exactdesign {BinGSD}

Compute sample size and boundaries using exact binomial distribution

Description

Compute sample size and boundaries of single-arm group sequential design with binary endpoint using exact binomial distribution

Usage

exactdesign(d)

Arguments

d

An object of the class asymdesign.

Details

Suppose X_{1}, X_{2}, \ldots are binary outcomes following Bernoulli distribution b(1,p), in which 1 stands for the case that the subject responds to the treatment and 0 otherwise. Consider a group sequential test with K planned analyses, where the null and alternative hypotheses are H_0: p=p_0 and H_1: p=p_1 respectively. Note that generally p_1 is greater than p_0. For k<K, the trial stops if and only if the test statistic Z_k crosses the futility boundary, that is, Z_k<=l_k. The lower bound for the last analysis l_K is set to be equal to the last and only upper bound u_K to make a decision. At the last analysis, the null hypothesis will be rejected if Z_K>=u_K.

The computation of lower bounds except for the last one is implemented with u_K fixed, thus the derived lower bounds are non-binding. Furthermore, the overall type I error will not be inflated if the trial continues after crossing any of the interim lower bounds, which is convenient for the purpose of monitoring. Let the sequence of sample sizes required at each analysis be n_{1}, n_{2}, \ldots, n_{K}. For binomial endpoint, the Fisher information equals n_k/p/(1-p) which is proportional to n_k. Accordingly, the information fraction available at each analysis is equivalent to n_k/n_K.

With exact test, the test statistic at analysis k is Z_k=\sum_{s=1}^{n_k}X_s which follows binomial distribution b(n_k,p). Actually, Z_k is the total number of responses up to the kth analysis.

Under the null hypothesis, Z_k follows a binomial distribution b(n_k,p_0). While under the alternative hypothesis, Z_k follows b(n_k,p_1). It may involve massive computation to simultaneously find proper n_K and u_K. In fact, the sample sizes obtained from asymptotic test ought to be close to those from exact test. Thus, we adopt n_K from asymptotic test as the starting value. The starting value of u_K is computed given the n_K. Iteratively update u_K and n_K until errors are limited to certain amount.

Like asymdesign, the lower boundaries for the first K-1 analyses are sequentially determined by a search method. However, if the actual overall type II error exceeds the desired level, not only sample sizes but also all the boundaries are updated, since the binomial distribution under H_0 involves with sample size.

Due to the discreteness of binomial distribution, in exact test, the type I and type II error actually spent at each analysis may not approximate the designated amount. With the only one upper bound, the whole type I error is spent at the final analysis. From some simulation studies, though not presented here, we found that carrying over unused type II error has minor influence on the resulting boundaries and sample sizes. However, in an attempt to reduce the false positive rate, we decided to recycle the unspent amount of desired type II error. Thus, the elements of betaspend in an exactdesign object may be greater than the amount pre-specified by the user.

Value

An object of the class exactdesign. This class contains:

• I: I used in computation, as in d.

• beta: The desired overall type II error level, as in d.

• betaspend: The desired type II error spent at each analysis used in computation, as in d.

• alpha: The desired overall type I error level, as in d.

• p_0: The response rate or the probability of success under null hypothesis, as in d.

• p_1: The response rate or the probability of success under alternative hypothesis, as in d.

• K: K used in computation, as in d.

• n.I: A vector of length K which contains sample sizes required at each analysis to achieve desired type I and type II error requirements. n.I equals sample size for the last analysis times the vector of information fractions.

• u_K: The upper boundary for the last analysis.

• lowerbounds: A vector of length K which contains lower boundaries for each analysis. Note that the lower boundaries are non-binding.

• problow: Probabilities of crossing the lower bounds under H_1 or the actual type II error at each analysis.

• probhi: Probability of crossing the last upper bound under H_0 or the actual type I error.

• power: power of the group sequential test with the value equals 1-sum(problow).

Reference

• Christopher Jennison, Bruce W. Turnbull. Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC, Boca Raton, FL, 2000.

See Also

exactprob, exactcp, asymdesign.

Examples

I=c(0.2,0.4,0.6,0.8,0.99)
beta=0.2
betaspend=c(0.1,0.2,0.3,0.3,0.2)
alpha=0.05
p_0=0.3
p_1=0.5
K=4.6
tol=1e-6
tt1=asymdesign(I,beta,betaspend,alpha,p_0,p_1,K,tol)
tt2=exactdesign(tt1)