Modeling Sage data with a truncated gamma-Poisson model

Background  

Serial Analysis of Gene Expressions (SAGE) produces gene expression measurements on a discrete scale, due to the finite number of molecules in the sample. This means that part of the variance in SAGE data should be understood as the sampling error in a binomial or Poisson distribution, whereas other variance sources, in particular biological variance, should be modeled using a continuous distribution function, i.e. a prior on the intensity of the Poisson distribution. One challenge is that such a model predicts a large number of genes with zero counts, which cannot be observed.     

Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution

We prove that the generalized Poisson distribution GP(theta, eta) (eta > or = 0) is a mixture of Poisson distributions; this is a new property for a distribution which is the topic of the book by Consul (1989). Because we find that the fits to count data of the generalized Poisson and negative binomial distributions are often similar, to understand their differences, we compare the probability mass functions and skewnesses of the generalized Poisson and negative binomial distributions with the first two moments fixed. They have slight differences in many situations, but their zero-inflated distributions, with masses at zero, means and variances fixed, can differ more. These probabilistic comparisons are helpful in selecting a better fitting distribution for modelling count data with long right tails. Through a real example of count data with large zero fraction, we illustrate how the generalized Poisson and negative binomial distributions as well as their zero-inflated distributions can be discriminated.     

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.     

Spatial distribution of nests of Acromyrmex crassispinus (Forel) (Hymenoptera: Formicidae) in Pinus taeda plantations

The spatial distribution of insects is essential to perform control strategies, to improve sample techniques and to estimate economic losses. We aimed to determine the spatial distribution of nests of Acromyrmex crassispinus (Forel) in Pinus taeda plantations. The experiments were carried out in P. taeda plantations with different ages (treatments: recently-planted, three and six-year old plants). The study took place in Rio Negrinho and in Três Barras, SC. Three plots of one hectare were delimited in each treatment, and plots were divided in 64 sample units. The analysis of the dispersion index [variance/mean relationship (I), index of Morisita (Iδ) and k exponent of negative binomial distribution] showed that the majority of the samplings presented random distribution. Among the three distributions of probabilities studied: Poisson, positive binomial and negative binomial, the Poisson distribution was the best model to fit the spatial distribution of A. crassispinus nests in all samplings. The result was a random distribution in the plantings of different ages.     

Estimation of sample zooplankton abundance from Folsom splitter sub-samples

Procedures are described for estimating the abundance of eachzooplanklon species in a sample after counting at least twosub-samples from a Folsom splitter. A multinomiai model is assumedfor the splitting process. The method may be summarized as follows:two sub-samples, balanced with respect to the left — rightsplits, are counted. The counts for each species are testedfor homogeneity using a x² test. If the counts are homogeneous,an estimator is given which permits the estimation of the numberof animals in the sample, and the variance of this estimate.If the subsample counts are heterogeneous, it is assumed thatclumping has occurred. A procedure is described in which additionalsub-samples are counted to locate the clump, and estimatorsof abundance (and variance estimators) are derived. If the splitterbias is <5%, bias of the abundance estimator is negligiblecompared to the binomial sampling error. Comparisons of speciesabundance between plankton samples is made more rigorous usingthe estimates of mean abundance, and the variance of this mean,provided by the methods described here. ^*Present address: Dunoon Road, Dorroughby, NSW 2480, Australia.     

Technical and biological variance structure in mRNA-Seq data: life in the real world

ABSTRACT: BACKGROUND: mRNA expression data from next generation sequencing platforms is obtained in the form of counts per gene or exon. Counts have classically been assumed to follow a Poisson distribution in which the variance is equal to the mean. The Negative Binomial distribution which allows for over-dispersion, i.e., for the variance to be greater than the mean, is commonly used to model count data as well. RESULTS: In mRNA-Seq data from 25 subjects, we found technical variation to generally follow a Poisson distribution as has been reported previously and biological variability was over-dispersed relative to the Poisson model. The mean-variance relationship across all genes was quadratic, in keeping with a Negative Binomial (NB) distribution. Over-dispersed Poisson and NB distributional assumptions demonstrated marked improvements in goodness-of-fit (GOF) over the standard Poisson model assumptions, but with evidence of over-fitting in some genes. Modeling of experimental effects improved GOF for high variance genes but increased the over-fitting problem. CONCLUSIONS: These conclusions will guide development of analytical strategies for accurate modeling of variance structure in these data and sample size determination which in turn will aid in the identification of true biological signals that inform our understanding of biological systems.     

Analysis of Variable Fractions Resulting from Microbial Counts

In microbiological studies various methods are employed to estimate fractions from paired counts of organisms. When the fraction (second count divided by first count) is constant among the paired counts, the maximum likelihood estimate is the ratio of the arithmetic means. In many practical applications this fraction may however not be constant, but vary substantially between pairs of counts.We discuss a statistical method that estimates the distribution of the fraction from pairs of counts, to allow for this variation. Four real data sets (concerning viability for growth and infection, recovery of a detection method, and removal in a treatment process) are analyzed by this method.Often, pairs of counts are not determined in the same physical sample, but the first count is made in one sample, and the second count in a second sample. We provide parametric models to deal with such a situation: the desired fraction is still estimated as a binomial probability, but the model includes sampling effects. This approach also allows for analysis of two distinct cases: paired observations, where the counts before and after are related in some way to each other, and unpaired observations, where they are not. The four models for separate samples: paired or unpaired observations, and binomial probability fixed or variable, are used to analyze the removal data.It is concluded that this approach of statistical analysis of fractions is more appropriate than often used calculations based on the ratio between the (geometric) means before and after. The implications for risk analysis are briefly discussed.     

Efficacy estimates from parasite count data that include zero counts

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.     

Inhibition Phenomenon of Staphylococci by Bacteriophages

In the study of the relationship between bacteriophage and strains of staphylococci showing inhibition, slight differences were observed in the ability to adsorb phage between staphylococci of full phage sensitivity and those showing inhibition by phage. Only a few plaques were produced by inhibitory phages adsorbed on strains showing inhibition, whereas almost all of the phages adsorbed on corresponding phage-propagating strains produced plaques. Some strains showing inhibition were converted to full sensitivity to certain phages by heat shock or trypaflavine treatment. Treated strains adsorbed inhibitory phages to almost the same degree as nontreated strains, but most of the phages adsorbed on treated strains produced plaques. Killing was not always observed in cells adsorbing inhibitory phages. These results suggest that inhibition is not due to low adsorption rates, but rather to plaque formation by a small number of the sensitive fraction of the population and overgrowth by nonlysed cells.     

The use of antibiotics to improve phage detection and enumeration by the double-layer agar technique

Background  

The Double-Layer Agar (DLA) technique is extensively used in phage research to enumerate and identify phages and to isolate mutants and new phages. Many phages form large and well-defined plaques that are easily observed so that they can be enumerated when plated by the DLA technique. However, some give rise to small and turbid plaques that are very difficult to detect and count. To overcome these problems, some authors have suggested the use of dyes to improve the contrast between the plaques and the turbid host lawns. It has been reported that some antibiotics stimulate bacteria to produce phages, resulting in an increase in final titer. Thus, antibiotics might contribute to increasing plaque size in solid media.     

Estimating abundance for a savanna elephant population using mark–resight methods: a case study for the Tembe Elephant Park, South Africa

Elephants living in dense woodlands are difficult to count. Many elephant populations in Africa occur in such conditions. Estimates of these populations based on total counts, aerial counts and dung counts often lack information on precision and accuracy. We use standard mark–recapture field methods to obtain estimates of population size with associated confidence limits. We apply this approach to a closed elephant population in the Tembe Elephant Park (300 km²), South Africa. A registration count completed in 4 months gives a known population size. We evaluate mark–recapture models against the known population size. Individual identification profiles obtained for elephants during the registration count and mark–recapture events indicate that at least 167 elephants live in the park. We consider this value as an estimate of the minimum number alive. We include 189 sightings of bulls and 37 sightings of breeding herds in the mark–recapture modelling. Of the models we test (Petersen, Schnabel, Schumacher, Jolly–Seber, Bowden's, Poisson and negative binomial), Bowden's gives an estimate closest to the registration count. Assumptions of the model are not violated. For all models except one (negative binomial), our estimates improve with increased sampling intensity. Confidence intervals do not improve with increased effort except for the Schnabel model. Mark–recapture methods should be considered as reliable estimators of population size for elephants occurring in dense woodlands and forests when other methods cannot be relied on.     

Detection of FRNA coliphages in groundwater: interference with the assay by somatic salmonella bacteriophages

Groundwater samples from two sites in Alabama, USA were plaque assayed for F-specific RNA (FRNA) coliphages using Salmonella typhimurium WG49 as the host bacterium. While numerous plaques were detected with WG49 (a strain possessing Escherichia coli F pili), plaques were also observed with an F^- control strain of Salm. typhimurium. Five isolates were plaque purified and examined by electron microscopy. All of the isolate particles were observed to be tailed, with five distinct particle types being differentiated. None of the isolate particles were consistent in morphology with the cubic FRNA coliphages. Host range evaluation supported classification of the isolates as salmonella phages. Somatic salmonella phages have previously been found to interfere with the assay of surface waters for FRNA coliphages. Their detection here demonstrates that such interference is also encountered in the assay of groundwaters.     

Restriction of bacteriophage plaque formation in Streptomyces spp.

Several Streptomyces species that produce restriction endonucleases were characterized for their ability to propagate 10 different broad host range bacteriophages. Each species displayed a different pattern of plaque formation. A restrictionless mutant of S. albus G allowed plaque formation by all 10 phages, whereas the wild-type strain showed plaques with only 2 phages. DNA isolated from three of the phages was analyzed for the presence of restriction sites for Streptomyces species-encoded enzymes, and a very strong correlation was established between the failure to form plaques on Streptomyces species that produced particular restriction enzymes and the presence of the corresponding restriction sites in the phage DNA. Also, the phages that lacked restriction sites in their DNA generally formed plaques on the corresponding restriction endonuclease-producing hosts at high efficiency. The DNAs from the three phages analyzed also generally contained either many or no restriction sites for the Streptomyces species-produced enzymes, suggesting a strong evolutionary trend to either eliminate all or tolerate many restriction sites. The data indicate that restriction plays a major role in host range determination for Streptomyces phages. Analysis of bacteriophage host ranges of many other uncharacterized Streptomyces hosts has identified four relatively nonrestricting hosts, at least two of which may be suitable hosts for gene cloning. The data also suggest that several restriction systems remain to be identified in the genus Streptomyces.     

Note on a problem in probit analysis

The statistical treatment of dosage-mortality data when the number of survivors is counted but not the total number of organisms in each sample, the latter being estimated from an untreated sample, has been discussed by Wadley (1949), on the assumption of Poisson variation in the number of organisms per sample. The procedure to be followed when a wider hypothesis is made that the number of organisms has a negative binomial distribution is described here; both the arithmetical analysis of the results and the optimum arrangement of the samples are considered.     

Serial enrichment of heteroduplex DNA using a MutS-magnetic bead system

Numerous applications in molecular biology and genomics require characterization of mutant DNA molecules present at low levels within a larger sample of non-mutant DNA. This is often achieved either by selectively amplifying mutant DNA, or by sequencing all the DNA followed by computational identification of the mutant DNA. However, selective amplification is challenging for insertions and deletions (indels). Additionally, sequencing all the DNA in a sample may not be cost effective when only the presence of a mutation needs to be ascertained rather than its allelic fraction. The MutS protein evolved to detect DNA heteroduplexes in which the two DNA strands are mismatched. Prior methods have utilized MutS to enrich mutant DNA by hybridizing mutant to non-mutant DNA to create heteroduplexes. However, the purity of heteroduplex DNA these methods achieve is limited because they can only feasibly perform one or two enrichment cycles. We developed a MutS-magnetic bead system that enables rapid serial enrichment cycles. With six cycles, we achieve complete purification of heteroduplex indel DNA originally present at a 5% fraction and over 40-fold enrichment of heteroduplex DNA originally present at a 1% fraction. This system may enable novel approaches for enriching mutant DNA for targeted sequencing.     

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.     

Isolation of Salmonella typhimurium mutants affecting the plaque morphology of phage P22

Summary Salmonella typhimurium mutants affecting the plaque morphology of P22 and other phages have been isolated. Using one such bacterial mutant phage mutants making turbid plaques have been isolated.     

A membrane filter direct count technique for enumeratingBdellovibrio

A membrane filter direct count method was devised for enumeratingBdellovibrio cells in “clean” suspensions. The procedure involves filtering a specified volume of a diluted, Trisbuffered suspension ofBdellovibrio cells through a known area of a 100-nm-pore-size Millipore brand membrane filter. A clarification solvent was used to render the filter transparent, so that the bdeloyvibrios on the filter could be photomicrographed and counted either visually or by means of a Quantimet 720 Image Analyzing Computer. The number ofBdellovibrio cells per milliliter in the undituted suspension could be calculated from the mean number of cells per unit area of the filter, the dilution factor, and the volume of diluted suspension filtered. TheBdellovibrio cells were distributed on the filters in a Poisson manner when there were not more than about 3.5 cells per 100 μm² of filter surface. The membrane filter direct counts correlated well with direct counts obtained by the Petroff-Hausser method. The correlation of direct counts with plaque (“viable”) counts showed that 80 to 95% of the direct-countedBdellovibrio cells in the clean suspensions were capable of forming plaques on lawns of suitable substrate bacteria. *** DIRECT SUPPORT *** A01R4002 00007     

Estimation of Poisson Rates with Misclassified Counts

The Poisson assumption is popular when data arises in the form of counts. In many applications such counts are fallible. Little research has been done on the Poisson distribution when both false positives and false negatives are present. We present a model in this paper that corrects for misclassification of count data. Bayesian estimators are developed. We provide the actual posterior distributions via integration. Markov Chain Monte Carlo results, which are more convenient for large sample sizes, are utilized for inference.