The power function for multinomial designs under intersection-union test (IUT)

Description

Calculate the type I error or power of a multinomial (response and disease progression) single- or two-stage design under IUT: H_0: p_1 ≤ p_{01} \ OR \ p_2 ≥ p_{02} \ versus \ H_1: p_1 ≥ p_{11} > p_{01} \ AND \ p_2 ≤ p_{12} < p_{02}

Usage

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IUT.power(method, s1.rej, t1.rej, s1.acc, t1.acc, n1, s2.rej, t2.rej, n2, p.s, p.t,
output.all)

Arguments

method

design methods according to number of stage and stopping rule, "s1" represents single-stage design stopping for both efficacy and futility, "s2" represents two-stage design stopping for both efficacy and futility, "s2.f" represents two-stage design stopping for futility only.

s1.rej

first stage responses threshold to stop the trial for efficacy. Applied for "s1" or "s2".

t1.rej

first stage disease progressions threshold to stop the trial for efficacy. Applied for "s1" or "s2".

s1.acc

first stage responses threshold to stop the trial for futility. Applied for "s2" or "s2.f".

t1.acc

first stage disease progressions threshold to stop the trial for futility. Applied for "s2" or "s2.f".

n1

first stage sample size. Applied for "s1", "s2" or "s2.f".

s2.rej

second stage responses threshold to stop the trial for efficacy. Applied for "s2" or "s2.f".

t2.rej

second stage disease progressions threshold to stop the trial for efficacy. Applied for "s2" or "s2.f".

n2

second stage sample size. Applied for "s2" or "s2.f".

p.s

pre-specified response rate, p.s=p_{01} for calculating type I error , p=p_{11} for calculating power.

p.t

pre-specified disease progression rate, p.s=p_{02} for calculating type I error, p=p_{12} for calculating power. Note: type I error calculation needs to take maximum of the power function with (p.s,p.t)=(p_{01},0) and (p.s,p.t)=(1-p_{02},p_{02})

output.all

logical, if FALSE (default), only output the value of power or type I error, otherwise, also output the probability of early termination (PET) and expected sample size (EN). Applied for "s2" or "s2.f".

Value

prob

the power function g(...,p.s,p.t): α = \max [g(...,p_{01},0), g(...,1-p_{02},p_{02}) ] or g(...,p_{11},p_{12})

References

Chang, M. N., Devidas, M., & Anderson, J. (2007). One- and two-stage designs for phase II window studies. Statistics in medicine , 26(13), 2604-2614.

Examples

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p01=0.1; p02=0.9
## Calculate type I error for single-stage design
max(IUT.power(method="s1", s1.rej=6, t1.rej=19, n1=25, p.s=p01, p.t=0),
IUT.power(method="s1", s1.rej=6, t1.rej=19, n1=25, p.s=1-p02, p.t=p02))
## Calculate power for single-stage design
IUT.power(method="s1", s1.rej=6, t1.rej=19, n1=25, p.s=p01+0.2, p.t=p02-0.2)

## Calculate type I error for two-stage design
max(IUT.power(method="s2", s1.rej=4, t1.rej=9, s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=p01, p.t=0),
IUT.power(method="s2", s1.rej=4, t1.rej=9, s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=1-p02, p.t=p02))
## Output PET and EN under null hypothesis
IUT.power(method="s2", s1.rej=4, t1.rej=9, s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=p01, p.t=p02, output.all=TRUE)[-1]
## Calculate power for two-stage design
IUT.power(method="s2", s1.rej=4, t1.rej=9, s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=p01+0.2, p.t=p02-0.2)

## Calculate type I error for two-stage design stopping for futility only,
## output PET and EN under null hypothesis
max(IUT.power(method="s2.f", s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=p01, p.t=0),
IUT.power(method="s2.f", s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=1-p02, p.t=p02))
## Output PET and EN under null hypothesis
IUT.power(method="s2.f", s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=p01, p.t=p02, output.all=TRUE)[-1]
## Calculate power for two-stage design
IUT.power(method="s2.f", s1.acc=0, t1.acc=13, n1=13,
s2.rej=6, t2.rej=18, n2=11, p.s=p01+0.2, p.t=p02-0.2)