A step-down design
step-down_design.RmdA direct improvement over the basic design can be achieved by using a
step-down version of the test. The function stepdown.mams implements
such a design that selects at interim either all promising treatments
(selection = "all.promising") or only the best performing
treatment (selection = "select.best") from those whose test
statistics are between the upper and lower boundaries (Stallard and Todd
2003). If the trial is stopped early, making a selection
becomes obsolete, but note that we consider stopping boundaries as
non-binding. The step-down procedure makes use of closed testing, as
described in Section 2.2 (Jaki, Pallmann, and Magirr 2019) and
(Magirr, Stallard, and
Jaki 2014).
We reuse the three-arm two-stage design m2, studied in
the Multi-stage design section
with 76 and 38 observations in stage 1 and cumulative sample sizes at
stage two of 152 and 76 on control and active treatments, respectively.
The sample size of the study can be specified through the matrix
nMat that has \(J\) rows
and \(K\) columns where the first
column contains the values for the control group. A lower boundary can
be set via lb, and we set it to \(0.786\) as in the original design. We can
then choose how much of the total family-wise error we want to spend at
each stage using the argument alpha.star, and we choose to
spend, in line with the triangular test, \(\alpha^*_1 = 0.026\) $ the first interim
analysis, with the remaining \(α\)
level being used at the second analysis. We compare the selection rules
all.promising and select.best:
m2.all <- stepdown.mams(nMat = matrix(c(76, 152, rep(c(38, 76), 3)),
nrow = 2, ncol = 4), lb = m2$l[1], alpha.star = c(0.026, 0.05),
selection = "all.promising")
m2.best <- stepdown.mams(nMat = matrix(c(76, 152, rep(c(38, 76), 3)),
nrow = 2, ncol = 4), lb = m2$l[1], alpha.star = c(0.026, 0.05),
selection = "select.best")The following table reports the upper boundary values for selected scenarios.
| Hypotheses | Stage 1 | Stage 2 (best) | Stage 2 (all) |
|---|---|---|---|
| \(H_1\),\(H_2\),\(H_3\) | 1.94 | 1.71 | 1.72 |
| \(H_{12}\),\(H_{13}\),\(H_{23}\) | 2.21 | 2.02 | 2.06 |
| \(H_{123}\) | 2.36 | 2.17 | 2.22 |
One can see that the option to proceed with more than one promising treatment comes at the cost of higher stopping boundaries for the intersection hypotheses at stage 2. So in order to reject the global null hypothesis at the second stage, a test statistic needs to exceed 2.22 while it only needs to be larger than 2.17 if only the best arm is chosen at the interim analysis. The boundaries are the same for the elementary hypotheses H1, H2, and H3 as well as for the intersection hypotheses H12, H13, and H23 because the sample sizes were chosen to be equal in all active treatment arms. A graphical display of the stopping boundaries using the plot function is shown in Figure.
Figure: Stopping boundaries for a three-arm two-stage step-down design
with selection of all promising treatments at interim (left) or just the
single best treatment (right).