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When analyzing Poisson-count data sometimes a lot of zeros are observed. When there are too many zeros a zero-inflated Poisson distribution can be used. A score test is presented to test whether the number of zeros is too large for a Poisson distribution to fit the data well.
The generalized Poisson distribution is a distribution which approximates various forms of mixtures of Poisson distributions. The mean and variance of the generalized Poisson distribution, which are simple functions of the two parameters of the distribution, are more useful than the original parameters in test data analysis. Therefore, we adopted two types of transformations of parameters. The first model has new parameters of mean and standard deviation. The second model contains new parameters of mean and variance/mean. An example indicates that the transformed parameters are convenient to understand the properties of data.
Use of the Poisson distribution to estimate the quality of random mixtures was evaluated as a measure of the highest attainable degree of dose uniformity of tablets. Ingredient A was assumed to have a large particle size as compared to diluent B. In contrast to the more precise binomial distribution, for the simple Poisson approach no experiments are necessary to investigate the mean proportions of the apparent volume, av and bv, which A and B assume within the powder samples in the die. The range of volume ratios was defined where the Poisson distribution is valid. Accepting an error of 5% of the random content variation of A per sample, av may amount to up to 0.1 (10%). In terms of the proportion by mass of A, a, this range is wider, and commonly of the order of 0.2 or higher. This approach was tested with tablets prepared from mixtures of coarse sucrose A and a fine Avicel/talc diluent B at A:B (m:m) ratios from 10:90 to 50:50. Even with the 30:70 tablets, the variations of the sucrose content were still in good agreement with the content variations of the random mixtures as estimated from the Poisson distribution. Estimates of the 50:50 ratio, however, deviated from the Poisson distribution.
A method was devised that employs deviations from the Poisson distribution to analyze the spatial arrangement of neurons and glia in human cerebral cortex. A field of randomly distributed points equal in number of a sample field of neuronal or glial cells is generated by computer, and the proportion of cells in the sample field that are closer to the nearest neighboring cells than to the nearest randomly distributed point is determined. We call this proportion the "Poisson ratio." When the cells are randomly distributed, the Poisson ratio is equal to 0.5. If the Poisson ratio is less than 0.5, the cells are farther away from one another than a random distribution would predict (exclusionary pattern); if the Poisson ratio is greater than 0.5, the cells are closer to one another than a random distribution would predict (clustering). A simple nonparametric statistical test is used to determine the significance of differences in the ratios. This method was applied to samples of human cerebral cortex in order to test the hypothesis that patients with schizophrenic psychosis may have an altered pattern of neuronal clustering. The analysis revealed that there is no difference in the nearest-neighbor distribution of either neurons or glia between psychotic patients and controls. It was found, however, that there is a highly significant difference in the spatial distribution of neurons versus glia in human cerebral cortex. Neurons of layers II to VI in the human cortex show greater-than-expected distances among them and are distributed according to an exclusionary pattern, while neurons in layer I show a clustering pattern.(ABSTRACT TRUNCATED AT 250 WORDS)
The value of Poisson distribution theory in describing and predicting the nature of sperm-egg interaction in vitro has been investigated using an interspecific in-vitro fertilization system, incorporating zona-free hamster oocytes and human spermatozoa. The frequency distribution of polyspermic oocyte penetrations in 72 experiments exhibited good agreement with the Poisson distribution at all levels of fertilization indicating that each oocyte must be of equal penetrability and that there can be no block to polyspermy in this interspecific system. Poisson distribution theory also accurately described the relationship between oocyte penetration and sperm motility in 50 out of 54 separate experiments spread across 10 serial dilution curves. For each dilution series the shape of the fitted curve was fixed but its location along the x-axis varied from donor to donor. The fixed nature of the relationship between sperm motility and egg penetration enables the results of such in-vitro fertilization experiments to be corrected for the number of motile spermatozoa in the incubation media. On the basis of these findings a protocol is described for assessing the results of the zona-free hamster oocyte penetration assay, which involves analysis of the degree of polyspermy followed by the application of Poisson distribution theory to correct the results to a standard concentration of motile spermatozoa. Changes in the penetrating ability of human spermatozoa after vasectomy and characterization of the degree of inter-ejaculate variation in penetrating potential are two clinical examples of such analyses given in the text. The statistical methods described in this paper should also be of general relevance to the study of fertilization mechanisms, in providing a rationale by which to analyse the quantitative nature of sperm-egg interaction in vitro.
Earlier reports indicated that sperm from 25% of patients from infertile couples, but not from normal or fertile donors, show deviations from the theoretical Poisson distribution of the number of sperm penetrating zona-free hamster ova. Using semen samples from 15 grandfathers (aged 60 to 84 years) and 24 young fathers (aged 25 to 36 years), this study analyzed whether age also has an effect on the distribution. It was found that the overall fit to the Poisson distribution of the samples from grandfathers was very poor; in contrast, the samples from young fathers fit well. The observed deviations from the Poisson distribution among grandfathers may be a consequence of their long periods of sexual abstinence. Decrease in sexual activity produces age-different populations of sperm that probably differ in penetrating ability. Samples from older fathers also show a worse fit to the Poisson distribution than do those from younger fathers. These results suggest that the duration of sperm storage in the genital tract after maturation has an effect on sperm function.
The common and convenient assumption in accident count analysis, that accidents are Poisson-distributed, is reexamined. Two statistical tests, for evaluating the assumption are described and compared. It is shown that a test based upon a combinatorial analysis is much more accurate than the alternative chi-square test when accident counts are expected to be small. The more accurate test is used to reinterpret data on accident count variability, the results indicating that the Poisson distribution is appropriate for the analysis of accidents at individual sites.
Since 1980 the German Children's Cancer Registry has documented all childhood malignancies in the Federal Republic of Germany. Various statistical procedures have been proposed to identify municipalities or other geographic units with increased numbers of malignancies. Usually the Poisson distribution, which requires the malignancies to be distributed homogeneously and uncorrelated, is applied. Other discrete statistical distributions (so-called cluster distributions) like the generalized or compound Poisson distributions are applicable more generally. In this paper we present a first explorative approach to the question of whether it is necessary to use one of these cluster distributions to model the data of the German Children's Cancer Registry. In conclusion, we find no indication that the Poisson approach is insufficient.
We have investigated the consequences of having multiple fusion complexes on exocytotic granules, and have identified a new principle for interpreting the calcium dependence of calcium-triggered exocytosis. Strikingly different physiological responses to calcium are expected when active fusion complexes are distributed between granules in a deterministic or probabilistic manner. We have modeled these differences, and compared them with the calcium dependence of sea urchin egg cortical granule exocytosis. From the calcium dependence of cortical granule exocytosis, and from the exposure time and concentration dependence of N-ethylmaleimide inhibition, we determined that cortical granules do have spare active fusion complexes that are randomly distributed as a Poisson process among the population of granules. At high calcium concentrations, docking sites have on average nine active fusion complexes.
Vital cells number (VCN) in the sampling of E. coli populations was experimentally measured and distribution histograms were obtained. In most cases distributions show considerable deviation from Poisson model. VCN distribution histograms are polymodal, dispersion/arithmetical mean ratio may essentially differ from 1. The more essential differences from Poisson distribution were observed for populations with the higher cell concentration. Computer simulation of the VCN histograms indicated that additional parameters (such as those describing cellular interaction of different nature and/or other factors that influence random behaviour of cells) should be introduced into Poisson model to explain observed variations in VCN distribution histograms.
An example of the use of the truncated Poisson distribution in an immunogold assay of dystrophin, a gene product of importance in the study of muscular dystrophies, is presented. The practical benefit of using minimum variance unbiased estimators of relevant functions of the parameter of the distribution is considered.
Transgenic mutation assays generally use mutant frequencies to estimate mutation frequencies but the degree to which clonal expansion inflates mutant frequencies is largely unknown. Mutant frequency is defined as the fraction of cells carrying mutations in the gene of interest and, according to the standard Big Blue protocol, is determined by dividing the number of mutant plaques by the total number of plaques screened. Mutation frequency is determined as the fraction of cells carrying definitely independent mutations and therefore requires correction for clonal expansion. Mutant and mutation frequencies were determined for brain, thymus and male germ cells of four mice from two age groups (3-versus 10-month old). The mutant frequency in thymus differed significantly between 3- and 10-month old mice (P < 0.05). By sequencing all mutants, the mutation frequency (i.e., corrected for jackpot mutations) in thymus was determined and was not significantly different between 3- and 10-month old mice. Mutant frequency does not fit a Poisson distribution, but mutation frequency corrected for jackpot mutations is substantially less variable and does fit a Poisson distribution.
Among probability distributions that are used to describe a chance mechanism whose observational apparatus becomes active only when at least one event occurs is the zero-truncated Poisson distribution (ZTPD). A modified version of the ZTPD, which we call an intervened Poisson distribution (IPD), is discussed in this paper. We give a genesis of IPD and obtain its statistical properties. A numerical example is included to illustrate the results.
Estimating the mutation rate from a Luria-Delbrück fluctuation experiment involves estimating the Poisson parameter in a compound Poisson distribution. The efficiency with which this can be estimated depends on how well the other random factors have been characterized. The assumption that cell growth can be represented as a stochastic pure birth or Yule process is biologically unrealistic but contributes little to the bias and variance of a maximum likelihood estimator of the mutation rate.
A truncated Poisson distribution provides a closer fit to observed caries in a sample of 10- to 11-year-old children than a previously reported chain binomial distribution. Operations research techniques are used in estimating a parameter.
Computer simulations are used to examine the significance levels and powers of several tests which have been employed to compare the means of Poisson distributions. In particular, attention is focused on the behaviour of the tests when the means are small, as is often the case in ecological studies when populations of organisms are sampled using quadrats. Two approaches to testing are considered. The first assumes a log linear model for the Poisson data and leads to tests based on the deviance. The second employs standard analysis of variance tests following data transformations, including the often used logarithmic and square root transformations. For very small means it is found that a deviance-based test has the most favourable characteristics, generally outperforming analysis of variance tests on transformed data; none of the latter appears consistently better than any other. For larger means the standard analysis of variance on untransformed data performs well.
Estimates of the prevalence of deviant behaviour, which are based on the usual survey methods are by far too low. Therefore, the use of capture-recapture methods or of the truncated Poisson distribution is to be preferred, provided appropriate data are available. Here an extended Poisson approach was applied in order to estimate the number of users of hard illegal drugs (heroin, cocaine) in Switzerland for each year from 1990 to 1993. These estimates indicate an increase by about 50% during the period 1990-1993.