Interpreting Cause-Specific Rates Across Populations

What a Cause-Specific Rate Comparison Compares

A cause-specific rate is conditional on a subject still being free of every first event immediately before the time of interest. Comparing that rate across populations therefore compares the instantaneous occurrence of a particular event among each population’s current event-free subjects. Those two groups need not have the same composition: their members may differ in baseline characteristics, and prior events may have removed different kinds of subjects from the risk sets.

This conditional feature is essential to interpretation. A higher cause-specific rate in one population does not, by itself, mean that a randomly selected member of that population has a higher probability of eventually experiencing the event. Nor does it establish that membership in the population caused the rate difference. It is a comparison of instantaneous rates in the groups that remain event-free at the time being considered.

Definition: Let $T$ be the time to the first event, $J$ its cause, and $G=g$ denote membership in population $g$. The cause-specific hazard for cause $k$ in population $g$ at time $t$ is $$ \lambda_k^{(g)}(t) = \lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t,\ G=g)}{\Delta t}. $$ It is an instantaneous rate, with units of inverse time. The conditioning event $T\geq t$ means that the subject has experienced no first event before $t$.

The condition $J=k$ identifies the event of interest; other first-event causes are not counted in its numerator. If another first event occurs, that subject is no longer event-free and cannot subsequently experience a different first event. The cause-specific hazard is thus defined using the event-free risk set, not a set that retains subjects after a competing first event.

In an empirical analysis, rates are estimated from observed event times and risk sets. Interpretation relies on correct event and time definitions, reliable cause classification, and suitable handling of censoring. In particular, a competing first event is an outcome that ends event-free follow-up, not ordinary censoring. Estimation also requires that censoring not selectively hide event experience in a way left unaddressed by the analysis; the relevant assumption depends on the design and method.

Interpreting a Contrast at a Fixed Time

For two populations, a cause-specific hazard ratio at time $t$ is $\lambda_k^{(1)}(t)/\lambda_k^{(0)}(t)$, provided the denominator is positive. A ratio above 1 indicates a higher instantaneous cause-$k$ rate in population 1 among its event-free subjects at that time. A difference, $\lambda_k^{(1)}(t)-\lambda_k^{(0)}(t)$, instead gives an absolute rate contrast, with the same inverse-time units as the rates.

Neither contrast is a ratio or difference of cumulative probabilities. For example, a rate ratio of 1.25 does not mean that 25% more people in one population will experience the event by a specified time. Cumulative incidence depends on how the cause-specific rates evolve over time and on the other causes of first events as well.

A reported single hazard ratio may summarize a time-varying comparison only under an appropriate model. In a proportional-hazards model, the hazard ratio is assumed constant over time, conditional on any covariates in the model. If the population-specific rates change relative to each other, one constant ratio can conceal that pattern; time-specific estimates or a model allowing time variation may be more informative.

Worked Example: Comparing Rates From Event-Free Person-Time

Suppose a hypothetical study follows two populations and records a target first event and other first-event causes. Over a specified follow-up window, population A contributes 1,600 person-years while event-free and has 64 target first events. Population B contributes 1,200 event-free person-years and has 60 target first events. Assume the event-free person-time is correctly calculated, events are classified correctly, and censoring is handled appropriately. For this example, summarize each population’s rate as target events divided by event-free person-time.

For A, the estimated rate is $$ \widehat{\lambda}_{k}^{(A)} = \frac{64\ \text{target events}}{1{,}600\ \text{person-years}} = 0.0400\ \text{per year}. $$

For B, it is $$ \widehat{\lambda}_{k}^{(B)} = \frac{60\ \text{target events}}{1{,}200\ \text{person-years}} = 0.0500\ \text{per year}. $$

The estimated rate ratio for B versus A is $0.0500/0.0400=1.25$, and the estimated rate difference is $0.0500-0.0400=0.0100$ per year. Thus, over this window, B has an estimated target-event rate 1.25 times A’s rate, or 0.0100 target events per person-year higher, among subjects while they are event-free. The calculation does not show that B’s cumulative probability of the target event is 1.25 times A’s. Nor does it establish that the population label explains the difference.

This pooled rate is an average over the window, not automatically the instantaneous hazard at every time within it. Treating it as a constant cause-specific rate over the window requires that a constant-rate summary be reasonable for the purpose at hand. If rates vary materially with time, estimates should be reported over suitable time intervals or with a time-varying model.

Why Risk-Set Composition Matters

An aggregate cause-specific hazard is a mixture of the hazards in subgroups represented among the event-free subjects at time $t$. The weights are the proportions of the current risk set in those subgroups—not necessarily the proportions in the population at baseline. If subgroup-specific rates differ, a population with more event-free subjects in a higher-rate subgroup can have a higher aggregate rate even when the subgroup-specific rates are the same across populations.

This follows directly by conditioning on subgroup membership. Let $Z$ denote a subgroup characteristic. Among subjects in population $g$ who are event-free at $t$, partition the probability of a cause-$k$ event in the next short interval by $Z$. Dividing by the interval length and taking its limit gives $$ \lambda_k^{(g)}(t) = \sum_z P(Z=z\mid T\geq t,G=g)\lambda_k^{(g)}(t\mid Z=z). $$ The subgroup weights are conditional on still being event-free. Consequently, even if baseline subgroup proportions match, the weights at time $t$ can differ after different event histories. This identity describes the aggregate rate; it does not imply that a population contrast is caused by composition alone.

Worked Example: Same Subgroup Rates, Different Risk-Set Weights

Consider two hypothetical populations at a fixed time $t$, each with a higher-rate subgroup and a lower-rate subgroup. Suppose the cause-specific rate is 0.0800 per year in the higher-rate subgroup and 0.0200 per year in the lower-rate subgroup in both populations. Assume these rates are constant within subgroups at $t$, and that the risk-set weights stated below are the proportions of each population’s event-free subjects in the subgroups at that time.

In population X, 0.2994329 of the event-free risk set is in the higher-rate subgroup and 0.7005671 is in the lower-rate subgroup. Its aggregate rate is $$ (0.2994329)(0.0800)+(0.7005671)(0.0200) = 0.023954632+0.014011342 = 0.037965974 \approx 0.0379660\ \text{per year}. $$

In population Y, the corresponding risk-set weights are 0.6000 and 0.4000. Its aggregate rate is $$ (0.6000)(0.0800)+(0.4000)(0.0200) = 0.04800+0.008000 = 0.05600\ \text{per year}. $$

The aggregate rate ratio, Y versus X, is $0.05600/0.037965974\approx1.475$. Yet the subgroup-specific rates are identical between populations. In this example, the aggregate contrast arises because the event-free risk sets have different subgroup weights. In real data, the weights and the subgroup rates may both differ, and observed data alone do not establish that composition is the sole explanation.

Worked Example: A Contrast After Risk-Set Composition Changes

Suppose two hypothetical populations began with the same proportion in each of two baseline risk groups. At a later time $t$, previous first events have left 20% of population A’s event-free risk set in the higher-rate group, compared with 60% of population B’s event-free risk set. Assume that at $t$, the cause-specific rate is 0.100 per year in the higher-rate group and 0.0200 per year in the lower-rate group in both populations.

For A, the aggregate rate is $$ (0.200)(0.100)+(0.800)(0.0200) = 0.0200+0.0160 = 0.0360\ \text{per year}. $$ For B, it is $$ (0.600)(0.100)+(0.400)(0.0200) = 0.0600+0.00800 = 0.0680\ \text{per year}. $$

The rate ratio, B versus A, is $0.0680/0.0360\approx1.889$. This is a valid contrast between the two current event-free populations at $t$, but it does not show that B’s group members have higher subgroup-specific rates. The stipulated subgroup rates are equal. Different risk-set composition is sufficient to produce the aggregate difference in this example. The example also shows why a comparison at a later time need not describe the same mixture of people as a comparison at baseline.

Assumptions, Diagnostics, and Alternatives

A meaningful cross-population comparison needs a common, clearly specified event definition, cause classification, time origin, time scale, and time point or follow-up interval. Otherwise, the two rates may not refer to comparable quantities. The cause-specific hazard must be defined from subjects who are actually event-free immediately before the time being analyzed. The estimation procedure must also account appropriately for censoring and the observed event history.

Useful checks include examining estimated rates over time rather than relying only on an overall summary; checking the number of subjects still event-free and the amount of person-time supporting each estimate; and comparing baseline and time-specific covariate composition where relevant. Sparse risk sets can make late-time rates unstable. A large rate ratio based on few events may be imprecise, and a smooth model fit does not remedy a lack of information. If a proportional-hazards model is used, check whether the observed relative rates are reasonably consistent with its constant-ratio assumption.

When differences in measured characteristics matter to the scientific question, one may report covariate-stratified or covariate-adjusted cause-specific rates. A standardized comparison can instead apply chosen common covariate weights to subgroup-specific rates. These approaches address measured composition according to the model and standardization target, but do not automatically remove differences due to unmeasured characteristics or make an observational comparison causal. The target population and weighting scheme should be stated.

The main alternative depends on the question. A cause-specific rate addresses the instantaneous rate of one event among subjects who remain event-free. If the question concerns the probability of experiencing a particular event by a specified time in the presence of competing events, the cumulative incidence function is the relevant probability-scale quantity. Subdistribution-rate comparisons use a different risk-set construction and answer a different rate-comparison question; they should not be substituted for a cause-specific interpretation merely because they yield a single ratio.

The central limitation is that the event-free groups are selected groups. Their membership changes over time as first events occur, and may differ across populations for reasons related to baseline composition, earlier event rates, or other factors. A cause-specific rate contrast is therefore informative about current event-free subjects at the stated time, but is not automatically a contrast for the original populations as a whole.

Common Mistakes

Calling a rate ratio a risk ratio. A cause-specific hazard ratio compares instantaneous rates among current event-free subjects; it is not a ratio of cumulative event probabilities.
Assuming the two risk sets are compositionally identical. Event-free subjects in the populations can differ at baseline and can become differently composed over time. Report or investigate this when it matters.
Reading a population association as a causal effect. A crude or adjusted rate contrast alone does not establish causation. Causal interpretation requires additional design and assumptions.
Ignoring time variation. A single rate or hazard ratio can hide changing rates or a changing relative contrast. State the time window and assess whether a constant-rate or proportional-hazards summary is justified.
Treating a competing first event as if it were the target event. It is not counted as a cause-$k$ event and the subject no longer remains in the event-free risk set.

Key takeaway: A cause-specific rate comparison is a comparison of instantaneous rates among each population’s event-free subjects. Because those risk sets may have different compositions, interpret the contrast at its stated time and avoid equating it with a cumulative-risk or causal comparison.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A cause-specific hazard ratio is 0.80 for population A versus population B at time $t$. What does this say, and what does it not say?
In one population, 25% of the event-free risk set is in a subgroup with rate 0.060 per year and 75% is in a subgroup with rate 0.020 per year. Calculate the aggregate rate.
Why might two populations have different aggregate cause-specific rates even if their subgroup-specific rates are equal?
What should be checked before interpreting one hazard ratio as a constant comparison over follow-up?

Solutions. (1) At $t$, A’s instantaneous rate for that cause among its event-free subjects is estimated to be 0.80 times B’s. This does not say A’s cumulative event probability is 20% lower or establish a causal effect. (2) The aggregate rate is $(0.25)(0.060)+(0.75)(0.020)=0.015+0.015=0.030$ per year. (3) The populations may have different subgroup proportions among their event-free subjects at that time, giving different weighted averages. (4) Check whether the relative rates are reasonably constant over time, whether the model’s proportional-hazards assumption is plausible, and whether the risk sets provide enough information across follow-up.