When doing cox proportional hazards regression one often has competing risks. The typical approach for this is to fit separate cox proportional hazards models for each risk, censoring the competing risk.
How do you do model selection in this scenario? Should the two models be the same? Do I build a model based on one of the risks and use the same model for the other risk? Do I build a model based off a completely different response (e.g. overall survival)?
As you say, one approach is to use cause-specific hazard functions, treating the observations experiencing a competing events as censored. The problem with such approach is that it doesn't have a direct interpretation in terms of survival probabilities, that are obviously reduced by the the other risk: you are basically diseregarding the information about the events due to it. However, such approach replies to an important etiological question, i.e. how each risk contributes to non-survival: it is epidemiologically interesting because it's based on real probabilities, i.e., in each moment, the one of dying among people that may actually die.
However, you wondered whether you should consider overall survival in modeling competing events. This is another approach, based on Fine and Gray (1999) cause-specific ${subdistribution}$ (i.e. the cumulative incidence function). Here, the drawback is that they consider in the risk set for a given event also individuals that may not have the event itself (because the competing one has already occurred previously), thus, their estimates are not based on probability in a proper sense. However, their approach has a direct interpretation in terms of survival probabilities: it allows to calculate the probability of each event. For example, a Fine and Gray approach may lead us to conclude that, in a given sample, with a given covariate pattern, $25\%$ of patients will die for cardiological reasons, $20\%$ of patients for other reasons, and $55\%$ will survive until the end of the follow-up.
Fine, J. P., & Gray, R.J. 1999. A proportional hazards model for the subdistribution of a competing risk. Journal of the American Statistical Association,94(446), 496–509.
