I am running two Cox Proportional Hazards Models:
Model 1: survival (1 or 0, with 1 as death) ~ biomarker (high or low)
Model 2: survival (1 or 0, with 1 as death) ~ biomarker (high or low) + treatment + biomarker:treatment
My question is - will the effect and hazard ratio for the biomarker alone (in bold) be the same in both models? If not, is it better to chose one model over another?
... will the effect and hazard ratio for the biomarker alone ... be the same in both models?
Almost certainly not, particularly if the association of the biomarker with outcome differs depending on the type of treatment. Say that there is no association of the biomarker with outcome under treatment R but there is under treatment C, and you use treatment R as the reference category for treatment. Then in the second model you could find an "insignificant" coefficient for biomarker (which is calculated for the reference treatment R) and a "significant" coefficient for the biomarker:treatment interaction. In that situation under the first model, the value and apparent "significance" of the biomarker coefficient will depend (among other things) on the fraction of cases that received each treatment.
If not, is it better to chose one model over another?
There is a substantial risk of omitted-variable bias with survival analysis. If you omit a predictor associated with outcome, even in standard linear regression you will bias the coefficients for included predictors that are associated with the omitted predictor. For example, you don't want to "discover a new biomarker" only to have a reviewer point out that it's just serving as a proxy for some correlated clinical variable having a well-established association with outcome.
In event-based analyses like logistic and survival regressions, you can get a downward bias in absolute magnitudes of coefficients for included predictors even if they aren't correlated with the omitted outcome-associated predictor. As a result, omitting outcome-associated covariates from your survival model could make it harder to establish a real association of your biomarker with outcome.
So you should go as far as possible, without overfitting, to include all clinical variables that might reasonably be associated with outcome in this type of analysis. Don't stop with just the biomarker and treatment.
