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The shape of the distribution of the number of sexual partners.

Information on the number of sexual partners that people have is necessary for predicting the likely long term course of the AIDS epidemic. More information is required than is provided by simply stating the mean and variance of the number of partners. It is useful to find a family of curves which approximate data on the proportion of people in the various 'number of partners' categories. Two Australian surveys of the sexual behaviour of first year university behavioural science students were analysed. It was found here that log-normal distributions gave good approximations to the distribution of number of partners amongst people who had more than one partner. Gamma and negative binomial distributions gave less satisfactory fits and the truncated normal and Poisson distributions were clearly unable to match the data.

Acquired Immunodeficiency Syndrome↗

SINGLE-DOSE ASSAY TECHNIQUE FOR VARIOLA VIRUS.

A biological assay for variola virus was needed that would (i) require a minimum of time and (ii) have acceptable precision. Preliminary titrations made in 11-day-old embryonated eggs and in suckling mice (6 to 24 hr of age) demonstrated a linear relationship between the concentration of variola virus injected and the mean reciprocal time to death (MTD) of both hosts. This linear response indicated that the injection of a single dose of virus suspension and the measurement of the MTD should result in an acceptable assay method. Seven replicate samples of liquid preparations (20% chorioallantoic membrane in Heart Infusion Broth) and the freeze-dried material obtained with these suspensions were assayed in triplicate for pock infectious units and for MTD. The variance of the pock counts was far greater than was expected from a Poisson distribution, and coefficients of variation ranged from 25 to 61%. Variances obtained with the single-dilution assay were all far below that expected from a Poisson distribution, and coefficients of variations ranged from 5.2 to 13.6%. The use of the MTD assay resulted in a saving of time, a saving in the number of hosts necessary per assay, and increased precision.

Animals↗

Bacterivory rate estimates and fraction of active bacterivores in natural protist assemblages from aquatic systems

Unlike the fraction of active bacterioplankton, the fraction of active bacterivores (i.e., those involved in grazing) during a specified time period has not been studied yet. Fractions of protists actively involved in bacterivory were estimated assuming that the distributions of bacteria and fluorescently labeled bacteria (FLB) ingested by protists follow Poisson distributions. Estimates were compared with experimental data obtained from FLB uptake experiments. The percentages of protists with ingested FLB (experimental) and the estimates obtained from Poisson distributions were similar for both flagellates and ciliates. Thus, the fraction of protists actively grazing on natural bacteria during a given time period could be estimated. The fraction of protists with ingested bacteria depends on the incubation time and reaches a saturating value. Aquatic systems with very different characteristics were analyzed; estimates of the fraction of protists actively grazing on bacteria ranged from 7 to 100% in the studied samples. Some nanoflagellates appeared to be grazing on specific bacterial sizes. Evidence indicated that there was no discrimination for or against bacterial surrogates (i.e., FLB); also, bacteria were randomly encountered by bacterivorous protists during these short-term uptake experiments. These analyses made it possible to estimate the ingestion rates from FLB uptake experiments by counting the number of flagellates containing ingested FLB. These results represent the first reported estimates of active bacterivores in natural aquatic systems; also, a proposed protocol for estimating in situ ingestion rates by protists represents a significant improvement and simplification to the current protocol and avoids the tedious work of counting the number of ingested FLB per protist.

Journal Article↗

Disease mapping using mixture distribution.

BACKGROUND & OBJECTIVES: Data on infectious diseases like tuberculosis (TB) have been analyzed in the past without giving adequate attention to spatial variations. Earlier studies also attempted to display disease status of sub regions, usually census tracts, by categorizing them into quartiles, that helps the authorities to identify high- or low-risk areas. This approach is based mainly on binomial and Poisson models for disease data, and the recent attempts focus on using mixture models of Poisson distribution. We carried out this study to find wards of Madurai Corporation having high risks for TB disease, to develop a model of mixture of Poisson distributions for the number of cases and to classify each ward to one of many risk groups for TB disease, and to represent spatial distribution of TB incidence in Madurai city. METHODS: produced the observed counts of TB patients in 72 wards of Madurai Corporation. The number of risk groups and the Poisson parameters of each group were found by maximum likelihood approach using the computer package C.A.MAN (Computer Assisted Mixture ANalysis). Bayesian methods were used to associate each ward to a particular risk group. The results were geographically presented in maps by using ArcView mapping software. RESULTS: Using binomial model, 26 wards were categorized as high risk wards, and with mixture model approach 15 wards showed standardized morbility ratio (SMR) >1. The wards along river Vaigai and densely populated wards had high risk. INTERPRETATION & CONCLUSION: Our findings demonstrate the usefulness of the mixture models for disease data with geographical variations.

Bayes Theorem↗

Effects of partial inbreeding on fixation rates and variation of mutant genes.

Diffusion methods were used to investigate the fixation probability, average time until fixation and extinction, and cumulative heterozygosity and genetic variance for single mutant genes in finite populations with partial inbreeding. The critical parameters in the approximation are the coefficient of inbreeding due to nonrandom mating (F) and the effective population size (Ne), which also depends on F and the variance of family size. For large Ns, the fixation probability (u) is u = 2(Ne/N)s (F + h - Fh), where N is the population census, s is the coefficient of selection of the mutant homozygote and h is the coefficient of dominance. For Poisson family size (independent Poisson distributions of selfed and nonselfed offspring with partial selfing, and independent Poisson distributions of male and female numbers with partial sib mating), Ne = N/(1 + F), and the time until fixation is approximately equal to Ne/N times the time to fixation with random mating, but this relation does not hold, however, for other distributions of family size. The cumulative nonadditive variance until fixation or loss for dominant genes is reduced with increasing F while for recessive genes it is increased with intermediate values of F. The average time until extinction of deleterious mutations is reduced by increasing F. This reduction, when expressed as a proportion, is approximately independent of the initial gene frequency as well as the selective disadvantage if this is large.

Animals↗

Impact of demographic distribution and population growth rate on haplotypic diversity linked to a disease gene and their consequences for the estimation of recombination rate: example of a French Canadian population.

A disease gene introduced into a rapidly growing population by a single individual remains in strong linkage disequilibrium with the surrounding molecular markers. Mapping strategies taking advantage of this phenomenon allow increased mapping resolution as compared to pedigree analysis. Demographic models underlying these strategies usually assume the population exponential growth approximated by Poisson distribution of the number of children per individual. Knowing the real demographic distribution in the studied French-Canadian population, we analyzed the validity of the Poisson approximation. We adapted the existing model of the Poisson branching process to the case of a rapidly growing population and to non-Poisson distributions. In consequence, we were able to apply maximum-likelihood methods to estimate the recombination rate under various demographic scenarios. Our analysis shows that the growth rate has a higher impact on the estimation of recombination rate than the shape of the demographic distribution. The choice of the demographic model (Poisson vs. non-Poisson) has little effect on the estimation of the recombination rate but affects the expected distribution of haplotype frequencies. This distribution, however, depends much more on the population growth rate. Finally, we also demonstrate the usefulness of the Luria-Delbrück method, which gives a correct estimation of the recombination rate in a growing population, provided the sampling error is taken into account in the confidence intervals.

Confidence Intervals↗

Bronchoalveolar lavage cellularity. The distribution in normal volunteers.

Bronchoalveolar lavage (BAL) is used to obtain inflammatory cells from the lung. For clinical research, parametric statistics are frequently used to compare cells present in BAL of patients with lung disease with cells present in BAL of normal subjects. To determine if these populations can be compared in this manner we performed BAL on 111 never-smoking, normal volunteers and determined: (1) the mean, median, standard deviation, and range of the cells in BAL; (2) whether the data are normally distributed and satisfy the criteria for use of parametric statistical analysis. The BAL cellularity was expressed as a percentage of total cells, cells per milliliter return, and total cells per lavage. Regardless of the means of expression, no measure of BAL cellularity (total cells, macrophages, lymphocytes, neutrophils, or eosinophils) conformed to the normal (bell-shaped) distribution when tested for goodness of fit with the G statistic (all p < 0.001). The lack of fit to the normal distribution was not substantially altered by either the method of expressing the data (i.e., cells per milliliter, total cells, or percent of cells) or log transformation of the data. The poor fit in all cases resulted from clumping of the data about the mean and large tails. The percent of cells were, therefore, tested for goodness of fit to the Poisson distribution, a distribution of discrete variables. The neutrophil and eosinophil percentages resulted in an excellent fit to the Poisson distribution, but macrophage and lymphocyte percentages did not.(ABSTRACT TRUNCATED AT 250 WORDS)

Adult↗

Usefulness and limits of biological dosimetry based on cytogenetic methods.

Damage from occupational or accidental exposure to ionising radiation is often assessed by monitoring chromosome aberrations in peripheral blood lymphocytes, and these procedures have, in several cases, assisted physicians in the management of irradiated persons. Thereby, circulating lymphocytes, which are in the G0 stage of the cell cycle are stimulated with a mitogenic agent, usually phytohaemagglutinin, to replicate in vitro their DNA and enter cell division, and are then observed for abnormalities. Comparison with dose-response relationships obtained in vitro allows an estimate of exposure based on scoring: Unstable aberrations by the conventional, well-established analysis of metaphases for chromosome abnormalities or for micronuclei; So-called stable aberrations by the classical G-banding (Giemsa-Stain-banding) technique or by the more recently developed fluorescent in situ hybridisation (FISH) method using fluorescent-labelled probes for centromeres and chromosomes. Three factors need to be considered in applying such biological dosimetry: (1) Radiation doses in the body are often inhomogeneous. A comparison of the distribution of the observed aberrations among cells with that expected from a normal poisson distribution can allow conclusions to be made with regard to the inhomogeneity of exposure by means of the so-called contaminated poisson distribution method; however, its application requires a sufficiently large number of aberrations, i.e. an exposure to a rather large dose at a high dose rate. (2) Exposure can occur at a low dose rate (e.g. from spread or lost radioactive sources) rendering a comparison with in vitro exposure hazardous. Dose-effect relationships of most aberrations that were scored, such as translocations, follow a square law. Repair intervening during exposure reduces the quadratic component with decreasing dose rate as exposure is spread over a longer period of time. No valid solution for this problem has yet been developed, although, in theory, both deterministic damage and aberrations might be repaired to a similar degree; a comparison of aberrations following a linear dose relationship might also help when the doses have been sufficiently large. (3) Investigations might have been possible only a certain time after the exposure. The relatively rapid disappearance of lymphocytes carrying unstable aberrations limits their use in retrospective dosimetry, years after exposure. Scoring stable aberrations, thought to persist in the circulating lymphocytes, might appear more appropriate in such situations. However, the examination of a representative number of cells by G-banding is extremely laborious, and the FISH method is not only expensive but has not yet been fully validated in different laboratories. In conclusion, biological dosimetry has serious limitations exactly for situations where the need for information is most urgent. It renders its most useful results when an individual has been exposed to a rather homogeneous high-level radiation over a short time interval, i.e. accidents at high-intensity radiation devices. On the other hand, it yielded less satisfactory information even when the most recent techniques were used for situations, where a low level, low dose rate exposure has occurred at some time in the past, for example for persons living in areas contaminated from the Chernobyl accident. Such negative experiences should be kept in mind in order to avoid futile and expensive investigations in the case of populations exposed from radioactivity and, notably, also from potentially clastogenic chemical agents.

Biological Assay↗

Photon counting statistics analysis of biophotons from hands.

The photon counting statistics of biophotons emitted from hands is studied with a view to test its agreement with the Poisson distribution. The moments of observed probability up to seventh order have been evaluated. The moments of biophoton emission from hands are in good agreement while those of dark counts of photomultiplier tube show large deviations from the theoretical values of Poisson distribution. The present results are consistent with the conventional delta-value analysis of the second moment of probability.

Hand↗

A comparison of two methods for estimating the effect of a countermeasure in the presence of regression effects.

The report studies the problem of estimating in a non-experimental before-and-after investigation the effect of a countermeasure on the number of traffic accidents at road junctions. The accidents are assumed to occur according to a Poisson process with different intensities at different junctions. The junctions studied in this investigation are assumed to have been selected with the blackspot-technique, i.e. junctions with high numbers of accidents during the before-period have been chosen for the investigation. In the mathematical model this has the consequence that the number of accidents occurring during the before-period at a selected junction has a truncated Poisson distribution. During the after-period the number of accidents has a Poisson distribution (without restrictions), so that the number of accidents on the average decreases between the periods even if the countermeasure has no effect. The magnitude of this regression effect is studied in the report. The observed numbers of accidents during the before and after period are used to estimate the pure effect of the countermeasure both with an intuitive method and with the maximum likelihood method. The characteristics of the two methods of estimation are illustrated with the aid of simulation studies. In general the maximum likelihood method appears preferable, mainly because it produces estimates with higher precision.

Accidents, Traffic↗

Iron catalyzed polyethylene chain growth on zinc: a study of the factors delineating chain transfer versus catalyzed chain growth in zinc and related metal alkyl systems.

The bis(imino)pyridine iron complex, [[2,6-(MeC=N-2,6-iPr2C6H3)2C5H)N]FeCl2] (1), in combination with MAO and ZnEt2 (> 500 equiv.), is shown to catalyze polyethylene chain growth on zinc. The catalyzed chain growth process is characterized by an exceptionally fast and reversible exchange of the growing polymer chains between the iron and zinc centers. Upon hydrolysis of the resultant ZnR2 product, a Poisson distribution of linear alkanes is obtained; linear alpha-olefins with a Poisson distribution can be generated via a nickel-catalyzed displacement reaction. Other dialkylzinc reagents such as ZnMe2 and ZniPr2 also show catalyzed chain growth; in the case of ZnMe2 a slight broadening of the product distribution is observed. The products obtained from Zn(CH2Ph)2 show evidence for chain transfer but not catalyzed chain growth, whereas ZnPh2 shows no evidence for chain transfer. The Group 13 metal alkyl reagents AlR3 (R = Me, Et, octyl, IBu) and GaR3 (R = Et, nBu) act as highly efficient chain transfer agents, whereas GaMe3 exhibits behavior close to catalyzed chain growth. LinBu, MgnBu2 and BEt3 result in very low activity catalyst systems. SnMe4 and PbEt4 give active catalysts, but with very little chain transfer to Sn or Pb. The remarkably efficient iron catalyzed chain growth reaction for ZnEt2 compared to other metal alkyls can be rationalized on the basis of: (1) relatively low steric hindrance around the zinc center, (2) their monomeric nature in solution, (3) the relatively weak Zn-C bond, and (4) a reasonably close match in Zn-C and Fe-C bond strengths.

Journal Article↗

Mathematical description of regenerative potentials recorded from circular smooth muscle of guinea pig antrum.

Regenerative potentials evoked by intracellular current injection in single bundles of circular smooth muscle taken from guinea pig antrum have the characteristics of the secondary regenerative component of the slow wave occurring in the same muscle layer. Such regenerative depolarizations might result from a mechanism that responds to membrane polarization with a delayed increase in the rate of production of unitary potentials detected in this tissue. To test this possibility, a two-stage reaction leading to the formation of an intracellular messenger was proposed. The first forward reaction was voltage-dependent, in the manner described by the Hodgkin-Huxley transient Na conductance formalism, allowing simulation of anode break excitation, stimulus threshold strength-duration characteristics, and refractory behavior. A conventional dose-effect relationship was proposed to describe the dependence of the mean rate of discharge of unitary potentials on messenger concentration. Unitary potentials were modeled as unitary membrane conductance modulations with an empirically derived amplitude distribution and Poisson-distributed intervals. The model reproduces a range of spontaneous and evoked membrane potential changes characteristic of antral circular muscle bundles.

Animals↗

LTD induction in adult visual cortex: role of stimulus timing and inhibition.

One Hertz stimulation of afferents for 15 min with constant interstimulus intervals (regular stimulation) can induce long-term depression (LTD) of synaptic strength in the neocortex. However, it is unknown whether natural patterns of low-frequency afferent spike activity induce LTD. Although neurons in the neocortex can fire at overall rates as low as 1 Hz, the intervals between spikes are irregular. This irregular spike activity (and thus, presumably, irregular activation of the synapses of that neuron onto postsynaptic targets) can be approximated by stimulation with Poisson-distributed interstimulus intervals (Poisson stimulation). Therefore, if low-frequency presynaptic spike activity in the intact neocortex is sufficient to induce a generalized LTD of synaptic transmission, then Poisson stimulation, which mimics this spike activity, should induce LTD in slices. We tested this hypothesis by comparing changes in the strength of synapses onto layer 2/3 pyramidal cells induced by regular and Poisson stimulation in slices from adult visual cortex. We find that regular stimulation induces LTD of excitatory synaptic transmission as assessed by field potentials and intracellular postsynaptic potentials (PSPs) with inhibition absent. However, Poisson stimulation does not induce a net LTD of excitatory synaptic transmission. When the PSP contained an inhibitory component, neither Poisson nor regular stimulation induced LTD. We propose that the short bursts of synaptic activity that occur during a Poisson train have potentiating effects that offset the induction of LTD that is favored with regular stimulation. Thus, natural (i.e., irregular) low-frequency activity in the adult neocortex in vivo should not consistently induce LTD.

Action Potentials↗

Experimental verification of overdispersion in radioassay data.

New evidence is provided suggesting that radioassay data are frequently overdispersed with respect to the Poisson distribution. Twelve cases of radioassay data were measured using commonly available detection systems. The data were analyzed using a limited version of the overdispersion model developed earlier. In that limit, the relationships between three overdispersed distributions were derived and discussed: beta-Poisson, negative binomial, and overdispersed Gaussian. Out of a total of 13 cases studied (12 measured plus one from the literature), 4 were consistent with the Poisson statistics at 90% confidence level while the remaining 9 were found overdispersed. This shows that the overdispersion is rather prevalent in radioassay. All three overdispersed distributions fitted the data very well. The overdispersion was attributed mostly to the excess fluctuations of the detection systems or, in 2 cases, sequential radioactive decay.

Models, Statistical↗

Multinucleation of cultured canine epithelial and lymphoma cell lines.

The frequency distribution patterns of monuclear and multinuclear giant cells were determined for two canine lymphoma cell lines (DT-5 and 11028), and a normal canine kidney epithelial cell line (DK). The proportion of multinuclear cells in the DK line (1.53%) was approximately twice those of the DT-5 (0.75%) and 11028 (0.73%) cell lines. The observed frequency distributions of cells with single and various numbers of multiple nuclei were compared to Poisson distributions using the chi-square test. For each cell line, the number of cells with three or more nuclei far exceeded the number predicted by the Poisson distribution. Hence, the occurrence of multinuclear cells in these canine cell lines does not follow a random distribution pattern. Possible explanations for the nonrandom accumulation of multinuclear giant cells are discussed.

Animals↗

Frequency distributions of Echinococcus granulosus and other helminths in stray dogs in Tunisia.

The intestinal helminths infecting stray dogs were investigated in the Testour and Bouzid regions of Tunisia. Overall, 21% of the dogs necropsied were found to be infected with Echinococcus granulosus. The prevalence among the animals from Testour (27.0%) was significantly higher than that in the dogs from Bouzid (6.9%). Although the prevalence of this species in the dog populations did give a significant fit to the age-prevalence models reported earlier, the best fit was not consistent with the known biological parameters of the parasite. The frequency distribution of E. granulosus was highly aggregated, with a mean abundance of infection of 538 parasites/host and a mean intensity in infected animals of 2534 parasites/host. Several Taenia species were also found in the dogs, but none of these showed evidence of an aggregated distribution and the frequency distribution of each fitted a Poisson distribution, which is unusual for naturally acquired infections. All the other cestode species and the nematode species encountered had aggregated distributions in the dogs. As there was relatively little correlation between the worm burdens of the different parasite species in the dogs, individual dogs may not be generally susceptible or resistant to intestinal helminths of several species.

Animals↗

Hypothesis testing of time-dependent recurrent events.

This paper examines the problem of hypothesis testing for comparison of time-dependent recurrent events between categorical exposure groups. We compare three methods (a summary chi-square test based on a generalization of the simple Poisson distribution, a chi-square test based on a generalization of the compound Poisson distribution, and a test on risk scores based on individual observed to expected ratios), when the dependent variable may be autocorrelated within an individual, and disease risk may be heterogeneous among subjects. We present a simulation study and an application to a cohort of sickle cell anaemia patients. With autocorrelation or heterogeneous risk present, the simple chi-square test is inappropriate, while the other two methods perform well. An attraction of the risk score method is its relative ease of application.

Anemia, Sickle Cell↗

Predictive Bayesian microbial dose-response assessment based on suggested self-organization in primary illness response: Cryptosporidium parvum.

The probability of illness caused by very low doses of pathogens cannot generally be tested due to the numbers of subjects that would be needed, though such assessments of illness dose response are needed to evaluate drinking water standards. A predictive Bayesian dose-response assessment method was proposed previously to assess the unconditional probability of illness from available information and avoid the inconsistencies of confidence-based approaches. However, the method uses knowledge of the conditional dose-response form, and this form is not well established for the illness endpoint. A conditional parametric dose-response function for gastroenteric illness is proposed here based on simple numerical models of self-organized host-pathogen systems and probabilistic arguments. In the models, illnesses terminate when the host evolves by processes of natural selection to a self-organized critical value of wellness. A generalized beta-Poisson illness dose-response form emerges for the population as a whole. Use of this form is demonstrated in a predictive Bayesian dose-response assessment for cryptosporidiosis. Results suggest that a maximum allowable dose of 5.0 x 10(-7) oocysts/exposure (e.g., 2.5 x 10(-7) oocysts/L water) would correspond with the original goals of the U.S. Environmental Protection Agency Surface Water Treatment Rule, considering only primary illnesses resulting from Poisson-distributed pathogen counts. This estimate should be revised to account for non-Poisson distributions of Cryptosporidium parvum in drinking water and total response, considering secondary illness propagation in the population.

Animals↗