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PubMed1995

Confidence limits for the population prevalence rate based on the negative binomial distribution.

This paper shows that the extension of the simple procedure of George and Elston in calculation of confidence limits for the underlying prevalence rate to accommodate any finite number of cases in inverse sampling is straightforward. To appreciate the fact that the length of the confidence interval calculated on the basis of the first single case may be too wide for general utility, I include a quantiative discussion on the effect due to an increase in the number of cases requested in the sample on the expected length of confidence intervals. To facilitate further the application of the results presented in this paper, I present a table that summarizes in a variety of situations the minimum required number of cases for the ratio of the expected length of a confidence interval relative to the underlying prevalence rate to be less than or equal to a given value. I also include a discussion on the relation between Clemans's confidence limits on the expected number of trials before the failure of a given device and those presented here.

Binomial Distribution↗
PubMed2002

An aspect of discrete data analysis: fitting a beta-binomial distribution to the hospitals' data.

Statistical analysis for discrete data, particularly for probability models such as the binomial, Poisson and multinomial, is by now very well understood, with a wealth of suitable software. It can happen that the standard generalized linear modelling (glm) software is not completely appropriate, since over-dispersion is present, relative to the standard distributions such as the Poisson or the binomial. Failure to take account of this over-dispersion, for example in fitting a model such as log(p/(1 - p)) = alpha + beta x (where the covariate x is the dose) will mean that our estimates of beta will be less precise than the binomial-based formula gives us. Thus for example we will be quoting confidence intervals for beta that are too narrow. One way of coping with this problem is to use a probability model which is more general than the binomial, and one such model is the beta-binomial. This paper discusses beta-binomial modelling (in S-Plus) in relation to the interesting data set given in the 1998 BMJ paper by Spiegelhalter and Marshall on success rates of 52 in vitro fertilisation clinics in the UK.

Data Interpretation, Statistical↗
PubMed1983

Early stopping for sequential restricted tests of binomial distributions.

In applications of restricted sequential procedures, the accumulating data can provide a strong indication that rejection of the null hypothesis is very unlikely if sampling is continued; however, trial termination with acceptance of the null hypothesis can require a large number of additional observations. Modifications of restricted sequential plans, based on the conditional probability of an inference contrary to that suggested by the data, are presented for trials with binary responses. The derived tests provide the potential for early stopping with acceptance of the null hypothesis and reduction of the expected stopping times. Properties of the stopping-time distributions are obtained exactly by considering the absorption probabilities of a binomial random walk on the lattice determined by the restricted sequential design. Relative efficiencies of the derived tests to the corresponding sequential probability ratio tests are presented, along with an example based on data from a clinical trial.

Statistics as Topic↗
PubMed1978

Use of the beta-binomial distribution in dominant-lethal testing for "weak mutagenic activity: part 2.

Experiments in Dominant-Lethal Testing have been simulated on the computer to estimate the type I error rates and the power of the Beta-Binomial test under various models. (1) The mating ratio is one; and p, the probability that an implant will die, is distributed over the couples. (2) The mating ratio is larger than one; and p is distributed over the males, the females mated to the same male being binomial observations of the value p supplied by the male. (3) The mating ratio is larger than one; and p is distributed over the females. The average rates of dead implants have been set at 0.08 and 0.10 for the control and treatment groups, respectively, and a nominal level of significance equal to 0.05 has been chosen. The type I error rate of the traditional chi-square test has also been estimated. A by-product of these simulations is the behaviour of the estimates alpha and beta of the beta-distribution parameters, which discloses that, in the actual experiments with mice, p is distributed over the females. Our results lead to the recommendations that, for a given number of animals per group, a mating ratio larger than one should be adopted and that the males should be considered as the experimental units for the calculations. With 300 and 450 animals per group, average powers of 0.72 and 0.85 are reached, respectively, for the chosen increment of 2% in the rate of dead implants. Under these models, the type I error rate of the traditional chi-square test may grow to 0.30 for the nominal level of 0.05.

Genes, Dominant↗
PubMed1988

Estimation bias using the beta-binomial distribution in teratology.

Kupper et al. (1986, Biometrics 42, 85-98) considered the fitting of dose-response regressions to litter proportions in teratology experiments. They found that the estimators which maximise the beta-binomial likelihood become biased when the intralitter correlation is incorrectly assumed to be homogeneous. This paper considers the nature of this bias and its implications for the analysis of litter proportions in toxicology.

Animals↗
PubMed1984

The analysis of chromosomally aberrant cells based on beta-binomial distribution.

Analysis carried out here generalized on earlier studies of chromosomal aberrations in the populations of Hiroshima and Nagasaki, by allowing extrabinomial variation in aberrant cell counts corresponding to within-subject correlations in cell aberrations. Strong within-subject correlations were detected with corresponding standard errors for the average number of aberrant cells that were often substantially larger than was previously assumed. The extrabinomial variation is accommodated in the analysis in the present report, as described in the section on dose-response models, by using a beta-binomial (beta-B) variance structure. It is emphasized that we have generally satisfactory agreement between the observed and the beta-B fitted frequencies by city-dose category. The chromosomal aberration data considered here are not extensive enough to allow a precise discrimination between competing dose-response models.

Chromosome Aberrations↗
PubMed1975

[The result-sequence method (sequence analysis) in leptospira research. 3. Communication: The bilateral sequential test (de Boer, Armitage) for testing the difference of mean values of two binomial distributions (author's transl)].

The influence of the liver-activated and non-activated cytostatic drug Cyclophosphamid on the respiration of Leptospira biflexa semaranga Veldrat S 173 was tested in three different concentrations by group-sequential testing of two relative frequencies in a two-sided test-reading. The conditioned probability theta = 0.82 results from thetan = 0.40 and thetaa = 0.75. On account of a practicable sample size, we choose theta' = 0.80 (activated form superior) and theta'' = 0.20 (non-activated form superior). The test-adjusted level of significance is alpha = 0.05 less than beta = 0.10. The expected values for the number of discordant pairs are Etheta' = Etheta'' = 13, Etheta = 1/1 = 14 and Ethetamax = 16. The acceptance inspection performed by control chart and tabulated reference figures results - in comparison with conventional procedures - in savings of discordant pairs of 65 per cent (concentration 10(-4) g/ml) in a decision in favour of activated form, of 65 per cent (concentration 10(-8) g/ml) in an equal efficiency of both states of Cyclophosphamid and of 10 per cent (concentration 10(-11) g/ml) in a non-significant difference. Taking into consideration all unrestrictedly selected pairs, the duration of the experiment takes only 4.25 months instead of 6.75 if the statistical data analysis is not performed conventionally, but group-sequentially instead.

Cyclophosphamide↗