ciss.liubailey {binomSamSize} R Documentation

## Sample size calculations using the Liu and Bailey (2002) approach

### Description

Calculate sample size for a binomial proportion based on the confidence interval width specification in Liu and Bailey (2002).

### Usage

ciss.liubailey(alpha, d, lambda.grid = 0:30)


### Arguments

 alpha a (1-alpha/2)*100% confidence interval is computed d half width of the confidence interval lambda.grid range of lambda values to try

### Details

The objective is to find the minimum sample size n so that the minimum coverage probability (aka. as the coverage coefficient) of the confidence interval for the binomial parameter is larger than 1-α. In the present approach the confidence interval is of form

(C_n(p.hat)-d,C_n(p.hat)+d)

as suggested in equation (3.1) of Liu & Bailey (2002):

(\hat{p}_l,\hat{p}_u) = \hat{p}_n + \frac{λ z^2 (0.5-\hat{p}_n)}{n+z^2} \pm d

where \hat{p}_n = x/n. The interval is then expanded to a full length of 2d using the following transformation:

\hat{p}_l^* = \max(0,\min( 1-2d, \hat{p}_l))

\hat{p}_u^* = \min(1,\max( 2d, \hat{p}_u))

As a consequence, the computed interval will always have length 2d.

Given d, fixed lambda and a sample size n, the proportion p in [0,1] where the coverage probability is minimum is computed. The sample size is then gradually increased until this minimum coverage probability becomes larger than 1-α. We then change the value of λ, and search the minimum sample size that guarantee the 1-α confidence level for this lambda value. The smallest minimum sample size over a set of lambda values in lambda.grid is then used as the required sample size; this sample size and the corresponding lambda value are used to calculate the confidence interval given above.

For a general overview of coverage probabilities of confidence intervals for a binomial proportion see Agresti and Coull (1998). Once actual binomial data are obtained the function binom.liubailey can be used to compute the actual confidence interval.

The R function code calls the original Fortran code developed for the Liu and Bailey (2002) article. NAG calls were replaced by R API calls and an R wrapper calling the code as a subroutine was created.

### Value

a vector containing the following three elements

 nstar sample size at most favorable lambda value in lambda.grid cp coverage probability lambda value in lambda.grid giving the lowest nstar value

### Author(s)

M. Höhle and W. Liu

### References

Agresti, A. and Coull, B.A. (1998), Approximate is Better than "Exact" for Interval Estimation of Binomial Proportions, The American Statistician, 52(2):119-126.

Liu, W. and Bailey, B.J.R. (2002), Sample size determination for constructing a constant width confidence interval for a binomial success probability. Statistics and Probability Letters, 56(1):1-5.

binom.liubailey
ciss.liubailey(alpha=0.1,d=0.05)