in South China: A Case of Poisson Lognormal Distribution

Abstract: Case studies on Poisson lognormal distribution of species abundance have been rare, especially in forest communities. We propose a numerical method to fit the Poisson lognormal to the species abundance data at an evergreen mixed forest in the Dinghushan Biosphere Reserve, South China. Plants in the tree, shrub and herb layers in 25 quadrats of 20 m× 20 m, 5 m× 5 m, and 1 m× 1 m were surveyed. Results indicated that: (i) for each layer, the observed species abundance with a similarly small median, mode, and a variance larger than the mean was reverse J-shaped and followed well the zero-truncated Poisson lognormal; (ii) the coefficient of variation, skewness and kurtosis of abundance, and two Poisson lognormal parameters (& and μ) for shrub layer were closer to those for the herb layer than those for the tree layer; and (iii) from the tree to the shrub to the herb layer, the α and the coefficient of variation decreased, whereas diversity increased. We suggest that: (i) the species abundance distributions in the three layers reflects the overall community characteristics; (ii) the Poisson lognormal can describe the species abundance distribution in diverse communities with a few abundant species but many rare species; and (iii) 1/α should be an alternative measure of diversity. (Managing editor: Ya-Qin HAN)

The lognormal distribution is not an appropriate null hypothesis for the species–abundance distribution

Summary Of the many models for species–abundance distributions (SADs), the lognormal has been the most popular and has been put forward as an appropriate null model for testing against theoretical SADs. In this paper we explore a number of reasons why the lognormal is not an appropriate null model, or indeed an appropriate model of any sort, for a SAD. We use three empirical examples, based on published data sets, to illustrate features of SADs in general and of the lognormal in particular: the abundance of British breeding birds, the number of trees > 1 cm diameter at breast height (d.b.h.) on a 50 ha Panamanian plot, and the abundance of certain butterflies trapped at Jatun Sacha, Ecuador. The first two are complete enumerations and show left skew under logarithmic transformation, the third is an incomplete enumeration and shows right skew. Fitting SADs by χ2 test is less efficient and less informative than fitting probability plots. The left skewness of complete enumerations seems to arise from a lack of extremely abundant species rather than from a surplus of rare ones. One consequence is that the logit‐normal, which stretches the right‐hand end of the distribution, consistently gives a slightly better fit. The central limit theorem predicts lognormality of abundances within species but not between them, and so is not a basis for the lognormal SAD. Niche breakage and population dynamical models can predict a lognormal SAD but equally can predict many other SADs. The lognormal sits uncomfortably between distributions with infinite variance and the log‐binomial. The latter removes the absurdity of the invisible highly abundant half of the individuals abundance curve predicted by the lognormal SAD. The veil line is a misunderstanding of the sampling properties of the SAD and fitting the Poisson lognormal is not satisfactory. A satisfactory SAD should have a thinner right‐hand tail than the lognormal, as is observed empirically. The SAD for logarithmic abundance cannot be Gaussian.

Cryptosporidium monitoring system at a water treatment plant, based on waterborne risk assessment

The water volume required for daily monitoring of Cryptosporidium (which can statistically ensure an annual risk of infection below 10(-4)), was assessed by evaluating the applicability of the Poisson lognormal (PLN) distribution in microbial risk assessment. PLN showed as good a fit to the observed data as to the negative binomial distribution. From the estimated PLN distributions for the source and finished water, the efficacy of the oocyst removal by the conventional water treatment process was estimated to follow log-normal distribution (median = 3.16 log10, 95% CI = 4.27-2.05 log10). The 365 consecutive negative results of daily monitoring for 180 L of finished water were found to be statistically equivalent to the annual risk of infection below 10(-4). This research also suggested the possibility of applying a qualitative detection method, such as CC-PCR, as a routine monitoring method for the quantitative risk management.

On mean scaled insurance risk models

A characterization problem for scale compound parametric families of distributions with the mean as scale parameter is analyzed. If the counting distribution of the compound family has the sample mean as maximum likelihood estimator of the counting mean, and if the maximum likelihood estimator of the mean scale parameter is the sample mean, then the compound family has necessarily a gamma secondary distribution. Necessary and sufficient conditions under which such a characterization holds are derived. Among the rich class of counting statistical models, which lead to such a characterization, one finds the Poisson, binomial, negative binomial, Hermite, Delaporte, extended Poisson-Pascal, mixed Poisson inverse Gaussian, Sichel, generalized Euler, and many others. The mixed Poisson lognormal is a counterexample for which the present characterization fails.