# Cox Proportional-Hazards Regression vs. Regression modeling of competing risk using Fine and Gray

30000 patients all have one diagnosis in common, 3000 of them have been exposed to a potential risk factor before the common diagnosis. 1500 patients develop a malignancy after the common diagnosis.

Ratios of patients that developed a malignancy between those exposed to risk factor and those who were not are similar (not statistically different using naive 2-sample test for equality of proportions):

                      Exposed      Not exposed

Malignancy              7,3%           6,7%
Time to malignancy      2yrs           4yrs


But there is a very significantly shorter time from common diagnosis to malignancy if the patient was exposed.

A Cox Proportional-Hazards Regression was performed where patients were followed up until malignancy, death or end of study period. The event was developing malignancy and adjusting factors were age, sex and time of common diagnosis. A check of the proportionality assumtion was performed and age did not adhere so a repeated model was performed with age stratification.

The result was that exposure had HR of 1.3 and a p-value of 0.001.

The same data was also prepared for a competing risk analysis using the crrs() function in R. A status code was encoded, where censored=0, malignancy=1 and death=2. The model was stratified for age as in the Cox regression.

Sample code:

x <- cbind(factor2ind(exposure, "False"), factor2ind(sex, "M"), factor2ind(timeOfDiagnosis, "1980-1990"))

res <- crrs(ftime = stime, fstatus = status, cov1 = x, strata = age, failcode = 1)


The result was that exposure had a SHR og 1.09 and a p-value of 0.32.

I'm new to using the competing risk analysis so I'm not sure what to conclude. Is the sample code on the right track or am I falling into a common pitfall? The Cox Proportional-Hazards Regression model has been used frequently and published in the domain of question and it is very significant. But on the other hand a patient can not develop a malignancy if he has already died which might constitute a competing risk.

Many thanks for any and all replies!

## 1 Answer

The crrs() function from the R crrSC package uses the Fine-Gray subdistribution hazard (SH) approach to modeling competing risks. @AdamO's answer on this page explains that approach nicely:

The interpretation of this subdistributional hazard function is the instantaneous risk of death from cause 1 given you are either still alive, or you've already died of something else. In effect, it averages across these two possibilities in such a way that a high risk of dying previously from other causes lowers your hazard for that specific failure.

In your case, replace "death from cause 1" in that answer with "developing malignancy."

This is to be distinguished from a cause-specific hazard function, in which cases are censored once there is an event of a type that is not of interest (e.g., death prior to developing malignancy in your case). That presumably was used in your standard Cox proportional hazards regression.

Austin and Fine explain the interpretation of these two types of hazard functions and associated regression coefficients, providing cautions on proper interpretation of SH results. Particularly relevant to your situation, they note (page 4397, top paragraph):

If the focus is on rates, authors may be better served by using the cause‐specific hazard model, which models the effect of covariates on the rate of the outcome in subjects who are event‐free (and thus who have not experienced any type of event). Rates may be of greater interest when the study has an etiological focus, while risks [as from SH analysis] may be of greater interest when the focus is on estimating patient prognosis and predicting patient outcome (eg, to inform the clinical management of patients).

More difficulty arises because censoring at event types other than the one of interest runs a risk of introducing informative censoring, a point made by Therneau et al. on page 9 of their vignette on competing risks for the R survival package:

... the computation for this hypothetical case [censored at times of other event types] is only correct if all of the competing endpoints are independent, a situation which is almost never true.

Therneau et al., however, "are not proponents of the [Fine-Gray] approach" for several reasons listed on page 25 of the vignette. They describe ways to analyze competing-risks data as multi-state models, using tools provided by the survival package. They work through in detail starting on page 14 a multi-state competing risk analysis of an example similar to yours (development of malignancy or death after a particular diagnosis), and compare that analysis against Fine-Gray.

So the analysis to use depends on which specific type of hypothesis you wish to test. That can require a good deal of careful thought in a competing-risks scenario.