I have run two models with the same covariates, one for cox proportional hazards model and another for Fine-Gray competing risk model. The C-index for the cox model is about 0.81 and that of the Fine-Gray model is around 0.79. Based on this alone, can I interpret that the cox model is better able to differentiate between high-risk and low-risk patients?
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First, the C-index isn't a very sensitive way to compare survival models. Frank Harrell, who introduced the C-index into survival analysis, agrees with that assessment.
Second, you can't compare point estimates (e.g., of C-indices) without also knowing the error in the estimates. In my experience, error estimates in C-indices are likely to be greater than the 0.02 difference between the values of 0.79 and 0.81. So it's not clear that there is a reliable difference here at all.
Third, and probably most important, the Cox model and the Fine-Gray model are modeling different things. Presumably you have a competing-risks scenario where you evaluated a single type of event with both a Cox model (right censoring at times of other event types) and a Fine-Gray model. I find the R competing risks vignette to be helpful in comparing the two approaches.
The Cox model evaluates the hazard of that event type if an individual is still at risk for that event type. In a combined Cox competing-risks model you can get proper estimates for the probability of experiencing each risk over time. But the losses of individuals to other types of events make it difficult to interpret the model coefficients in terms of the overall risk of a particular event type.
The Fine-Gray model evaluates the "subdistribution hazard" for that event type in a way that includes individuals who have already had a different type of event. That tries to estimate the overall risk of having that particular event type, with regression coefficients directly related to that estimate. But if you model all of the competing risks that way, it's possible to have the estimated sum of event probabilities exceed 1.
As they model different things, the two types of models really shouldn't be compared that way at all.
