Proportional Hazards in Cause-Specific Cox Models

What Proportionality Means for a Cause-Specific Hazard

A Cox model for a particular event cause describes how covariates relate to the instantaneous rate of that cause among individuals who have not yet experienced any event. The proportional hazards assumption is a precise restriction on that relationship: for a fixed contrast in covariates, the cause-specific hazard ratio is constant over follow-up time. It is not a claim that event probabilities, cumulative incidence, or the baseline hazard remain constant.

Let $T$ be the time to the first event of any cause, $J$ its cause when an event occurs, and $X$ a covariate vector. For cause $k$, the cause-specific hazard at time $t$, conditional on covariates $X=x$, is the instantaneous rate of cause $k$ at $t$ among people still event-free just before $t$. In a cause-specific analysis, a competing event removes a person from that event-free risk set at its occurrence.

Definition: A proportional cause-specific hazards Cox model for cause $k$ specifies $$ \lambda_k(t\mid X=x)=\lambda_{0k}(t)\exp(\beta_k^\top x). $$ Here $\lambda_{0k}(t)$ is an unspecified, nonnegative baseline cause-specific hazard, and $\beta_k$ is a vector of constant regression coefficients. Proportionality means that, for any two fixed covariate vectors $x_a$ and $x_b$, the ratio $\lambda_k(t\mid x_a)/\lambda_k(t\mid x_b)$ does not depend on $t$, wherever the baseline hazard is positive.

The baseline hazard may change freely with time; the model does not require a constant event rate. The restriction concerns how the hazard for one covariate profile compares with the hazard for another. A separate cause-specific model may have its own baseline hazard and coefficients for another cause. Proportionality for cause $k$ does not imply proportionality for every other cause.

Why a Constant Coefficient Implies a Constant Hazard Ratio

For two people or profiles with covariates $x_a$ and $x_b$, the model gives a hazard ratio by dividing their cause-specific hazards at the same time. The shared baseline hazard cancels. The resulting contrast depends on the coefficient vector and the covariate difference, but not on follow-up time.

\frac{\lambda_k(t\mid x_a)}{\lambda_k(t\mid x_b)} = \frac{\lambda_{0k}(t)\exp(\beta_k^\top x_a)} {\lambda_{0k}(t)\exp(\beta_k^\top x_b)} = \exp\!\left\{\beta_k^\top(x_a-x_b)\right\}.

This is the central implication of proportional hazards. For a binary covariate $X$, comparing $X=1$ with $X=0$ while holding other covariates fixed gives a cause-specific hazard ratio of $\exp(\beta_k)$ at every time. For a continuous covariate, a one-unit increase gives $\exp(\beta_k)$; a change of $d$ units gives $\exp(d\beta_k)$. The coefficient is on the log-hazard scale, so its sign determines whether the modeled cause-specific hazard is higher or lower for an increase in the covariate.

Worked Example: A Binary Covariate and a Twofold Hazard Ratio

Suppose, hypothetically, that a cause-specific Cox model for event 1 has one binary exposure $X$, with $X=1$ for exposed participants and $X=0$ otherwise. The fitted coefficient is $\hat{\beta}_1=0.6931$. At one year, suppose the baseline cause-specific hazard is $0.020$ events per person-year; at three years it is $0.008$ events per person-year. Find the modeled hazard ratio at both times and the exposed hazards.

The model is $\lambda_1(t\mid X)=\lambda_{01}(t)\exp(0.6931X)$. Since $\exp(0.6931)\approx2.000$, the hazard ratio comparing exposed with unexposed participants is 2.000 at every follow-up time.

At one year, the unexposed hazard is $0.020$ events per person-year and the exposed hazard is $0.020\times2.000=0.040$ events per person-year. Their ratio is $0.040/0.020=2.000$.

At three years, the unexposed hazard is $0.008$ events per person-year and the exposed hazard is $0.008\times2.000=0.016$ events per person-year. Their ratio is $0.016/0.008=2.000$. The baseline hazard has fallen, but the modeled hazard ratio remains the same. This is a comparison of instantaneous cause-specific rates among participants still event-free, not a statement that twice as many exposed participants will experience the event.

The same cancellation applies to a continuous covariate. The scale and units matter: a coefficient per one unit may be numerically small while a contrast over several units is consequential. Exponentiating the coefficient after multiplying by the intended unit difference gives the interpretable hazard ratio.

Worked Example: Interpreting a Coefficient per Year of Age

In a hypothetical model for a particular cause, age is recorded in years and has coefficient $\hat{\beta}_k=-0.0400$ per year. Compare the cause-specific hazards for two otherwise identical participants aged 70 and 60 years.

The age difference is 10 years, so the modeled hazard ratio for age 70 versus age 60 is $$ \exp\{(-0.0400)(70-60)\}=\exp(-0.400)=0.6703. $$

Thus, at any fixed follow-up time where the baseline hazard is positive, the 70-year-old profile has 0.6703 times the cause-specific hazard of the 60-year-old profile under this model, conditional on the other modeled covariates. Equivalently, the modeled hazard is about 32.97% lower, since $1-0.6703=0.3297$. This is an association in the specified model; it does not establish that age causes the difference.

The coefficient is negative because the modeled hazard decreases with increasing age. The result concerns a 10-year contrast, not a 1-year contrast: for one year the hazard ratio would be $\exp(-0.0400)=0.9608$.

Assumptions and the Scope of the Restriction

Proportionality is a model assumption about the covariate effect over time, not a property guaranteed by fitting a Cox model. The model also requires that the specified covariate effects adequately represent the comparisons of interest. For the basic form above, covariate values are treated as fixed over follow-up; if values change, the model must specify how the covariate history enters the hazard. The recorded cause and event times must be appropriate for the intended cause-specific analysis, and censoring must be conditionally independent of the event process given the modeled covariates for standard Cox partial-likelihood inference to be valid.

The cause-specific hazard has a selected population at each time: people who have experienced neither the event of interest nor a competing event. Consequently, a cause-specific hazard ratio is conditional on remaining event-free up to that time. It does not directly compare eventual event probabilities in the original groups. Those probabilities also depend on the hazards of competing causes and on how the event-free population changes over time.

The model leaves $\lambda_{0k}(t)$ unspecified, which is useful because it avoids selecting a parametric shape for the baseline hazard. But that flexibility does not make the covariate restriction flexible: a single constant $\beta_k$ still requires a constant hazard ratio for each fixed covariate contrast. A changing baseline hazard is allowed; a changing covariate hazard ratio is not represented by this model as written.

Worked Example: A Time-Changing Ratio Is Not Proportional

Consider hypothetical cause-specific hazards for two groups, with the unexposed group having hazard $\lambda_{01}(t)$. Suppose the exposed-to-unexposed hazard ratio is 1.5 during the first year but 0.8 during the second year. For example, the unexposed hazards might be $0.010$ and $0.020$ events per person-year in those respective periods, giving exposed hazards $0.015$ and $0.016$.

The ratios are $0.015/0.010=1.5$ in the first period and $0.016/0.020=0.8$ in the second. Under a one-coefficient binary-covariate Cox model, both ratios would have to equal the same value $\exp(\beta_k)$. Since $1.5\ne0.8$, no single constant coefficient can represent these stipulated hazards at both periods.

This example identifies a structural mismatch, not a statistical test result. In data, estimated ratios from separate periods are noisy, so visible variation alone does not prove that proportionality fails. Formal assessment and diagnostic procedures require attention to uncertainty and are considered separately.

Derivation for a Covariate Contrast

The proportionality result follows directly from the model, but it is useful to make the logic explicit. Suppose the covariates are held fixed at $x_a$ and $x_b$, and the same cause-specific baseline hazard applies to both profiles. Then:

Write the two modeled hazards.
$\lambda_k(t\mid x_a)=\lambda_{0k}(t)\exp(\beta_k^\top x_a)$, while $\lambda_k(t\mid x_b)=\lambda_{0k}(t)\exp(\beta_k^\top x_b)$.

Form their ratio at the same time.
Dividing the first hazard by the second cancels the common $\lambda_{0k}(t)$, leaving $\exp(\beta_k^\top x_a)/\exp(\beta_k^\top x_b)$.

Combine the exponentials.
Using $\exp(u)/\exp(v)=\exp(u-v)$, the ratio becomes $\exp\{\beta_k^\top(x_a-x_b)\}$, which contains no $t$.

The cancellation requires a shared baseline hazard for the two profiles, as in the stated model. If the coefficient were allowed to vary with time, the contrast would instead involve a time-dependent coefficient and need not be constant. That possibility is outside the constant-coefficient model defined here.

Comparing the Interpretation with Other Approaches

Cause-specific Cox regression answers a cause-specific rate question: among people who are event-free at time $t$, how does the instantaneous rate of cause $k$ differ across covariate profiles? Fine-Gray regression instead models a subdistribution hazard associated with the cumulative incidence function and uses a different risk-set construction. Their hazard ratios are not interchangeable. The choice between them depends on the scientific target; a constant cause-specific hazard ratio is not automatically a constant contrast in cumulative incidence.

For absolute risk prediction, cause-specific hazards for all relevant causes must be considered together to obtain cumulative incidence. Even when a cause-specific hazard ratio is constant, the cumulative incidence curves can have different shapes and their ratio need not be constant. Competing causes affect how many people remain event-free and therefore contribute to the probability of experiencing the event of interest by a given horizon.

The Cox form also contrasts with fully parametric survival models, which specify a particular baseline hazard distribution. The semiparametric Cox model leaves the baseline unspecified while retaining a specific covariate-effect structure. This separation is useful, but the constant hazard-ratio restriction remains an assumption to scrutinize rather than a consequence of avoiding a parametric baseline.

Limitations, Warning Signs, and Common Mistakes

A fitted constant coefficient summarizes the model’s assumed covariate effect over the observed follow-up. If the true association changes with time, the estimate may be a compromise that does not describe every period well. Warning signs can include systematic time patterns in residual-based checks or materially different effect estimates across follow-up intervals. Such signs motivate further assessment; they do not, by themselves, identify the reason for a mismatch. A detailed diagnostic procedure should account for sampling variability, censoring, and the number of events.

Confusing hazard with risk. A hazard is an instantaneous rate among those still event-free. A hazard ratio of 2 does not mean twice the probability of the event by a chosen horizon.
Assuming the baseline hazard is constant. Proportional hazards permits $\lambda_{0k}(t)$ to vary with time. What is constant is the modeled ratio for a fixed covariate contrast.
Ignoring the cause in the interpretation. The coefficient pertains to the specified cause-specific hazard, not automatically to every event type or to the overall probability of any event.
Treating a constant estimate as proof of proportionality. The Cox model imposes a constant coefficient in its stated form. A fitted coefficient does not establish that the assumption is scientifically adequate.

The main limitation is therefore interpretive as well as statistical. The model gives a concise relative-rate summary, but it does not alone provide absolute risks, prove causality, or guarantee that the covariate effect is stable over time. Predictions and conclusions should match the estimand and the assumptions actually used.

Key takeaway: In a cause-specific Cox model, a constant coefficient $\beta_k$ implies a time-constant cause-specific hazard ratio $\exp\{\beta_k^\top(x_a-x_b)\}$ for any fixed covariate contrast. It does not imply a constant baseline hazard or a constant ratio of cumulative incidence.

Check Your Understanding

Answer each question, then compare with the solutions.

A binary covariate has coefficient $0.4055$ in a cause-specific Cox model. What is the modeled hazard ratio comparing covariate value 1 with value 0, and what does it mean?
A continuous biomarker has coefficient $-0.12$ per unit. Find the modeled hazard ratio for a 3-unit increase.
Does proportional hazards require the baseline cause-specific hazard to be constant over time? Explain.
A model has a cause-specific hazard ratio of 1.8. Does this establish that the event probability by five years is 1.8 times as large? Why or why not?
For a fixed covariate contrast, estimated hazard ratios are 1.4 early and 0.9 late. What does this suggest, and what can you not conclude from those estimates alone?

Solutions. (1) $\exp(0.4055)\approx1.500$. At each time, the modeled instantaneous cause-specific hazard among participants still event-free is 1.5 times as high for value 1 as for value 0, holding other covariates fixed. (2) The ratio is $\exp\{(-0.12)(3)\}=\exp(-0.36)\approx0.6977$; the modeled hazard is about 0.698 times as high for the profile three units higher. (3) No. The baseline hazard can vary freely with time; the covariate hazard ratio is the quantity constrained to be constant. (4) No. A hazard ratio is not a risk ratio, and cumulative incidence also depends on follow-up and competing causes. (5) The difference may warn that a constant hazard ratio is inadequate, but separate estimates are subject to sampling variability. Those estimates alone do not prove a violation or establish its cause.