Table 4 Results of the simulation study for 25% (\(p_M=0.25\)), 50% (\(p_M=0.5\)) and 75% (\(p_M=0.75\)) of patients with high marker level. The treatment HRs were \(\text{ HR}_{TM_{low}}=1\) and \(\text{ HR}_{TM_{high}}=0.75\), the interaction HR was \(\text{ HR}_{I}=0.75\), the marker effect among patients treated with the standard treatment was \(\text{ HR}_M=0.8\), the OR between marker and treatment was \(\text{ OR}_{MT}=0.5\), 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\(_{c}\) | |
p\(_{M}=0.25\) | ||||||||||
Cox | 200 | 0 | \(-\) 2.7 | 0.1 | 20.4 | 33.8 | 15.0 | 97.2 (96.6) | 1.8 (3.2) | 8692 |
400 | 0 | \(-\) 1.5 | 0 | 2.3 | \(-\) 15.7 | 2.3 | 96.8 (95.3) | 3.8 (6.6) | 9855 | |
600 | 0 | \(-\) 0.2 | 0 | \(-\) 2.8 | \(-\) 18.4 | \(-\) 2.0 | 95.9 (94.6) | 6.2 (8.8) | 9978 | |
Firth | 200 | 0 | 3.0 | 0 | 14.0 | 11.2 | 12.2 | 97.6 (96.0) | 1.4 (4.2) | 9946 |
400 | 0 | 0.9 | 0 | 3.1 | 3.4 | 3.3 | 97.0 (95.5) | 3.1 (6.0) | 9999 | |
600 | 0 | 1.3 | 0 | \(-\) 0.1 | \(-\) 0.8 | 0.7 | 96.3 (95.1) | 5.0 (7.4) | 10000 | |
p\(_{M}=0.5\) | ||||||||||
Cox | 200 | 0 | \(-\) 1.4 | 0 | \(-\) 1.4 | \(-\) 20.3 | \(-\) 1.7 | 95.9 (94.7) | 4.9 (7.1) | 9900 |
400 | 0 | \(-\) 3.0 | 0 | \(-\) 2.6 | \(-\) 11.9 | \(-\) 3.1 | 95.0 (94.5) | 7.9 (9.2) | 9999 | |
600 | 0 | \(-\) 1.3 | 0 | \(-\) 2.4 | \(-\) 8.2 | \(-\) 2.0 | 94.9 (94.5) | 10.2 (11.1) | 10,000 | |
Firth | 200 | 0 | 4.9 | 0 | 2.7 | 2.9 | 3.6 | 96.8 (95.1) | 3.5 (6.0) | 9998 |
400 | 0 | \(-\) 0.1 | 0 | 1.4 | 1.7 | 0.5 | 95.7 (94.7) | 6.3 (7.9) | 10,000 | |
600 | 0 | 0.7 | 0 | 0.2 | 0.5 | 0.3 | 95.3 (94.9) | 8.7 (10.1) | 10,000 | |
p\(_{M}=0.75\) | ||||||||||
Cox | 200 | 0 | 5.2 | 0 | \(-\) 1.5 | 0.6 | 6.6 | 97.2 (96.2) | 2.7 (4.6) | 9477 |
400 | 0 | \(-\) 2.2 | 0 | \(-\) 0.8 | \(-\) 8.2 | \(-\) 0.5 | 95.7 (95.0) | 5.9 (7.6) | 9911 | |
600 | 0 | \(-\) 2.3 | 0 | \(-\) 0.8 | \(-\) 9.6 | \(-\) 1.8 | 95.4 (94.9) | 8.8 (10.1) | 9984 | |
Firth | 200 | 0 | 6.3 | 0 | 5.5 | 20.7 | 10.9 | 97.6 (96.3) | 2.1 (4.0) | 9792 |
400 | 0 | 1.7 | 0 | 2.3 | 6.7 | 3.0 | 96.3 (95.3) | 4.9 (6.6) | 9947 | |
600 | 0 | 0.6 | 0 | 1.2 | 1.1 | 0.9 | 95.9 (95.2) | 7.5 (9.1) | 9988 | |
