*! version 1.1 3Dec2010
help simon2stage

Title

Finding Simon two stage designs with extensions

Syntax

simon2stage
[, options]

Main
p0(#) specifies the null proportion, the default is 0.1.

p1(#) specifies the alternative proportion, the default is 0.4.

alpha(#) specifies the type I error, the default is 0.05.

beta(#) specifies the type 2 error, the default is 0.2.

minn(#) specifies the start sample size in the initial grid search, the default is 1.

maxn(#) specifies the largest sample size in the initial grid search, the default is 35.

optimal specifies that the interest is in the optimal design rather than the minimax.

optp(#) specifies the true proportion to optimize the design for, the default is p0.

eff specifies that the design can stop for efficacy as well as futility.

deltaminimax specifies that the delta-minimax design is found.

admiss specifies that the set of admissible designs should be found.

Description

The Simon two stage design is a single arm study with an interim analysis. The main purpose of this design is to
investigate whether an intervention works or not and to stop the study early for futility.
Under the null hypothesis the probability of a success is p0, this is usually taken as the probability of success
for the current standard treatment.
The probability of a success in this study, p, is tested using the null hypothesis
H0:p=p0 versus the alternative hypothesis H1:p>=p1. The probability of success for the alternative
hypothesis is fixed to be a pre-specified value p1, where p1>p0.

The Simon two stage design consists of studying n1 participants in a first stage and the study stops if there
are r1 or fewer responders to the intervention. If there are more than r1 responders in the first
stage then the study continues until n participants in total are studied. Then the null hypothesis is
not rejected if there are r or fewer responders. Each design must satisfy the type 1 (alpha) and type 2 (beta)
errors.

The probability of not rejecting H0 can be calculated conditional on any p and let this function be R(p).
The design must therefore satisfy the constraints R(p0)>=1-alpha and R(p1)<=beta.
The minimax design is one that satisfies these
constraints with the smallest total sample size n and the smallest expected sample size under H0.
The alternative is to find the "optimal" design which is
the design with the smallest expected sample size conditional on the true proportion being specified by the optp() option.
The default is optimising the design under the null proportion p0 and this is the classical Simon two stage design, however
this command allows greater flexibility.

The Simon two stage design has been extended here to allow for stopping for efficacy. The design can be indexed
by 5 numbers (r1 r2/n1, r/n). Now the study stops for futility if there are r1 or fewer responders
OR stops for efficacy if there are more than r2 responders in the first stage. If the study continues
to the second stage then the null hypothesis is no rejected if there are r or fewer responders. Again
the type 1 and type 2 errors must be satisfied for a design to be considered.

There are two additional designs that are of interest firstly the delta-minimax design as described in Shuster 2002
and secondly the set of admissible designs as described by Mander et al. (2010). The set of admissible designs
are displayed by using a novel figure to help with decision making.

Latest Version

The latest version is always kept on the SSC website. To install the latest version click
on the following link

ssc install simon2stage, replace.

Options

Main

p0(#) specifies the null proportion, the default is 0.1.

p1(#) specifies the alternative proportion, the default is 0.4.

alpha(#) specifies the type I error, the default is 0.05.

beta(#) specifies the type II error, the default is 0.2.

minn(#) specifies the start total sample size in the initial grid search, the default is 1.

maxn(#) specifies the largest total sample size in the initial grid search, the default is 35.

optimal specifies that the interest is in the optimal design rather than the minimax.

optp(#) specifies the true proportion to optimize the design for, the default is p0.

eff specifies that the design can stop for efficacy as well as futility.

deltaminimax specifies that the delta-minimax design is found.

admiss specifies that the set of admissible designs should be found.

Examples

simon2stage

The minimax design is {1/8 3/13}, if there are no responders or one responder out of the first 8 participants
then the study stops and the null hypothesis is not rejected. If the study proceeds to the second stage then
the null hypothesis is rejected if there are more than 3 responders. Under the null hypothesis there is a 0.813
chance that the study stops at the first stage and hence the average sample size is 8*0.813+13*(1-0.813) = 8.934.

simon2stage, optimal

The optimal design is {1/7 3/15}, this is similar to the previous design but there is now a 0.85 chance that the
study finishes at stage 1 and the average sample size under the null hypothesis is 8.198 which is slightly smaller than the minimax
desgin.

simon2stage, optimal optp(0.2)

The optimal design is still {1/7 3/15} but the average sample size is much greater at 10.386 as a direct result of
a smaller chance 0.577 of early termination.

simon2stage, optimal optp(0.2) eff

The optimal design is {(1 2)/7 3/15} the expected sample size, 9.202, is smaller than the previous design
because of a greater chance of an early termination 0.725.

simon2stage,optimal eff deltaminimax

The delta-minimax design is {(1 2)/7 3/15} the expected sample size, 9.550, is larger than the previous designs
because this is the maximum expected sample size.

simon2stage, optp(0.2)

The minimax design optimised at a true response of 0.2 is {1/8 3/13}. The expected sample size, 10.483,
much greater than when calculating the expected sample size at the null value. This has not altered the minimax
design.

simon2stage,optimal eff admiss maxn(16)

The set of admissible designs is plotted. Setting the maximum to be 16 means that the set of admissible designs
are found by fixing the maximum sample size to be 16 or lower (this limits the designs to search through).
Interestingly the design {(1 2)/7 3/15} is only good if there is little weight given to the overall
sample size.

Author

Adrian Mander, MRC Biostatistics Unit, Cambridge, UK.

Email adrian.mander@mrc-bsu.cam.ac.uk

References

R.P.A'Hern (2001) Sample size tables for exact single-stage phase II designs. Statistics in Medicine 20:859-866.

T.R. Fleming (1982) One-sample multiple testing procedure for phase II clinical trials. Biometrics 38:143-151.

Richard Simon (1989) Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials 10:1-10.

A.P. Mander and S.G. Thompson (2010) Two-stage designs optimal under the alternative hypothesis for phase II cancer trials. Contemporary Clinical Trials 31(6):572-578.

A.P. Mander, J.M. Wason, M.J. Sweeting and S.G. Thompson (2010) Admissible two-stage designs for phase II cancer clinical trials. (Submitted)

Also see

Related commands

HELP FILES SSC installation links Description

samplesize (if installed) (ssc install samplesize) Sample Size graphics

sampsi_fleming (if installed) (ssc install sampsi_fleming) Sample Size for Fleming design

sampsi_reg (if installed) (ssc install sampsi_reg) Sample Size for linear regression

sampsi_mcc (if installed) (ssc install sampsi_mcc) Sample Size for matched case/control studies

sampsi_rho (if installed) (ssc install sampsi_rho) Sample Size for Pearson correlation