Reporting Cause-Specific Cox Estimates

What a Cause-Specific Cox Estimate Describes

A reported hazard ratio is interpretable only if readers can tell which event it concerns, who is being compared, and which other variables were included in the model. In competing-risk analyses, the event definition and risk set matter especially: a cause-specific Cox model concerns the instantaneous rate of one specified cause among people who have not yet experienced any event.

Definition: Let $T$ be time to the first event of any cause, and let $J$ denote its cause. For a specified cause $k$, the cause-specific hazard at time $t$, conditional on covariates $X$, is the instantaneous event rate for cause $k$ among individuals who are event-free immediately before $t$. A cause-specific Cox model is $$ h_k(t\mid X)=h_{0k}(t)\exp(\beta_k^\mathsf{T}X), $$ where $h_{0k}(t)$ is the unspecified baseline cause-specific hazard and $\beta_k$ is the vector of regression coefficients for cause $k$. The risk set at time $t$ consists of individuals still under observation and free of every event type just before $t$.

For the specified cause, a person who experiences a competing event leaves this event-free risk set at that time. In the usual cause-specific Cox analysis, that competing event is treated as a censoring time for the cause-$k$ model. This describes the cause-specific hazard; it does not make the competing event equivalent to ordinary loss to follow-up or establish that competing events are independent of the outcome.

If $X$ changes from $x_0$ to $x_1$, the model's cause-specific hazard ratio is $ \exp\{\beta_k^\mathsf{T}(x_1-x_0)\} $, comparing the instantaneous cause-$k$ hazards at the same time among individuals with those covariate values who remain event-free. For a one-unit increase in a single covariate, the hazard ratio is $\exp(\beta_k)$. The proportional-hazards assumption makes this ratio constant over time, conditional on the model covariates.

What to State When Reporting the Estimate

A complete report connects the numerical estimate to its model and comparison. State the cause modeled, the population or analysis cohort, the exposure contrast and its reference category, the adjustment variables, and the estimate with an uncertainty interval. Give the interval's confidence level and, when useful, the test and p-value. Include the number of cause-$k$ events and the analysis sample size so readers can assess how much information supports the estimate.

A methods sentence might say: “We fitted a cause-specific Cox proportional-hazards model for time to [cause $k$]; individuals experiencing another first event left the event-free risk set at that time. The model was adjusted for [listed covariates], and [category] was the reference for [exposure].” The results sentence can then report the contrast, hazard ratio, confidence interval, and p-value. Name adjustment variables rather than relying on “adjusted model,” which does not reveal what was controlled.

For a Wald interval, if $\hat\beta_k$ is a coefficient estimate and $SE(\hat\beta_k)$ its standard error, the approximate 95% interval is calculated on the log-hazard-ratio scale and then exponentiated:

\left[ \exp\{\hat\beta_k-1.96SE(\hat\beta_k)\}, \exp\{\hat\beta_k+1.96SE(\hat\beta_k)\} \right].

This follows because the approximate normal interval for the coefficient is $\hat\beta_k\pm1.96SE(\hat\beta_k)$, and exponentiation converts the coefficient limits to hazard-ratio limits. The interval is generally asymmetric around the hazard-ratio estimate because the transformation is nonlinear. A 95% confidence interval describes the long-run coverage of the procedure under its assumptions; it is not a 95% probability statement about this fixed parameter.

Worked Example: Reporting a Treatment Contrast

Consider hypothetical follow-up of 600 patients assigned to treatment A or B. The prespecified outcome for this model is cause-specific failure $k$; a different first event removes a patient from the event-free risk set. Suppose 120 cause-$k$ events occurred. The fitted Cox model adjusts for age and disease stage, and treatment A is the reference. The treatment B coefficient is $\hat\beta=0.336$, with $SE=0.145$. Assume the proportional-hazards and covariate-form diagnostics do not show substantial problems, and censoring from loss to follow-up is conditionally independent given the modeled covariates.

The adjusted hazard ratio for B versus A is $\exp(0.336)=1.399$, or 1.40 to three significant figures. Its 95% Wald interval is

\exp\{0.336\pm1.96(0.145)\} =\exp(0.336\pm0.2842) =\left[\exp(0.0518),\exp(0.6202)\right] =(1.053,\ 1.859).

The Wald statistic is $z=0.336/0.145=2.317$. The two-sided standard-normal p-value is approximately $0.02049$. A suitable results sentence is: “Among 600 patients, with 120 cause-$k$ events, treatment B was associated with a higher cause-specific hazard of cause $k$ than treatment A (adjusted HR 1.40, 95% CI 1.05–1.86; Wald $p=0.02049$), adjusting for age and disease stage.” The estimate compares instantaneous cause-$k$ rates in the event-free risk sets; it does not state that B increases the cumulative probability of cause $k$ by 40%.

Reference Categories and Adjustment

For a categorical predictor, report the reference category explicitly and give each contrast in relation to it. The reference is encoded in the model, but a table containing only category names and estimates may not make the comparisons clear. For a continuous predictor, state the unit of change represented by the estimate. If the unit is not clinically helpful, rescale it before fitting or clearly explain the scale used.

Adjustment means that the reported coefficient is conditional on the other covariates included in the specified model. It does not by itself show that confounding has been eliminated. Readers need the covariate list and, where relevant, the way continuous predictors were represented. Do not describe an adjusted hazard ratio as a causal effect unless the study design and causal assumptions support that interpretation.

Worked Example: Reporting Multiple Stage Comparisons

Suppose a hypothetical model of cause-specific mortality includes disease stage and adjusts for age and treatment group. Stage I is the reference. The estimates are $\hat\beta_{\mathrm{II}}=0.405$, $SE=0.180$, and $\hat\beta_{\mathrm{III}}=0.693$, $SE=0.210$. Assume the model's proportional-hazards and censoring assumptions are reasonable for this illustration.

For Stage II versus Stage I, the adjusted hazard ratio is $\exp(0.405)=1.499$. The 95% interval is $\exp\{0.405\pm1.96(0.180)\}=\exp(0.0522,0.7578)=(1.054,2.134)$. The Wald statistic is $0.405/0.180=2.250$, giving a two-sided $p$-value of approximately $0.02445$.

For Stage III versus Stage I, the adjusted hazard ratio is $\exp(0.693)=2.000$. The 95% interval is $\exp\{0.693\pm1.96(0.210)\}=\exp(0.2814,1.1046)=(1.325,3.018)$. Here $z=0.693/0.210=3.300$, with a two-sided $p$-value of approximately $0.0009676$.

A clear table would label both contrasts “versus Stage I (reference)” and identify age and treatment group as adjustment variables in its caption or a note. The estimates describe cause-specific hazard contrasts for patients with the same modeled age and treatment group who remain event-free. They do not compare absolute risks of cause-specific mortality.

Diagnostics, Failure Modes, and Alternatives

Before presenting a single hazard ratio as a time-constant summary, check whether proportional hazards is plausible for the covariate effects. A time-varying effect, nonlinearity in a continuous predictor, influential observations, sparse events in a category, or convergence problems can make the point estimate misleading or unstable. Report meaningful diagnostics and any model changes; an interval alone does not reveal these failures. Also verify that the reference coding, cause definition, event counts, and reported adjustment set match the fitted model.

The censoring assumption for ordinary loss to follow-up should be described precisely: conditional on modeled covariates, censoring must not selectively remove people in a way that carries additional information about their future cause-specific event process. Competing events are different: they end the event-free time at risk for every cause-specific hazard, by definition. Estimating a cause-specific hazard does not require treating a competing event as if it were independent loss to follow-up for the purpose of estimating cumulative incidence.

Cause-specific Cox regression and cumulative incidence summaries answer different questions. The cause-specific hazard ratio is a relative instantaneous rate among those still event-free; cumulative incidence is an absolute probability of experiencing a cause by a specified time in the presence of competing events. A Fine–Gray model instead targets a subdistribution hazard and has a different risk-set interpretation. These measures should not be given interchangeable labels or interpreted as if they estimated the same quantity.

Common mistake: “The hazard ratio is 1.40, so treatment B causes a 40% higher probability of the event.” This changes both the estimand and the claim: the cause-specific hazard ratio compares instantaneous cause-$k$ rates among those currently event-free, not cumulative probabilities, and an observational association alone does not establish causation. Instead name the cause, the event-free risk set, the comparison, and the adjustment set, and call the result an adjusted cause-specific hazard ratio.

A reporting checklist can prevent common omissions:

Name the modeled cause and specify that the risk set is event-free from all causes.
Give the comparison and reference category, including the unit for continuous predictors.
List adjustment variables and report the analysis sample size and cause-specific event count.
Report the hazard ratio, confidence interval, and any stated test or p-value consistently.
Describe relevant model diagnostics and avoid translating a hazard ratio into an absolute-risk difference.

Worked Example: Rescaling a Continuous Predictor

In a hypothetical model for cause-specific hospitalization, age is entered in units of 10 years. The model adjusts for treatment group and disease stage. The estimated coefficient for age per 10 years is $\hat\beta=0.182$, with $SE=0.070$. Assume the proportional-hazards assumption is reasonable over the observed follow-up, the age relationship is adequately represented as linear on the log-hazard scale, and censoring from loss to follow-up is conditionally independent.

The adjusted hazard ratio per 10-year increase is $\exp(0.182)=1.200$. Its 95% Wald interval is $\exp\{0.182\pm1.96(0.070)\}=\exp(0.0448,0.3192)=(1.046,1.376)$. The Wald statistic is $0.182/0.070=2.600$, giving a two-sided p-value of approximately $0.009322$.

A suitable report is: “For each 10-year increase in age, the adjusted cause-specific hazard of hospitalization was estimated to be 1.20 times as high (95% CI 1.05–1.38; Wald $p=0.009322$), conditional on treatment group and disease stage, among patients who remained event-free.” The specified unit is essential: this is a comparison per decade, not per year. The estimate is an association in the fitted model and does not imply that the cumulative hospitalization probability rises by 20%.

Check Your Understanding

Answer each item, then compare your reasoning with the solutions below.

A cause-specific Cox model reports HR 0.72 for treatment B versus A, with A as reference. State the comparison and the appropriate risk-set interpretation.
A reported estimate says “HR 1.5, adjusted.” Name two pieces of information needed to make this statement interpretable.
For a coefficient $\hat\beta=0.20$ with $SE=0.10$, calculate the hazard ratio and approximate 95% Wald confidence interval.
Explain why a cause-specific hazard ratio of 1.40 is not a statement that the cumulative incidence is 40% higher.

Solutions. (1) Treatment B has an estimated cause-specific hazard 0.72 times that for treatment A among individuals with the same modeled covariates who are still free of every event type immediately before time $t$. (2) State the modeled cause, the reference category or contrast, and the adjustment variables; ideally also report the confidence interval, sample size, and cause-specific event count. (3) The HR is $\exp(0.20)=1.221$. The log-scale limits are $0.20\pm1.96(0.10)=(0.004,0.396)$, so the HR limits are $(\exp(0.004),\exp(0.396))=(1.004,1.486)$. (4) A cause-specific HR compares instantaneous rates within event-free risk sets. Cumulative incidence is an absolute probability over time and also depends on the competing causes, so it cannot be read directly from this HR.

Reporting Cause-Specific Cox Estimates

What You'll Learn