Function to update boundaries based on observed sample sizes

The function determines updated boundaries of a multi-arm multi-stage study based on observed number of observations per arm.

Usage

new.bounds(K=3, J=2, alpha=0.05, nMat=matrix(c(10, 20), nrow=2, ncol=4), 
           u=NULL, l=NULL, ushape="obf", lshape="fixed", ufix=NULL,
           lfix=0, N=20, parallel=TRUE, print=TRUE)

Arguments

K: Number of experimental treatments (default=3).
J: Number of stages (default=2).
alpha: One-sided familywise error rate (default=0.05).
nMat: Jx(K+1) dimensional matrix of observed/expected sample sizes. Rows correspond to stages and columns to arms. First column is control (default: 2x4 matrix with 10 subjects per stage and arm).
u: Vector of previously used upper boundaries (default=NULL).
l: Vector of previously used upper boundaries (default=NULL).
ushape: Shape of upper boundary. Either a function specifying the shape or one of "pocock", "obf" (the default), "triangular" and "fixed". See details.
lshape: Shape of lower boundary. Either a function specifying the shape or one of "pocock", "obf", "triangular" and "fixed" (the default). See details.
ufix: Fixed upper boundary (default=NULL). Only used if shape="fixed".
lfix: Fixed lower boundary (default=0). Only used if shape="fixed".
N: Number of quadrature points per dimension in the outer integral (default=20).
parallel: if TRUE (default), allows parallelisation of the computation via a user-defined strategy specified by means of the function future::plan(). If not set differently, the default strategy is sequential, which corresponds to a computation without parallelisation.
print: if TRUE (default), indicate at which stage the computation is.

Details

This function finds the boundaries for a given matrix of sample sizes in multi-arm multi-stage study with K active treatments plus control. The vectors u and l are the boundaries used so far while u.shape and l.shape specify the shape to the boundaries for the remaining analysis. By specifying u and l as NULL, a design using only the shapes given by ushape and lshape can be found for any sample sizes per stage and arm.

The shape of the boundaries (ushape, lshape) are either using the predefined shapes following Pocock (1977), O'Brien & Fleming (1979) or the triangular Test (Whitehead, 1997) using options "pocock", "obf" or "triangular" respectively, are constant (option "fixed") or supplied in as a function. If a function is passed it should require exactly one argument specifying the number of stages and return a vector of the same length. The lower boundary shape is required to be non-decreasing while the upper boundary shape needs to be non-increasing. If a fixed lower boundary is used, lfix must be smaller than \(\Phi^{-1}(1-\alpha)/2\) to ensure that it is smaller than the upper boundary.

Value

An object of the class MAMS containing the following components:

l: Lower boundary.
u: Upper boundary.
n: Sample size on control in stage 1.
N: Maximum total sample size.
K: Number of experimental treatments.
J: Number of stages in the trial.
alpha: Familywise error rate.
power: Power under least favorable configuration.
rMat: Matrix of allocation ratios. First row corresponds to control and second row to experimental treatments.

References

Jaki T., Pallmann P. and Magirr D. (2019), The R Package MAMS for Designing Multi-Arm Multi-Stage Clinical Trials, Journal of Statistical Software, 88(4), 1-25. Link: doi:10.18637/jss.v088.i04

Magirr D., Jaki T. and Whitehead J. (2012), A generalized Dunnett test for multi-arm multi-stage clinical studies with treatment selection, Biometrika, 99(2), 494-501. Link: doi:10.1093/biomet/ass002

Magirr D., Stallard N. and Jaki T. (2014), Flexible sequential designs for multi-arm clinical trials, Statistics in Medicine, 33(19), 3269-3279. Link: doi:10.1002/sim.6183

Pocock S.J. (1977), Group sequential methods in the design and analysis of clinical trials, Biometrika, 64(2), 191-199.

O'Brien P.C., Fleming T.R. (1979), A multiple testing procedure for clinical trials, Biometrics, 35(3), 549-556.

Whitehead J. (1997), The Design and Analysis of Sequential Clinical Trials, Wiley: Chichester, UK.

Author

Thomas Jaki, Dominic Magirr and Dominique-Laurent Couturier

Examples

# \donttest{
# Note that some of these examples may take a few minutes to run

# 2-stage design with O'Brien & Fleming efficacy and zero futility boundary with 
# equal sample size per arm and stage. Results are equivalent to using 
# mams(K=4, J=2, alpha=0.05, power=0.9, r=1:2, r0=1:2, ushape="obf", lshape="fixed", 
#      lfix=0, sample.size=FALSE)

new.bounds(K=4, J=2, alpha=0.05, nMat=matrix(c(10, 20), nrow=2, ncol=5), u=NULL, l=NULL, 
           ushape="obf", lshape="fixed", lfix=0)
#>    i) find new lower and upper boundaries
#>       
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
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#> 
#>   ii) define alpha star
#> Design parameters for a 2 stage trial with 4 treatments
#> 
#>              Stage 1 Stage 2
#> Upper bound:   3.068   2.169
#> Lower bound:   0.000   2.169

# A 2-stage design that was designed to use an O'Brien & Fleming efficacy and 
# zero futility boundary with equal sample size per arm and stage (n=14).
# The observed sample size after stage one are 10, 10, 18, 10, 13 for each arm 
# while the original upper bounds used are (3.068, 2.169) for stage 1. 
# The updated bounds are (3.068, 2.167).

new.bounds(K=4, J=2, alpha=0.05, 
           nMat=matrix(c(10, 28, 10, 28, 18, 28, 10, 28, 13, 28), nrow=2, ncol=5), 
           u=3.068, l=0, ushape="obf", lshape="fixed", lfix=0)
#>    i) find new lower and upper boundaries
#>       
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> 
#>   ii) define alpha star
#> Design parameters for a 2 stage trial with 4 treatments
#> 
#>              Stage 1 Stage 2
#> Upper bound:   3.068   2.167
#> Lower bound:   0.000   2.167
           
# same using parallelisation via separate R sessions running in the background
future::plan(multisession)
new.bounds(K=4, J=2, alpha=0.05, 
           nMat=matrix(c(10, 28, 10, 28, 18, 28, 10, 28, 13, 28), nrow=2, ncol=5), 
           u=3.068, l=0, ushape="obf", lshape="fixed", lfix=0)
#>    i) find new lower and upper boundaries
#>       
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> .
#> 
#>   ii) define alpha star
#> Design parameters for a 2 stage trial with 4 treatments
#> 
#>              Stage 1 Stage 2
#> Upper bound:   3.068   2.167
#> Lower bound:   0.000   2.167
future::plan("default")
           
# }