Fig. 2

Results of the simulation study for treatment assignment independent of the marker level, i.e., \(\text{ OR}_{MT}=1\), and a harmful (\(\text{ HR}_M=3\), left panel; \(\text{ HR}_M=6\), right panel) marker effect among patients treated with the standard treatment. The treatment HRs were \(\text{ HR}_{TM_{low}}=1\) and \(\text{ HR}_{TM_{high}}=0.5,\) i.e., \(\beta _{TM_{low}}=0\) and \(\beta _{TM_{high}}=-0.69\), the interaction HR was \(\text{ HR}_{I}=0.5\), i.e., \(\beta _{I}=-0.69\), and the proportion of patients with high marker level was \(p_{M}=0.25\). Case-only results were obtained with a Firth-corrected logistic regression, while full cohort results were obtained with a Firth-corrected Cox proportional hazards model. Number of patients was the number of patients per dataset in full cohort HR, hazard ratio; M, marker; OR, odds ratio; PL, profile likelihood; Rel., relative; SE, standard error.