# Cox model vs. Fine-Gray: hazard ratios & predicted cumulative incidence under competing risks

I am unclear about deriving both hazard ratios and a predicted cumulative incidence curves under competing risks. After reading this interesting article:

https://statisticalhorizons.com/for-causal-analysis-of-competing-risks/

I was under the impression that the most approriate way to estimate hazard ratios (for effect estimation / causal inference) under competing risks, is to perform an oridnary Cox model, but censor at the time of the competing event. So if the focal event is diagnosis of x, but competing risk is death, then we truncate the time to x variable at time of death i.e. censor on death. I believe this is then simply called the cause-specific hazard model.

Often we also want to estimate adjusted (predicted) cumulative incidence, which can be done after fitting the Cox model, see here:

Can we compute predicted cumulative incidence after fitting a Cox model?

However, since the Fine Gray method (as I understand it) models the cumulative incidence (i.e. as the response variable), isn't that the best way to get a predicted cumulative incidence curve under competing risks? Indeed, in this linked article, it says in the third from last paragraph that "Cumulative incidence functions are particularly useful for prediction", also see post from @FrankHarrell:

... since the Fine Gray method (as I understand it) models the cumulative incidence (i.e. as the response variable), isn't that the best way to get a predicted cumulative incidence curve under competing risks?

Whether that's "the best way" depends on the specific question you're trying to answer and the assumptions that you need to make and verify. The R vignette on "Multi-state models and competing risks" is a superb overview of the issues. Working through its examples will illustrate much more than can be shown here. A few points to emphasize:

First, although censoring at other event types provides correct hazard ratios for Cox models, it doesn't directly model the probability of each event type over time. That requires a joint fit of all events, as explained in Section 3.1 of the vignette. In the example there on monoclonal gammopathy of undetermined significance (MGUS), treating progression to plasma cell malignancy (PCM) and death as competing risks, sex has no association with progression to PCM in the Cox proportional hazards model. Nevertheless:

The effect of sex on the lifetime probability of PCM is not zero, however. Because females live longer, a female with MGUS will on average spend more total years at risk for PCM than the average male, and so has a larger lifetime risk of PCM.

That illustrates the fundamental issue: with competing risks, the instantaneous hazard of an event type in a Cox model doesn't directly translate to the probability of observing that event type. If you want to know the relationship between covariate values and the instantaneous relative hazard of an event type, a Cox model with censoring at times of other events will do. If you want to know the relationship between covariate values and the probability of observing that event type over time, additional work is required.

Second, you can estimate those probabilities over time with a joint Cox model of all competing risks, as illustrated in Section 3.1 of the vignette. That constrains the lifetime probabilities summed over all competing events to equal 1. Those probabilities, however, typically don't show simple relationships with individual covariate values. The probability of observing one type of event depends on the probability of not observing an earlier event of a different type, which depends on the values of all covariates.

Third, the Fine-Gray approach (outlined, with references, here) makes a tradeoff. Yes, it does model the cumulative incidence of observing each event type separately and directly as a function of covariate values, in a set of Cox-type models applied to specially formatted data. But the set of Fine-Gray models need not be consistent: the sum of the lifetime probabilities of the individual events can exceed 1. Furthermore, the assumptions of a Fine-Gray model can be harder to meet in practice than those of a multi-state Cox model. As the vignette explains near the end of Section 4 on Fine-Gray (FG) models:

The primary strength of the Fine-Gray model with respect to the Cox model approach is that if lifetime risk is a primary question, then the model has given us a simple and digestible answer to that question: “females have a 1.2 fold higher lifetime risk of PCM, after adjustment for age and serum m-spike”. This simplicity is not without a price, however, and these authors are not proponents of the approach... The attempt to capture a complex process as a single value is grasping for a simplicity that does not exist for many (perhaps most) data sets. The necessary assumptions in a multivariate Cox model of proportional hazards, linearity of continuous variables, and no interactions are strong ones. For the FG model these need to hold for a combined process — the mixture of transition rates to each endpoint — which turns out to be a more difficult barrier.

In terms of meeting the assumptions of additivity across covariates, linearity, and proportional hazards, Section 4 of the vignette also notes:

In a multi-state model, however, these assumptions cannot hold for both the per-transition and Fine-Gray models formulations at the same time; if true for one, they will not be true for the other.

A Fine-Gray model thus might provide useful, simple summaries of associations between covariates and particular competing outcomes in some circumstances, but it's not necessarily "the best way to get a predicted cumulative incidence curve under competing risks."

• Much appreciated as always @EdM! Unless I'm mistaken, I think we can account for violations of the assumptions you mention (additivity across covariates, linearity, and proportional hazards) using e.g. interactions between covariates, splines and time interaction effects respecively? I guess that you are referring to the Cox model in its basic form? Are you saying that its basically harder to get the model structure "correct" in the FG model due to the combined process you mentioned? Commented Nov 26, 2022 at 12:02
• @user167591 it's possible to try to deal with violations of assumptions of Cox models in the ways you note, although you can end up with so many parameters to estimate that you end up overfitting the data. The last sentence of your comment summarizes one important difficulty with FG modeling. A standard way of dealing with violation of PH in a Cox model, stratifying by an outcome-associated covariate that violates PH so you can get estimates of coefficients for covariates of interest, wouldn't seem to make much sense in a FG model.
– EdM
Commented Nov 26, 2022 at 14:19
• why would stratifying make less sense in a FG model? A quick google search suggests its possible: pubmed.ncbi.nlm.nih.gov/21155744 Commented Nov 26, 2022 at 15:20
• @user167591 What's described there is stratification by institution, among which baseline hazards due to un-modeled predictors could vary. In that situation stratification might make sense for FG. But what if sex violated PH and was associated with outcome? If sex wasn't of primary interest you could stratify by it in a Cox model to get hazard ratios for other predictors. But would it make sense (even if technically possible) to stratify by sex for FG analysis in the MGUS-PCM example in the vignette? How would you interpret the results in practice?
– EdM
Commented Nov 26, 2022 at 15:36
• I see thanks @EdM! Commented Nov 26, 2022 at 16:01