An improved approximate formula for calculating sample sizes for comparing two binomial distributions.
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Methods↗
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An approach to numerical identification of bacterial species.
The distribution of matching coefficients (M values) for strains of a species to their own 'hypothetical mean organism' (HMO) or to HMO patterns of other organisms was studied in 754 strains of 19 mycobacterial species, testing for 91 discriminating characters. The M values of strains of a species to the HMO of other species usually showed a normal distribution, and M values to their own HMO showed either a normal distribution of a binomial distribution, depending on the mean of M values. If the number of test characters was large, the binomial distribution usually resembled the normal distribution. After preparation of the HMO for every species and estimation of the mean of the M values (M) and the standard deviation (s), numerical identification could be carried out: if a test strain had an M value to the HMO of species chi that only fell within the range (M +/2s) for species chi, the strain would be identified as a member of that species.
Bacteria↗
Multichannel 18-test panels: are 60% of panels abnormal by chance?
Current teaching concerning the frequency of abnormal results secondary to chance alone in a multichannel panel is theoretically based on the binomial distribution. However, this distribution can be used only when the probability of an abnormal result (pi) is the same for each test in the panel. In modern-day multichannel testing, pi varies from test to test and most often is less than the usually reported 0.05. On the other hand, a test such as cholesterol may have a pi level as high as 0.55. Theoretically the only distribution that can take this variability into consideration is the Lexis distribution, a form of the binomial distribution that allows for varying pi s. Since no formula is available to calculate this distribution, we wrote a computer program to generate it. We arranged 18-test panels from 203 normal patients in a frequency distribution. This was then compared with the theoretical Lexis and binomial distributions. This analysis showed that although there was a 50% chance of having one abnormality per panel and a 16% chance of having two abnormalities per panel, there was less than 4% chance of having three or more abnormalities per 18-test panel. In addition, most of the abnormalities noted were minor and were thought to be clinically unimportant.
Adult↗
[Family clustering analysis of HBV infection].
A family clustering analysis of HBsAg, anti-HBs, anti-HBc positive and total infected persons in 148 families on a farm was carried out by the methods of G statistic, binomial distribution and negative binomial distribution. The results consistently showed that there was significant clustering of HBsAg carriers in families, whereas there were not clustering was seen an overall HBV infected persons in families. The clustering of HBsAg carriers in families is due probably to the effect of some genetic factors. The results of G statistic analysis indicate that anti-HBc also have significant clustering in families.
Carrier State↗
Statistics of neuromuscular transmitter release in young and old mouse muscle.
1. It was reported previously that in limb muscles of old (27-30 months) CBF-1 mice, quantal content (m) of evoked transmitter release was increased compared to that in young (9-12 months) mice. In diaphragm muscles there was no change with age. The object of the present study was to determine whether the age-related increase in transmitter release was due to increase in the binomial parameter n or the parameter p. The analysis also involved consideration of goodness-of-fit between observed and expected binomial distribution of the data. 2. Spontaneous miniature end-plate potentials (m.e.p.p.s) and evoked end-plate potentials (e.p.p.s) were recorded with intracellular techniques from soleus and diaphragm muscles bathed in low-Ca high-Mg medium. The goodness-of-fit between the observed e.p.p amplitude distribution and that expected from a binomial distribution was evaluated by chi 2 test. 3. In different muscles and at different ages, the percentage of fibres with binomial e.p.p. distributions varied from 17 to 44%, even though in all fibres there was a similar proportionality between direct quantal content and the reciprocal of the square of the coefficient of variation of e.p.p. amplitudes. In addition, apparent graphical agreement between observed and theoretical binomial e.p.p. distributions was often not substantiated by the chi 2 criterion. 4. In soleus muscles from young mice, lowering the stimulus frequency from 10 to 0.5 Hz and shortening the train length from 250 to 100 pulses increased the prevalence of binomial e.p.p. distributions, but the same result was not obtained in diaphragm or soleus muscles from old mice. If the mean amplitude of groups of 10 e.p.p.s in any train showed any drift (greater than 10%) then that train was excluded from the results. Thus, in order to make valid age comparisons, only fibres with binomial e.p.p. distributions were analysed further. 5. There was no change with age in m, n or p in diaphragm muscles, but in soleus muscles from old animals a nearly 2-fold increase in n entirely accounted for the increase in m. 6. If, as proposed by others, n represents the number of release sites, then the ageing soleus neuromuscular junction may have increased numbers or length of active zones or associated membrane components.
Action Potentials↗
The negative binomial model and the denominator problem in a rural family practice.
Much accurate health services and epidemiologic research in primary care is impeded by an inability to determine or estimate practice size in many countries. While the number of patients attending a practice is known, the number of non-attending members is not. Much of the work aimed at solving the denominator problem has focused on mathematical models for estimating the non-attenders. Of these, the negative binomial distribution of illness episodes received considerable early attention because of its success in British general practices. Enthusiasm for the negative binomial distribution waned when its application in non-capitated practices was largely unsuccessful. However, these American studies contained major biases, particularly in terms of independent knowledge of practice size and the potential for frequent usage of other primary care services. The negative binomial distribution was re-evaluated in a rural Canadian practice where there was only limited opportunity for external service utilization and where the practice's patient list had been subjected to separate validation. It fits well with the distribution of illness episodes in each of two consecutive 12-month periods and provided accurate estimates of practice size. It did not function well with the distribution of patient visits. This observation regarding the negative binomial distribution is discussed in the context of further searches for a solution to the denominator problem.
Adolescent↗
The micronucleus test: statistical design and analysis.
Alternative statistical procedures are discussed which may be employed to compare the incidences among treatment groups of micronucleated polychromatic and normochromatic erythrocytes and their ratios. Comparison of incidences of micronucleated polychromatic erythrocytes using a sequential sampling strategy based on the negative binomial distribution is shown to require fewer animals for the same sensitivity of test than a similar procedure based on the binomial distribution. The sequential test is superior, both in power and number of animals required, to an alternative 1-stage test based on the same distribution. The procedure described permits the investigator to optimize the number of animals in each test group and the number of cells counted per animal to detect a predetermined increase in the incidence of micronucleated cells over that observed in the control population within chosen limits of type I and type II error. An alternative sequential approach based on the binomial distribution is presented, which is applicable when the number of cells analyzed per animal is variable.
Animals↗
Bilateral distribution of implantation sites in small mammals of 22 North American species.
The distribution of activity between the left and right sides of the reproductive tract, as measured by numbers of CL, embryos and placental scars, was studied in small mammals of 22 species. Shrews ovulate from the two ovaries in a distribution that does not differ from the binomial. Implantation of blastocysts in the two uterine horns is more nearly even ('balanced') than would be predicted from the binomial distribution. Balance in this group apparently is achieved by transuterine migration of blastocysts, perhaps in conjunction with some spacing mechanism within the uterus. Some cricetid rodents show little or no balance, but in others the distribution of activity sites (embryos, CL and placental scars) departs significantly from the binomial distribution. Reproductive activity sites of heteromyid and geomyid rodents (Geomyoidea) are highly balanced; uterine balance apparently is achieved by means of ovarian rather than uterine control. We know of no previous reports of ovarian balance and suggest that physiological mechanisms controlling numbers of ovulations in the species exhibiting this characteristic may differ from those in species exhibiting a random distribution of ovulation sites. Hypotheses regarding evolutionary aspects of balance are considered in phylogenetic and ecological terms, generating several testable research questions for physiologists, anatomists, and evolutionary ecologists.
Animals↗
The distribution of numbers of open channels in multi-channel patches.
Assumptions inherent in use of the binomial distribution are relaxed in order to identify alternative distributions with different variances, for fitting to the frequency distributions of numbers of open channels in multi-channel patches. The variance is less than that for the binomial distribution with the same expectation when either there is a negative interaction between channels or channels have different probabilities of being open. When the variance is greater, this may be because of positive interaction between channels or non-stationary behaviour. Use of the distributions and lack-of-fit statistics is illustrated.
Ion Channels↗
Frequency of elevated urinary beta 2-microglobulin levels in relatives of patients with asymptomatic low-molecular-weight proteinuria.
We studied urinary beta 2-microglobulin levels in a total of 29 apparently healthy relatives (aged 0.8-70 years) of 8 male patients with asymptomatic low-molecular-weight proteinuria in six families. The frequency of levels above the age- and sex-associated 95% confidence limit was 7 of 29 (24%), 4 of 12 (33%) in first-degree relatives, 2 of 6 (33%) fathers, and 2 of 6 (33%) mothers. These frequencies were significantly above those in the general population (P less than 0.01, by a normal distribution test, a binomial distribution, and Poisson distribution test for the sample proportion). The increased frequency in fathers argues against an X-linked pattern of inheritance for this entity, suggesting that there is heterogeneity in the inheritance.
Adolescent↗
Word recognition functions for the CID W-22 test in multitalker noise for normally hearing and hearing-impaired subjects.
Word recognition functions for Auditec recordings of the CID W-22 stimuli in multitalker noise were obtained using subjects with normal hearing and with mild-to-moderate sensorineural hearing loss. In the first experiment, word recognition functions were generated by varying the signal-to-noise ratio (S/N); whereas in the second experiment, a constant S/N was used and stimulus intensity was varied. The split-half reliability of word recognition scores for the normal-hearing and hearing-impaired groups revealed variability that agreed closely with predictions based on the simple binomial distribution. Therefore, the binomial model appears appropriate for estimating the variability of word recognition scores whether they are obtained in quiet or in a competing background noise. The reliability for threshold (50% point) revealed good stability. The slope of the recognition function was steeper for normal listeners than for the hearing-impaired subjects. Word recognition testing in noise can provide insight into the problems imposed by hearing loss, particularly when evaluating patients with mild hearing loss who exhibit no difficulties with conventional tests. Clinicians should employ a sufficient number of stimuli so that the test is adequately sensitive to differences among listening conditions.
Adult↗
Characterization of a cation channel on the apical surface of the frog lens epithelium.
The properties of a single conductance pathway of the apical (fibre-facing) surface of the frog lens epithelium are reported. Using the patch-clamp technique (Hamill, Marty, Neher, Sakmann & Sigworth, 1981), the most common single-channel currents had an amplitude of 1.9 pA, the mean open time 2.1 ms and a conductance of 25 pS. One open-state time constant (to = 3.3 ms) and two closed-state time constants (tau c1 = 0.9 ms, tau c2 = 23.1 ms) were resolved. The channel current and the mean open time were both increased when Ca2+ was removed from the external solution and the open time distribution was no longer fitted by a single exponential. Multiple-channel events in cell-attached patches containing two or more identical channels were distributed in a binomial fashion and the probability that an individual channel was open, obtained by fitting the binomial distribution, was 0.039. The channel was found to have a Na+:K+ selectivity ratio of 3:1. When Ca2+ was removed from the pipette solution the probability that an individual channel was open increased to 0.137 and the Na+:K+ selectivity ratio increased to 4:1. Channel activity was observed in the presence of tetrodotoxin (10(-6) M) in the bathing medium and the pipette solution but was abolished by internal perfusion of the patch pipette with 0.5 x 10(-4) M amiloride. this apical conductance pathway is identified as an amiloride-sensitive cation channel. These channels are clustered in groups on the apical membrane, spontaneously active at the resting potential and with the possibility of altering their Na+:K+ selectivity. They represent a distinct type of channel, that differ from nerve and muscle Na+ channels in their manner of activation, but do share some common features with both Na+ and Ca2+ channels in excitable cells.
Amiloride↗
[Natural microbial control of crickets populations (Orthoptera: Gryllotalpidae: Scapteriscus borellii): regulation of populations aggregated in time and space].
From 1983 through 1988, a total of 1,762 collections, containing 31,312 individuals of the mole cricket, Scapteriscus borellii, were made, principally in the State of São Paulo, Brazil. Collections were found to fit a negative binomial distribution both as whole and when divided into monthly collections. In these collections, an iridovirus, a entomogenous nematode, and the fungi Metarhizium anisopliae, Beauveria bassiana, Paecilomyces sp., and Entomophtora sp., were found to be agents of natural mortality, although usually as endozootics and relatively rarely as epizootics and panzootics. As a group, these diseases were also distributed in a binomial negative. These data suggest that the temporal and spatial aggregations of the mole crickets, produced by high rates of migration among suitable habitats, are adaptations to outbreaks of epidemics, which also serve as mole cricket population regulators. These ideas are develop and derived from simple mathematical models of population change.
Animals↗
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.
Antineoplastic Agents↗
On the regularities of distribution of Hypoderma bovis De Geer larvae parasitizing cattle herds in different parts of the range of this warble fly.
The type and parameters of the distribution of the second and third instar larvae of Hypoderma bovis in cattle herds in Czechoslovakia (54 herds, 7233 head), Mongolia (20 herds, 1809 head) and the USSR (48 herds, 4978 head) were studied. A statistical analysis showed a) that in the majority of cases negative binomial distribution serves as a model of the distribution of larvae with sufficient reliability; b) that the regularity of dependence of the distribution exponent k of the negative binomial distribution on the incidence of infestation remains constant in different parts of the range of this warble fly. The latter fact indicates that the regulatory systems limiting the population numbers of this warble fly are associated with host-parasite relationships and do not depend on a complex of conditions specific for various natural zones. In order to understand the regulatory processes in parasite populations it is necessary to study equally the regulatory mechanisms operating primarily on the level of specimens as well as the regulatory systems operating on the level of populations.
Animals↗
On the regularities of distribution of Hypoderma bovis De Geer larvae parasitizine cattle herds in different parts of the range of this warble fly.
The type and parameters of the distribution of the second and third instar larvae of Hypoderma bovis in cattle herds in Czechoslovakia (54 herds, 7233 head), Mongolia (20 herds, 1809 head) and the USSR (48 herds, 4978 head) were studied. A statistical analysis showed a) that in the majority of cases negative binomial distribution serves as a model of the distribution of larvae with sufficient reliability; b) that the regularity of dependence of the distribution exponent k of the negative binomial distribution on the incidence of infestation remains constant in different parts of the range of thes warble fly. The latter fact indicates that the regulatory systems limiting the population numbers of this warble fly are associated with host-parasite relationships and do not depend on a complex of conditions specific for various natural zones. In order to understand the regulatory processes in parasite populations it is necessary to study equally the regulatory mechanisms operating primarily on the level of specimens as well as the regulatory systems operating on the level of populations.
Animals↗
Doxapram dosage regimen in apnea of prematurity based on pharmacokinetic data.
To 18 premature apneic patients refractory to theophylline, doxapram (0.5-2.5 mg/kg/h) was administered in combination with therapeutic doses of theophylline. Doxapram concentrations in serum were measured 48 h after commencement of the infusion and then in 2-hour intervals during a 6-8 h withdrawal. Total body clearance (dose/Css) of the drug ranged from 0.20 to 0.56 liter/h in 13 patients and 1.14 to 1.75 liter/h in 4 patients suggesting a binomial distribution in the disposition kinetics of the drug. Other pharmacokinetic indices, although variable, did not exhibit binomial distribution. The mean volume of distribution and half-life of doxapram were 7.33 +/- 4.55 liter/kg and 8.17 +/- 4.13 h, respectively. Based on our calculations to accelerate the attainment of a steady-state plasma concentration (Css) of approximately 1.5 mg/l, a loading dose of 5.5 mg/kg and a maintenance dose of 1 mg/kg/h along with serum concentration monitoring are recommended.
Apnea↗
Estimation and analysis of the concentration-response surfaces associated with multiple-agent combinations.
Chinese hamster cells (V79) were treated with ethylnitrosourea (ENU) and cis-diamminedichloroplatinum(II) (DDP) alone and in combination. Sister chromatid exchanges (SCEs) were quantified as measures of genotoxicity of the two agents. The combination experiment employed a factorial design in which cells were treated, in various concentration combinations, with both agents simultaneously. Response surface methodology, using a polynomial model based on a negative binomial distribution of SCE events, was employed for analysis of the interactions of the two genotoxic agents. The negative binomial distribution, a generalization of the Poisson distribution, is required since SCEs are discrete variables which, under the conditions of these experiments, have a distribution which exhibits extra-Poisson variability. The model of the ENU/DDP combinations indicated an increasingly less-than-additive effect resulting from increasing concentrations of each agent in the combination. The analysis of these experiments demonstrates the usefulness of a powerful statistical procedure for evaluating the biological effects resulting from exposure to multiple cytotoxic agents. The methodology can be used with many other types of endpoints and is not limited by the number of treatment agents.
Cells, Cultured↗