exactdesign {BinGSD} | R Documentation |

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

exactdesign(d)

`d` |
An object of the class asymdesign. |

Suppose *X_{1}, X_{2}, …* 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}, …, 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=∑_{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.

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).

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

`exactprob`

, `exactcp`

,
`asymdesign`

.

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)

[Package *BinGSD* version 0.0.1 Index]