Deriving the Cause-Specific Cox Partial Likelihood

Why the Partial Likelihood Uses Cause-Specific Failures

A Cox model estimates covariate effects without specifying the shape of the baseline hazard. In a competing-risks setting, the derivation must also distinguish the event type: for a model of cause $k$, the event terms correspond to failures from cause $k$, not failures from every cause. The event-time risk sets and the conditional probability of the failing subject give a direct route to that likelihood.

The coding convention is that each subject has a follow-up time, a cause label for an observed event, or a censoring label. At a particular event time, the risk set consists of subjects still under observation and free of any event immediately before that time. For the target cause, the cause-specific Cox model specifies its hazard among those still at risk.

Definition: Let $T_i$ be the observed follow-up time for subject $i$, $\delta_i$ the event-type indicator ($\delta_i=0$ denotes censoring), and $x_i$ the covariate vector. For target cause $k$, define the risk set just before time $t$ by $R(t)=\{j:T_j\ge t\}$, subject to the convention that a subject is at risk immediately before an event at $t$. The cause-specific Cox model is $$ \lambda_k(t\mid x_i)=\lambda_{0k}(t)\exp(\beta_k^\top x_i), $$ where $\lambda_k(t\mid x_i)$ is the instantaneous hazard of cause $k$ at $t$, conditional on being event-free and under observation just before $t$, $\lambda_{0k}(t)$ is an unspecified baseline cause-specific hazard, and $\beta_k$ is the target-cause regression coefficient vector.

For now suppose event times are distinct. Let $D_k=\{i:\delta_i=k\}$ denote the subjects who experience the target cause, and let $t_i=T_i$ for $i\in D_k$. The partial likelihood is built by multiplying a factor for each such event time. It does not include a failure factor for a competing cause; that subject can be in risk sets before its event, but its event does not supply a target-cause failure factor.

Deriving One Event-Time Contribution

Consider a time $t$ at which exactly one subject experiences cause $k$. Conditional on the risk set immediately before $t$, the model gives each subject $j$'s cause-$k$ event rate as $\lambda_{0k}(t)\exp(\beta_k^\top x_j)$. Over a very short interval of length $dt$, the probability that subject $j$ experiences cause $k$ is approximately this rate times $dt$. Thus, conditional on a cause-$k$ failure occurring in the risk set at $t$, the probability that $i$ is the failing subject is proportional to its cause-$k$ rate.

P(i\text{ fails from cause }k\mid \text{one cause-}k\text{ failure in }R(t)) = \frac{\lambda_{0k}(t)\exp(\beta_k^\top x_i)} {\sum_{j\in R(t)}\lambda_{0k}(t)\exp(\beta_k^\top x_j)} = \frac{\exp(\beta_k^\top x_i)} {\sum_{j\in R(t)}\exp(\beta_k^\top x_j)}.

The last equality follows because the same baseline hazard $\lambda_{0k}(t)$ multiplies every rate in that risk set, so it cancels. This conditional probability is the event-time contribution to the partial likelihood. A competing cause has its own hazard, but the conditioning event here is specifically a cause-$k$ failure; only cause-$k$ rates appear in the probability of which at-risk subject had that failure.

Result: With distinct event times, the cause-specific Cox partial likelihood for cause $k$ is $$ L_k(\beta_k)= \prod_{i:\delta_i=k} \frac{\exp(\beta_k^\top x_i)} {\sum_{j\in R(T_i)}\exp(\beta_k^\top x_j)}. $$ Its log form is $$ \ell_k(\beta_k)= \sum_{i:\delta_i=k} \left[ \beta_k^\top x_i- \log\left\{\sum_{j\in R(T_i)}\exp(\beta_k^\top x_j)\right\} \right]. $$

The product is called partial because it captures the information about $\beta_k$ in the ordering of target-cause failures relative to the covariates in their risk sets, without specifying or estimating the baseline hazard as part of this likelihood. Maximizing it gives the partial-likelihood estimate when a finite maximizer exists.

Worked Examples

Worked Example: One Target-Cause Failure

Suppose a hypothetical risk set at time $t=4$ contains three people with a binary covariate $x$, and one experiences the target cause at that time. Their covariate values are $x_A=1$, $x_B=0$, and $x_C=1$; person A has the target-cause event. Assume the cause-specific Cox model $\lambda_k(t\mid x)=\lambda_{0k}(t)e^{\beta_k x}$. Find the event contribution at $\beta_k=\log(2)$.

The risk set is $R(4)=\{A,B,C\}$, and the target-cause failure is A. The contribution is $$ \frac{e^{\beta_k x_A}}{e^{\beta_k x_A}+e^{\beta_k x_B}+e^{\beta_k x_C}} = \frac{e^{(\log 2)(1)}}{e^{(\log 2)(1)}+e^{(\log 2)(0)}+e^{(\log 2)(1)}} = \frac{2}{2+1+2} = 0.4000. $$

This is the conditional model probability assigned to A being the cause-$k$ failure in this risk set at the specified coefficient value. It is not the probability that A experiences the target cause by time 4.

Worked Example: Two Target-Cause Events

In a hypothetical cohort, three subjects are under observation at time 2, with covariates $x_A=0$, $x_B=1$, and $x_C=2$. A experiences the target cause at time 2. Subject C is censored at time 3. At time 5, B experiences the target cause, and the risk set is then $\{B\}$. Calculate the partial likelihood at $\beta_k=\log(2)$.

At time 2, the contribution is $$ \frac{e^{\beta_k x_A}}{\sum_{j\in R(2)}e^{\beta_k x_j}} = \frac{1}{1+2+4} = \frac{1}{7}. $$

At time 5, only B is at risk, so its contribution is $$ \frac{e^{\beta_k x_B}}{\sum_{j\in R(5)}e^{\beta_k x_j}} = \frac{2}{2} = 1. $$

Therefore, $$ L_k(\log 2)=\frac{1}{7}\times 1=0.1429. $$ The second event contributes a factor of 1 because there is no other subject in its risk set to compare with B. It adds no information about $\beta_k$.

Worked Example: Event Types Determine the Factors

Consider a hypothetical study with three subjects, all having $x=0$. Subject A experiences the target cause at time 2. Subject B experiences a competing cause at time 3. Subject C is censored at time 4. All three are at risk just before time 2. Identify which event times supply factors to the cause-$k$ partial likelihood and calculate it.

At time 2, A has a cause-$k$ event. Its factor is $e^0/(e^0+e^0+e^0)=1/3$. At time 3, B has a competing event, not a cause-$k$ event, so time 3 supplies no event factor for this likelihood. At time 4, censoring also supplies no failure factor.

Thus the complete partial likelihood is $L_k(\beta_k)=1/3$ for every $\beta_k$ in this example. Since every covariate value is identical, the data provide no covariate contrast for estimating a coefficient. This example isolates the event-type rule; in a nontrivial risk set, a competing event's subject can have affected earlier denominators by being at risk before that event.

Why Only Target-Cause Failures Contribute

The event term arises by asking which at-risk subject experienced a cause-$k$ failure at a time when such a failure occurred. If subject $i$ instead experiences cause $m\ne k$, the observed event is not a cause-$k$ failure, so it does not answer that conditional question and contributes no numerator for the cause-$k$ partial likelihood. Likewise, a censoring time does not identify a failure from cause $k$. In contrast, before a competing event or censoring time, that subject may be part of the risk set and therefore part of a denominator at an earlier cause-$k$ event time.

This distinction follows from the estimand and model: the cause-specific hazard is the instantaneous rate of cause $k$ among subjects still free of all events. The partial likelihood uses those at-risk subjects as the comparison set, but it uses a failure numerator only when the observed event is cause $k$. The details of how competing events affect later risk sets are a separate risk-set construction issue.

Assumptions, Ties, and Diagnostics

The derivation relies on a correctly defined time origin, event times, event types, and covariates measured for the people at risk. It also uses the cause-specific proportional-hazards model: the covariate effect for cause $k$ is represented by a time-constant coefficient vector $\beta_k$, while the baseline cause-specific hazard may vary freely over time. For the usual interpretation of the observed risk sets, censoring must be non-informative conditional on the modeled covariates and relevant observed history: after conditioning on that information, censoring should not selectively remove subjects according to their unobserved future event process. Independent sampling units and accurate event-type classification are also required for standard likelihood-based inference.

In continuous time, distinct event times are expected under regular conditions, but recorded times can be tied because of rounding or genuinely grouped measurement. The distinct-time formula does not directly describe a tied group as though its failures happened in a known order. Common implementations use approximations such as Breslow or Efron, or an exact conditional method; the appropriate choice depends on the tie structure and analysis convention. The formula above is the derivation for distinct event times, not a claim that ties can always be ignored.

Check the event coding.
Confirm that the event label for each subject distinguishes cause $k$, other causes, and censoring. A mislabeled competing event can create an incorrect numerator.

Reconstruct event-time risk sets.
At each target-cause event, verify who was still observed and event-free immediately beforehand. A subject's inclusion or exclusion changes the denominator.

Check covariate variation and support.
If target-cause failures have little covariate contrast within their risk sets, the likelihood may contain little information about some coefficients. Sparse failures or separation can also lead to unstable estimates or no finite maximizer.

Inspect the time and model setup.
Investigate unexpected ties, inconsistent time units, and whether a constant coefficient over follow-up is plausible. The partial-likelihood derivation alone does not verify proportional hazards or correct functional form.

A common mistake is to put every observed failure in the product because every failure is an event. That is wrong for a cause-$k$ partial likelihood: only cause-$k$ events supply event-time factors. The correction is to use a numerator for each target-cause failure, while constructing its denominator from the relevant at-risk set. Another mistake is to remove a subject from earlier risk sets because that person later has a competing event; the later outcome cannot change who was at risk at an earlier event time.

What the Derivation Does—and Does Not—Provide

The partial likelihood isolates estimation of covariate effects from the unspecified baseline hazard. It is not itself an absolute-risk prediction formula: a cause-specific hazard ratio or coefficient does not by itself give the probability of experiencing cause $k$ by a chosen time. Absolute-risk calculations require baseline hazard information and must account for the competing causes as well. Nor does deriving a partial likelihood establish a causal interpretation; the fitted association is conditional on the model and covariates, and causal claims require additional design and assumptions.

The key comparison is with a full event-time likelihood, which models the event-time distribution and baseline hazards more explicitly, and with a Fine–Gray regression, which targets a different hazard and uses a different risk-set construction. Those alternatives do not change the derivation here: the cause-specific Cox partial likelihood follows from conditioning on target-cause failures within their event-time risk sets.

Key takeaway: For cause $k$, each distinct target-cause failure contributes its modeled cause-$k$ rate divided by the sum of cause-$k$ rates over the event-time risk set. The common baseline hazard cancels, and other event types do not create target-cause failure terms.

Check Your Understanding

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

At an event time, the risk set has covariate values $0$ and $1$, and the subject with value $1$ experiences cause $k$. Write that event's partial-likelihood contribution.
Why does a competing-cause event not supply a numerator in the cause-$k$ partial likelihood?
There are two target-cause failures with event-time contributions $1/4$ and $2/5$. What is the partial likelihood at the coefficient value used to compute those contributions?
What happens to a target-cause event-time contribution if the failing subject is the only person in its risk set?
Name one reason the distinct-event-time formula may need modification in practice.

Solutions. (1) The contribution is $e^{\beta_k}/(1+e^{\beta_k})$. (2) The numerator represents the subject experiencing cause $k$ at that time; a different cause is not an observed cause-$k$ failure. (3) Multiply the factors: $L_k=(1/4)(2/5)=2/20=0.1000$. (4) It is $e^{\beta_k^\top x_i}/e^{\beta_k^\top x_i}=1$, so it gives no covariate contrast. (5) Tied event times may require a tie method such as Breslow, Efron, or an exact conditional calculation.