Table 3 Results of the simulation study for a protective (\(\text{ HR}_M=0.6\)), a null (\(\text{ HR}_M=1\)) and a harmful (\(\text{ HR}_M=3\)) marker effect among patients treated with the standard treatment. The treatment HRs were \(\text{ HR}_{TM_{low}}=1\) and \(\text{ HR}_{TM_{high}}=0.25\), the interaction HR was \(\text{ HR}_{I}=0.25\), the OR between marker and treatment was \(\text{ OR}_{MT}=1\), the proportion of patients with high marker level was \(p_{M}=0.25\), and the proportion of censored patients with low marker level receiving standard treatment was \(p_c=0.2\).
From: Cox proportional hazards regression in small studies of predictive biomarkers
Bias\(^\text {a}\) | Coverage (%) | Power (%) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
n | \(\widehat{\beta }_{TM_{low}}\) | 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) | N\(_{c}\) | |
HR\(_M=0.6\) | ||||||||||
Cox | 200 | 0 | \(-\) 1.2 | 0.6 | 60.2 | 46.5 | 45.7 | 96.6 (96.6) | 2.5 (5.0) | 4912 |
400 | 0 | \(-\) 1.2 | 0.2 | 35.3 | 16.4 | 31.0 | 96.8 (96.9) | 12.6 (23.1) | 7671 | |
600 | 0 | \(-\) 0.7 | 0 | 22.8 | 1.6 | 21.2 | 97.2 (97.3) | 33.0 (46.0) | 8895 | |
Firth | 200 | 0 | 3.6 | 0.3 | 42.8 | 19.0 | 37.3 | 97.6 (96.7) | 2.4 (11.3) | 9703 |
400 | 0 | 0.8 | 0.1 | 22.8 | 5.2 | 21.1 | 96.9 (96.8) | 12.2 (29.3) | 9995 | |
600 | 0 | 1.2 | 0 | 12.7 | 0.5 | 12.3 | 97.0 (96.6) | 31.6 (46.6) | 10,000 | |
HR\(_M=1\) | ||||||||||
Cox | 200 | 0 | \(-\) 1.1 | 0.3 | 39.3 | 23.9 | 30.0 | 96.6 (96.7) | 8.5 (15.2) | 6950 |
400 | 0 | \(-\) 1.4 | 0 | 19.4 | 0 | 16.8 | 97.2 (97.1) | 35.6 (46.3) | 9129 | |
600 | 0 | \(-\) 0.6 | \(-\) 0.1 | 6.9 | \(-\) 5.7 | 6.5 | 96.8 (96.0) | 61.6 (68.1) | 9750 | |
Firth | 200 | 0 | 3.2 | 0.1 | 28.3 | 8.6 | 24.8 | 97.2 (96.6) | 7.5 (21.5) | 9969 |
400 | 0 | 0.6 | 0 | 11.0 | 0.8 | 9.9 | 97.0 (96.1) | 33.7 (46.5) | 9999 | |
600 | 0 | 1.1 | 0 | 4.3 | \(-\) 0.5 | 4.2 | 96.5 (95.3) | 59.3 (65.8) | 10,000 | |
HR\(_M=3\) | ||||||||||
Cox | 200 | 0 | \(-\) 1.9 | \(-\) 0.1 | 7.8 | \(-\) 4.2 | 4.8 | 96.4 (95.6) | 44.7 (50.2) | 9724 |
400 | 0 | \(-\) 1.7 | 0 | \(-\) 2.4 | \(-\) 3.7 | \(-\) 2.1 | 95.5 (94.7) | 79.4 (81.0) | 9991 | |
600 | 0 | \(-\) 0.5 | 0 | \(-\) 1.2 | \(-\) 3.2 | \(-\) 0.4 | 95.7 (95.1) | 93.8 (94.3) | 10,000 | |
Firth | 200 | 0 | 2.5 | 0 | 6.5 | 0.7 | 5.2 | 96.6 (95.1) | 41.5 (48.8) | 10,000 |
400 | 0 | 0.2 | 0 | 1.0 | 0.4 | 0.9 | 95.7 (94.9) | 77.5 (80.0) | 10,000 | |
600 | 0 | 0.8 | 0 | 1.8 | \(-\) 0.5 | 2.2 | 96.1 (95.5) | 93.3 (94.0) | 10,000 | |
