Interpreting Cause-Specific Hazards in Competing Risks

Defining the Cause-Specific Hazard

Let's start with the central concept. In standard survival analysis, the hazard function, $h(t)$, represents the instantaneous potential for an event to occur at time $t$, given survival up to that time. When competing risks are present, we can no longer talk about a single hazard; we must consider the hazard for each specific type of event.

The cause-specific hazard is the instantaneous rate of a particular event type (e.g., cardiovascular death) in the group of subjects who have not yet experienced any event (i.e., they are still alive).

Think of it this way: if you have a group of 100 people who are currently event-free, the cause-specific hazard for cardiovascular death at this moment is the rate at which they are currently dying specifically from cardiovascular causes.

The following video clip provides a clear definition and introduces the mathematical notation, which builds directly on the hazard function we've already learned.

The key takeaway is the composition of the risk set. For the cause-specific hazard of event $k$, the risk set at time $t$ consists of every individual who has not yet experienced event $k$ or any of its competing events. This is a very intuitive definition: we are modeling the rate of a specific event among those who are still alive and capable of experiencing it.

The following reading provides a concise summary and a helpful formula.

Fitting a Cause-Specific Hazard Model

One of the most appealing features of the cause-specific approach is its ease of implementation. You can fit a cause-specific hazard model for a particular event using the exact same software you would for a standard Cox proportional hazards model.

The method involves a simple but clever analytical "trick": to model the cause-specific hazard for your event of interest, you treat all competing events as if they were censored observations. For instance, if your event of interest is "death from cancer" and the competing event is "death from heart disease," you would run a Cox model where all patients who die from heart disease are coded as censored at their time of death.

The following clip from the Q&A section of the Population Data BC video explains why this seemingly counter-intuitive step is valid.

Because the CSH model is essentially a series of standard Cox models (one for each event type), the interpretation of the hazard ratio (HR) is also very similar. A cause-specific hazard ratio (CS-HR) quantifies how a covariate affects the instantaneous rate of a specific cause of failure among those who are currently event-free.

For example, if a CSH model for cardiovascular death yields a CS-HR of 1.5 for a specific drug, it means that, among patients who are still alive, those on the drug have a 50% higher instantaneous rate of dying from cardiovascular causes compared to those not on the drug.

The Crucial Caveat: Hazard vs. Incidence

Here we arrive at the most important—and subtle—point in interpreting a CSH model. In a standard Cox model without competing risks, a hazard ratio greater than 1 implies both an increased rate of the event and an increased overall incidence (or probability) of the event over time.

This is not true for cause-specific models.

A covariate can increase the cause-specific hazard of an event but simultaneously decrease its cumulative incidence. How is this possible?

Imagine a new treatment for a severe disease. The treatment has a side effect that slightly increases the rate of cardiovascular death (CS-HR > 1 for CV death). However, the treatment is so effective at curing the primary disease that it dramatically reduces the rate of death from that disease (a competing event). By preventing so many deaths from the primary disease, more people are now alive and at risk of other things. But the effect on the competing risk can be so strong that, overall, the probability of dying from cardiovascular causes might not change or could even decrease, as people are removed from the risk set by the competing event at a different rate.

The video below explains this critical disconnect.

This disconnect is why cause-specific models are primarily used for etiologic questions—that is, questions about the direct causal mechanism or biological pathway. The CS-HR tells you about a covariate's effect on the instantaneous rate of an event, which is useful for understanding how a factor works. It is not the right tool for predicting a patient's overall probability of experiencing that event over time.

This image shows a comparison of different models for kidney graft failure, with donor age as a covariate. The solid lines represent the cumulative incidence as estimated by a cause-specific model. Notice how these estimates differ from the other models. The CSH model helps us understand the rate of failure at any given time for a specific cause, but it doesn't directly give us the cumulative probability (the curve itself) without considering the hazards of all other competing events.

Simulation: Seeing the CSH in Action

To make this concrete, let's examine a simulation. In the following article, the author simulates data where a variable X only causes event A, and a variable Z only causes event B.

The simulation shows the strength of the CSH model: it correctly attributes the cause of each event, making it the superior tool for understanding causal relationships. The author's main point is that if your question is "Does X cause Y?", the cause-specific hazard model is the appropriate choice.

Conclusion

In this lesson, we dissected the cause-specific hazard model, the first of our two main approaches for regression analysis with competing risks. It's a powerful tool, but only when used to answer the right question.

Here are the key takeaways:

• The cause-specific hazard is the instantaneous rate of a specific event occurring in a population that is still event-free (i.e., alive).
• CSH models can be fit using standard Cox regression software by treating competing events as censored.
• The resulting cause-specific hazard ratio (CS-HR) measures the effect of a covariate on the rate of that specific event. It is a measure of etiology.
• Crucially, an increased CS-HR does not automatically mean an increased cumulative incidence (overall probability) of the event, because the covariate's effect on competing risks also matters.

You now understand how to analyze competing risks when your goal is to uncover the underlying causal effects on event rates. But what if your goal is different? What if you want to predict a patient's overall probability of an event, accounting for the fact that they might experience a competing event first? For this prognostic question, we need a different tool.

In our next lesson, we will explore the Fine-Gray subdistribution hazard model, which is designed specifically for that purpose.