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PubMed1992

A statistical approach to drug sampling: a case study.

In many countries it is left to the discretion of the court to accept or reject conclusions based on sampling procedures as applied to the total drug exhibit. As an alternative to this subjective approach, a statistical basis is presented using binomial and hypergeometric distributions to determine a lower limit for the proportion of units in a population which contains a drug, at a given confidence level. A method for calculating the total weight of a drug present in a population within a given confidence interval is also presented. In the event of no failures (all units sampled contain a drug), a sample size of six or seven units is generally sufficient to state that a proportion of at least 0.70 of the population contains a drug at a confidence level of at least 90%. When failures do occur in the sample, point estimation is used as the basis for selecting the appropriate sample size.

Forensic Medicine↗
PubMedDate not supplied

Scoring in genetically modified organism proficiency tests based on log-transformed results.

The study considers data from 2 UK-based proficiency schemes and includes data from a total of 29 rounds and 43 test materials over a period of 3 years. The results from the 2 schemes are similar and reinforce each other. The amplification process used in quantitative polymerase chain reaction determinations predicts a mixture of normal, binomial, and lognormal distributions dominated by the latter 2. As predicted, the study results consistently follow a positively skewed distribution. Log-transformation prior to calculating z-scores is effective in establishing near-symmetric distributions that are sufficiently close to normal to justify interpretation on the basis of the normal distribution.

Data Interpretation, Statistical↗
PubMed1983

Modeling denominator populations for mental health care.

Difficulties in using basic epidemiologic survey techniques in estimating the prevalence of disease in the community have led health services researchers to develop alternative methods using routinely collected health care data. The purpose of this paper is to illustrate the application of a model that estimates the prevalence of diagnosable mental disorder in a community, including the proportion of untreated cases. A mathematical model is developed based on a well-known probability distribution (the negative binomial) and simple calculation formulae are presented. This model has been used to estimate total practice population size. This paper extends that work by illustrating the utility of the model vis-a vis a specific type of disorder and shows pitfalls when caseness is not well established. An application of the model to data from two studies in an organized health care setting shows that the estimated period prevalence (one year) of mental disorder is between 17.4% and 21.8%. Of those individuals, it is estimated that 28.6% seek no health care during the year. The implications of these findings and the utility of the model are discussed.

Catchment Area, Health↗
PubMed1994

The resampling cross-validation technique in exercise science: modelling rowing power.

The past 10-15 yr have witnessed a rapid increase in the development of new (and not so new) statistical methods that capitalize on recent advances in high-speed computing. These computer-intensive methods are often broadly referred to as resampling techniques and take several forms depending on the specific details of the procedure and the information of interest. Resampling techniques can be used both for inferential hypothesis testing as well as exploratory data description. Regardless of which method is employed, the central unifying theme is based upon the computer's power to rapidly resample many pseudosamples from a known (in-hand) data set (e.g., randomization tests, jackknife, boot-strap, cross-validation) or to randomly generate many pseudosamples from a theoretical probability distribution (e.g., normal, binomial, Poisson) with some known parameters (Monte Carlo method). This paper is not intended as a detailed description of computer-intensive methods, but only as an introduction to the resampling approach in cross-validation. A brief discussion of the motivation and an example in an exercise science context will be presented.

Humans↗
PubMed1990

Frequency distribution of lymphatic filariasis microfilariae in human populations: population processes and statistical estimation.

This paper uses simple mathematical models and statistical estimation techniques to analyse the frequency distribution of microfilariae (mf) in blood samples from human populations which are endemic for lymphatic filariasis. The theoretical analysis examines the relationship between microfilarial burdens and the prevalence of adult (macrofilarial) worms in the human host population. The main finding is that a large proportion of observed mf-negatives may be 'true' zeros, arising from the absence of macrofilarial infections or unmated adult worms, rather than being attributable to the blood sampling process. The corresponding mf distribution should then follow a Poisson mixture, arising from the sampling of mf positives, with an additional proportion of 'true' mf-zeros. This hypothesis is supported by analysis of observed Wuchereria bancrofti mf distributions from Southern India, Japan and Fiji, in which zero-truncated Poisson mixtures fit mf-positive counts more effectively than distributions including the observed zeros. The fits of two Poisson mixtures, the negative binomial and the Sichel distribution, are compared. The Sichel provides a slightly better empirical description of the mf density distribution; reasons for this improvement, and a discussion of the relative merits of the two distributions, are presented. The impact on observed mf distributions of increasing blood sampling volume and extraction efficiency are illustrated via a simple model, and directions for future work are identified.

Animals↗
PubMed1986

Molecular basis of the oxygen exchange from CO2 catalyzed by carbonic anhydrase III from bovine skeletal muscle.

The exchange of 18O from CO2 to H2O in aqueous solution is caused by the hydration-dehydration cycle and is catalyzed by the carbonic anhydrases. In our previous studies of 18O exchange at chemical equilibrium catalyzed by isozymes I and II of carbonic anhydrase, we observed simple first-order depletion of 18O from CO2 with the 18O distribution among the species C18O18O, C16O18O, and C16O16O described by the binomial expansion (i.e., a random distribution of 18O). Using membrane-inlet mass spectrometry, we have measured 18O exchange between CO2 and H2O catalyzed by native zinc-containing and cobalt(II)-substituted carbonic anhydrase III from bovine skeletal muscle near pH 7.5. The distributions of 18O in CO2 deviate from the binomial expansion and are accompanied by biphasic 18O-exchange patterns; moreover, we observed regions in which 18O loss from CO2 was faster than 18O loss from HCO3-. These data are interpreted in terms of a model that includes 18O loss from an enzyme-substrate or intermediate complex. We conclude that more than one 18O can be lost from CO2 per encounter with the active site of isozyme III, a process that requires scrambling of oxygens in a bicarbonate-enzyme complex and cycling between intermediate complexes. This suggests that the rate of dissociation of H2(18)O (or 18OH-) from isozyme III is comparable to or faster than substrate and product dissociation.

Animals↗
PubMed1971

The binomial nature of transmitter release at the crayfish neuromuscular junction.

1. Transmitter release at excitatory junctions on the opener muscle of the crayfish dactyl was studied by recording junctional potentials with extracellular micro-electrodes.2. At low temperatures, evoked release was dispersed sufficiently in time for potentials produced by individual quanta to be counted, and the mean (m) and variance (sigma(2)) of the quantum content distribution for a series of trials measured directly. These values were used to calculate the average probability of quantal release (p), assuming a binomial distribution.3. For all values of m and p, the observed release pattern (number of 0, 1, 2, 3,... quantal releases during a series of trials) was approximated closely by the corresponding binomial distribution. However, Poisson predictions differed significantly from the observed quantal distribution for values of p > 0.2.

Animals↗
PubMed2003

The life cycle and seasonal changes in the occurrence of Pomphorhynchus laevis (Palaeacanthocephala, Pomphorhynchidae) in a small isolated lake.

In a small isolated lake in Slovakia, the fish acanthocephalan Pomphorhynchus laevis using Gammarus balcanicus and the minnow Phoxinus phoxinus, respectively, as its intermediate and final hosts, represented a dominant helminth species. Its prevalence and intensity of infection in fish showed no significant variation during a year fluctuating above the mean values of 89% and 6.6 worms per fish. The mean prevalence of P. laevis larvae in Gammarus was 41.4% with a maximum in the late summer and autumn; individual crustaceans were infected by 1-9 larvae. There was one generation of P. laevis per year. Following up an annual cycle, an occurrence of new infections of Gammarus culminated in October and in the next May for Phoxinus. The sex ratios of both the adults and larvae of acanthocephalans were near unity but favoured slightly males in spring and autumn. The distribution of P. laevis in minnows and crustaceans was highly aggregated and fitted with the negative binomial model. The spatial distribution analysis of parasites along the fish alimentary tract showed a clear preference of P. laevis for its proximal half, with the maximum numbers in the site of the first intestinal loop. Immature worms of both sexes predominated in the proximal region and moved slightly down the alimentary tract during their growth and maturation.

Acanthocephala↗
PubMed2000

On variability in the density of corticocortical and thalamocortical connections.

Variability is an important but neglected aspect of connectional neuroanatomy. The quantitative density of the 'same' corticocortical or thalamocortical connection may vary by over two orders of magnitude between different injections of the same tracer. At present, however, the frequency distribution of connection densities is unknown. Therefore, it is unclear what kind of sampling strategies or statistical methods are appropriate for quantitative studies of connectivity. Nor is it clear if the measured variability represents differences between subjects, or if it is simply a consequence of intra-individual differences resulting from experimental technique and the exact placement of tracers relative to local spatial and laminar variation in connectivity. We used quantitative measurements of the density of a large number of corticocortical and thalamocortical connections from our own laboratories and from the literature. Variability in the density of given corticocortical and thalamocortical connections is high, with the standard deviation of density proportional to the mean. The frequency distribution is close to exponential. Therefore, analysis methods relying on the normal distribution are not appropriate. We provide an appendix that gives simple statistical guidance for samples drawn from exponentially distributed data. For a given corticocortical or thalamocortical connection density, between-individual standard deviation is 0.85 to 1.25 times the within-individual standard deviation. Therefore, much of the variability reported in conventional neuroanatomical studies (with one tracer deposited per animal) is due to within-individual factors. We also find that strong, but not weak, corticocortical connections are substantially more variable than thalamocortical connections. We propose that the near exponential distribution of connection densities is a simple consequence of 'patchy' connectivity. We anticipate that connection data will be well described by the negative binomial, a class of distribution that applies to events occurring in clumped or patchy substrates. Local patchiness may be a feature of all corticocortical connections and could explain why strong corticocortical connections are more variable than strong thalamocortical connections. This idea is supported by the columnar patterns of many corticocortical but few thalamocortical connections in the literature.

Animals↗
PubMed1975

Binomial analysis of quantal transmitter release at glycerol treated frog neuromuscular junctions.

1. Transmitter release was analysed at frog neuromuscular junctions pre-treated with 400 mM glycerol Ringer. In the absence of added drugs, end-plate potentials (e.p.p.s) and miniature e.p.p.s (m.e.p.p.s) could be recorded at selected junctions. 2. E.p.p.s were unusually large and calculations of quantal content indicated a high level of release. Also recorded were anomalous action potentials resembling e.p.p.s but these could be distinguished using a summation test. 3. Plots of coefficient of variation of e.p.p.s (0-5 Hz stimulation) versus direct quantal content (M1) showed a marked deviation from Poisson expectations with high M1. Analysis of these results with amplitude-frequency histograms showed a progressively better fit to binomial predictions with increasing M1. 4. The use of binomial statistics allowed direct calculations of the mean probability of release (p) and the readily available store (n). Increasing Ca/Mg caused increases in both n and p. 5. Plots of M1 vs. Ca/Mg showed a power relationship of 3.58. Maximum m.e.p.p. amplitude occurred at control Ca/Mg. Both results were consistent with studies in muscles not treated with glycerol and indicated that glycerol treatment caused no major alterations pre- or post-junctionally. 6. Estimates of quantum size using Poisson assumptions showed an over-all increase when stimulus frequency was raised, indicating a shift from binomial to a Poisson distribution. 7. The combined findings demonstrate that the glycerol treated preparation can be used to examine the release process during high output. Such release conforms to binomial statistics and allows direct determinations of the parameters n and p.

Action Potentials↗
PubMed1993

"Within patient"-dependent outcomes in graft occlusion after coronary artery bypass. SINBA Group.

In clinical trials aimed at assessing the efficacy of drugs after coronary artery grafting, statistical analysis is usually carried out on a per-patient basis and on a per-graft basis. Owing to the independence of responses among patients, the former outcome is analyzed by assuming a binomial model. However, this model cannot be directly adopted in analyzing the latter outcome because multiple vein grafts in the same patient do not act independently. It has been suggested that patients be considered as clusters of distal anastomoses, subsequently comparing the frequencies of occlusion under different treatments after adjusting the variance of each frequency for the size of the cluster. Alternatively, one can measure the intraclass (intrapatient) correlation coefficient and adopt distribution functions that include this quantity as one of the parameters. This approach has been adopted by many authors although the studies differed in the statistical model used to represent this kind of data. Two probability functions, Altham's and beta-binomial, have found wide application in different biomedical fields. Both are able to model the extrabinomial variability since their tails tend to zero more slowly than those of the binomial distribution. An alternative to adopting a specific probabilistic model consists of specifying a mixed model or a Markov-like susceptibility model. These models follow the same rationale since they assume the existence of two processes, one that causes an individual to become susceptible to occlusion, the other that determines the subsequent probability of occlusion in the individual. After presenting these models in a unified framework, this paper compares the estimates obtained by fitting the models to data gathered at coronary angiography 1 year after surgery by the SINBA group on a total of 847 saphenous vein anastomoses in 349 patients. Finally, the issue concerning the contribution of each patient to the effective sample size is discussed.

Coronary Angiography↗
PubMed1997

Schistosoma mansoni: distribution patterns of miracidia among Biomphalaria glabrata snail as related to host susceptibility and sporocyst regulatory processes.

Parasite prevalences, miracidia developmental capacity, mother sporocyst mean intensities, sporocyst distribution patterns, and cercarial production levels were determined after individual exposure of Biomphalaria glabrata snails to increased doses of Schistosoma mansoni miracidia for two geographical strains (Brazilian, BRE, and Guadeloupean, GUA) of host and parasite. For a high level (100%) of host-parasite susceptibility and in the absence of mother sporocyst regulatory processes for the BRE combination, parasites were randomly dispersed among snail hosts with a frequency distribution conforming to a positive binomial. In contrast, for a moderate level (65%) of host-parasite susceptibility and in the presence of mother sporocyst regulatory processes for the GUA combination, parasites were overdispersed among snail hosts with a frequency distribution conforming to the negative binomial. Levels of cercarial production were found to be strain dependent, to be determined during early development of mother sporocysts, and to be correlated with the number of developed mother sporocysts. Results were analyzed in the general context of the infrapopulation dynamics of the intramolluscan stages of trematode and are discussed in terms of their consequences on the distribution of the genetic diversity of adult schistosomes among the definitive host population.

Animals↗
PubMedDate not supplied

Stochastic modeling of facilitated neurosecretion.

Two models of neurosecretion were evaluated in terms of their ability to predict the dependency of quantal content (m) on the frequency of repetitive stimulation of a lobster motoneuron. First, the hypothesis that neurosecretion is limited by a fixed number of release sites was tested by the fit of the distribution of m by uniform and nonuniform binomial statistics. The obtained release probabilities suggest that frequency facilitation can be due to activation of a group of sites with high release probabilities. However, the fit obtained using this model is not statistically significant due to a large number of fitting parameters. Second, the hypothesis that neurosecretion is limited by the rates of exchange between the releasable pool and the total store of quanta and that each stimulus enhances quantal mobilization was tested. Monte Carlo simulation was carried out in accordance with this model and reproduced the observed distribution of m with very few fitting parameters and therefore with a high level of significance (>0.1). This result demonstrates that mobilization of extra vesicles with each stimulus is a mechanism that allows a very accurate and parsimonious quantitative description of frequency facilitation.

Action Potentials↗
PubMed1979

Frequency Distribution Of Ascaris Lumbricoides In Rural Koreans With Special Reference On The Effect Of Changing Endemicity.

This study was undertaken to figure out the basic patterns of the frequency distribution of Ascaris lumbricoides in rural Koreans and to find out the effects of changing endemicity on the frequency distribution pattern. Total 853 cases in 6 areas were surveyed from November 1975 to February 1978. The mean ova positive rate was 33.1% and in the range from 23.9 to 66.7% by the surveyed unit. From the analyses of the collected data, the following results were obtained. 1. The overdispersion pattern of worm burden per case was mathematically fitted both to the theoretical values of the negative binomial and Poly-Eggenberger distributions. However, when "0" and lower burden cases were considered, the better fitness was observed in the negative binomial than in Polya-Eggenberger distribution. 2. Although the endemicity of Ascaris is changed, the relative frequency of the lower burden cases is rather constant. 3. The false negative rate of stool examination and unfertilized ova passers in the population were not changed significantly according to the endemicity. And the rate of unfertilized ova passers in infected cases was increased when the endemicity was lowered. From the above results, it is concluded that the overdispersed frequency distribution pattern of Ascaris in an endiemic area is changed by the relatively rapid decrease of heavy burden cases and increase of non-infected cases according to the lowered endemicity due to the impact of the control.

Journal Article↗
PubMed1995

Estimating the degree of heterogeneity between event rates using likelihood.

The observed variability between mortality or morbidity rates in epidemiologic studies is partly due to random fluctuations. The same is true for rate ratios relative to reference rates. A method for estimating the distribution of true rate ratios is applied to a data set of perinatal mortality in 515 small areas in the North West Thames Health Region, England, in the period 1986-1990. Combining the random Poisson variability with the assumption that the true rate ratios are drawn from a gamma distribution (a family of positive unimodal distributions) produces a negative binomial log-likelihood for the dispersion parameter of the gamma. The maximum likelihood estimate of this parameter and its confidence interval are then found via direct numerical methods; alternatively, the hypothesis of no heterogeneity is tested by a likelihood ratio. The standardized mortality ratios (SMRs) for the data have an empirical distribution with 5th percentile at 0 and 95th percentile at 1.92, but their true variability, as described by the 5th to 95th percentiles of the fitted gamma distribution, is from 0.72 to 1.32. The likelihood ratio test confirmed this result, rejecting the hypothesis that the true rates are homogeneous (p = 0.015). The method requires only modest computing resources and is useful when assessing the need for more detailed study.

England↗
PubMed1994

Estimating confidence limits on a standardised mortality ratio when the expected number is not error free.

OBJECTIVE: The aim was to demonstrate how the beta distribution may be used to find confidence limits on a standardised mortality ratio (SMR) when the expected number of events is subject to random variation and to compare these limits with those obtained with the standard exact approach used for SMRs and with a Fieller-based confidence interval. DESIGN: The relationship of the binomial and the beta distributions is explained. For cohort studies in which deaths are counted in exposed and unexposed groups exact confidence limits on the relative risk are found conditional on the total number of observed deaths. A similar method for the SMR is justified by analogy between the SMR and the relative risk found from such cohort studies, and the fact that the relevant (beta) distribution does not require integer parameters. SOURCE OF DATA: Illustrative examples of hypothetical data were used, together with a MINITAB macro (see appendix) to perform the calculations. MAIN RESULTS: Exact confidence intervals that include error in the expected number are much wider than those found with the standard exact method. Fieller intervals are comparable with the new exact method provided the observed and expected numbers (taken to be means of Poisson variates) are large enough to approximate normality. As the expected number is increased, the standard method gives results closer to the new method, but may still lead to different conclusions even with as many as 100 expected. CONCLUSIONS: If there is reason to suppose the expected number of deaths in an SMR is subject to sampling error (because of imprecisely estimated rates in the standard population) then exact confidence limits should be found by the methods described here, or approximate Fieller-based limits provided enough events are observed and expected to approximate normality.

Adolescent↗
PubMed2025

Generating correlated data for omics simulation.

Simulation of realistic omics data is a key input for benchmarking studies that help users obtain optimal computational pipelines. Omics data involves large numbers of measured features on each sample and these measures are generally correlated with each other. However, simulation too often ignores these correlations, perhaps due to computational and statistical hurdles of doing so. To alleviate this, we describe three approaches for generating omics-scale data with correlated measures which mimic real datasets. These approaches are all based on a Gaussian copula approach with a covariance matrix that decomposes into a diagonal part and a low-rank part. This decomposition allows for extremely efficient simulation, overcoming a hurdle for adoption of past methods. We use these approaches to demonstrate the importance of including correlation in two benchmarking applications. First, we show that variance of results from the popular DESeq2 method increases when dependence is included. Second, we demonstrate that CYCLOPS, a method for inferring circadian time of collection from transcriptomics, improves in performance when given gene-gene dependencies in some circumstances. We provide an R package, dependentsimr, that has efficient implementations of these methods and can generate dependent data with arbitrary marginal distributions, including discrete (binary, ordered categorical, Poisson, negative binomial), continuous (normal), or with an empirical distribution.

Computer Simulation↗
PubMed1996

An analytical framework for relating dose, risk, and incidence: an application to occupational tuberculosis infection.

An adverse health impact is often treated as a binary variable (response vs. no response), in which case the risk of response is defined as a monotonically increasing function R of the dose received D. For a population of size N, specifying the forms of R(D) and of the probability density function (pdf) for D allows determination of the pdf for risk, and computation of the mean and variance of the distribution of incidence, where the latter parameters are denoted E[SN] and Var[SN], respectively. The distribution of SN describes uncertainty in the future incidence value. Given variability in dose (and risk) among population members, the distribution of incidence is Poisson-binomial. However, depending on the value of E[SN], the distribution of incidence is adequately approximated by a Poisson distribution with parameter mu = E[SN], or by a normal distribution with mean and variance equal to E[SN] and Var[SN]. The general analytical framework is applied to occupational infection by Mycobacterium tuberculosis (M. tb). Tuberculosis is transmitted by inhalation of 1-5 microns particles carrying viable M. tb bacilli. Infection risk has traditionally been modeled by the expression: R(D) = 1 - exp(-D), where D is the expected number of bacilli that deposit in the pulmonary region. This model assumes that the infectious dose is one bacillus. The beta pdf and the gamma pdf are shown to be reasonable and especially convenient forms for modeling the distribution of the expected cumulative dose across a large healthcare worker cohort. Use of the the analytical framework is illustrated by estimating the efficacy of different respiratory protective devices in reducing healthcare worker infection risk.

Cohort Studies↗