Competing risks are common in the analysis of event time data. The classic example is death, with distinctions among different kinds of death: if you die of a heart attack, you can’t then die of cancer or suicide. But examples also abound in other fields. A marriage can end either by divorce or by the death of a spouse, but not both. If a loan is terminated by pre-payment, it is no longer at risk of a default.
In this post, I argue that one of the more popular methods for regression analysis of competing risks—the analysis of subdistribution hazards, introduced by Fine and Gray (1999)—is not valid for causal inference. In fact, it can produce completely misleading results. Instead, I recommend the analysis of cause-specific hazards, a long-standing and easily implemented method.
Let’s review that classic method first. The estimation of cause-specific hazard functions (Kalbfleish and Prentice 2002) can be accomplished with standard methods for single kinds of events. You simply treat all competing events as though the individual were right censored at the time the competing event occurs. For example, if you want to study the effect of obesity on the risk of death due to heart disease, just estimate a Cox proportional hazards model in which all causes of death other than heart disease are treated as censoring. Or if you want to estimate the effect income on divorce, estimate a Cox model in which spousal death is treated as censoring.
By contrast, the method of Fine and Gray (1999) does not treat competing events in the same way as censored observations. Based on cumulative incidence functions, their method estimates a proportional hazards model for something they call the subdistribution hazard.
The definition of the subdistribution hazard is similar to that for a cause-specific hazard, with one key difference: the cause-specific hazard removes an individual from the risk set when any type of event occurs; the subdistribution hazard removes an individual from the risk set when an event of the focal type occurs or when the individual is truly censored. However, when a competing event occurs, the individual remains in the risk set. Fine and Gray acknowledge that this is “unnatural” because, in fact, those who experience competing events are no longer actually at risk of the focal event. But it’s necessary in order to get a model that correctly predicts cumulative incidence functions. (More on that later).
According to Google Scholar, the Fine and Gray paper has been cited more than 5,000 times. It is now widely available in most major software packages, including Stata (with the stcrreg command), SAS (with the EVENTCODE option in PROC PHREG) and R (with the ‘cmprsk’ package). In some fields, it has become the standard method for analyzing competing risks. In the minds of many researchers, it is the only proper way to analyze competing risks.
But there’s one big problem: the subdistribution method doesn’t isolate distinct causal effects on the competing risks. In fact, it confounds them in predictable and alarming ways. Specifically, any variable that increases the cause-specific risk of event A will appear to decrease the subdistribution hazard for event B. Why? Because whenever a type A event occurs, it eliminates the possibility that a type B event will happen.
Here’s a simple simulation that demonstrates this phenomenon. I generated 10,000 observations on each of two event types, labeled A and B. Event times for type A were generated by a Weibull regression model (a parametric version of the proportional hazards model). The only predictor was variable X, which had a standard normal distribution.
Event times for type B were generated by an identical Weibull regression model in which the only predictor was variable Z, also standard normal. X and Z had a correlation of .50.
If event A occurred before event B, the event time for B was not observed. Similarly, if B occurred before A, the event time for A was not observed. Any event times greater than 8 were treated as right censored. SAS code for generating the data and performing the analysis is appended below.
In the resulting data set, there were 4350 type A events, 4277 type B events, and 1391 truly censored observations (neither A nor B was observed). Given the data generating process, any attempt to estimate causal parameters should find no effect of X on B and no effect of Z on A.
In Table 1 below, I show the results from applying the two different methods: cause-specific Cox regression models with competing events treated as censoring; and subdistribution proportional hazards models. For type A, the cause-specific estimate of .494 for the effect of X is close to the true value of .500 and highly significant. The coefficient of -.008 for Z is close to the true value of 0 and far from statistically significant. Overall, pretty good performance.
The subdistribution estimates for type A, on the other hand, are clearly unsatisfactory. The coefficient for X appears to be biased downward by about 10%. The coefficient for Z (-.216) is far from the true value of 0, and highly significant. Thus, the subdistribution estimates would lead one to conclude that an increase in Z reduced the risk of event A. What it actually did is reduce the likelihood that type A events would be observed because it increased the risk of event B.
The results for event B in the lower panel of Table 1 are the mirror image of those for event A. For both X and Z, the cause-specific estimates are close to the true values. The subdistribution estimates are biased, and would lead to incorrect causal inferences.
Table 1. Results from Two Methods for Estimating Competing Risks Models, Non-Informative Censoring.
|Cause-Specific Hazards||Subdistribution Hazards|
So why would anyone seriously consider the subdistribution method? Well, there’s one big problem with the cause-specific hazards approach. Virtually all methods based on cause-specific hazards implicitly assume that censoring is non-informative. Roughly speaking, that means that if an individual is censored at a particular point in time, that fact tells you nothing about the individual’s risk of the event.
If the censoring times are determined by the study design (as when all event times beyond a certain calendar time are censored), that’s not usually an issue. But if censoring times are not under the control of the investigator, the censoring may be informative. And that can lead to bias.
If competing risks are treated as a form of censoring, then we need to be concerned about whether that censoring is informative or non-informative. How might competing events be informative? If a spouse dies, does that tell us anything about the risk that the couple would have divorced? Maybe, but probably not. Does the fact that a person dies of heart disease tell us anything about that person’s risk of dying of cancer? Maybe so. That would definitely be the case if cancer and heart disease had common risk factors, and those factors were not included in the regression model.
Unfortunately, there’s no way to test the assumption that censoring is non-informative (Tsiatis 1975). And even if there were, there’s no good method available for estimating causal effects when censoring is informative.
By contrast, the subdistribution hazard method does not explicitly assume that competing risks are non-informative. And that has been one of its major attractions. However, as I now show, the subdistribution method does no better (and actually somewhat worse) than the cause-specific method when competing events are informative.
In the previous simulation, the assumption of non-informative censoring was satisfied by the data generating process. To model informative censoring, I added an unobserved common risk factor to the regression equations for the two event times. This was simply a normally distributed random variable with a standard deviation of 2. This “random intercept” induced a correlation of .28 between the uncensored event times for type A and type B. I then re-estimated the models, with results shown in Table 2 below.
The message of Table 2 is this: Yes, informative censoring leads to cause-specific estimates that are biased. And unlike in the previous table, the cause-specific estimates might lead one to conclude, incorrectly, that Z affects the risk for event A and that X affects the risk for event B. But the subdistribution estimates are also biased. And, with one minor exception, the biases are worse for the subdistribution method than for the cause-specific method.
Table 2. Results from Two Methods for Estimating Competing Risks Models, Informative Censoring.
|Cause-Specific Hazards||Subdistribution Hazards|
To repeat my earlier question: Why would anyone ever seriously consider using the subdistribution method? To be fair to Fine and Gray, they never claimed that subdistribution regression would accurately estimate causal parameters. Instead, they introduced the method as a way to model the impact of covariates on the cumulative incidence functions. These functions are often preferable to Kaplan-Meier estimates of the survivor function because they do a better job of describing the empirical distribution of events rather than some hypothetical distribution that would apply only in the absence of competing risks.
Cumulative incidence functions are particularly useful for prediction. Suppose you have a cohort of newly diagnosed cancer patients. Based on the experience of earlier patients, you want to predict in five years what percentage will have died of cancer, what percentage will have died of other causes, and what percentage will still be alive. Cumulative incidence functions will give you that information. Kaplan-Meier survivor functions will not.
The Fine and Gray method provides a way to introduce covariate information into those predictions, potentially making them more accurate for individual patients. It’s important to note, however, that one can also calculate cumulative incidence functions based on cause-specific hazard functions. Most commercial packages for Cox regression don’t have that capability, but there are downloadable SAS macros that will accomplish the task.
In sum, the subdistribution method may be useful for generating predicted probabilities that individuals will be in particular states at particular times. But it is not useful for estimating causal effects of covariates on the risks that different kinds of events will occur. For that task, the analysis of cause-specific hazards is the way to go. Unfortunately, both methods are vulnerable to competing events that are informative for each other. The only effective way to deal with that problem is to estimate cause-specific hazard models that include common risk factors as covariates.
SAS Code for Simulation
data finegraytest; do i=1 to 10000; *Generate a common risk factor. To include it in the model, change 0 to 1; comrisk=0*rannor(0); *Generate x and z, bivariate standard normal with r=.5; x=rannor(0); z=.5*x + sqrt(1-.25)*rannor(0); *Generate w with Weibull distribution depending on x; logw=2 + -.75*x + 1.5*log(ranexp(0)) + 2*comrisk; w=exp(logw); *Generate y with Weibull distribution depending on z; logy=2 + -.75*z + 1.5*log(ranexp(0)) + 2*comrisk; y=exp(logy); *Allow events to censor each other; if y>w then do; type=1; t=w; end; else if w>y then do; type=2; t=y; end; *Censor all event times at 8; if t>8 then do; type=0; t=8; end; output; end; run; proc freq; table type; run; /*Estimate cause-specific hazard regressions */ *Model for type1 event; proc phreg data=finegraytest; model t*type(0 2)=x z; run; *Model for type2 event; proc phreg data=finegraytest; model t*type(0 1)=x z; run; /*Estimate subdistributions hazard regressions */ *Model for type1 event; proc phreg data=finegraytest; model t*type(0)=x z / eventcode=1; run; *Model for type2 event; proc phreg data=finegraytest; model t*type(0)=x z / eventcode=2; run;