Dealing with unforeseen design modifications

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.