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Given the ratio, k, of two sample sizes, sample sizes for comparing two independent proportions can be approximated from the sample size for the equal-sample-size design. The proposed method is compared to the method of Fleiss , Tytun and Ury (1980, Biometrics 36, 343-346). Methods for choosing the ratio are also presented.
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It is thought that, in a consignment of discrete units, a certain proportion of the units contain illegal material. A sample of the consignment is to be inspected. Various methods for the determination of the sample size are compared. The consignment will be considered as a random sample from some super-population of units, a certain proportion of which contain drugs. For large consignments, a probability distribution, known as the beta distribution, for the proportion of the consignment which contains illegal material is obtained. This distribution is based on prior beliefs about the proportion. Under certain specific conditions the beta distribution gives the same numerical results as an approach based on the binomial distribution. The binomial distribution provides a probability for the number of units in a sample which contain illegal material, conditional on knowing the proportion of the consignment which contains illegal material. This is in contrast to the beta distribution which provides probabilities for the proportion of a consignment which contains illegal material, conditional on knowing the number of units in the sample which contain illegal material. The interpretation when the beta distribution is used is much more intuitively satisfactory. It is also much more flexible in its ability to cater for prior beliefs which may vary given the different circumstances of different crimes. For small consignments, a distribution, known as the beta-binomial distribution, for the number of units in the consignment which are found to contain illegal material, is obtained, based on prior beliefs about the number of units in the consignment which are thought to contain illegal material. As with the beta and binomial distributions for large samples, it is shown that, in certain specific conditions, the beta-binomial and hypergeometric distributions give the same numerical results. However, the beta-binomial distribution, as with the beta distribution, has a more intuitively satisfactory interpretation and greater flexibility. The beta and the beta-binomial distributions provide methods for the determination of the minimum sample size to be taken from a consignment in order to satisfy a certain criterion. The criterion requires the specification of a proportion and a probability.
Skeletal abnormalities in rabbit, rat and mouse foetuses are found to follow closely beta-binomial distributions rather than simple binomial distributions, i.e., foetuses from the same litter are found to have closer chances of being abnormal than foetuses from different litters. Control and treated groups of 12 litters were simulated using beta-binomial distributions and were used to compare the performances of three types of analysis on two-group teratological studies: Student's t test on the litter proportions, transformed or weighted where necessary; the Wilcoxon distribution-free on ranked litter proportions; and likelihood ratio tests assuming a beta-binomial distribution of abnormalities. It was found that the likelihood ratio tests do not have Type I errors equal to the nominal level and are not more powerful than the Student's t test or the Wilcoxon test. The latter two tests produce similar results but the Student's t test is marginally more powerful.
The beta-binomial distribution introduced by Skellam has been applied in many teratology problems for modelling the litter effect. Recently, Morel and Nagaraj proposed a new distribution for modelling cluster multinomial data when the clustering is believed to be caused by clumped sampling. It turns out that the distribution is a mixture of two binomial distributions and accommodates the estimation of an additional parameter to account for intra-litter effect. The new distribution arises from a cluster mechanism in which some individuals within a cluster exhibit the same behaviour while the remaining individuals from the cluster react independently of each other. Such a mechanism is a natural model in teratology problems, where typically a genetic trait is passed with a certain probability to the foetuses of the same litter. In this article, we use the new distribution to model binary responses with logistic regression. We analyse data from a teratology experiment to demonstrate that the new model provides a useful addition to current methodology. The experiment investigates the synergistic effect of the anticonvulsant phenytoin and trichloropopene oxide on the prenatal development of inbred mice. In a simulation study we investigate the type I error rate and the power of the maximum likelihood ratio test when the data follow a finite mixture distribution.
PURPOSE: To evaluate the assumptions on which the American College of Medical Genetics (ACMG) Standards and Guidelines for detecting mosaicism in amniotic fluid cultures are based. METHODS: Data from 653 cases of amniotic fluid mosaicism were collected from 26 laboratories. A chi-square goodness-of-fit test was used to compare the observed number of mosaic cases with the expected number based on binomial distribution theory. RESULTS: Comparison of observed data from the in situ colony cases with the expected distribution of cases detected based on the binomial distribution did not reveal a significant difference (P = 0.525). CONCLUSIONS: The empirical data fit the binomial distribution. Therefore, binomial theory can be used as an initial discussion point for determining whether ACMG Standards and Guidelines are adequate for detecting mosaicism.
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.
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.
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.
Usual approaches for estimating the variance of a standardized rate may not be applicable to rates of recurrent events. Where individuals are prone to repeated health events, Greenwood and Yule (J R Stat Soc [A], 1920;83:255-79) advocated use of the negative binomial distribution to account for departures from the assumption of randomness of recurrent events required by the Poisson distribution. In this paper, the authors implemented the negative binomial distribution in the computation of annual hospitalization rates within certain hospital market areas. Data used were from 1,549,915 New England residents aged 65 years or more who were enrolled in Medicare between October 1, 1988, and September 30, 1989, and who had 458,593 hospital admissions during that year. New England was partitioned into 170 hospital market areas ranging in population size from 162 to 70,821 elderly Medicare enrollees. The negative binomial distribution demonstrated substantially better fits than the Poisson distribution to the numbers of hospitalizations within hospital market areas. Estimated standard errors for indirectly standardized rates based on the negative binomial distribution were 25-51 percent higher than estimated standard errors that assumed an underlying Poisson distribution. Using regression analysis to smooth overdispersion parameters across hospital market areas produced similar results. The approach described in this paper may be useful in estimation of confidence intervals for standardized rates of recurrent events when these events do not recur randomly.
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.
The assumptions underlying the use of the Poisson distribution are essentially that the probability of an event is small but nearly identical for all occurrences and that the occurrence of an event does not alter the probability of recurrence of such events. These assumptions do not seem to be met for evolutionary events since (i) the probability of fixing nucleotide codon substitutions is not equal for all substitutions at a codon, and probably varies for the same substitution in different lineages; (ii) the probability of fixing codon substitutions varies among positions of a cistron; and (iii) the fixation of a nucleotide codon substitution at one position in a cistron modifies, and may even promote, the fixation of a codon substitution elsewhere along the cistron. Natural selection presumably is the causative factor that acts to modify the probability of a nucleotide codon substitution's being fixed in a population. The use of the negative binomial distribution is consistent with the evidence that selective pressure on amino acid or nucleotide codon positions varies both among codon positions of a cistron and at a particular position during evolutionary time. If the number of fixations of nucleotide codon substitutions per position of cistrons encoding cytochromes c are phyletically inferred (phylogeny based on a paleontological record) rather than phenetically inferred (based on paired comparisons of extant species' differences in the absence of a phylogeny) the distribution of these fixation data cannot be described adequately by a single Poisson distribution. The fit of these same data to a negative binomial distribution is very satisfactory. It has been argued that the fit of phenetically inferred fixation data, which do not take account of parallel or reverse fixations, to the Poisson distribution was supportive evidence for the hypothesis that protein evolution results from the fixation of selectively neutral codon substitutions. This argument now appears to be undercut by the evidence that data on nucleotide codon fixation are more probably distributed according to the negative binomial distribution. The fact that fixation data can be described by a particular discrete probability distribution does not of itself provide insight into the mechanisms of the evolutionary process. However, the facts-(i) that the assumptions underlying the use of the negative binomial distribution adequately deal with the varying probability of fixing amino acid or nucleotide codon substitutions at and among the positions of a cistron and (ii) that the negative binomial distribution provides an excellent fit for the phyletically inferred fixation data-suggest that the negative binomial is a very appropriate discrete probability distribution for describing evolutionary events. Amino acids or their nucleotide codon substitutions may be fixed at a position of a cistron as though selectively neutral relative to the codon being replaced, even though the codon position will not be selectively neutral, since many amino acids cannot function there. The negative binomial distribution treats this situation well whereas a single Poisson distribution could only be satisfactory if all codon positions that could vary were selectively neutral.