Fig 1.
An example for independent parallel group trials (gold standard trial design; upper panel) and the corresponding umbrella parallel group subtrials (lower panel). All patients in the Bi trial and in the Bi subtrial, respectively, exhibit a positive test result for biomarker Bi (i = 1, …, m). Patients in the B− subtrial exhibit negative test results for all Bi. Screen: screening platform, R: randomisation, EXPi: experimental treatment related to Bi, STD: standard treatment.
Table 1.
Assumed biomarker status- and (sub-) trial-specific individual outcomes.
Fig 2.
Ratio of discarded to included patients in the independent trial design and in the umbrella design with the random allocation scheme.
πi (i = 1, 2) denotes the prevalence of a positive test result for biomarker Bi. A “discarded patient” is a patient that was screened but not included in a (sub-) trial. The ratio is derived as E[Nscreen]−2N divided by 2N with E[Nscreen] from Eqs (3) and (9), respectively. #: number of.
Fig 3.
Proportion of patients with a positive test result for both biomarkers in the independent trial design and in the umbrella design with the random allocation scheme.
πi (i = 1, 2) denotes the prevalence of a positive test result for biomarker Bi. The underlying formulae are given in Eqs (10) and (12). For each (sub-) trial, the associated prevalence is indicated in brackets next to the (sub-) trial number above the main plot region.
Fig 4.
Probability that subtrial 1 closes earlier than subtrial 2 of the umbrella design with the random allocation scheme for varying prevalence of the two biomarkers.
πi (i = 1, 2) denotes the prevalence of a positive test result for biomarker Bi. The underlying formula is given in Eq (8).
Fig 5.
Difference in the estimated treatment effects between an independent trial and the corresponding umbrella subtrial with the random allocation scheme.
The impact of the biomarker status is larger in (sub-) trial 1. The difference in the treatment effect estimate (ΔXi, i = 1, 2) is given in Eq (14). The treatment effect estimate of subtrial i equals δ+ ΔXi. N denotes the (sub-) trial size and δ the true treatment effect in the corresponding independent trial. πi denotes the prevalence of a positive test result for biomarker Bi. For each subtrial, the associated prevalence is indicated in brackets next to the (sub-) trial number above the main plot region.
Table 2.
Characteristics of the patients in the reduced data set in the real data application—Overall as well as treatment group-specific.
Table 3.
Ratio of discarded to included patients in the independent trial design as well as in both umbrella designs.
Table 4.
Proportion of patients with a positive test result for both biomarkers in the independent trial design as well as in both umbrella designs.
Fig 6.
Difference in the estimated treatment effects between an independent trial and the corresponding umbrella subtrial in the application of different analysis methods in the simulation study.
The weights for the weighted linear regression are given in equation (S25) in S2 Note. The biomarker status impact on the treatment response is larger in (sub-) trial 1. ΔXi (i = 1, 2) is the difference between the mean treatment effect estimate across the simulation runs and δ. δ denotes the true treatment effect in the corresponding independent trial. πi denotes the prevalence of a positive test result for biomarker Bi. For each subtrial, the associated prevalence is indicated in brackets next to the (sub-) trial number below the main plot region. indep.: independent trial design, random: umbrella trial design with the random allocation scheme, pragm.: umbrella trial design with the pragmatic allocation scheme.
Fig 7.
Estimated treatment effects in the independent trial and in both umbrella trial designs in the application of different analysis methods in the real data application.
The estimated treatment effect corresponds to the regression coefficient. The calculation of the weights for the weighted linear regression are given in S2 Note. The trial size N, the trial design, the respective subtrial and the analysis method are indicated. The distribution across the 1, 000 bootstrap runs are provided.