BinGSD (version 1.1)

exactcp: Conditional power computation using exact test.

Description

Compute conditional power of single-arm group sequential design with binary endpoint based on binomial distribution.

Usage

exactcp(d, p_1, i, z_i)

Value

A list with the elements as follows:

• K: As in d.

• n.I: As in d.

• u_K: As in d.

• lowerbounds: As in d.

• i: i used in computation.

• z_i: As input.

• cp: A matrix of conditional powers under different response rates.

• p_1: As input.

• p_0: As input.

Arguments

d

An object of the class exactdesign or exactprob.

p_1

A scalar or vector representing response rate or probability of success under the alternative hypothesis. The value(s) should be within (p_0,1).

i

Index of the analysis at which the interim statistic is given. Should be an integer ranges from 1 to K-1. i will be rounded to its nearest whole value if it is not an integer.

z_i

The interim statistic at analysis i.

Reference

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

Details

Conditional power quantifies the conditional probability of crossing the upper bound given the interim result \(z_i\), \(1\le i<K\). Having inherited sample sizes and boundaries from exactdesign or exactprob, given the interim statistic at \(i\)th analysis \(z_i\), the conditional power is defined as

\(\alpha _{i,K}(p|z_i)=P_{p}(Z_K\ge u_K, Z_{K-1}>l_{K-1}, \ldots, Z_{i+1}>l_{i+1}|Z_i=z_i)\)

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.

The increment statistic \(Z_k-Z_{k-1}\) also follows a binomial distribution \(b(n_k-n_{k-1},p)\) independently of \(Z_{1}, \ldots, Z_{k-1}\). Then the conditional power can be easily obtained using the same procedure for deriving unconditional boundary crossing probabilities.

Note that \(Z_{1}, \ldots, Z_{K}\) is a non-decreasing sequence, thus the conditional power is 1 when the interim statistic \(z_i>=u_K\).

See Also

exactprob, asymcp, exactdesign.

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)
tt3=exactprob(p_1=c(0.4,0.5,0.6,0.7,0.8,0.9),d=tt2)
exactcp(tt2,p_1=c(0.4,0.5,0.6,0.7,0.8,0.9),1,2)
exactcp(tt3,p_1=c(0.4,0.5,0.6,0.7,0.8,0.9),3,19)