Table 2 Results of the simulation study for treatment assignment independent of the marker level, i.e., \(\text{ OR}_{MT}=1\), and a harmful (\(\text{ HR}_M=3\) and \(\text{ HR}_M=6\)) 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.
From: Case-only analysis in small studies of predictive biomarkers
Bias | Coverage (%) | Power (%) | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
n | e | \(\widehat{\beta }_{TM_{low}}\) | SE(\(\widehat{\beta }_{TM_{low}}\)) | \(\widehat{\beta }_{TM_{high}}\) | SE(\(\widehat{\beta }_{TM_{high}}\)) | \(\widehat{\beta }_{I}\) | SE(\(\widehat{\beta }_{I}\)) | Wald (PL) | Wald (PL) | Nc | |
HR\(_M=3\) | |||||||||||
Case-only | 200 | 45 | 0 | 0.1 | 0.1 | 0.8 | 22.5 | 0.8 | 95.1 ( 94.6 ) | 11.0 ( 12.8 ) | 10000 |
400 | 89 | 0 | \(-\)0.8 | 0.1 | \(-\)0.4 | 21.6 | \(-\)0.2 | 93.7 ( 93.6 ) | 21.4 ( 22.6 ) | 10000 | |
600 | 133 | 0 | \(-\)0.3 | 0.1 | 0.3 | 20.8 | \(-\)0.3 | 93.2 ( 93.2 ) | 31.8 ( 32.7 ) | 10000 | |
Full cohort | 200 | 45 | 0 | 2.3 | 0 | 2.6 | 1.1 | 2.8 | 96.2 ( 95.1 ) | 15.6 ( 18.6 ) | 10000 |
400 | 89 | 0 | 0.2 | 0 | 0.2 | 0 | 0.4 | 95.3 ( 94.7 ) | 32.4 ( 34.2 ) | 10000 | |
600 | 133 | 0 | 0.8 | 0 | 1.8 | \(-\)1.2 | 1.6 | 95.8 ( 95.4 ) | 47.3 ( 48.8 ) | 10000 | |
HR\(_M=6\) | |||||||||||
Case-only | 200 | 56 | \(-\)0.1 | 0 | 0.2 | 0.1 | 39.5 | 0.2 | 92.4 ( 92.1 ) | 10.8 ( 11.6 ) | 10000 |
400 | 110 | \(-\)0.1 | \(-\)0.8 | 0.2 | 0.7 | 39.6 | 0.6 | 88.9 ( 89.0 ) | 18.2 ( 18.7 ) | 10000 | |
600 | 165 | \(-\)0.1 | \(-\)0.2 | 0.2 | 0.9 | 39.2 | \(-\)0.2 | 86.0 ( 86.2 ) | 26.1 ( 26.5 ) | 10000 | |
Full cohort | 200 | 56 | 0 | 1.8 | 0 | \(-\)0.1 | \(-\)0.1 | 1.0 | 95.5 ( 94.8 ) | 22.2 ( 24.0 ) | 10000 |
400 | 110 | 0 | \(-\)0.1 | 0 | 0 | \(-\)0.2 | 0.4 | 95.4 ( 95.1 ) | 42.1 ( 43.2 ) | 10000 | |
600 | 165 | 0 | 0.6 | 0 | 0.9 | \(-\)1.1 | 0.7 | 95.5 ( 95.2 ) | 59.6 ( 60.2 ) | 10000 | |
