Fitting Separate Cox Models for Multiple Causes

Why Fit a Model for Each Cause?

When more than one event type can occur, a covariate need not relate to every cause in the same way. A treatment might be associated with a lower cause-specific hazard for one outcome and little change, or a different association, for another. Fitting one cause-specific Cox model per event type makes those distinct associations explicit. The models are separate regressions, not a claim that the event types or their risks are unrelated.

For a cause-specific hazard, the population at risk at time $t$ consists of individuals who have not yet experienced any event and are still under observation just before $t$. Once an individual experiences a competing event, that individual leaves this risk set. The cause-specific hazard describes the instantaneous rate of the event of interest among individuals who remain event-free up to that time; it is not itself the probability of that event by a future horizon.

Definition: Let $T_i$ be the event time, $J_i\in\{1,\ldots,K\}$ the event type, and $C_i$ the censoring time for individual $i$. The observed time is $\tilde T_i=\min(T_i,C_i)$. For cause $k$, the cause-specific Cox model is $$ \lambda_k(t\mid X_i)=\lambda_{0k}(t)\exp(\beta_k^\mathsf{T}X_i), $$ where $\lambda_k(t\mid X_i)$ is the cause-$k$ hazard among individuals event-free just before $t$, $\lambda_{0k}(t)$ is an unspecified baseline hazard, and $\beta_k$ is the cause-specific coefficient vector. Each cause has its own baseline hazard and may have its own coefficient vector.

Setting Up and Fitting the Separate Models

For the model of cause $k$, mark an observed event as an event only if its type is $k$. Individuals who experience another cause are treated as leaving the cause-$k$ risk set at that time; they are not retained as though they could later experience cause $k$. Individuals who are censored contribute follow-up through their censoring time. This is the cause-specific analysis convention: it defines the risk set for the cause-specific hazard, rather than asserting that a competing event is ordinary, independent loss to follow-up.

For distinct cause-$k$ event times $t_{kj}$, let $R_k(t_{kj})$ be the individuals event-free and under observation immediately before that time. If there are no tied event times, the cause-specific partial likelihood is

L_k(\beta_k)= \prod_j \frac{\exp(\beta_k^\mathsf{T}X_{i_{kj}})} {\sum_{i\in R_k(t_{kj})}\exp(\beta_k^\mathsf{T}X_i)},

where $i_{kj}$ is the individual who experiences cause $k$ at $t_{kj}$. With tied event times, a fitting procedure must use an appropriate tie method, such as the Breslow or Efron method, rather than apply the distinct-time expression unchanged.

The reason a cause-specific fit can be constructed is that, conditional on a cause-$k$ event occurring at a particular time among its risk set, the relative probability that individual $i$ is the one experiencing it is proportional to $\exp(\beta_k^\mathsf{T}X_i)$. The common baseline hazard cancels from this conditional comparison. Maximizing the resulting partial likelihood estimates the covariate effects for that cause; the baseline hazard is not needed to estimate these coefficients.

The standard interpretation requires a correctly specified log-linear covariate relationship for the fitted model and proportional cause-specific hazards over the modeled time range: for two covariate vectors, the hazard ratio is constant over time. The analysis also requires appropriate handling of censoring, typically independent of the event process conditional on included covariates, and valid event-type and time measurements. The data should support the parameters being estimated: very few events for a cause or sparse covariate patterns can make estimates unstable or separated. Formal proportional-hazards diagnostics are a separate subject, but the assumption should not be silently taken for granted.

Worked Example: Constructing a Cause-Specific Risk Set

Consider six hypothetical individuals, with a binary exposure $X$ (1 = exposed, 0 = unexposed). Their observed records are: A, $X=1$, cause 1 at time 2; B, $X=0$, cause 2 at time 3; C, $X=1$, cause 2 at time 4; D, $X=0$, cause 1 at time 5; E, $X=0$, censored at time 6; and F, $X=1$, censored at time 7. For illustration, fit the cause-1 model with coefficient $\beta$.

At time 2, all six individuals are in the cause-1 risk set. The numerator for A's cause-1 event is $\exp(\beta)$, and the denominator is $3\exp(\beta)+3$, because three individuals have $X=1$ and three have $X=0$. At time 5, B and C have already left the risk set after their cause-2 events. D, E, and F remain, so the numerator for D's event is 1 and the denominator is $\exp(\beta)+2$. Thus

L_1(\beta)= \frac{\exp(\beta)}{3\exp(\beta)+3} \cdot \frac{1}{\exp(\beta)+2}.

Writing $y=\exp(\beta)$, the log-likelihood is $\ell(\beta)=\beta-\log(3y+3)-\log(y+2)$. Its derivative is $\ell'(\beta)=1-y/(y+1)-y/(y+2)$. Setting this equal to zero gives $1-y/(y+1)-y/(y+2)=0$, or $y^2=2$. Therefore the maximum occurs at $y=\sqrt{2}$, so $\hat\beta=\log(\sqrt{2})=0.3466$, and the estimated cause-1 hazard ratio is $\exp(\hat\beta)=1.414$. This tiny example is only an arithmetic illustration; it is far too small for dependable inference. It also shows why B and C leave the cause-1 risk set at their competing-event times, rather than remaining in it until their later observed or hypothetical cause-1 time.

Organizing and Interpreting Cause-Specific Effects

For $K$ event types, fit a model for each cause with its own baseline hazard and coefficient vector. A useful report puts estimates side by side, names the outcome and covariate for each estimate, and gives uncertainty measures. A cause-specific hazard ratio compares instantaneous rates among individuals still event-free at the time in question. It does not directly compare cumulative probabilities of experiencing the event by a fixed time.

Worked Example: Reporting Two Treatment Effects

Suppose a hypothetical randomized study records two possible first events. Separate cause-specific Cox fits give a treatment coefficient of $-0.400$ with standard error $0.180$ for cause 1, and $0.150$ with standard error $0.200$ for cause 2. The proportional-hazards assumption is being used for each fitted cause-specific model, and assume for this illustration that the event counts and follow-up support estimation. For the cause-1 estimate, the hazard ratio is $\exp(-0.400)=0.6703$: at a given time, among participants still event-free, the estimated instantaneous cause-1 rate in the treatment group is 0.6703 times the rate in the reference group.

A Wald 95% confidence interval for the cause-1 coefficient is $-0.400\pm1.96(0.180)=(-0.7528,-0.0472)$, which rounds to $(-0.753,-0.047)$. Exponentiating the unrounded endpoints gives a hazard-ratio interval of approximately $(0.471,0.954)$. In repeated sampling under the model and its assumptions, the procedure used to form such intervals has 95% coverage; this interval is not a statement that the fixed parameter has a 95% probability of lying inside it.

For cause 2, the estimated hazard ratio is $\exp(0.150)=1.162$. Its Wald statistic is $z=0.150/0.200=0.750$, giving a two-sided $p$-value of about $0.453$. The data do not provide strong evidence against a zero cause-2 treatment coefficient at conventional significance levels. This is not evidence that the cause-2 effect is exactly zero. Nor does the difference between the cause-1 and cause-2 point estimates by itself test whether the effects differ; that requires a direct comparison with its uncertainty accounted for.

The covariate sets need not be identical across causes. Selection should be based on the scientific question, subject-matter knowledge, and a prespecified analysis plan—not only on which covariates happen to be statistically significant in each model. A covariate may be included in all cause models for comparability or confounding control, while another may be relevant only to a particular cause. If a predictor is omitted from a model, the reported coefficients describe associations conditional on the predictors that are included; omitted variables can matter if they are related to both the included covariates and the cause-specific outcome process.

Worked Example: Different Predictors for Different Causes

In a hypothetical study, investigators consider age, treatment, and a disease-specific biomarker. Based on the scientific question, they prespecify the cause-1 model to include age and treatment, and the cause-2 model to include age, treatment, and the biomarker. Suppose the fitted treatment coefficients are $-0.250$ for cause 1 and $0.050$ for cause 2. The corresponding hazard ratios are $\exp(-0.250)=0.7788$ and $\exp(0.050)=1.051$. These are cause-specific conditional associations, interpreted among individuals still event-free, and conditional on the covariates in each respective model.

Suppose the age coefficients are $0.030$ and $0.080$, respectively. A one-year increase in age is associated with cause-specific hazard ratios of $\exp(0.030)=1.030$ for cause 1 and $\exp(0.080)=1.083$ for cause 2, under the linear log-hazard and proportional-hazards assumptions. If the cause-2 biomarker coefficient is $0.400$, its hazard ratio is $\exp(0.400)=1.492$ per one-unit increase in the biomarker, with the unit defined by its measurement scale.

This layout makes the scientific structure visible: the biomarker is part of the cause-2 model but not the cause-1 model. It does not justify saying that the biomarker has no cause-1 association; that model did not estimate one. The coefficient estimates also should not be compared as though they came from identical conditioning sets. If cross-cause contrasts are central, specify models and contrasts that directly address those comparisons.

Assumptions, Diagnostics, and Alternatives

Before interpreting a set of separate fits, check whether the modeling choices are credible for each cause. Useful checks include verifying that each individual has one valid observed time and event status, tabulating the number of events by cause, checking the number of events relative to the parameters, and examining whether covariate patterns are sparse. A very small cause-specific event count can lead to large standard errors, extreme coefficients, or estimates that change substantially when a few observations are removed. Convergence warnings and implausibly large estimates are signs to investigate, not results to report without qualification.

Also inspect whether the chosen covariates and functional forms are defensible and whether proportional hazards appear plausible for each cause. A model may be adequate for one cause but not another. Censoring assumptions and coding errors can affect all cause-specific fits; coding a competing event as ordinary censoring for the event-free risk-set construction is standard, but coding it as a cause-$k$ event is not. Missing covariate values, changing follow-up support, and a cause that is rare in important covariate strata can further limit interpretation.

An alternative is a single cause-stratified Cox analysis with cause-specific baseline hazards. If it includes cause-by-covariate interactions, it can allow the coefficients to differ by cause and can represent the same broad modeling strategy as separate fits when covariates, risk sets, and fitting conventions align. A model that instead constrains selected coefficients to be common across causes imposes a substantive equality assumption and may gain precision if that assumption is reasonable. Separate cause-specific fits are often clearer when predictor sets differ. Fine-Gray regression targets a different hazard and should not be treated as a direct replacement with the same coefficient interpretation.

Common pitfall: “A competing event was censored in the cause-1 Cox fit, so it should be treated like ordinary independent loss to follow-up, and the cause-1 hazard ratio is the cause-1 risk ratio.” Both steps are wrong. The competing event removes the individual from the cause-1 risk set because the individual is no longer event-free; this defines the cause-specific hazard. A cause-specific hazard ratio is an instantaneous rate comparison in that risk set, not a ratio of cumulative event probabilities. Absolute risk depends on the hazards for all competing causes and on follow-up time.

Key takeaway: Fit and interpret a cause-specific Cox model for each event type, with that cause’s risk sets and a clearly stated covariate set. Cause-specific effects may differ, and predictor inclusion should reflect the scientific aims and model assumptions rather than an automatic requirement that every cause use the same variables.

Check Your Understanding

Answer each question, then compare with the solutions.

At time 4, an individual experiences cause 2. Should that individual remain in the cause-1 risk set after time 4?
A fitted cause-1 coefficient is $0.20$. What is the estimated cause-specific hazard ratio, and what does it compare?
Why can a cause-2 model include a predictor that is absent from the cause-1 model?
A report gives a cause-specific hazard ratio of 0.80 and calls it a 20% reduction in cumulative cause-specific risk. What is the problem with that interpretation?

Solutions. (1) No. The person leaves the cause-1 risk set at the competing event time because they are no longer event-free. (2) $\exp(0.20)=1.221$. Under the model, this compares instantaneous cause-1 rates for individuals differing by one unit in the covariate, at the same time and with other modeled covariates held fixed. (3) Predictor sets can reflect cause-specific scientific questions and prespecified modeling aims; omission from one model does not establish absence of association for that cause. (4) The hazard ratio compares instantaneous cause-specific rates among those still event-free, not cumulative probabilities. Cumulative risk also depends on follow-up and the hazards of competing event types.

Fitting Separate Cox Models for Multiple Causes

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