Diagnosing Misleading Hazard Comparisons

Why a Hazard Comparison Can Mislead

A hazard ratio or rate difference can look straightforward while comparing quantities that do not answer the same question. One group may have a different mix of people still at risk, more competing events, a different time origin, or a different definition of the focal event. In each case, a numerical contrast may be correct for its stated hazard yet misleading as evidence about a different clinical question.

The diagnostic task is not simply to ask whether a hazard is larger. First identify the event, the time scale, and who contributes to the risk set at each time. Then ask whether the comparison is meant to describe the instantaneous rate among people still event-free, the rate under a subdistribution risk-set convention, or the probability of experiencing the event by a specified time.

Definition: Let $T$ be the time to the first event and $J$ its cause, with $J=k$ denoting a focal cause. The cause-specific hazard for cause $k$ is the instantaneous rate of cause-$k$ events among people who have had no event just before time $t$: $$ \lambda_k(t)=\lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}. $$ The subdistribution hazard uses a different risk-set convention, retaining people who have already had a competing event. If the focal-event cumulative incidence $F_k(t)=P(T\leq t,J=k)$ is absolutely continuous, then, where its density $f_k(t)$ is defined and $F_k(t)<1$, $$ \lambda_k^*(t)=\frac{f_k(t)}{1-F_k(t)}. $$ These hazards condition on different risk sets and are not interchangeable.

A cause-specific hazard describes the event rate among those currently free of every event type. A subdistribution hazard describes a rate using a constructed risk set that continues to represent people with a prior competing event. Neither hazard alone is the probability of the focal event by a future time. Interpretation also depends on the time scale: time since treatment, attained age, and time since diagnosis need not yield comparable rates.

Assumptions to Check Before Interpreting a Contrast

A hazard comparison requires a precise event process and observation scheme. The first-event time and event type must be defined consistently across groups, including what happens if event types occur simultaneously. Censoring must be distinguished from competing events: censoring ends observation without establishing an event, whereas a competing event is an observed event that prevents the focal first event from occurring later in the defined process.

When hazards are estimated from observed follow-up, the assumptions used to handle censoring must also be plausible. In particular, independent censoring, possibly conditional on measured covariates, is a separate assumption from the definition of the hazard. It does not make a comparison causal. A causal interpretation would need additional design and confounding assumptions.

For a comparison across groups, ask whether each group’s hazard is evaluated on the same time scale and with the same cause labels. Then inspect the risk sets: even if hazards within clinically relevant strata are similar, the groups can have different marginal hazards because the distribution of strata among people still event-free changes over time.

Risk-Set Composition Can Change a Marginal Hazard

The following justification makes the composition issue explicit. Suppose the population is divided into strata $s$, and let $w_s(t)$ be the proportion of the group’s event-free risk set in stratum $s$ just before $t$. For a short interval of length $\Delta t$, the probability of a focal event, conditional on being event-free and in stratum $s$, is approximately $\lambda_{k,s}(t)\Delta t$. Averaging across the risk set gives:

\begin{aligned} P(\text{cause }k\text{ in }[t,t+\Delta t)\mid T\geq t) &\approx \sum_s w_s(t)\lambda_{k,s}(t)\Delta t,\\ \lambda_k(t)&=\sum_s w_s(t)\lambda_{k,s}(t). \end{aligned}

The second line follows by dividing the first by $\Delta t$ and taking the limit as $\Delta t$ decreases to zero. Thus a marginal hazard is a risk-set-weighted average of stratum-specific hazards. Its weights can differ between groups and change over time. A marginal contrast can therefore reflect different risk-set composition even when the corresponding stratum-specific hazards agree.

Worked Example: Different Marginal Hazards With Identical Stratum Hazards

Hypothetical snapshot. Consider two groups at the same time $t$. Each contains low-risk and high-risk strata. At this instant, the cause-specific hazard is $0.020$ events per person-year in the low-risk stratum and $0.100$ events per person-year in the high-risk stratum, in both groups. Among people still event-free, Group A is 80% low-risk and 20% high-risk; Group B is 20% low-risk and 80% high-risk.

For Group A, the marginal hazard is the risk-set-weighted average:

0.80(0.020)+0.20(0.100)=0.016+0.020=0.036 \quad\text{events per person-year.}

For Group B, it is:

0.20(0.020)+0.80(0.100)=0.004+0.080=0.084 \quad\text{events per person-year.}

The marginal hazards differ by $0.048$ events per person-year, although the specified within-stratum hazards are identical. The difference at this snapshot is due to the composition of the event-free risk sets. This calculation does not establish why the groups have different compositions, nor does it establish a causal effect. It diagnoses why a pooled hazard contrast should not automatically be read as a within-stratum contrast.

Competing Events Change What a Hazard Contrast Implies

A focal cause-specific hazard can be the same in two populations while their focal-event probabilities differ, because competing events alter how many people remain able to experience the focal first event. The next calculation assumes constant cause-specific hazards, no loss to follow-up, and a first event that is either the focal event or one competing event. These assumptions make the calculations explicit; the result is an illustration, not a general model for every clinical setting.

Worked Example: Same Focal Hazard, Different Competing Hazards

Hypothetical model. In both populations, the focal cause-specific hazard is $\lambda_1=0.10$ per year. In Population A, the competing-event hazard is $\lambda_2=0.02$ per year; in Population B, it is $\lambda_2=0.10$ per year. Under these constant hazards, event-free survival to time $t$ is $S(t)=\exp[-(\lambda_1+\lambda_2)t]$. The focal-event density is $f_1(t)=\lambda_1 S(t)$, so integrating it from 0 to 5 gives the focal-event probability by year 5.

F_1(5)=\int_0^5 \lambda_1 e^{-(\lambda_1+\lambda_2)u}\,du =\frac{\lambda_1}{\lambda_1+\lambda_2} \left[1-e^{-(\lambda_1+\lambda_2)5}\right].

For Population A:

F_{1,A}(5)=\frac{0.10}{0.12}(1-e^{-0.60}) =0.8333(0.4512)=0.3760.

For Population B:

F_{1,B}(5)=\frac{0.10}{0.20}(1-e^{-1.00}) =0.5000(0.6321)=0.3161.

The focal cause-specific hazard is $0.10$ per year in both populations, but the model gives a 5-year focal-event probability of about 37.6% in A and 31.6% in B. The higher competing hazard in B leaves fewer people event-free long enough to have the focal first event. Thus the equal cause-specific hazards do not imply equal focal-event probabilities.

The subdistribution hazards also differ in this example. At year 5, the focal density in A is $0.10e^{-0.60}=0.05488$ per year, and $1-F_{1,A}(5)=0.6240$, giving $\lambda^*_{1,A}(5)\approx0.08795$ per year. In B, the density is $0.10e^{-1.00}=0.03679$ per year and $1-F_{1,B}(5)=0.6839$, giving $\lambda^*_{1,B}(5)\approx0.05379$ per year. These subdistribution hazards reflect both the focal-event density and the competing-event pattern through their definition; they are not evidence that the focal cause-specific hazard differs.

The calculation does not make the subdistribution hazard a direct probability or a simple measure of biological mechanism. It shows why the intended question matters. A question about the instantaneous rate among people still event-free calls for a cause-specific interpretation. A question about the distribution of focal first events in the presence of competing events concerns absolute event probability, and a subdistribution hazard is a different rate-based summary of that distribution.

Time Scale and Cause Definition Are Part of the Comparison

Hazards are rates indexed by time, so changing the time origin can change the comparison. A rate at five years after enrollment is not necessarily comparable to a rate at attained age 70. Before interpreting a contrast, state the clock, its origin, and how delayed entry is handled. If groups enter at different ages, comparing the same follow-up duration may compare people at different attained ages.

Worked Example: Follow-Up Time Versus Attained Age

Hypothetical model. Suppose the same cause-specific hazard function applies to everyone as a function of attained age $a$, with age measured in years:

\lambda(a)=0.020+0.010(a-60) \quad\text{events per person-year},\qquad 60\leq a\leq75.

Group X enters observation at age 60 and Group Y at age 70. At five years after entry, X is age 65 and has hazard $0.020+0.010(65-60)=0.070$ per year; Y is age 75 and has hazard $0.020+0.010(75-60)=0.170$ per year. A comparison at the same follow-up duration therefore shows a difference of $0.100$ per year, even though the model specifies a common age-specific hazard for the two groups.

At attained age 70, the specified hazard is $0.020+0.010(70-60)=0.120$ per year for either group, provided the person is still event-free. A comparison at that age aligns the time scale, though it does not by itself address differences in who remains event-free or other sources of confounding. The example shows that “at five years” and “at age 70” are distinct comparisons.

Cause labels also define the event being analyzed. A clinical endpoint such as “cardiovascular death” can depend on adjudication rules, while “death” is broader. Reclassifying ambiguous events between focal and competing causes changes the focal-event counts and therefore the hazard being described. The same caution applies when the endpoint changes from first hospitalization to first hospitalization-or-death: these are different event definitions, not interchangeable labels for one hazard.

Worked Example: Ambiguous Events Change the Focal Rate

Hypothetical rate illustration. In a specified follow-up interval, a group contributes 100 person-years at risk and has six unequivocal focal events plus four deaths whose cause is uncertain. If the four deaths are classified as competing events, the crude focal rate is $6/100=0.060$ per person-year. If the protocol classifies all four as focal, it is $10/100=0.100$ per person-year. The arithmetic is correct under either coding rule, but the two rates refer to different focal-event definitions.

These are crude event rates for the stated interval, not exact instantaneous hazard estimates: an instantaneous hazard is defined by a limit over a short interval conditional on being event-free. The example isolates the coding issue. A defensible comparison requires consistent, clinically justified cause rules across groups, transparent handling of uncertain causes, and awareness that differential misclassification can create or obscure a contrast.

Diagnostics, Alternatives, and Limits

A hazard comparison is more informative when its definition and risk-set behavior can be checked directly. Useful diagnostic questions include:

Are event-free risk-set sizes and covariate composition changing differently across groups?
Do the groups have meaningfully different competing-event patterns?
Is time measured from the same origin, and does the chosen scale match the clinical question?
Were focal and competing causes classified with the same rules and quality of ascertainment?
Could censoring depend on prognosis after accounting for the variables used in the analysis?

A single hazard ratio can conceal time variation: the group ordering may change over follow-up, so a constant-hazard summary can be difficult to interpret. Inspecting time-specific estimates or plots can reveal such patterns, but the diagnostic display must use clearly defined risk sets and time scales. Sparse risk sets late in follow-up also make estimates unstable. A large late hazard ratio may be based on very few people and should not be treated as precise merely because it is numerically large.

The main alternatives answer related but distinct questions. A cause-specific hazard comparison describes rates among people still event-free. A subdistribution hazard comparison uses its competing-event-inclusive risk-set convention. A cumulative incidence comparison targets the probability of the focal event by a specified time, taking competing events into account. Standardized absolute risks can help make group comparisons on a common covariate distribution, but they require a stated target population and appropriate modeling assumptions. No one summary replaces the others, and none alone resolves confounding or poor event classification.

Common mistake: “The focal cause-specific hazards are equal, so the groups have the same focal-event probability.” This is not generally true: different competing-event patterns can change the number of people who remain able to experience the focal first event. The correction is to name the estimand, report the competing-event context, and use an absolute event-probability summary when the clinical question concerns event probability by a specified time.

A second frequent error is treating a hazard contrast as a causal effect merely because it is adjusted for measured covariates. Hazards condition on the evolving risk set, whose membership may differ between groups over time. Adjustment does not automatically make these selected risk sets comparable, and causal conclusions need assumptions beyond the hazard definition. Reporting should therefore specify the event, risk set, time scale, censoring assumptions, and comparison being made.

Check Your Understanding

Answer each question, then compare with the solutions below.

Two groups have identical cause-specific hazards within every baseline stratum but different stratum proportions among people still event-free. Can their marginal cause-specific hazards differ? Why?
Two populations have equal focal cause-specific hazards, but one has a higher competing-event hazard. Does equality of focal hazards establish equal focal-event probabilities by a fixed time?
A study compares event rates at three years after enrollment in groups entering at ages 50 and 70. Name one concern about the time scale.
Four ambiguous deaths are classified as competing events in one analysis and focal events in another. What must be clarified before interpreting the rate difference?

Solutions. (1) Yes. The marginal hazard is the risk-set-weighted average of stratum-specific hazards; different weights can give different averages even when each stratum’s hazards agree. (2) No. A higher competing-event hazard can reduce the number of people who remain event-free to experience the focal event, so focal-event probabilities can differ. (3) Three years after enrollment corresponds to different attained ages; the groups may be compared on different underlying time scales. (4) The cause-definition and adjudication rules, whether applied consistently across groups, and the handling of uncertain causes must be stated; the two analyses define different focal events.