mams: Function to design multi-arm multi-stage studies with normal endpoints

Description

The function determines the boundaries of a multi-arm multi-stage study for a given boundary shape and finds the required number of subjects.

Usage

mams(obj = NULL, K=NULL, J=NULL, alpha=NULL, power=NULL, r=NULL, 
                     r0=NULL, p=NULL, p0=NULL, delta=NULL, delta0=NULL, 
                     sd=NULL,ushape=NULL, lshape=NULL, ufix=NULL, lfix=NULL, 
                     nstart=NULL, nstop=NULL, sample.size=NULL, Q=NULL,
                     type=NULL, parallel=NULL, print=NULL, nsim=NULL, 
                     H0=NULL, method=NULL)

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.
alpha.star: Cumulative familywise error rate spent by each analysis.
power: Power under least favorable configuration.
rMat: Matrix of allocation ratios. First row corresponds to control while subsequent rows are for the experimental treatments.
sim: a list indicating, for each hypothesis of interest (null and/or alternative), the expected sample size and standard deviation per group (ess), the cumulated probability of efficacy (efficacy) and futility (futility) per treatment arm and look
input: the list of all input parameters except K and J

Arguments

obj: object of class MAMS
K: Number of experimental treatments (default=4).
J: Number of stages (default=2).
alpha: One-sided familywise error rate (default=0.05).
power: Desired power (default=0.9).
r: Vector of allocation ratios (default=1:2).
r0: Vector ratio on control (default=1:2).
p: Interesting treatment effect on the probability scale. See Details (default=0.75).
p0: Uninteresting treatment effect on the probability scale. See Details (default=0.5). Note that this parameter affects the sample size estimation to ensure that Treatment 1 is best only when selecting the simultaneous stopping rule (i.e., 'method = "simultaneous"') and, for all methods, as values for treatment arms 2 to K in the simulations under H1.
delta: Interesting treatment effect on the traditional scale. See Details (default=NULL).
delta0: Uninteresting treatment effect on the traditional scale. See Details (default=NULL). Note that this parameter affects the sample size estimation to ensure that Treatment 1 is best only when selecting the simultaneous stopping rule (i.e., 'method = "simultaneous"') and, for all methods, as values for treatment arms 2 to K in the simulations under H1.
sd: Standard deviation, assumed to be known. See Details (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".
nstart: Starting point for finding the sample size (default=1).
nstop: Stopping point for finding the sample size (default=NULL).
sample.size: Logical if sample size should be found as well (default=TRUE).
Q: Number of quadrature points per dimension in the outer integral (default=20).
type: Will be changed automatically by the wrappers tite.mams() (to "tite") and ordinal.mams() (to "ordinal") to customise the output.
parallel: if TRUE (default), allows parallelization 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 parallelization.
print: if TRUE (default), indicate at which stage the computation is.
nsim: a scalar indicating the number of simulations (default = 50'000, minimum = 1000)
H0: if TRUE (default), the simulation also considers the case with all effect sizes set to 0.
method: Type of the desired design: simultaneous(default) for simultaneous stopping rules, sep for separate stopping, and dtl for drop-the-losers design.

Author

Thomas Jaki, Dominic Magirr, Dominique-Laurent Couturier and Nikita Mozgunov

Details

This function finds the boundaries and sample size of a multi-arm multi-stage study with K active treatments plus control in which all promising treatments are continued at interim analyses as described in Magirr et al (2012). At each interim analysis the test statistics are compared to the lower (futility) bound and any treatment whose corresponding test statistic falls below that bound is discontinued. Similarly if any test statistic exceeds the upper (efficacy) bound the null hypothesis corresponding to that treatment can be rejected and superiority of that treatment over control claimed. At the same time the study is stopped. If at least one test statistic exceeds the lower bound and none exceeds the upper bound the study is continued and further patients are recruited to all remaining experimental treatments plus control.

The design is found under the least favorable configuration, which requires an interesting treatment effect p that if present we would like to find with high probability and an uninteresting effect p0. Both p and p0 are parameterized as \(P(X_k > X_0 ) = p\), that is the probability of a randomly selected person on treatment k observing a better outcome than a random person on control. For p=0.5 the experimental treatment and control perform equally well. The advantage of this parameterization is that no knowledge about the variance is required. To convert traditional effect sizes, d\(\delta\) to this format use \(p=\Phi(\frac{\delta}{\sqrt{2}\sigma})\). Alternatively, the interesting and uninteresting effect size can also be specified directly on the traditional scale of delta and delta with an additional specification of the standard deviation sd assumed to be known.

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.

The default starting point for finding the sample size is nstart=1, and the default point where the search is stopped (when nstop=NULL) is 3 times the sample size of the corresponding fixed single-stage design.

Computation of designs with more than four stages are very time consuming and not advised. The parameter sample.size controls whether the required sample size is computed as well. Setting to FALSE approximately halves the computation time.

For designs with more than 2 stages, parallelization of the computation by means of the packages future and future.apply lead to decreased computation times when choosing a parallelization strategy like, for example, multicore (using separate forked R processes, available to unix/osx users) or multisession (using separate R sessions, available to all users) (refer to future::plan() for detail).

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: tools:::Rd_expr_doi("https://doi.org/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: tools:::Rd_expr_doi("https://doi.org/10.1093/biomet/ass002")

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

O'Brien P.C., and 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. Wason J, Stallard N, Bowden J, Jennison C. A multi-stage drop-the-losers design for multi-arm clinical trials. Statistical Methods in Medical Research. 2017;26(1):508-524. doi:10.1177/0962280214550759

Examples

Run this code

# \donttest{
## A fixed sample (single stage) design specified on the p scale
mams(K=4, J=1, alpha=0.05, power=0.9, r=1, r0=1, p=0.65, p0=0.55)

## The same design specified on the delta scale

mams(K=4, J=1, alpha=0.05, power=0.9, r=1, r0=1, p=NULL, p0=NULL,
     delta=0.545, delta0=0.178, sd=1)

## An example in Table 1 of Magirr et al (2012)
# 2-stage design with O'Brien & Fleming efficacy and zero futility boundary

mams(K=4, J=2, alpha=0.05, power=0.9, r=1:2, r0=1:2, p=0.65, p0=0.55,
     ushape="obf", lshape="fixed", lfix=0, nstart=40)

## An example of separate stopping rules
# 2-stage design with O'Brien & Fleming efficacy and zero futility boundary

mams(method = "sep",K=4, J=2, alpha=0.05, power=0.9, r=1:2, r0=1:2, 
       p=0.65, p0=0.55, ushape="obf", lshape="fixed", lfix=0, nstart=40)

# An example of running drop-the-losers design 
# `K` should be defined as vector length of J defining allocation arms per 
# stages with final element equal to 1.
mams(method = "dtl", K=c(4,1), J=2, alpha=0.05,  
         power=0.9, r=1:2, r0=1:2, p=0.65, p0=0.55, ushape="obf", 
         lshape="fixed", lfix=0, nstart=40)



# Note that these examples may take a few minutes to run

## 3-stage design with Triangular efficacy and futility boundary
mams(K=4, J=3, alpha=0.05, power=0.9, r=1:3, r0=1:3, p=0.65, p0=0.55,
  ushape="triangular", lshape="triangular", nstart=30)

## Different allocation ratios between control and experimental treatments.
## Twice as many patients are randomized to control at each stage.
  mams(K=4, J=2, alpha=0.05, power=0.9, r=1:2, r0=c(2, 4), p=0.65, 
  p0=0.55, ushape="obf", lshape="fixed", lfix=0, nstart=30)


  ##
  ## example considering different parallelization strategies
  ##


  # parallel = FALSE (future framework not used)
  set.seed(1)
  system.time(
  print(mams(K=4, J=3, alpha=0.05, power=0.9, r=1:3, r0=1:3,
          p=0.65, p0=0.55, ushape="triangular", lshape="triangular",
          nstart=30, parallel = FALSE))
          )
 # parallel = TRUE (default) with default strategy (sequential computation)
 plan(sequential)
 set.seed(1)
 system.time(
 print(mams(K=4, J=3, alpha=0.05, power=0.9, r=1:3, r0=1:3,
          p=0.65, p0=0.55, ushape="triangular", lshape="triangular", 
          nstart=30))
          )
 # parallel = TRUE(default) with multisession strategy (parallel computation)
 plan(multisession)
 set.seed(1)
 system.time(
 print(mams(K=4, J=3, alpha=0.05, power=0.9, r=1:3, r0=1:3, 
          p=0.65, p0=0.55, ushape="triangular", lshape="triangular",
           nstart=30))
          )
          plan("default")
          # }

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