binomSamSize (version 0.1-3)

ciss.liubailey: 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 $(1-\alpha/2)\cdot 100%$ confidence interval is computed

d

half width of the confidence interval

lambda.grid

range of lambda values to try

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

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 $1-\alpha$. In the present approach the confidence interval is of form $$(C_n(\hat{p}_n)-d,C_n(\hat{p}_n)+d)$$ as suggested in equation (3.1) of Liu & Bailey (2002): $$(\hat{p}_l,\hat{p}_u) = \hat{p}_n + \frac{\lambda z^2 (0.5-\hat{p}_n)}{n+z^2} \pm d$$ 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 $1-\alpha$. We then change the value of $\lambda$, and search the minimum sample size that guarantee the $1-\alpha$ 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 subroutine was created.

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.

See Also

binom.liubailey

Examples

ciss.liubailey(alpha=0.1,d=0.05)
ciss.liubailey(alpha=0.1,d=0.05,lambda.grid=5)