The number distribution for involved lymph nodes in cancer

The number of involved lymph nodes exhibits considerable heterogeneity within populations. Here, the implications of population heterogeneity are explored with respect to the kinematics of nodal metastases. Data from the National Cancer Institute's Surveillance, Epidemiology, and End Results program for 224656 breast, 12404 gastric, 18015 rectal, 4117 cervical and 2443 laryngeal cancers as well as 9118 melanomas were used to construct frequency distributions for the number of involved nodes which were then fitted to the negative binomial distribution. The negative binomial distribution described the heterogeneity in nodal involvement well. The patterns of nodal involvement can be explained by either of two models: one where involved nodes could seed further nodal metastases, the other where the number of nodal metastases in any individual was randomly distributed, with the deviations between patients accounted for by population heterogeneity. Since the number of sampled nodes similarly approximated a negative binomial distribution, random involvement with superimposed population heterogeneity would more credibly explain both sets of observations.     

Test Based on Independent,but Non-identically Distributed Random Variables

The binomial test is applied for the problem of testing a hypothesis based on a sample of independent, but non-identically distributed random variables. The used basic idea is that each random variable indicates the presence of the hypothesis. Hence each random variable is transformed such that the binomial test can be used as a simple procedure.     

On the Theory of Partially Inbreeding Finite Populations. I. Partial Selfing

Some stochastic theory is developed for monoecious populations of size N in which there are probabilities beta and 1 - beta of reproduction by selfing and by random mating. It is assumed that beta much greater than N-1. Expressions are derived for the inbreeding coefficient of one random individual and the coefficient of kinship of two random separate individuals at time t. The mean and between-lines variance of the fraction of copies of a locus that are identical in two random separate individuals in an equilibrium population are obtained under the assumption that there is an infinite number of possible alleles. It is found that the theory for random mating populations holds if the effective population number is Ne = N'/(1 + FIS), where FIS is the inbreeding coefficient at equilibrium when N is infinite and N' is the reciprocal of the probability that two gametes contributing to random separate adults come from the same parent. When there is a binomial distribution of successful gametes emanating from each adult, N' = N. An approximation to the probability that an allele A survives if it is originally present in one AA heterozygote is found to be 2(N'/N)(FISS1 + (1 - FIS)S2), where S1 and S2 are the selective advantages of AA and AA in comparison with AA. In the last section it is shown that if there is partial full sib mating and binomial offspring distributions Ne = N/(1 + 3FIS).     

An exact test for detecting inconsistency in readers interpretation over time in screening mammograms

Radiologists' interpretation on screening mammograms is measured by accuracy indices such as sensitivity and specificity. The hypothesis that radiologists' interpretation on screening mammograms is constant across time can be tested by measuring overdispersion. However, small sample sizes are problematic for the accuracy of asymptotic approaches. In this article, we propose an exact conditional distribution for testing overdispersion of the binomial assumption that is assumed for the accuracy indices. An exact p -value can be defined from the developed distribution. We also describe an algorithm for computing this exact test. This proposed method is applied to data from a study in reading screening mammograms in a population of US radiologists (Beam et al., 2003). The exact method is compared analytically with a currently available method based on large sample approximations.     

On the power of some binomial modifications of the Bonferroni multiple test

Widely used in testing statistical hypotheses, the Bonferroni multiple test has a rather low power that entails a high risk to accept falsely the overall null hypothesis and therefore to not detect really existing effects. We suggest that when the partial test statistics are statistically independent, it is possible to reduce this risk by using binomial modifications of the Bonferroni test. Instead of rejecting the null hypothesis when at least one of n partial null hypotheses is rejected at a very high level of significance (say, 0.005 in the case of n = 10), as it is prescribed by the Bonferroni test, the binomial tests recommend to reject the null hypothesis when at least k partial null hypotheses (say, k = [n/2]) are rejected at much lower level (up to 30-50%). We show that the power of such binomial tests is essentially higher as compared with the power of the original Bonferroni and some modified Bonferroni tests. In addition, such an approach allows us to combine tests for which the results are known only for a fixed significance level. The paper contains tables and a computer program which allow to determine (retrieve from a table or to compute) the necessary binomial test parameters, i.e. either the partial significance level (when k is fixed) or the value of k (when the partial significance level is fixed).     

Predictive probability early termination plans for phase II clinical trials

A phase II clinical trial is designed to gather data to help decide whether an experimental treatment has sufficient effectiveness to justify further study. In a one-arm trial with dichotomous outcome, we wish to test a simple null hypothesis on the Bernoulli parameter against a one-sided alternative in a sample of N patients. It is advisable to have a rule to terminate the trial early when evidence accumulates that the treatment is ineffective. Predictive probabilities based on the binomial distribution and beta and uniform prior distributions for the binomial parameter are found to be useful as the basis of group sequential designs. Size, power and average sample size for these designs are discussed. A process for the specification of an early termination plan, advice on the quantification of prior beliefs, and illustrative examples are included.     

Prevalence and evolutionary relationships of haematozoan parasites in native versus introduced populations of common myna Acridotheres tristis

The success of introduced species is frequently explained by their escape from natural enemies in the introduced region. We tested the enemy release hypothesis with respect to two well studied blood parasite genera (Plasmodium and Haemoproteus) in native and six introduced populations of the common myna Acridotheres tristis. Not all comparisons of introduced populations to the native population were consistent with expectations of the enemy release hypothesis. Native populations show greater overall parasite prevalence than introduced populations, but the lower prevalence in introduced populations is driven by low prevalence in two populations on oceanic islands (Fiji and Hawaii). When these are excluded, prevalence does not differ significantly. We found a similar number of parasite lineages in native populations compared to all introduced populations. Although there is some evidence that common mynas may have carried parasite lineages from native to introduced locations, and also that introduced populations may have become infected with novel parasite lineages, it may be difficult to differentiate between parasites that are native and introduced, because malarial parasite lineages often do not show regional or host specificity.     

Functional heterogeneity at the frog sensory-motor synapse

This article compares four models of amplitude fluctuations in postsynaptic potentials. The convolution of two binomial distributions and the beta model proved the best fit with experimentally obtained data (as compared with the binomial model). The beta model is based on the assumption that the probability of quantal transmitter release is a random variable with a beta distribution. Numbers of quantal generators as estimated by the beta model were found to resemble numbers of morphological identifiable synaptic boutons. Findings from research using this model showed that the binomial parameter n may be interpreted as the number of transmitter release sites functioning with a probability in excess of 0.2. The findings obtained confirm the postulated functional diversity of release sites at interneuronal synapses.I. M. Sechenov Institute of Evolutionary Physiology and Biochemistry, Academy of Sciences of the USSR, Leningrad. Translated from Neirofiziologiya, Vol. 21, No. 6, pp. 780–788, November–December, 1989.     

Fitting of a binomial distribution to the number of chiasmata per bivalent

Statistical tests on the distribution of the number of chiasmata per chromosome, collected from literatures, showed that they can be approximated by binomial distributions with one obligatory chiasma, i.e., B(N-1, p). N is proportional to the average number of chiasmata, while p is nearly constant for the species tested.     

Extended moment analysis for binomial parameters of transmitter release.

The quantum hypothesis proposes that a binomial distribution should fit the amplitude distribution for synaptic potentials. Since importance is now being attached to significant changes in the n and p parameters of the binomial model during various treatments of synaptic preparations, this paper describes an important extension of the method of moments which can be used to extract binomial parameters in difficult experimental circumstances. Essentially, the skewness (third moment) of the observed amplitude distribution of synaptic responses is used to provide the additional information needed in cases where spontaneous miniature responses are absent. Computer simulations are used to assess the reliability of the proposed new estimators. The estimator bias due to non-uniform unit responses is also evaluated. Other applications of the extended method of moments, including a new test of the binomial hypothesis, are also described.     

Misleading interpretation of shifting baseline syndrome in the conservation of European mink

The European mink is a critically endangered mustelid with only three declining and isolated populations. Based on particular interpretation of genetic studies, the origin of SW population (France and Spain) and its conservation measures were recently questioned, and considered an example of managed relocation. The four published genetic studies show similar results, revealing a bottleneck structure. The SW population was probably established by very few individuals, which can be explained by dispersal of some-long distance migrants or by human introduction. The historical records evidence a long-distance dispersal capacity of the species and the temporal trend expansion in SW Europe is spatially collated with the idea of an expanding population. On contrast, most support for the introduction hypothesis comes from critics to other hypotheses, while it lacks of structure to stand on its own as hypothesis: no ecological barrier preventing natural expansion identified, no reports or reasons for captivity breeding or translocation programs, and lacks of supporting documents among others. Arguing shifting baseline syndrome in European mink conservation has weak basis and can result perilously misleading for a species in the brink of extinction.     

Probabilistic secretion of quanta from nerve terminals at synaptic sites on muscle cells: non-uniformity, autoinhibition and the binomial hypothesis

A model of the secretion of a quantum at a release site is proposed in which, following the influx of calcium ions, synaptic vesicles are made available for release by the activation of kappa phosphorylation steps with rate alpha. At any time during this process the vesicles may become unavailable for secretion at rate gamma. On completion of the kappa phosphorylation steps the vesicles participate in the formation of a fusion pore with the terminal membrane to give exocytosis at rate delta. Changes in alpha, delta and kappa are shown to produce characteristic changes in the number and timecourse of quantal secretions following a nerve impulse, which are similar to those observed following drug treatments that are thought to act selectively on each of these processes. The number of quanta secreted from nerve terminals that consist of many release sites does not fluctuate much during a low frequency train of impulses: the variance is small compared with the mean level, so secretion follows binomial rather than Poisson statistics. A theory is derived that shows that variations in the probability of secretion amongst these release sites of any particular kind fails to reduce the variance of the total secretion from the terminal; Poisson rather than binomial statistics then still apply. The theory shows that an interaction between release sites is required to reduce this variance and such an effect is provided if secretion at a site inhibits secretion at nearby sites. Simulations show that incorporating this process of autoinhibition into the model reproduces the experimental observations on the effects of calcium ions on the binomial parameters p and n as well as on the relative constancy of p during facilitation and depression of quantal secretion. Methods for estimating the timecourse of changes in the probability of secretion at release sites following an impulse, by using either the time of occurrence of first, second, third or later quantal latencies, are given. These procedures show that current methods for estimating the time-dependent probability changes are inadequate for detecting interaction between release sites, such as autoinhibition, unless this is relatively large. Therefore, estimates from third quantal latencies are used.     

Quantal Characteristics of Synaptic GABA Release under Conditions of Stimulation of Single Synaptic Terminals of Cultured Cortical Cells of the Rat

We studied evoked inhibitory postsynaptic currents (eIPSC) using local electrical stimulation of single presynaptic terminals of cultured rat neocortical neurons. According to pharmacological and kinetic properties, these currents were qualified as GABA_A-activated. Using autocorrelation analysis of distributions of the eIPSC amplitudes, which were in all cases polymodal, we examined quantal characteristics of the above eIPSC. These results were compared with the values of quantal parameters (N, p, Q, and m) of the current families obtained using approximation by binomial distribution. Amplitude histograms of spontaneous miniature IPSC recorded under conditions of the minimum quantal release of the neurotransmitter were normal (close to Gaussian) with the mode within a 10 pA range, which is very close to analogous parameters calculated using autocorrelation and binomial techniques.     

Statistical methods to decide whether mutagenicity test data from Drosophila assays indicate a positive, negative, or inconclusive result

Two alternative hypotheses are used to distinguish among the possibilities of a positive, inconclusive, or negative result in Drosophila mutagenicity tests. In the null hypothesis one assumes that there is no difference in the mutation frequency between control and treated series. The alternative hypothesis postulates a priori that the treatment results in an increased mutation frequency that is m times the spontaneous frequency. To test against the hypotheses, the conditional binomial test according to Kastenbaum and Bowman or the chi 2 test for proportions may be applied. These 2 methods are in principle equivalent. An alternative method which is based on determining confidence limits of observed mutation frequencies also leads to the same conclusions. The practical calculations are formulated and an application is shown with a test example demonstrating the genotoxicity of the pyrrolizidine alkaloid 7-acetylintermedine in the somatic wing mosaic test. In the Appendix, the calculus for the 3 testing methods is explained with a numerical example.     

The parasite capacity of the host population

The estimation of parasitic pressure on the host populations is frequently required in parasitological investigations. The empirical values of prevalence of infection are used for this, however the latter one as an estimation of parasitic pressure on the host population is insufficient. For example, the same prevalence of infection can be insignificant for the population with high reproductive potential and excessive for the population with the low reproductive potential. Therefore the development of methods of an estimation of the parasitic pressure on the population, which take into account the features the host population, is necessary. Appropriate parameters are to be independent on view of the researcher, have a clear biological sense and be based on easily available characteristics. The methods of estimation of parasitic pressure on the host at the organism level are based on various individual viability parameters: longevity, resistance to difficult environment etc. The natural development of this approach for population level is the analysis of viability parameters of groups, namely, the changing of extinction probability of host population under the influence of parasites. Obviously, some critical values of prevalence of infection should exist; above theme the host population dies out. Therefore the heaviest prevalence of infection, at which the probability of host population size decreases during the some period is less than probability of that increases or preserves, can serve as an indicator of permissible parasitic pressure on the host population. For its designation the term "parasite capacity of the host population" is proposed. The real parasitic pressure on the host population should be estimated on the comparison with its parasite capacity. Parasite capacity of the host population is the heaviest possible prevalence of infection, at which, with the generation number T approaching infinity, there exists at least one initial population size ni(0) for which the probability of size decrease through T generations is less than the probability of its increase. [formula: see text] The estimation of the probabilities of host population size changes is necessary for the parasite capacity determination. The classical methods for the estimation of extinction probability of population are unsuitable in this case, as these methods require the knowledge of population growth rates and their variances for all possible population sizes. Thus, the development methods of estimate of extinction probability of population, based on the using of available parameters (sex ratio, fecundity, mortality, prevalence of infection PI) is necessary. The population size change can be considered as the Markov process. The probabilities of all changes of population size for a generation in this case are described by a matrix of transition probabilities of Markov process (pi) with dimensions Nmax x Nmax (maximum population size). The probabilities of all possible size changes for T generations can be calculated as pi T. Analyzing the behaviour matrix of transition at various prevalence of infection, it is possible to determine the parasite capacity of the host population. In constructing of the matrix of transition probabilities, should to be taken into account the features the host population and the influence of parasites on its reproductive potential. The set of the possible population size at a generation corresponds to each initial population size. The transition probabilities for the possible population sizes at a generation can be approximated to the binomial distribution. The possible population sizes at a generation nj(t + 1) can be calculated as sums of the number of survived parents N1 and posterities N2; their probabilities--as P(N1) x P(N2). The probabilities of equal sums N1 + N2 and nj(t + 1) > or = Nmax are added. The number of survived parents N1 may range from 0 to (1-PI) x ni(t). The survival probabilities can be estimated for each N1 as [formula: see text] The number of survived posterities N2 may range from 0 to N2max (the maximum number of posterities). N2max is [formula: see text] and the survival probabilities for each N2, is defined as [formula: see text] where [formula: see text], ni(t) is the initial population size (including of males and infected specimens of host), PI is the prevalence of infection, Q1 is the survival probabilities of parents, Pfemales is the frequency of females in the host population, K is the number of posterities per a female, and Q2 is the survival probabilities of posterities. When constructing matrix of transition probabilities of Markov process (pi), the procedure outlined above should be repeated for all possible initial population size. Matrix of transition probabilities for T generations is defined as pi T. This matrix (pi T) embodies all possible transition probabilities from the initial population sizes to the final population sizes and contains a wealth of information by itself. From the practical point of view, however, the plots of the probability of population size decrease are more suitable for analysis. They can be received by summing the probabilities within of lines of matrix from 0 to ni--1 (ni--the population size, which corresponds to the line of the matrix). Offered parameter has the number of advantages. Firstly, it is independent on a view of researcher. Secondly, it has a clear biological sense--this is a limit of prevalence, which is safe for host population. Thirdly, only available parameters are used in the calculation of parasite capacity: population size, sex ratio, fecundity, mortality. Lastly, with the availability of modern computers calculations do not make large labour. Drawbacks of this parameter: 1. The assumption that prevalence of infection, mortality, fecundity and sex ratio are constant in time (the situations are possible when the variability of this parameters can not be neglected); 2. The term "maximum population size" has no clear biological sense; 3. Objective restrictions exist for applications of this mathematical approach for populations with size, which exceeds 1000 specimens (huge quantity of computing operations--order Nmax 3*(T-1), work with very low probabilities). The further evolution of the proposed approach will allow to transfer from the probabilities of size changes of individual populations to be probabilities of size changes of population systems under the influence of parasites. This approach can be used at the epidemiology and in the conservation biology.     

On the Theory of Partially Inbreeding Finite Populations. III. Fixation Probabilities under Partial Selfing When Heterozygotes Are Intermediate in Viability

In a previous paper by the senior author, an approximation to the probability of survival was given for a mutant, which is originally present in a single heterozygote, in a population that reproduces partly by selfing and partly by random mating. The population was assumed to be very large, but the result obtained is general with regard to the level of dominance in viability. In this paper two errors which were made in that earlier work are corrected. A general approximate expression is then derived for the probability that an allele A is fixed in a partially self fertilizing population of size N, if its initial frequency is p, selection is weak and heterozygotes with the allele are exactly intermediate in viability compared with genotypes AA and AA. A rigorous proof is given for a special case that is a generalization of the classical binomial sampling model. In this case, but not in general, the approximate fixation probability is independent of the probability of reproduction by selfing. Some implications are discussed.     

Sampling of Nesidiocoris tenuis (Heteroptera: Miridae) in tomato crops

The number of N. tenuis on the upper, middle, and bottom leaves of tomato plants was correlated; nymphs were most abundant on middle and adults on upper leaves. The dispersion pattern of N. tenuis is aggregated and can be described using a negative binomial distribution. A binomial sampling method is proposed.     

Estimating the population size and colony boundary of subterranean termites by using the density functions of directionally averaged capture probability

Marked termites were released in a linear-connected foraging arena, and the spatial heterogeneity of their capture probabilities was averaged for both directions at distance r from release point to obtain a symmetrical distribution, from which the density function of directionally averaged capture probability P(x) was derived. We hypothesized that as marked termites move into the population and given sufficient time, the directionally averaged capture probability may reach an equilibrium P(e) over the distance r and thus satisfy the equal mixing assumption of the mark-recapture protocol. The equilibrium capture probability P(e) was used to estimate the population size N. The hypothesis was tested in a 50-m extended foraging arena to simulate the distance factor of field colonies of subterranean termites. Over the 42-d test period, the density functions of directionally averaged capture probability P(x) exhibited four phases: exponential decline phase, linear decline phase, equilibrium phase, and postequilibrium phase. The equilibrium capture probability P(e), derived as the intercept of the linear regression during the equilibrium phase, correctly projected N estimates that were not significantly different from the known number of workers in the arena. Because the area beneath the probability density function is a constant (50% in this study), preequilibrium regression parameters and P(e) were used to estimate the population boundary distance 1, which is the distance between the release point and the boundary beyond which the population is absent.     

Introduced plants of the invasive Solidago gigantea (Asteraceae) are larger and grow denser than conspecifics in the native range

Introduced plant species that became successful invaders appear often more vigorous and taller than their conspecifics in the native range. Reasons postulated to explain this better performance in the introduced range include more favourable environmental conditions and release from natural enemies and pathogens. According to the Evolution of Increased Competitive Ability hypothesis (EICA hypothesis) there is a trade‐off between investment into defence against herbivores and pathogens, and investment into a stronger competitive ability. In this study, we conducted field surveys to investigate whether populations of the invasive perennial Solidago gigantea Ait (Asteraceae) differ with respect to growth and size in the native and introduced range, respectively. We assessed size and morphological variation of 46 populations in the native North American range and 45 populations in the introduced European range. Despite considerable variation between populations within continents, there were pronounced differences between continents. The average population size, density and total plant biomass were larger in European than in American populations. Climatic differences and latitude explained only a small proportion of the total variation between the two continents. The results show that introduced plants can be very distinct in their growth form and size from conspecifics in the native range. The apparently better performance of this invasive species in Europe may be the result of changed selection pressures, as implied by the EICA hypothesis.