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PubMed1976

Nomograms for obtaining a necessary, minimum sample size. II. When distribution of proportions from binomial data is approximately normal.

Two different nomograms corresponding to finite and infinite populations were devised to obtain a necessary, minimum sample size at a 95% confidence rate when the distribution of sample proportions calculated from binomial data was approximately normal. They could also be used for obtaining the 95% confidence limits of the population proportion from a sample proportion after having carried out a work.

Ecology↗
PubMed1989

The extremal quotient in small-area variation analysis.

This article reviews the current small-area variation analysis (SAVA) approach to population-based rates of surgery, and describes a new method for ascertaining variance based on the beta-binomial probability distribution of small-area rates. The critical review of the current SAVA approach focuses (1) on how incidence rates are calculated, and (2) on how the significance of the observed magnitude between the largest and smallest rates (i.e., the external quotient) is ascertained. While reducing the problems of calculating rates by considering only certain operative procedures, the new method addresses the current inadequacies of ascertaining significant differences among small areas. Not only does it correctly assess likelihood of an extermal quotient, it also can determine the particular area's rate, producing an unlikely extermal quotient. The method evaluates the probability that the observed magnitude of the extremal quotient is due solely to chance and study design effects, and tables of these probabilities are available for the method's application. A mathematical model, based on a combination of the binomial and beta distributions, uses (1) the sample size, (2) the average of the areas' rates, (3) the variance among the rates, and (4) a specific quotient level to determine the probability of observing the quotient by chance. After computerizing this calculation, probability tables for reasonable values of these four parameters are generated. In addition to looking at just one quotient for each sample, the probability tables facilitate the easy examination of intermediate quotients when the extremal quotient is unlikely due to chance. By alternatively ignoring the highest and lowest rates, two new quotients can be produced and tested. Given that one of these two quotients is likely due to chance, the excluded rate (i.e., producing the unlikely extremal quotient) can be classified as an outliner, and the associated small area should be the focus of more detailed investigation. The probability tables reveal that the external quotient is not the appropriate statistic to be applied in studies where many small areas are to be included. The probability of seeing even a "large" extremal quotient simply by chance rapidly approaches one as the sample size increases. However, an extremal quotient modeled from a beta-binomial distribution can be useful for studies with small sample sizes (e.g., six counties). The use of this beta-binomial model for small-area rates provides a new method of designing and evaluating small-area studies where costs or domain limit the number of areas under consideration.(ABSTRACT TRUNCATED AT 400 WORDS)

Analysis of Variance↗
PubMed1998

Measurement and biological correlates of antibody bioactivity during antibody immunotherapies.

An important factor in the effectiveness of antibody immunotherapies is the antibody bioactivity, or the availability of free binding sites. Bioactivity may be decreased in settings where the target antigen exists in a soluble form capable of binding to circulating antibody. Because many antigens have soluble forms, we developed a method for determining if antibody is bound by soluble antigen in vivo. As a model system, we studied the interaction of soluble interleukin-2 receptor alpha (Tac; IL2R alpha; CD25) and anti-Tac antibody. We show first that HPLC readily separates free antibody from antibody which is monovalently or bivalently bound by soluble antigen. Further, we demonstrate that the distribution of the three forms of antibody accords with predictions of mass action and the binomial probability distribution. These methods were used to examine the bioactivity and concentration of free antibody in 14 patients undergoing therapeutic trial with Humanized anti-Tac antibody in leukemia and lymphoma. Results of two contrasting patients are highlighted. Low bioactivities correlated with reduced targeting of tumor cells and reduced therapeutic effectiveness. This report highlights the importance of soluble antigen in antibody therapies and demonstrates a simple method for evaluating in vivo bioactivity of antibody after therapeutic administration.

Adult↗
PubMed1981

Statistical methods for the analysis of tumor multiplicity data.

The fit of tumor multiplicity data from 93 mouse skin, lung, and liver carcinogenicity experiments to Poisson, negative binomial, and normal distributions was studied. The data were fitted well by the negative binomial distribution. This distribution has two parameters, the mean tumor multiplicity and an exponent determined by the interanimal homogeneity of tumor response. The value of the latter parameter was related to animal strain and the target tissue studied in the carcinogenicity experiments. The null distribution of the two-sample likelihood ratio test based on the negative binomial with common exponent model for tumor multiplicity data was shown by simulation studies to be approximately chi 2 with 1 d.f. Simulation also indicated that the likelihood ratio test has sufficiently better performance when the negative binomial model is valid to make its use more attractive than the more commonly used Wilcoxon test or Student t test. Charts for estimating the number of animals per group that are required to detect specified differences in tumor multiplicities are provided for several commonly used assays.

Adenoma↗
PubMedDate not supplied

[Probabilistic models to analyze psychiatric contacts].

OBJECTIVE: To present three probabilistic models (Poisson, negative binomial, Waring) to analyze the distribution of the number of contacts of patients followed using a psychiatric case register. DESIGN: Longitudinal to obtain the distribution of the number of contacts (during 91 days following that of the first contact) observed on patients followed using the South-Verona Psychiatric Case Register during the period 1/1/79-31/12/91. RESULTS: There were a total number of 6913 contacts on 3454 subjects. The chi-square test for the goodness of fit yields a significant result both for the Poisson distribution (3580 with 6 degrees of freedom, p < 0.001) and for the negative binomial distribution (65.47 with 18 degrees of freedom, p < 0.001); on the other hand a non significant result was obtained for the Waring distribution (25.31 with 19 degrees of freedom, p = 0.151). CONCLUSIONS: The Poisson distribution gave a very poor fit for the distribution of contacts. The negative binomial distribution could be employed to analyze the pattern of contacts when the right tail of the distribution is not important. The Waring distribution is the best of the three presented. Moreover, the variance of the Waring distribution can be decomposed in three components: a random component, a component which accounts for endogenous factors and another component which accounts for esogenous factors. Therefore the Waring distribution is useful when one wants to make comparisons between psychiatric case registers of the same country or of different countries.

Humans↗
PubMed2005

Activity levels and drug use in a sample of Spanish adolescents.

The importance of adolescents taking part in various activities as a protective factor in substance consumption has been demonstrated over time. The objective of this paper is to analyze a sample of 1378 adolescents and the explanatory capacity of participating in different activities in relation to present-day alcohol, cannabis and tobacco consumption. Count variables were used as response variables, thus the Poisson Regression Model was applied in the analysis, within the context of a Generalized Linear Model. The results of the research demonstrate the explanatory value and differences in explanatory roles of each activity in connection with each the substances studied.

Adolescent↗
PubMed2003

A longitudinal study of a natural lice infestation in growing cattle over two winter periods.

A group of 61 cattle which were naturally infested with lice was followed over two winter periods. Data were collected on the number of lice found at various body sites over this period. Summary statistics were produced and due to the repeated measured and hierarchical structure of the data, multi-level analysis was used to model the population dynamics of Bovicola bovis and assess the influence of the various hierarchical levels. A four level Poisson model was produced-level one, the individual number of lice at each parting; level two, the body site examined (shoulder, midline or rump); level three, the time of the inspection and level four, the animal. Seasonal fluctuation in lice numbers was modelled using a cosine function transformation of time. A seasonal pattern was seen in both years with lice counts higher in the first than the second year. The midline area was the most sensitive to detection of B. bovis. Variance contributed by the individual animal was less than that contributed by the body site examined and the time of the inspection. The model suggested that lice numbers within the shoulder and rump sites were near random following a Poisson distribution, but aggregation of lice occurred at the midline site with the distribution between animals following a Negative Binomial Pattern. The midline site was the most sensitive site for detecting B. bovis. Infestation numbers were higher in the first winter when cattle were younger. In the second winter, there was no difference in lice numbers between cattle exposed to infection for the first or second time.

Animals↗
PubMed1993

Mathematical models for the population biology of Ostertagia ostertagi and the significance of aggregated parasite distributions.

Parasite frequency distributions are frequently aggregated. Such distributions are conveniently described using the negative binomial frequency distribution. This distribution is completely characterized by two parameters: the mean of the distribution and an exponent, k. The degree of aggregation is inversely proportional to the value of k. Aggregated parasite distributions enhance the ability of regulatory processes to maintain parasite populations at or near their equilibrium level but incorporating parasite frequency distributions in realistic models of parasite population biology is fraught with difficulty (several simplified examples are given). Indeed, it is not always possible to incorporate parasite frequency distributions and the question arises whether this compromises model performance. Trichostronglyid nematodes, including Ostertagia ostertagi, are aggregated but the estimated value of k for such populations is usually greater than 1. This is typical of populations with high population means (tens of thousands). It is shown that when the degree of aggregation is such that k > 1, the results of a model which recognizes parasite frequency distribution is not much different from the results of a model which assumes all hosts contain exactly the same number of parasites.

Animals↗
PubMed2002

Evidence for two distinct processes in the final stages of neurotransmitter release as detected by binomial analysis in calcium and strontium solutions.

The statistical parameters underlying acetylcholine (ACh) release were studied using Ca(2+) and Sr(2+) ions to promote ACh secretion. Experiments were performed at frog neuromuscular junctions using electrophysiological recording techniques. Increases in asynchronous ACh release, reflected as the frequency of occurrence of miniature end-plate potentials (MEPP(f)), were evoked by high potassium depolarization in either Ca(2+) or Sr(2+) solutions. Increases in MEPP(f) mediated by Ca(2+) were of very low probability and well-described by a Poisson distribution whilst similar MEPP(f) increases mediated by Sr(2+) were best described as a simple binomial distribution. From the binomial distribution in Sr(2+) solutions, values for the average probability of release (p) and the number of releasable ACh quanta (n) may be determined (whereby mean MEPP(f) = np). In Sr(2+) solutions, values of p were independent of both bin width and of the value of n, suggesting that both n and p were stationary. Calculations of p using the simple binomial distribution in Sr(2+) solutions gave theoretical values for the third moment of the mean which were indistinguishable from the experimental distribution. These results, in conjunction with Monte Carlo simulations of the data, suggest that spatial and temporal variance do not measurably affect the analysis. Synchronous ACh release evoked by nerve impulses (end-plate potentials, EPPs) follow a simple binomial distribution in both Ca(2+) and Sr(2+) solutions. Similar mean levels of synchronous ACh release (m, where m = np) were produced by lower values of p and higher values of n in Ca(2+) as compared to Sr(2+). The statistical analyses suggest the presence of two different Ca(2+)-dependent steps in the final stages of neurotransmitter release. The results are discussed in accordance with (i) statistical models for quantal neurotransmitter release, (ii) the role of Sr(2+) as a partial agonist for evoked ACh release, and (iii) the specific loci that may represent the sites of Ca(2+) and Sr(2+) sensitivity.

Acetylcholine↗
PubMed1993

Heterogeneity in patterns of malarial oocyst infections in the mosquito vector.

Oocyst prevalence and intensity have been recorded in 349 laboratory infections of Anopheles stephensi with Plasmodium berghei. Intensity and prevalence of infection are shown to be predictably related. The structure and heterogeneity in the infections has been analysed with the objective of describing the biological mechanisms by which the observed negative binomial oocyst distributions are generated. The analysis has revealed that the most likely processes lie within the population dynamic events of malaria within the mosquito, namely gametogenesis, fertilization and mortality. The distribution is similar in all Plasmodium-mosquito combinations examined so far, whether they are of laboratory (P. gallinaceum in Aedes aegypti) or field (P. vivax in An. albimanus and P. falciparum in An. gambiae s.l. and An. funestus) origin. Further we conclude that there is competition between parasites in the vector. Oocyst frequency distribution analysis shows that under natural conditions of transmission intensity, and even under the best laboratory conditions, significant numbers (> 10%) of fully susceptible mosquitoes will not be infected under conditions where the mean infection is as high as 250 oocysts. Failure to infect is not therefore an absolute indicator of refractoriness. In assessing transmission data it is shown that sample sizes should not be less than 50, and ideally 100 mosquitoes, if reliable data are to be obtained. In field it is suggested that difficulties in determining the low natural intensity of oocyst infections indicate that prevalence estimates are a useful and accessible parameter to measure.(ABSTRACT TRUNCATED AT 250 WORDS)

Aedes↗
PubMed1990

[Distribution pattern of Dirofilaria immitis microfilariae in peripheral blood of a dog and their intake by mosquitoes].

Experiments were made in which Aedes aegypti mosquitoes fed on a dog infected with Dirofilaria immitis, in order to determine the distribution pattern of microfilariae (Mf) in the dog's periphery blood and the number of Mf taken by the mosquitoes. Mosquitoes fed on an anesthetized dog at some 50 shaved skin spots and they were weighed immediately before and after feeding. This enabled us to examine not only the Mf number taken by each mosquito but also the Mf intake density in terms of Mf number per mg of blood. The Mf intake density showed a wide variation at the different parts of the dog's body, and the frequencies of Mf showed uneven distributions which were fitted best by the negative binomial. Significant differences were detected in the Mf intake density at different parts of the skin, but these differences were concluded as a non-consistent result because a similar result was not reproduced in the next experiment. It was presumed that not only the Mf number per mosquito but also the Mf intake density increased as the blood intake by the mosquitoes increased. This increase of both the Mf number per mosquito and the Mf intake density was also supported by another experiment in which mosquitoes were fed at the same skin spots consecutively one after another.

Aedes↗
PubMed1975

[Sterility control: basic mathematical considerations (author's transl)].

The subject of control of sterility in homogenous lots by means of random sampling is discussed. The probability that non-sterile lots may be declared as accepted, was calculated by means of hypergeometric, binomial, and Poisson distribution. The following factors were considered: size of samples, size of lots, and degree of contamination of lots. The values obtained were reproduced in graphs (Figs. 1, 2, and 4). Reference is made to the conditions that have to be met in each case to enable a use of the various distribution functions for a calculation of the acceptance probability of non-sterile lots (Table 1). Furthermore, the necessary size of samples to contain at least one contaminated unit was calculated by means of binomial distribution for probabilities of 90, 95, and 99% (Fig. 3). For example, from a lot having a degree of contamination of 10%, 30 units would have to be assayed for sterility to recognize non-sterility of such a lot in 95% of cases. A degree of contamination of 1% would reguire already 300 units to be examined...

Mathematics↗
PubMed1998

A model of multiple tumorigenesis allowing for cell death: quantitative insight into biological effects of urethane.

This paper considers the utility of a stochastic model of carcinogenesis proposed by Yakovlev and Polig [Math. Biosci. 132 (1996) 1-33] in the analysis of experimental data on multiple tumors induced by chemical carcinogens. The model provides a good description of published data on multiple tumors developing in the lungs of mice in response to different schedules of urethane. The distribution of pulmonary tumor counts appears to be negative binomial for each period of time after exposure to urethane. Our results suggest that the rate of administration of urethane has little effect both on the mean number of initiated cells per unit dose and on the rate of formation of lesions responsible for cell death. As our estimates show, more than 80% of initiated cells die in the course of tumor promotion. The model is robust to variations in the rate of urethane excretion given a fixed total dose of the carcinogen. Some prospects for further development of the model to allow for expansion of promoted cell clones are discussed.

Animals↗
PubMed1994

The evaluation of potential global morbidity attributable to intestinal nematode infections.

This paper presents a method of estimating the potential global morbidity due to human intestinal nematode infections (Ascaris lumbricoides, Trichuris trichiura and hookworms), based on the observed prevalence of infection. The method relies on the observed relationships between prevalence and intensity of infection, and between worm burden and potential morbidity. This approach is shown to be sensitive to the precision of the original prevalence estimates and, in particular, to the degree of spatial heterogeneity in levels of infection. The estimates presented here indicate that several tens of millions of children are likely to suffer developmental consequences from infection, and suggest that the global disease burden of geohelminthiasis may be significantly greater than was supposed.

Adolescent↗