Why a Cause-Specific Hazard Can Trend
A cause-specific hazard describes the instantaneous rate of a particular first-event cause among individuals who are still event-free. That risk set changes over time: people leave it when any event occurs. If subgroups have different overall event rates, their representation among those still event-free can shift, changing the pooled cause-specific hazard even when the cause-specific rate within every subgroup stays constant.
This distinction matters when interpreting a trend. A rising pooled hazard does not, by itself, show that the underlying rate has risen for individuals in every subgroup—or even in any subgroup. The trend may instead reflect which subgroups remain at risk. The calculations below separate these explanations.
For a simple two-cause setting, write the overall event hazard in subgroup \(g\) as \(\lambda_{\cdot}(t\mid g)=\lambda_1(t\mid g)+\lambda_2(t\mid g)\). The same reasoning applies with more causes, by summing all cause-specific hazards to obtain the overall hazard. A person leaves the event-free risk set after an event of any cause.
How the Risk-Set Weights Arise
Let \(\pi_g=P(G=g)\) be the subgroup’s proportion at the start of follow-up, and let \(S_g(t)=P(T\geq t\mid G=g)\) be the probability of remaining event-free through time \(t\) in that subgroup. Among people who are still event-free at time \(t\), the fraction in subgroup \(g\) is
This is conditional probability: the starting subgroup proportion is multiplied by the probability of surviving event-free to \(t\), then divided by the corresponding total across subgroups. The weights describe the current risk set, so they can differ from the starting proportions.
Within each subgroup, the cause-\(j\) hazard applies to the subgroup members still event-free. Averaging those rates using the subgroup proportions in the current risk set gives the pooled hazard:
This identity follows directly from conditioning on subgroup membership. Over a short interval of length \(dt\), the probability of a cause-\(j\) event, conditional on being event-free at \(t\), is approximately \(\lambda_j(t\mid g)dt\) for subgroup \(g\). Averaging over the subgroups currently at risk gives \(\sum_g w_g(t)\lambda_j(t\mid g)dt\). Dividing by \(dt\) and taking the small-interval limit yields the displayed hazard identity.
If subgroup hazards are constant, then \(S_g(t)=\exp\{-t\lambda_{\cdot}(g)\}\), where \(\lambda_{\cdot}(g)\) is the subgroup’s total hazard across all event causes. A subgroup with a larger total hazard generally loses event-free members more quickly. Whether the pooled hazard for a particular cause rises or falls depends on how that depletion changes the weights and on the cause-specific rates of the subgroups.
Worked Examples
Worked Example: A Pooled Hazard Rises Under Constant Subgroup Rates
Consider a hypothetical population split equally between subgroups A and B at the start of follow-up. Suppose the two cause-specific hazards are constant within each subgroup, in person-years\(^{-1}\):
| Subgroup | Cause 1 hazard | Cause 2 hazard | Overall hazard |
|---|---|---|---|
| A | 0.04 | 0.24 | 0.28 |
| B | 0.12 | 0.08 | 0.20 |
At time zero, the event-free risk set has weights \(w_A(0)=w_B(0)=0.50\). Thus the pooled cause 1 hazard is
At five years, the subgroup-specific event-free probabilities are
Using the initial proportions of 0.50, the risk-set weights at five years are
Use the unrounded weights for the hazard calculation:
The pooled cause 1 hazard has risen from 0.08000 to about 0.08790 per person-year. Yet neither subgroup’s cause 1 hazard changed. Subgroup A has the larger overall hazard and is depleted more quickly; the remaining risk set is therefore more heavily weighted toward B, which has the larger cause 1 hazard.
Worked Example: A Pooled Hazard Falls Under Constant Subgroup Rates
Now suppose the hypothetical starting population is again divided equally, but subgroup A has cause 1 and cause 2 hazards of 0.12 and 0.28 per person-year, while subgroup B has hazards of 0.04 and 0.06 per person-year. Each subgroup’s rates are constant. Initially, the pooled cause 1 hazard is
The overall hazards are 0.40 per year in A and 0.10 per year in B. At four years, the event-free probabilities are \(S_A(4)=e^{-1.6}=0.201897\) and \(S_B(4)=e^{-0.4}=0.670320\). The risk-set weights are therefore
The pooled cause 1 hazard at four years is
Here the pooled hazard falls, again with no within-subgroup rate change. A is depleted faster, but A has the higher cause 1 hazard. As its weight shrinks, the pooled cause 1 hazard moves toward B’s lower rate. Depletion does not automatically make a pooled hazard rise: its direction depends on which subgroup is depleted and which has the larger rate for the cause of interest.
Worked Example: Separating Rate Change From Composition Change
A hypothetical analysis reports two subgroups at two times. The weights and cause 1 hazards are \(w_A=0.60,\ w_B=0.40\) with subgroup hazards 0.05 and 0.10 per year at the first time; at the second time they are \(w_A=0.50,\ w_B=0.50\) with subgroup hazards 0.06 and 0.11 per year. The pooled hazards are
For this comparison, add and subtract \(\sum_g w_g(t_1)\lambda_1(t_2\mid g)\). The total difference can then be separated into a within-subgroup rate component, using the first-time weights, and a composition component, using the second-time rates:
The within-subgroup rate changes account for 0.010 per year of the increase, and the changing composition accounts for 0.005 per year under this decomposition. The split depends on the stated reference convention; the total difference, 0.015 per year, does not. This calculation is a way to describe the observed change, not proof that either component has a causal interpretation.
How to Diagnose a Pooled Trend
When a pooled cause-specific hazard changes over time, examine both its component rates and its risk-set weights. Where subgroup membership is measured, calculate or plot the subgroup-specific hazards alongside the pooled hazard. If the subgroup rates are stable but the weights shift, changing risk-set composition is a direct explanation for the pooled trend. If subgroup rates also change, both mechanisms may contribute.
A useful comparison is standardization to fixed subgroup weights. Choose reference weights \(q_g\) that sum to one, then calculate \(\sum_g q_g\lambda_j(t\mid g)\) at each time. Unlike the pooled hazard, this standardized summary does not change solely because the observed risk-set composition changes. Its interpretation depends on the chosen reference weights; it is not the hazard in the actual event-free risk set unless those weights match the current weights.
This comparison requires sufficiently reliable subgroup-specific estimates. Sparse risk sets can make estimated hazards unstable, particularly late in follow-up. Also check how censoring is handled: if censoring changes subgroup representation and is not adequately accounted for, the observed risk set may not represent the intended event-free population. In practice, estimates and uncertainty should be assessed with methods appropriate to the study design and censoring assumptions.
| Question | Useful quantity | What it helps distinguish |
|---|---|---|
| What is the rate in the risk set as it exists at time \(t\)? | Pooled hazard \(\sum_g w_g(t)\lambda_j(t\mid g)\) | The actual marginal event rate among those still event-free |
| Are subgroup-specific rates changing? | \(\lambda_j(t\mid g)\) by subgroup | Within-subgroup trends versus a trend found only after pooling |
| How would the summary change with a fixed mix? | \(\sum_g q_g\lambda_j(t\mid g)\) | Rate trends separated from shifts in chosen subgroup weights |
Common Mistakes and Limitations
- Calling a pooled trend a within-subgroup change. A pooled hazard is an average using weights from the current event-free risk set. Check subgroup rates before attributing its trend to changes within subgroups.
- Assuming depletion always raises the hazard. The direction depends on the relationship between depletion and the cause-specific rates. The two worked examples show that the pooled hazard can rise or fall.
- Using the starting proportions at every time. The relevant weights are \(P(G=g\mid T\geq t)\), not generally \(P(G=g)\). Substituting starting proportions ignores event-free depletion.
- Treating a hazard as an event probability. A hazard is a rate, with units of inverse time, conditional on being event-free at that time. It is not itself the probability of an event by a specified follow-up time.
The mixture identity is descriptive and exact when subgroup membership partitions the population and the hazards are defined for the same event-free condition. It does not identify why subgroup rates differ, and a trend can remain difficult to interpret if relevant subgroup characteristics are unmeasured. A pooled hazard and a fixed-weight standardized summary answer different questions; neither replaces the other in every analysis.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two equally sized subgroups have constant cause 1 hazards of 0.03 and 0.09 per year. What is the initial pooled cause 1 hazard?
- In that setting, suppose the first subgroup’s weight among those event-free later becomes 0.70. What is the pooled cause 1 hazard then, and why did it change?
- Can a pooled cause-specific hazard fall even when every subgroup-specific cause-specific hazard is constant? Explain using the risk-set weights.
- What does standardizing subgroup hazards to fixed weights remove, and what does it not remove?
Solutions. (1) The initial weights are each 0.50, so the pooled hazard is \(0.50(0.03)+0.50(0.09)=0.060\) per year. (2) The second subgroup’s weight is 0.30, so the pooled hazard is \(0.70(0.03)+0.30(0.09)=0.048\) per year. It fell because the lower-rate subgroup now makes up more of the event-free risk set; neither subgroup rate needed to change. (3) Yes. If the higher-rate subgroup is depleted more quickly, its weight can shrink and pull the pooled hazard toward the lower subgroup rate. (4) Fixed-weight standardization removes changes due solely to the selected subgroup weights. It does not remove within-subgroup rate changes, and its value depends on which reference weights are chosen.