When computing a confidence interval for a binomial proportion p one must choose between using an exact interval, which has a coverage probability of at least 1-{\alpha} for all values of p, and a shorter approximate interval, which may have lower coverage for some p but that on average has coverage equal to 1-\alpha. We investigate the cost of using the exact one and two-sided Clopper--Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper--Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper--Pearson interval and Bayesian intervals based on noninformative priors.

The cost of using exact confidence intervals for a binomial proportion

Background 
Animal transmission studies can provide important insights into host, viral and environmental factors affecting transmission of viruses including influenza A. The basic unit of analysis in typical animal transmission experiments is the presence or absence of transmission from an infectious animal to a susceptible animal. In studies comparing two groups (e.g. two host genetic variants, two virus strains, or two arrangements of animal cages), differences between groups are evaluated by comparing the proportion of pairs with successful transmission in each group. The present study aimed to discuss the significance and power to estimate transmissibility and identify differences in the transmissibility based on one-to-one trials. The analyses are illustrated on transmission studies of influenza A viruses in the ferret model.


Methodology/Principal Findings 
Employing the stochastic general epidemic model, the basic reproduction number, R0, is derived from the final state of an epidemic and is related to the probability of successful transmission during each one-to-one trial. In studies to estimate transmissibility, we show that 3 pairs of infectious/susceptible animals cannot demonstrate a significantly higher transmissibility than R0 = 1, even if infection occurs in all three pairs. In comparisons between two groups, at least 4 pairs of infectious/susceptible animals are required in each group to ensure high power to identify significant differences in transmissibility between the groups.


Conclusions 
These results inform the appropriate sample sizes for animal transmission experiments, while relating the observed proportion of infected pairs to R0, an interpretable epidemiological measure of transmissibility. In addition to the hypothesis testing results, the wide confidence intervals of R0 with small sample sizes also imply that the objective demonstration of difference or similarity should rest on firmly calculated sample size.

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Sample Size Considerations for One-to-One Animal Transmission Studies of the Influenza A Viruses

When computing a confidence interval for a binomial proportion $p$ one must choose between using an exact interval, which has a coverage probability of at least $1-\alpha$ for all values of $p$, and a shorter approximate interval, which may have lower coverage for some $p$ but that on average has coverage equal to $1-\alpha$. We investigate the cost of using the exact one and two-sided Clopper–Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper–Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper–Pearson interval and Bayesian intervals based on noninformative priors.

/pdf/the-cost-of-using-exact-confidence-intervals-for-a-binomial-z4mq5a36hk.pdf

The shorthand X ∼ extreme value(α, β) is used to indicate that the random variable X has the extreme value distribution with positive scale parameter α and positive shape parameter β. An extreme value random variable X with parameters α and β has probability density function f (x) = (β/α)e x β−e xβ /α − ∞ < x < ∞. The extreme value distribution can be used to model water levels associated with floods and earthquake magnitudes. The probability density function with three different parameter settings is illustrated below.

Extreme Value Distribution

The most common method for calculating the confidence interval is sometimes called the Wald method, and is presented in nearly all statistics textbooks. It is so widely accepted and applied, that for many it is the only method they have used. For most others it is the technique of first choice. Careful study however reveals that it is flawed and inaccurate for a large range of n and p, to such a degree that it is ill-advised as a general method 1,2 . Because of this many statisticians have reverted to the exact Clopper-Pearson method, which is based on the exact binomial distribution, and not a large sample normal approximation (as is the Wald method). Studies have shown however that this confidence interval is very conservative, having coverage levels as high as 99% for a 95% CI, and requiring significantly larger sample sizes for the same level of precision 1,2,3 . An alternate method, called the Wilson Score method is often suggested as a compromise. It has been shown to be accurate for most parameter values and does not suffer from being over-conservative, having coverage levels closer to the nominal level of 95% for a 95% CI. In this discussion a brief review of the Wald, Wilson-Score, and exact Clopper Pearson methods of calculating confidence intervals for binomial proportions will be presented, focusing on differences between the Wald and Wilson Score methods. Sample size calculations for the Wald and Wilson Score methods will also be discussed. SAS programs for these formulas will be also presented and applied to a worked out example, which can be readily modified for other data. Finally the differences between the methods will be discussed in general.

/pdf/confidence-interval-calculation-for-binomial-proportions-1g4m5gm1xf.pdf

Confidence Interval Calculation for Binomial Proportions

In this article, we point out some interesting relations between the exact test and the score test for a binomial proportion p. Based on the properties of the tests, we propose some approximate as well as exact methods of computing sample sizes required for the tests to attain a specified power. Sample sizes required for the tests are tabulated for various values of p to attain a power of 0.80 at level 0.05. We also propose approximate and exact methods of computing sample sizes needed to construct confidence intervals with a given precision. Using the proposed exact methods, sample sizes required to construct 95% confidence intervals with various precisions are tabulated for p = .05(.05).5. The approximate methods for computing sample sizes for score confidence intervals are very satisfactory and the results coincide with those of the exact methods for many cases.

Jie Peng

Papers

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation