Stratified Cause-Specific Cox Regression

Why Stratify a Cause-Specific Cox Model?

In a competing-risks analysis, a cause-specific Cox model describes the instantaneous rate of a particular event among people who have not yet experienced any event. A covariate may have a common association with that rate across several groups even when the groups have very different underlying event rates. For example, treatment effects might be modeled as shared across hospitals while each hospital has its own baseline rate.

Stratification handles this situation by allowing the baseline hazard to vary freely between strata, while estimating one shared coefficient for each covariate. It is useful when a variable defines important baseline differences but its own hazard ratio is not the main target, or when a common baseline hazard across groups is implausible.

Definition: For cause $k$, a stratified cause-specific Cox model with stratum $g$ is $$ \lambda_{kg}(t\mid X)=\lambda_{0kg}(t)\exp(\beta^\mathsf{T}X). $$ Here $\lambda_{kg}(t\mid X)$ is the cause-specific hazard at time $t$, $\lambda_{0kg}(t)$ is the otherwise unrestricted baseline hazard for cause $k$ in stratum $g$, $X$ is the covariate vector, and $\beta$ is shared across strata. The model allows different baseline hazards across strata, but assumes the covariate effects are common across them.

Let $T$ denote the observed time to the first event or censoring, $J$ the event type (with $J=0$ for censoring), and $G$ the stratum. For a model of cause $k$, the risk set $R_g(t)$ consists of individuals in stratum $g$ who are still event-free and under observation immediately before $t$. A person who later experiences a competing event is in the risk set before that event; after it, that person is no longer at risk for any first event.

Partial Likelihood Within Strata

Assume first that event times are distinct. At a time when one person in stratum $g$ experiences cause $k$, the model assigns that person a relative event rate proportional to $\lambda_{0kg}(t)\exp(\beta^\mathsf{T}X)$. Conditional on one cause-$k$ event occurring among the people at risk in that stratum, the probability that individual $i$ is the one who experiences it is proportional to their rate divided by the sum of rates in that risk set. The shared baseline hazard cancels from that ratio.

\frac{\lambda_{0kg}(t)\exp(\beta^\mathsf{T}X_i)} {\sum_{j\in R_g(t)}\lambda_{0kg}(t)\exp(\beta^\mathsf{T}X_j)} = \frac{\exp(\beta^\mathsf{T}X_i)} {\sum_{j\in R_g(t)}\exp(\beta^\mathsf{T}X_j)}.

Multiplying these conditional contributions over cause-$k$ events gives the stratified partial likelihood. Only people in the event subject’s stratum appear in its denominator. Thus, estimation compares covariate values among people in the same stratum who were at risk at that event time; it does not require comparing baseline event rates between strata.

L_p(\beta)= \prod_g\ \prod_{\substack{i:\,J_i=k\\G_i=g}} \frac{\exp(\beta^\mathsf{T}X_i)} {\sum_{j\in R_g(T_i)}\exp(\beta^\mathsf{T}X_j)}.

This derivation uses distinct event times. Tied failures require a specified tie-handling method; the simple expression above should not be applied unchanged to a tied event set. The stratification structure itself does not remove the need to handle ties appropriately.

Worked Example: Writing the Model for Several Hospitals

Suppose a hypothetical study follows patients at three hospitals and models the cause-specific hazard of cardiovascular death. The covariates are treatment (1 for the new treatment, 0 for control) and age in years. Hospital is the stratum. A suitable model is $$ \lambda_{k g}(t\mid X)=\lambda_{0kg}(t) \exp\{\beta_1\text{Treatment}+\beta_2\text{Age}\}, $$ where $g$ indexes the three hospitals and $k$ denotes cardiovascular death.

Each hospital has its own baseline hazard, so the model does not require the hospitals to have the same underlying rate over time. It does require a shared treatment effect and a shared age effect. If $\exp(\beta_1)=0.80$, then, at the same age and within any one hospital, the fitted cause-specific hazard for treated patients is 0.80 times that for control patients. This is a 20% lower cause-specific hazard, not a 20% lower probability of cardiovascular death.

Hospital itself has no estimated coefficient in this model: its effect is absorbed into the hospital-specific baseline hazards. The model is appropriate only if the common covariate effects are a reasonable assumption and the study design and censoring process support cause-specific Cox estimation.

A Numerical Partial-Likelihood Calculation

The following hypothetical risk sets illustrate how the common coefficient is estimated. There is one binary covariate $X$. Assume no tied event times, correct risk sets, and independent censoring conditional on modeled information. Each listed event is of the cause being modeled.

Worked Example: Estimating a Shared Effect From Two Strata

In stratum A, a cause-$k$ event occurs for a person with $X=1$. The risk set at that time contains that person and two others with $X=0$ and $X=1$. In stratum B, a cause-$k$ event occurs for a person with $X=0$; the risk set contains that person and one person with $X=1$. Write $y=\exp(\beta)$.

The contribution from stratum A is $y/(2+y)$: the event subject contributes $y$, and the risk-set denominator is $1+1+y=2+y$. The contribution from stratum B is $1/(1+y)$: the event subject contributes $1$, and the denominator is $1+y$. Therefore, $$ L_p(\beta)=\frac{y}{2+y}\frac{1}{1+y}. $$

The log partial likelihood is $\ell(\beta)=\log y-\log(2+y)-\log(1+y)$. Since $dy/d\beta=y$, its derivative is $$ \frac{d\ell}{d\beta} =1-\frac{y}{2+y}-\frac{y}{1+y} =\frac{2-y^2}{(2+y)(1+y)}. $$

Setting the score to zero gives $2-y^2=0$, so $y=\sqrt{2}=1.414$, and $\hat{\beta}=\log(\sqrt{2})=0.3466$. The fitted shared cause-specific hazard ratio for $X=1$ versus $X=0$ is therefore 1.414. This estimate comes from within-stratum risk-set comparisons; it does not require the baseline hazards in A and B to be equal. With such small risk sets, the estimate is only an arithmetic illustration, not a reliable clinical conclusion.

Estimating Baselines and Checking the Shared-Effect Assumption

After estimating $\beta$, a stratum-specific baseline cumulative hazard can be estimated from that stratum’s event times. Under the Breslow estimator, the increment at time $t$ is the number $d_{kg}(t)$ of cause-$k$ events in stratum $g$ at $t$, divided by the risk-set sum of fitted relative hazards. For distinct event times, $d_{kg}(t)=1$. This estimates a baseline for the covariate reference value $X=0$, or for whatever reference values define the zero of the linear predictor.

\Delta\widehat{\Lambda}_{0kg}(t) = \frac{d_{kg}(t)} {\sum_{j\in R_g(t)}\exp(\hat{\beta}^\mathsf{T}X_j)}.

Worked Example: Different Baseline Increments, Shared Hazard Ratio

Consider a hypothetical model with one binary covariate and a fitted shared coefficient $\hat{\beta}=\log(2)$. At one event time in stratum A, one cause-$k$ event occurs for a person with $X=1$; the risk set also contains two people with $X=0$. At one event time in stratum B, one cause-$k$ event occurs for a person with $X=0$; one other person with $X=0$ is also at risk. Assume distinct event times and the stated risk sets.

In A, the risk-set sum is $2+1+1=4$, because $\exp(\hat{\beta})=2$. The baseline cumulative hazard increment is $1/4=0.2500$. In B, the risk-set sum is $1+1=2$, so the increment is $1/2=0.5000$. These are stratum-specific increments for the baseline covariate value. At either stratum, the fitted cause-specific hazard ratio comparing $X=1$ with $X=0$ remains $\exp(\hat{\beta})=2$.

The two increments illustrate that separate baseline hazard functions are estimated; a single shared baseline curve is not imposed. They do not, by themselves, establish that one stratum has a truly higher underlying hazard: increments also depend on the observed event and risk-set information.

The key modeling restriction is not equality of baselines but equality of covariate effects. Useful diagnostics include examining cause-specific proportional-hazards diagnostics, such as scaled Schoenfeld residuals over time, and checking whether they suggest time-varying covariate effects. Also compare covariate-effect patterns across strata where data permit. Large differences can indicate that a shared coefficient is too restrictive, although stratum-specific estimates may be imprecise when event counts are low. Inspect risk-set overlap as well: if a covariate has almost no variation within the strata and event times that contribute information, its effect may be weakly identified or not estimable.

Assumptions, Alternatives, and Limitations

A stratified cause-specific Cox analysis relies on several conditions. The covariate effects must be proportional over time within the modeled strata unless time-varying effects are explicitly included. Covariate functional forms must be adequate, and the strata must be defined for the analysis rather than chosen to create unsupported comparisons. Loss to follow-up censoring must be independent of the event process conditional on the modeled information, or otherwise handled by an appropriate method. The usual partial-likelihood interpretation also assumes correctly recorded event times and risk sets. Stratification does not correct confounding, informative censoring, or a misspecified covariate effect.

The main alternatives differ in what they assume or estimate:

Approach	Baseline and covariate structure	Key implication
Unstratified Cox model	One baseline hazard shared across groups; covariate effects are modeled.	More efficient if the common baseline is plausible, but can be too restrictive when baseline hazards differ.
Stratified Cox model	Separate baseline hazards by stratum; shared covariate coefficients.	Allows flexible baseline differences but does not estimate a coefficient for the stratifying variable.
Separate Cox models by stratum	Separate baselines and separate covariate coefficients.	Allows effects to differ but uses less information for each fit and may yield imprecise estimates.
Model with stratum-by-covariate interactions	Baseline structure is specified, with covariate effects allowed to vary by stratum.	Directly models effect variation, but requires enough events to estimate the additional parameters.

Stratification is not a method for estimating the effect of the stratum itself, and it does not produce a single common baseline survival curve. A cause-specific hazard ratio is a comparison of instantaneous event rates among people still event-free at time $t$; it is not a risk ratio or a direct comparison of cumulative event probabilities. If the scientific target is an event probability in the presence of competing events, a hazard ratio alone does not answer that question.

Common Pitfalls

Pooling risk sets across strata. This wrongly compares people under different baseline hazards. Use only the event subject’s own stratum in each partial-likelihood denominator.
Assuming stratification makes effects stratum-specific. It does not. The standard stratified model has distinct baselines but common $\beta$. Use an interaction or separate models if effect variation is the target.
Reporting a coefficient for the stratifying variable. Its effect is not separately estimated when each stratum has its own unrestricted baseline hazard.
Calling a cause-specific hazard ratio a probability ratio. It describes an instantaneous rate among those still event-free, not the probability of eventually experiencing the cause.
Ignoring sparse strata or tied failures. Sparse risk sets can provide little information about shared effects; tied events require an appropriate tie method rather than the distinct-time formula unchanged.

Key takeaway: Stratified cause-specific Cox regression trades estimation of a common baseline hazard for flexibility: each stratum gets its own baseline hazard, while the modeled covariate effects remain shared. Interpret the estimated coefficients as cause-specific hazard ratios and check whether the shared-effect and proportional-hazards assumptions are credible.

Check Your Understanding

Answer each question, then compare with the solutions below.

What differs across strata in a standard stratified cause-specific Cox model, and what remains shared?
Why does a partial-likelihood denominator include only members of the event subject’s stratum?
A shared treatment coefficient is $\log(0.75)$. State the hazard-ratio interpretation and one interpretation it does not support.
What should be considered if treatment effects appear substantially different across strata?

Solutions. (1) The baseline cause-specific hazard differs by stratum; covariate coefficients are shared. (2) Stratification permits arbitrary baseline differences, so valid conditional comparisons for that event are made within the same stratum; the baseline cancels there. (3) The fitted cause-specific hazard for treated versus control individuals at the same modeled covariate values is 0.75, a 25% lower instantaneous cause-specific hazard. It does not imply a 25% lower cumulative probability of the event. (4) Assess whether the pattern is supported by adequate events and risk-set overlap, and consider a model with stratum-by-treatment interaction or separate stratum-specific fits; the standard shared-effect model may be too restrictive.