# Dealing with unforeseen design modifications

`unforeseen_design_modifications.Rmd`

Two functions of the package: `new.bounds`

and
`stepdown.update`

, allow for unexpected design modifications
to be taken into account. The function `new.bounds`

recalculates the boundary values when the sample sizes achieved are not
as planned in advance. We consider again the two-stage design
`m2`

studied in the Multi-stage design section where 76
patients were required per stage in the control arm and 38 patients per
stage for each of the three experimental treatment arms. Now assume
these requirements could not be met and the observed sample sizes at the
interim analysis were 75 for control and 40, 35, and 41 for the
experimental treatments. We can recalculate the final boundary value
with new.bounds in which we specify the interim bounds \(u = 2.359\) and \(l = 0.786\) (as obtained for
`m2`

). The sample sizes as observed at stage 1 and planned
for stage 2 are given in the \(J × (K +
1)\) matrix `nMat`

.

```
m2.nb <- new.bounds(K = 3, J = 2,
nMat = matrix(c(75, 152, 40, 76, 35, 76, 41, 76),
nrow = 2, ncol = 4), alpha = 0.05, u = m2$u[1], l = m2$l[1],
ushape = "triangular", lshape = "triangular")
m2.nb
```

```
## Design parameters for a 2 stage trial with 3 treatments
##
## Stage 1 Stage 2
## Upper bound: 2.359 2.224
## Lower bound: 0.786 2.224
```

We find that as a result of the deviation from the planned sample
size at the interim analysis, the final boundary value has been lowered
from 2.225 in the original `m2`

design to 2.224. The function
`stepdown.update`

uses the conditional error approach to
incorporate unplanned sample size reassessment and/or treatment
selection (e.g., elimination of treatment arms due to safety issues)
while maintaining control of the desired FWER. We once again base this
example on the original three-arm two-stage design but consider the
step-down version (`m2.all`

) studied in the Step-down design section and assume
there were some unforeseen design changes during the course of the
trial. Initially the sample sizes at interim were planned to be 76 for
the control group and 38 per active treatment arm. At the interim
analysis, we now wish to take into account three deviations from the
planned study.

Firstly, we want to account for the deviation from the desired sample
size which, as in the previous example, turned out to be 75 for control
and 40, 35, and 41 for the experimental treatments, which translates to
`nobs = c(75, 40, 35,41)`

in the function
`stepdown.update()`

. Secondly, treatment 2 has been dropped
from the study due to safety, so that only treatment arms 1 and 3
(`selected.trts = c(1, 3)`

) are to be continued.Finally,
following a reassessment of the sample size, we wish to increase the
cumulative sample size at the second stage by 50% from 152 to 228 in the
control arm and 76 to 114 in the active arms. We can specify this using
`nfuture = matrix(c(228,114, 35, 114), 1, 4)`

.

Notice that since treatment arm 2 has already been abandoned, no
additional patients are recruited beyond the 35 already in the study.
Further supposing the interim evaluation yielded Z statistics of
`zscores = c(1.1, 0.9, 0.9)`

, we can calculate the modified
design.

```
m2.update <- stepdown.update(current.mams = m2.all,
nobs = c(75, 40, 35, 41), zscores = c(1.1, 0.9, 0.9),
selected.trts = c(1, 3), nfuture = matrix(c(228, 114, 35, 114),
nrow = 1, ncol = 4))
```

We summarise in Table 3 (column “Updated”) the output of
`m2.update`

. The boundaries for the elementary hypothesis
\(H_1\) and \(H_3\) have been slightly increased to
account for the change in sample size while the boundary for \(H_2\) has been slightly decreased.
Similarly, the boundary for the intersection hypothesis involving only
the remaining treatments (\(H_{13}\))
has been increased while the others have been decreased. No change in
the threshold for the global null hypothesis (\(H_{123}\)) is observed in this example. As
before we can also illustrate the updated design using the plot
function.

Hypotheses | Initial | Updated (cond. error) |
---|---|---|

\(H_1\) | 1.72 | 1.73(0.088) |

\(H_2\) | 1.72 | 2.02(0.069) |

\(H_3\) | 1.72 | 2.17(0.058) |

\(H_{12}\) | 2.06 | 1.71(0.056) |

\(H_{13}\) | 2.06 | 2.02(0.051) |

\(H_{23}\) | 2.06 | 2.17(0.043) |

\(H_{123}\) | 2.22 | 1.71(0.041) |

Initial and updated upper boundaries (with conditional errors) for the elementary (\(H_1\), \(H_2\), \(H_3\)), intersection (\(H_{12}\), \(H_{13}\), \(H_{23}\)), and global (\(H_{123}\)) hypotheses of a three-arm twostage step-down design involving selection of all promising treatments at interim. Treatment 2 has been dropped at the interim analysis and the sample size for the remaining comparisons increased.