Background
Methods
Assessing the biomarker–treatment interaction
Strategies for covariate inclusion
-
Main: A model including only the main effects of treatment T and the biomarker B and their interaction T×B, ignoring all other possible prognostic covariates.
-
True: A model including the main effects of treatment T, the biomarker of interest B and their interaction T×B, as well as all covariates that are truly associated with the outcome, indicating perfect prior knowledge of relevant covariates.
-
AIC A : A model that includes the main effects of treatment T and the biomarker B and their interaction T×B and additionally all covariates that were selected in a forward variable selection procedure based on Akaike’s information criterion (AIC) [14] given T, B and T×B are included (a model including T, B and T×B was used as a starting and minimal model). Additional covariates were selected as long as the AIC criterion$$ \operatorname{AIC} = 2\operatorname{ll}(\hat{\boldsymbol{\beta}}) - 2p $$(2)was increased, where \(\operatorname {ll}(\hat {\boldsymbol {\beta }})\) is the partial log-likelihood evaluated at the maximum likelihood estimator \(\hat {\boldsymbol {\beta }}\) and p is the number of estimated regression coefficients.
-
AIC B : A modelling strategy similar to AIC A described above, but prognostic factors were selected based on the AIC criterion considering just the main effect of treatment T as a starting model and not including B or T×B in the variable selection process. After prognostic factors were chosen according to the AIC criterion, B and T×B were added to the model to estimate the biomarker–treatment interaction.
-
Significance: A model that includes the main effects of treatment T, the covariate of interest and their interaction, as well as all covariates that were significantly associated with the outcome in a Cox regression model including only one covariate (often referred to as univariate Cox models in the medical literature). While this strategy is generally not recommended from a statistical point of view [18], it appears to be a quite popular approach in practice.
-
Full: A model that includes the treatment T, the biomarker B and their interaction T×B as covariates as well as the main effects of all K potential predictors X1,…,X K .
Data generation and simulation settings
Analysis and presentation of results
Results
K
| Interaction | Censoring | Main | True | AIC
A
| AIC
B
| Significance | Full |
---|---|---|---|---|---|---|---|---|
12 | No | Low | 0.058 | 0.060 | 0.062 | 0.056 | 0.059 | 0.060 |
12 | No | High | 0.054 | 0.051 | 0.056 | 0.048 | 0.053 | 0.054 |
12 | Quantitative | Low |
0.118
|
0.133
|
0.139
|
0.131
|
0.130
|
0.134
|
12 | Quantitative | High |
0.095
|
0.100
|
0.108
|
0.099
|
0.096
|
0.099
|
12 | Qualitative | Low |
0.579
|
0.663
|
0.663
|
0.654
|
0.653
|
0.661
|
12 | Qualitative | High |
0.393
|
0.437
|
0.441
|
0.424
|
0.432
|
0.436
|
24 | No | Low | 0.065 | 0.058 | 0.067 | 0.056 | 0.057 | 0.058 |
24 | No | High | 0.057 | 0.062 |
0.073
| 0.054 | 0.059 | 0.063 |
24 | Quantitative | Low |
0.115
|
0.135
|
0.150
|
0.131
|
0.131
|
0.136
|
24 | Quantitative | High |
0.094
|
0.100
| 0.114 |
0.092
|
0.100
|
0.103
|
24 | Qualitative | Low |
0.465
|
0.634
|
0.645
|
0.616
|
0.610
|
0.633
|
24 | Qualitative | High |
0.349
|
0.411
| 0.426 |
0.383
|
0.399
|
0.415
|
36 | No | Low | 0.065 | 0.059 |
0.078
| 0.056 | 0.061 | 0.063 |
36 | No | High | 0.067 | 0.068 |
0.085
| 0.057 | 0.066 |
0.071
|
36 | Quantitative | Low |
0.114
|
0.132
| 0.153 |
0.118
|
0.130
|
0.134
|
36 | Quantitative | High |
0.093
|
0.102
| 0.127 |
0.093
|
0.101
| 0.108 |
36 | Qualitative | Low |
0.412
|
0.618
| 0.629 |
0.578
|
0.576
|
0.610
|
36 | Qualitative | High |
0.302
|
0.406
| 0.431 |
0.367
|
0.382
| 0.402 |
K
|
Σ
|
β
k
| Censoring | Main | True | AIC
A
| AIC
B
| Significance | Full |
---|---|---|---|---|---|---|---|---|---|
No interaction | |||||||||
12 |
Σ
1
|
β
eq
| Low | 0.066 | 0.069 |
0.072
| 0.066 | 0.067 | 0.069 |
12 |
Σ
1
|
β
v
| Low | 0.051 | 0.059 | 0.059 | 0.055 | 0.052 | 0.058 |
12 |
Σ
2
|
β
eq
| Low | 0.050 | 0.051 | 0.054 | 0.048 | 0.051 | 0.051 |
12 |
Σ
2
|
β
v
| Low | 0.060 | 0.047 | 0.052 | 0.047 | 0.052 | 0.052 |
12 |
Σ
3
|
β
eq
| Low | 0.060 | 0.065 | 0.066 | 0.061 | 0.063 | 0.065 |
12 |
Σ
3
|
β
v
| Low | 0.063 | 0.066 | 0.067 | 0.060 | 0.067 | 0.065 |
12 |
Σ
1
|
β
eq
| High | 0.057 | 0.056 | 0.060 | 0.054 | 0.055 | 0.056 |
12 |
Σ
1
|
β
v
| High | 0.038 | 0.040 | 0.037 | 0.034 | 0.040 | 0.044 |
12 |
Σ
2
|
β
eq
| High | 0.057 | 0.046 | 0.054 | 0.044 | 0.046 | 0.046 |
12 |
Σ
2
|
β
v
| High | 0.060 | 0.047 | 0.058 | 0.049 | 0.055 | 0.055 |
12 |
Σ
3
|
β
eq
| High | 0.053 | 0.059 | 0.060 | 0.050 | 0.062 | 0.059 |
12 |
Σ
3
|
β
v
| High | 0.057 | 0.056 | 0.064 | 0.055 | 0.061 | 0.065 |
Quantitative interaction | |||||||||
12 |
Σ
1
|
β
eq
| Low |
0.122
|
0.137
| 0.143 |
0.121
|
0.123
|
0.137
|
12 |
Σ
1
|
β
v
| Low |
0.119
|
0.134
|
0.139
|
0.132
|
0.134
|
0.136
|
12 |
Σ
2
|
β
eq
| Low |
0.105
|
0.131
|
0.136
|
0.129
|
0.131
|
0.131
|
12 |
Σ
2
|
β
v
| Low |
0.113
|
0.131
|
0.141
|
0.135
|
0.132
|
0.132
|
12 |
Σ
3
|
β
eq
| Low |
0.129
|
0.146
|
0.148
|
0.146
|
0.141
|
0.146
|
12 |
Σ
3
|
β
v
| Low |
0.121
|
0.121
|
0.127
|
0.120
|
0.116
|
0.120
|
12 |
Σ
1
|
β
eq
| High |
0.098
|
0.109
|
0.120
|
0.109
|
0.100
|
0.109
|
12 |
Σ
1
|
β
v
| High |
0.108
|
0.112
|
0.118
|
0.111
|
0.108
|
0.111
|
12 |
Σ
2
|
β
eq
| High |
0.077
|
0.088
|
0.099
|
0.086
|
0.088
|
0.088
|
12 |
Σ
2
|
β
v
| High |
0.104
|
0.095
|
0.106
|
0.096
|
0.093
|
0.093
|
12 |
Σ
3
|
β
eq
| High |
0.086
|
0.093
|
0.095
|
0.091
|
0.088
|
0.093
|
12 |
Σ
3
|
β
v
| High |
0.095
|
0.101
|
0.107
|
0.098
|
0.100
|
0.101
|
Qualitative interaction | |||||||||
12 |
Σ
1
|
β
eq
| Low |
0.625
|
0.685
| 0.673 |
0.662
|
0.644
|
0.685
|
12 |
Σ
1
|
β
v
| Low |
0.605
|
0.664
|
0.664
|
0.661
|
0.649
|
0.661
|
12 |
Σ
2
|
β
eq
| Low |
0.517
|
0.641
|
0.641
|
0.634
|
0.641
|
0.641
|
12 |
Σ
2
|
β
v
| Low |
0.521
|
0.646
|
0.648
|
0.643
|
0.644
|
0.644
|
12 |
Σ
3
|
β
eq
| Low |
0.621
|
0.678
|
0.686
|
0.673
|
0.680
|
0.678
|
12 |
Σ
3
|
β
v
| Low |
0.583
|
0.661
|
0.664
|
0.652
|
0.661
|
0.658
|
12 |
Σ
1
|
β
eq
| High |
0.427
|
0.462
|
0.464
|
0.446
|
0.433
|
0.462
|
12 |
Σ
1
|
β
v
| High |
0.424
|
0.438
|
0.447
|
0.432
|
0.440
|
0.443
|
12 |
Σ
2
|
β
eq
| High |
0.338
|
0.403
|
0.410
|
0.389
|
0.403
|
0.403
|
12 |
Σ
2
|
β
v
| High |
0.359
|
0.431
|
0.429
|
0.413
|
0.433
|
0.433
|
12 |
Σ
3
|
β
eq
| High |
0.394
|
0.424
|
0.425
|
0.407
|
0.420
|
0.424
|
12 |
Σ
3
|
β
v
| High |
0.418
|
0.466
|
0.471
|
0.456
|
0.465
|
0.449
|
Mean number of prognostic covariates included | |||||||||
β
eq
| Low | 0 | 12 | 6.8 | 7.3 | 8.7 | 12 | ||
β
v
| Low | 0 | 8 | 6.4 | 6.9 | 8.7 | 12 | ||
β
eq
| High | 0 | 12 | 5.1 | 5.7 | 8.0 | 12 | ||
β
v
| High | 0 | 8 | 5.3 | 5.8 | 8.2 | 12 |
K
|
Σ
|
β
k
| Censoring | Main | True | AIC
A
| AIC
B
| Significance | Full |
---|---|---|---|---|---|---|---|---|---|
No interaction | |||||||||
24 |
Σ
1
|
β
eq
| Low | 0.044 | 0.049 | 0.065 | 0.053 | 0.048 | 0.049 |
24 |
Σ
1
|
β
v
| Low | 0.055 | 0.069 |
0.074
| 0.065 | 0.060 | 0.069 |
24 |
Σ
2
|
β
eq
| Low |
0.087
| 0.052 | 0.063 | 0.046 | 0.052 | 0.052 |
24 |
Σ
2
|
β
v
| Low | 0.068 |
0.071
|
0.081
| 0.066 |
0.071
|
0.071
|
24 |
Σ
3
|
β
eq
| Low | 0.068 | 0.049 | 0.061 | 0.052 | 0.055 | 0.049 |
24 |
Σ
3
|
β
v
| Low | 0.066 | 0.056 | 0.061 | 0.053 | 0.054 | 0.056 |
24 |
Σ
1
|
β
eq
| High | 0.035 | 0.056 | 0.069 | 0.054 | 0.046 | 0.056 |
24 |
Σ
1
|
β
v
| High | 0.051 |
0.071
|
0.076
| 0.059 | 0.068 |
0.076
|
24 |
Σ
2
|
β
eq
| High |
0.073
| 0.066 |
0.074
| 0.056 | 0.066 | 0.066 |
24 |
Σ
2
|
β
v
| High | 0.060 | 0.054 | 0.066 | 0.048 | 0.056 | 0.056 |
24 |
Σ
3
|
β
eq
| High | 0.062 | 0.058 |
0.073
| 0.047 | 0.059 | 0.058 |
24 |
Σ
3
|
β
v
| High | 0.059 | 0.068 |
0.079
| 0.060 | 0.057 | 0.066 |
Quantitative interaction | |||||||||
24 |
Σ
1
|
β
eq
| Low |
0.114
|
0.142
|
0.158
|
0.137
|
0.122
|
0.142
|
24 |
Σ
1
|
β
v
| Low |
0.106
|
0.138
| 0.150 |
0.132
|
0.125
|
0.143
|
24 |
Σ
2
|
β
eq
| Low | 0.119 |
0.135
|
0.148
|
0.130
|
0.135
|
0.135
|
24 |
Σ
2
|
β
v
| Low |
0.111
| 0.124 | 0.145 |
0.117
| 0.126 | 0.126 |
24 |
Σ
3
|
β
eq
| Low |
0.121
|
0.136
|
0.145
|
0.128
|
0.141
|
0.136
|
24 |
Σ
3
|
β
v
| Low |
0.121
|
0.136
|
0.157
|
0.141
|
0.134
|
0.136
|
24 |
Σ
1
|
β
eq
| High |
0.088
|
0.100
|
0.117
|
0.091
|
0.093
|
0.100
|
24 |
Σ
1
|
β
v
| High |
0.085
| 0.104 | 0.110 |
0.096
|
0.094
| 0.111 |
24 |
Σ
2
|
β
eq
| High | 0.094 |
0.109
| 0.122 |
0.094
|
0.109
|
0.109
|
24 |
Σ
2
|
β
v
| High |
0.113
|
0.096
|
0.116
|
0.088
|
0.100
|
0.100
|
24 |
Σ
3
|
β
eq
| High |
0.082
|
0.098
| 0.109 |
0.097
|
0.103
|
0.098
|
24 |
Σ
3
|
β
v
| High |
0.100
|
0.091
| 0.109 |
0.086
|
0.098
|
0.098
|
Qualitative interaction | |||||||||
24 |
Σ
1
|
β
eq
| Low |
0.630
|
0.697
|
0.686
|
0.658
|
0.632
|
0.697
|
24 |
Σ
1
|
β
v
| Low |
0.547
|
0.678
| 0.688 |
0.656
|
0.615
|
0.685
|
24 |
Σ
2
|
β
eq
| Low | 0.349 |
0.610
|
0.619
|
0.595
|
0.610
|
0.610
|
24 |
Σ
2
|
β
v
| Low |
0.358
| 0.590 | 0.608 |
0.584
| 0.578 | 0.578 |
24 |
Σ
3
|
β
eq
| Low |
0.443
|
0.596
|
0.620
|
0.582
|
0.587
|
0.596
|
24 |
Σ
3
|
β
v
| Low |
0.465
|
0.632
|
0.651
|
0.621
|
0.636
|
0.631
|
24 |
Σ
1
|
β
eq
| High |
0.448
|
0.457
|
0.463
|
0.424
|
0.412
|
0.457
|
24 |
Σ
1
|
β
v
| High |
0.384
| 0.453 | 0.466 |
0.425
|
0.420
| 0.464 |
24 |
Σ
2
|
β
eq
| High | 0.276 |
0.364
| 0.387 |
0.340
|
0.364
|
0.364
|
24 |
Σ
2
|
β
v
| High |
0.292
|
0.397
|
0.423
|
0.378
|
0.411
|
0.411
|
24 |
Σ
3
|
β
eq
| High |
0.355
|
0.387
| 0.405 |
0.356
|
0.382
|
0.387
|
24 |
Σ
3
|
β
v
| High |
0.338
|
0.408
| 0.409 |
0.377
|
0.404
|
0.408
|
Mean number of prognostic covariates included | |||||||||
β
eq
| Low | 0 | 24 | 13.2 | 13.7 | 17.5 | 24 | ||
β
v
| Low | 0 | 16 | 12.7 | 13.1 | 17.8 | 24 | ||
β
eq
| High | 0 | 24 | 10.2 | 10.7 | 16.5 | 24 | ||
β
v
| High | 0 | 16 | 10.6 | 11.0 | 16.8 | 24 |
K
|
Σ
|
β
k
| Censoring | Main | True | AIC
A
| AIC
B
| Significance | Full |
---|---|---|---|---|---|---|---|---|---|
No interaction | |||||||||
36 |
Σ
1
|
β
eq
| Low | 0.047 | 0.065 |
0.080
| 0.054 | 0.056 | 0.065 |
36 |
Σ
1
|
β
v
| Low | 0.059 | 0.067 |
0.084
| 0.066 | 0.069 |
0.073
|
36 |
Σ
2
|
β
eq
| Low |
0.077
| 0.053 |
0.073
| 0.051 | 0.053 | 0.053 |
36 |
Σ
2
|
β
v
| Low |
0.075
| 0.054 |
0.074
| 0.052 | 0.056 | 0.056 |
36 |
Σ
3
|
β
eq
| Low | 0.067 | 0.060 |
0.083
| 0.057 | 0.064 | 0.060 |
36 |
Σ
3
|
β
v
| Low | 0.063 | 0.055 |
0.074
| 0.053 | 0.069 | 0.069 |
36 |
Σ
1
|
β
eq
| High | 0.052 |
0.071
|
0.086
| 0.059 | 0.055 |
0.071
|
36 |
Σ
1
|
β
v
| High | 0.047 | 0.063 |
0.080
| 0.058 | 0.050 | 0.066 |
36 |
Σ
2
|
β
eq
| High |
0.085
|
0.080
|
0.082
| 0.057 |
0.080
|
0.080
|
36 |
Σ
2
|
β
v
| High |
0.085
| 0.064 |
0.086
| 0.054 | 0.070 | 0.070 |
36 |
Σ
3
|
β
eq
| High | 0.057 | 0.069 |
0.094
| 0.056 | 0.063 | 0.069 |
36 |
Σ
3
|
β
v
| High |
0.075
| 0.063 |
0.081
| 0.057 |
0.076
|
0.071
|
Quantitative interaction | |||||||||
36 |
Σ
1
|
β
eq
| Low |
0.103
|
0.150
| 0.165 |
0.130
|
0.138
|
0.150
|
36 |
Σ
1
|
β
v
| Low |
0.095
|
0.131
| 0.150 |
0.120
|
0.125
| 0.136 |
36 |
Σ
2
|
β
eq
| Low | 0.121 |
0.120
| 0.141 |
0.109
|
0.120
|
0.120
|
36 |
Σ
2
|
β
v
| Low | 0.128 |
0.128
| 0.147 |
0.121
|
0.134
|
0.134
|
36 |
Σ
3
|
β
eq
| Low |
0.115
|
0.121
| 0.142 |
0.103
|
0.122
|
0.121
|
36 |
Σ
3
|
β
v
| Low |
0.119
|
0.143
| 0.172 |
0.125
|
0.140
|
0.143
|
36 |
Σ
1
|
β
eq
| High |
0.081
| 0.108 | 0.133 |
0.098
|
0.094
| 0.108 |
36 |
Σ
1
|
β
v
| High |
0.095
|
0.112
| 0.132 |
0.101
|
0.102
|
0.118
|
36 |
Σ
2
|
β
eq
| High | 0.100 | 0.085 | 0.103 |
0.072
| 0.085 | 0.085 |
36 |
Σ
2
|
β
v
| High | 0.092 |
0.109
| 0.127 |
0.093
|
0.118
|
0.118
|
36 |
Σ
3
|
β
eq
| High |
0.091
|
0.104
| 0.134 |
0.099
|
0.105
|
0.104
|
36 |
Σ
3
|
β
v
| High | 0.097 |
0.093
| 0.132 |
0.093
| 0.102 | 0.115 |
Qualitative interaction | |||||||||
36 |
Σ
1
|
β
eq
| Low |
0.551
|
0.652
| 0.657 |
0.603
|
0.558
|
0.652
|
36 |
Σ
1
|
β
v
| Low |
0.517
|
0.700
| 0.688 |
0.658
|
0.599
| 0.669 |
36 |
Σ
2
|
β
eq
| Low | 0.280 |
0.570
| 0.582 |
0.518
|
0.570
|
0.570
|
36 |
Σ
2
|
β
v
| Low | 0.266 |
0.555
| 0.575 |
0.517
|
0.542
|
0.542
|
36 |
Σ
3
|
β
eq
| Low |
0.408
|
0.609
| 0.637 |
0.582
|
0.581
|
0.609
|
36 |
Σ
3
|
β
v
| Low |
0.451
|
0.623
| 0.637 |
0.592
|
0.605
|
0.620
|
36 |
Σ
1
|
β
eq
| High |
0.389
| 0.447 | 0.456 |
0.403
|
0.385
| 0.447 |
36 |
Σ
1
|
β
v
| High |
0.390
|
0.472
| 0.486 |
0.437
|
0.418
|
0.453
|
36 |
Σ
2
|
β
eq
| High | 0.219 | 0.368 | 0.411 |
0.334
| 0.368 | 0.368 |
36 |
Σ
2
|
β
v
| High | 0.228 |
0.385
| 0.419 |
0.353
|
0.389
|
0.389
|
36 |
Σ
3
|
β
eq
| High |
0.282
|
0.364
| 0.396 |
0.328
|
0.352
|
0.364
|
36 |
Σ
3
|
β
v
| High | 0.303 |
0.402
| 0.416 |
0.350
| 0.382 | 0.391 |
Mean number of prognostic covariates included | |||||||||
β
eq
| Low | 0 | 36 | 19.6 | 19.9 | 24.5 | 36 | ||
β
v
| Low | 0 | 24 | 19.0 | 19.4 | 25.2 | 36 | ||
β
eq
| High | 0 | 36 | 15.2 | 15.7 | 23.1 | 36 | ||
β
v
| High | 0 | 24 | 16.0 | 16.4 | 23.9 | 36 |