Poisson Lognormal Distribution Research Articles

Confronting different models of community structure to species‐abundance data: a Bayesian model comparison

Species abundances are undoubtedly the most widely available macroecological data, but can we use them to distinguish among several models of community structure? Here we present a Bayesian analysis of species-abundance data that yields a full joint probability distribution of each model's parameters plus a relatively parameter-independent criterion, the posterior Bayes factor, to compare these models. We illustrate our approach by comparing three classical distributions: the zero-sum multinomial (ZSM) distribution, based on Hubbell's neutral model, the multivariate Poisson lognormal distribution (MPLN), based on niche arguments, and the discrete broken stick (DBS) distribution, based on MacArthur's broken stick model. We give explicit formulas for the probability of observing a particular species-abundance data set in each model, and argue that conditioning on both sample size and species count is needed to allow comparisons between the two distributions. We apply our approach to two neotropical communities (trees, fish). We find that DBS is largely inferior to ZSM and MPLN for both communities. The tree data do not allow discrimination between ZSM and MPLN, but for the fish data ZSM (neutral model) overwhelmingly outperforms MPLN (niche model), suggesting that dispersal plays a previously underestimated role in structuring tropical freshwater fish communities. We advocate this approach for identifying the relative importance of dispersal and niche-partitioning in determining diversity of different ecological groups of species under different environmental conditions.

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.

Analyzing Spatial Structure of Communities Using the Two‐Dimensional Poisson Lognormal Species Abundance Model

The joint spatial and temporal fluctuations in the community structure of tropical butterflies are analyzed by fitting the bivariate Poisson lognormal distribution to a large number of observations in space and time. By applying multivariate dependent diffusions for describing the fluctuations in the abundances, the environmental variance is estimated to be very large and so is the strength of local density regulation. The variance in the lognormal species abundance distribution is partitioned into components expressing the heterogeneity between the species, independent noise components for the different species, a demographic stochastic component, and a component due to overdispersion in the sampling. In disagreement with the neutral theory, the estimates show that the heterogeneity component is the dominating one, representing 81% of the total variance in the lognormal model. Different spatial components of diversity, the alpha, beta, and gamma diversity, are also estimated. The spatial scale of the autocorrelation function for the community is of order 1 km, while sampling of a quadrat would need to be 10 km on a side to yield the total diversity for the community.

Analyzing Spatial Structure of Communities Using the Two-Dimensional Poisson Lognormal Species Abundance Model

The joint spatial and temporal fluctuations in the community structure of tropical butterflies are analyzed by fitting the bivariate Poisson lognormal distribution to a large number of observations in space and time. By applying multivariate dependent diffusions for describing the fluctuations in the abundances, the environmental variance is estimated to be very large and so is the strength of local density regulation. The variance in the lognormal species abundance distribution is partitioned into components expressing the heterogeneity between the species, independent noise components for the different species, a demographic stochastic component, and a component due to overdispersion in the sampling. In disagreement with the neutral theory, the estimates show that the heterogeneity component is the dominating one, representing 81% of the total variance in the lognormal model. Different spatial components of diversity, the alpha, beta, and gamma diversity, are also estimated. The spatial scale of the autocorrelation function for the community is of order 1 km, while sampling of a quadrat would need to be 10 km on a side to yield the total diversity for the community.

Efficacy Estimates from Parasite Count Data That Include Zero Counts

Cox, J. L., Heyse, J. F., and Tukey, J. W. 2000. Efficacy estimates from parasite count data that include zero counts. Experimental Parasitology96, 1–8. Because of the positive skewness of parasite distributions and the greater constancy of percentage of response of therapy in animal populations, parasite count data are conventionally transformed logarithmically before combining results from different animals, either all controls or all treated. Observations of zero counts raise difficulties, since the logarithm of zero is not useful. In this study, several types of zero count adjustments are compared. Two systems for assigning values to zero counts were considered: a fixed system, which assigns the same value to all zero counts regardless of the proportion of such counts in a treatment group, and a variable system, which replaces zero counts with a value based on the proportion of zero counts in the group. The values assigned by either system are then adjusted to reflect aliquot size. An evaluation was performed by using 32 compound Poisson lognormal distributions, three sample sizes, and three representatives of each zero count adjustment system. The Poisson lognormal distribution provides a convenient method with which to provide variability greater than Poisson. Expected values of the sample estimate of the (known) population mean were calculated for each of the 576 combinations of these factors, and the bias associated with each combination was derived. The bias associated with the three representatives of the variable adjustment system was similar. The variable adjustment system had a lower overall bias than any representatives of the fixed adjustment system.

On random placement and species-area relations

Consider a collection C of individuals from several species residing ina region R which is the union of a number of nonoverlapping subregions. Let n i be the number of individuals from species i which belong to C and hence reside somewhere in R . The simplest hypothesis about the spatial distribution throughout R of the members of C is that their placement in dwelling sites is random and noninteractive, with the probability of a given individual of C residing in a particular subregion r of R equal to the ratio α ofthe area of r to the area of R . It is shown here that when this hypothesis of random placement holds, the mean s̄ and variance σ 2 of the number of species from C represented in r are given by explicit functions of α, provided the number n i are known. Thus, if all the species in C have been censured, species-area data permit a test of the hypothesis of random placement. The nature of the dependence of s̄ and σ 2 on α is discussed in detail for special cases in which the numbers n i are given by such theoritical abundance relations as the logarithmic series distribution, the “broken stick” distribution, the lognormal distribution, the Poisson lognormal distribution, and the gamma distribution. The arguments employed here to deduce consequences of the hypothesis of random choice are rigorous and exact. No use is made of the assumption, commonly made heretofore (but not in general correct, even under the hypothesis of random placement), that a species-area curve (giving the number of species expected to be found in a sample of known area) must have the same form as the corresponding collector's curve (giving the number of species expected in a sample of a known number of individuals). Nor is it assumed in advance, as is often done in the theory of island biogeography, that the distribution of individuals throughout the subregions of R is such thatthe species abundance relations for subregions of different areas must be of a preassigned type, i.e., must share a common form, such as that associated with a truncated lognormal distribution.

Study of Caligus minimus (Otto, 1821), (Caligidae Copepoda) infections of the sea Bass Dicentrarchus labrax (L.) in Bardawil lagoon.

Caligus minimus (Otto, 1821) is a common parasite of Dicentrarchus labrax in Bardawil Lagoon and it also occurs sporadically on D. punctatus in this lagoon. It has also been found on D. labrax which were recently established in the Gulf of Suez. The incidence as well as the intensity of infection showed a distinct pattern of seasonal variation with maximum levels during winter and early spring and lowest levels of infection during summer and early autumn. Overall infection levels also showed distinct annual variation. Occurrence of larval stages on the fish followed the seasonal pattern observed in infections by adults copepods. The highest occurrence was observed every year in May. However, at any month of the year ovigerous females, and at least a few copeptides or chalimi may be found. Spatial distribution of infection among the fish was significantly contagious and with some exceptions followed the Poisson log normal distribution. However, the magnitude of overdispersion showed distinct seasonal fluctuations, with more extreme overdispersion in summer and autumn. These fluctuations apparently resulted from the diversity of temperature and salinity conditions in the lagoon which were most extreme during the summer and autumn. Positive correlation between host size and infection was evident only in the winter and spring months with highest levels of infection in the largest fish. Female copepods usually out number male copepods. The dominance of female over males was also evident in the preadult stages.

On Fitting the Poisson Lognormal Distribution to Species-Abundance Data

An extension of MacArthur's broken stick model is proposed to explain why species abundances should be lognormally distributed. A method of fitting the compound Poisson lognormal distribution by maximum likelihood is described; a computer program is available for performing the calculations. It is shown how the information theory measure of species diversity can be estimated from the parameters of the fitted distribution.