Stratified Fine-Gray Regression

Why Stratify a Fine-Gray Model?

A Fine-Gray regression can relate covariates to the subdistribution hazard for a particular event type. Sometimes a grouping variable is expected to correspond to distinct underlying event patterns, but its effect is not well described by a single proportional-hazards coefficient. Stratification allows the baseline subdistribution hazard to differ across groups without requiring a coefficient for group membership. The model can still impose a common effect for other covariates.

For example, a clinical analysis might compare a treatment effect across hospitals while allowing each hospital to have its own baseline subdistribution hazard for the event of interest. The shared treatment coefficient then describes a common proportional subdistribution-hazard association across hospitals; it does not make the hospitals’ cumulative incidence functions identical.

Definition: Let $T$ be the time to the first event, $J$ its event type, $k$ the event type of interest, and $S$ a categorical stratum. The cause-$k$ subdistribution hazard in stratum $s$, conditional on covariates $Z$, is $$ \lambda^*_{k}(t\mid Z,S=s) = \lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t\ \text{or}\ (T<t,J\ne k),Z,S=s)} {\Delta t}. $$ A stratified Fine-Gray proportional subdistribution hazards model specifies $$ \lambda^*_{k}(t\mid Z,S=s) = \lambda^*_{0ks}(t)\exp(\beta^\mathsf{T}Z), $$ where $\lambda^*_{0ks}(t)$ is a separate baseline subdistribution hazard for each stratum and $\beta$ is a common coefficient vector.

The defining Fine-Gray risk set retains individuals who have already experienced a competing event, because those individuals remain relevant to the subdistribution hazard. The risk-set calculation also accounts for right censoring using the appropriate censoring weights. Stratification changes which subjects share a risk-set denominator: comparisons are made within the event subject’s stratum. The earlier material on the weighted partial likelihood for Fine-Gray regression provides the unstratified foundation for this calculation.

What Is Shared, and What Is Not?

The model permits the baseline subdistribution hazard to vary freely with time from one stratum to another. It does not estimate a regression coefficient for the stratum itself. Instead, coefficients in $\beta$ are constrained to be common across strata. For a one-unit increase in covariate $Z_r$, holding the other modeled covariates fixed, the cause-$k$ subdistribution hazard ratio is $\exp(\beta_r)$ in every stratum.

This common hazard ratio is not a common cumulative-incidence ratio. Different stratum-specific baselines can produce different cumulative incidence functions, even when the covariate coefficients are the same. In particular, a shared treatment coefficient does not imply equal absolute treatment differences across strata. Nor does a subdistribution hazard ratio directly state a ratio of event probabilities at a fixed time.

The common-effect restriction is substantive. If the treatment association differs by stratum, the stratified model without interactions does not represent that difference. A model with stratum-by-treatment interactions can allow stratum-specific coefficients, but it adds parameters and answers a different question. If estimating the association of the stratum itself is a primary goal, stratification is not the way to obtain that coefficient; a model with an explicit stratum term or another suitable structure is needed.

Likelihood Within Strata

Assume a specified target event type, correctly observed event types and times, and censoring that is independent of the event process conditional on the variables used to define the censoring model. The censoring weights must be appropriate for that assumption and have adequate support over the analysis period. For the expression below, assume event times are distinct; tied failures require a tie-handling method rather than silently applying the distinct-time expression.

Let $D_s$ be the set of observed cause-$k$ event times in stratum $s$. At an event time $t$, let $i(t)$ be the subject experiencing the target event. Write $a_j(t)$ for the Fine-Gray risk-set weight of subject $j$, including the censoring adjustment where needed. Subjects with competing events before $t$ can remain in this weighted risk set; subjects censored before $t$ are handled through the weighting scheme rather than treated as observed event-free through $t$.

The stratified weighted partial likelihood is

L(\beta) = \prod_s\prod_{t\in D_s} \frac{a_{i(t)}(t)\exp\{\beta^\mathsf{T}Z_{i(t)}\}} {\displaystyle\sum_{j:S_j=s}a_j(t)\exp\{\beta^\mathsf{T}Z_j\}}.

Only subjects in the event subject’s stratum appear in that event-time denominator. In the usual construction, the observed target-event subject has weight one at their event time. The weights for other risk-set members encode the Fine-Gray treatment of censoring and competing events; they are not generally all one.

Why the Baseline Cancels

The reason this likelihood permits separate baselines is the same cancellation principle that motivates partial likelihood, applied separately within each stratum. At a target-event time $t$ in stratum $s$, the event subject’s modeled hazard is proportional to $\lambda^*_{0ks}(t)\exp\{\beta^\mathsf{T}Z_{i(t)}\}$. The weighted total hazard contribution across that stratum’s Fine-Gray risk set is proportional to $\lambda^*_{0ks}(t)\sum_{j:S_j=s}a_j(t)\exp\{\beta^\mathsf{T}Z_j\}$. Their ratio cancels the stratum-specific baseline factor:

\frac{a_{i(t)}(t)\lambda^*_{0ks}(t)\exp\{\beta^\mathsf{T}Z_{i(t)}\}} {\lambda^*_{0ks}(t)\sum_{j:S_j=s}a_j(t)\exp\{\beta^\mathsf{T}Z_j\}} = \frac{a_{i(t)}(t)\exp\{\beta^\mathsf{T}Z_{i(t)}\}} {\sum_{j:S_j=s}a_j(t)\exp\{\beta^\mathsf{T}Z_j\}}.

Multiplying these conditional contributions over observed target-event times gives the stratified weighted partial likelihood. Because each event contribution uses only its own stratum, no comparison forces two strata to share a baseline. Because the same $\beta$ appears in every contribution, the covariate effects remain common.

Worked Example: Writing the Stratified Model

Hypothetical setting. Suppose patients are grouped by hospital $S$, and treatment $Z$ is coded 1 for the new treatment and 0 for control. The event of interest is disease progression; death before progression is a competing event. Consider a stratified Fine-Gray model with a common treatment coefficient.

For hospital $s$, the model is $\lambda^*_{k}(t\mid Z,S=s)=\lambda^*_{0ks}(t)\exp(\beta Z)$. Thus, hospital 1 has baseline $\lambda^*_{0k1}(t)$, hospital 2 has baseline $\lambda^*_{0k2}(t)$, and so on. The model makes no requirement that these baseline functions be equal.

If the fitted coefficient were $\hat\beta=-0.4055$, the estimated treatment subdistribution hazard ratio would be $\exp(-0.4055)=0.6667$. Under the model, this is the same estimated ratio in each hospital: at a given time, the treatment group’s progression subdistribution hazard is estimated to be two-thirds that of control, conditional on modeled covariates and hospital stratum. It is not a claim that progression probability is two-thirds as large at every time.

This interpretation relies on a common proportional treatment effect being reasonable across hospitals, adequate censoring adjustment, and sufficient within-hospital information to compare treatment groups. If treatment is given only in one hospital and control only in another, the within-stratum comparisons needed to estimate a treatment effect are absent.

Worked Likelihood Contributions and Estimation

Worked Example: One Weighted Event Contribution

Hypothetical data. At time $t=2$ in stratum A, subject 1 has the target event, with covariate $Z_1=1$ and risk-set weight $a_1(t)=1$. Subject 2 has $Z_2=0$, weight 1, and remains in the risk set. Subject 3 previously experienced a competing event, has $Z_3=2$, and has Fine-Gray weight 0.8 at this time. Suppose the trial coefficient value is $\beta=0.4$.

The weighted partial-likelihood contribution for this event is

\frac{1\cdot\exp(0.4\cdot1)} {1\cdot\exp(0.4\cdot1)+1\cdot\exp(0.4\cdot0)+0.8\cdot\exp(0.4\cdot2)} = \frac{1.4918}{1.4918+1+1.7804} = 0.3491.

The competing-event subject appears in the denominator with weight 0.8; removing that subject as if the competing event were ordinary censoring would change the Fine-Gray risk set. Only stratum A belongs in this denominator. A subject from stratum B is not added, even if that subject is otherwise at risk at time 2.

Worked Example: Estimating a Common Coefficient From Strata

Hypothetical simplified data. Consider three strata, each with one distinct target-event time and two subjects in its Fine-Gray risk set: one with $Z=0$ and one with $Z=1$. Suppose censoring weights are all 1 at these times, and the target event occurs to the $Z=1$ subject in strata 1 and 2, but to the $Z=0$ subject in stratum 3. This illustrates the stratified partial-likelihood calculation without competing-event or censoring weight adjustments.

For each stratum, the denominator is $1+\exp(\beta)$. The first two event contributions have numerator $\exp(\beta)$; the third has numerator 1. Therefore

L(\beta) = \left\{\frac{\exp(\beta)}{1+\exp(\beta)}\right\}^2 \left\{\frac{1}{1+\exp(\beta)}\right\} = \frac{\exp(2\beta)}{\{1+\exp(\beta)\}^3}.

The log likelihood is $\ell(\beta)=2\beta-3\log\{1+\exp(\beta)\}$. Its score is

\frac{d\ell}{d\beta} = 2-\frac{3\exp(\beta)}{1+\exp(\beta)}.

Setting the score to zero gives $\exp(\hat\beta)/(1+\exp(\hat\beta))=2/3$, so $\exp(\hat\beta)=2$ and $\hat\beta=\log(2)=0.6931$. The estimated common subdistribution hazard ratio for a one-unit increase in $Z$ is 2. This toy calculation estimates one shared coefficient from within-stratum comparisons; it does not estimate separate stratum baselines or a stratum effect.

Diagnostics, Alternatives, and Limitations

Stratification is useful only when the grouping and the common-effect restriction are defensible. Check whether each stratum has target events and enough covariate variation to contribute information. A stratum with no target events may contribute little or no information about the regression coefficients. If treatment or another covariate barely varies within strata, the common effect may be weakly identified even when the overall sample is large. Very small strata can also make estimates unstable.

The proportional subdistribution hazards assumption applies to each shared coefficient within every stratum. Use the established diagnostics for proportionality in Fine-Gray regression, examining whether there is evidence of time-varying covariate effects and whether patterns differ by stratum. A pooled diagnostic can obscure a violation concentrated in one stratum. Also inspect whether the stratification variable creates many sparse groups, whether censoring weights become extreme, and whether the result depends heavily on a few event contributions.

A useful comparison is with an unstratified Fine-Gray model that includes stratum as an ordinary covariate. That approach estimates a coefficient for the stratum and imposes a common baseline shape up to the modeled proportional effect. Stratification instead allows arbitrary stratum-specific baseline functions but does not estimate a stratum coefficient. A model with stratum-by-covariate interactions directly allows effects to differ, but can be imprecise when strata or events are sparse. The appropriate choice depends on the scientific question, not simply on which model gives the preferred result.

Approach	Baseline structure	Covariate effects	Main trade-off
Common-baseline Fine-Gray model	One baseline, with modeled covariate terms	Common unless interactions are included	More efficient if its baseline structure is adequate; may constrain baseline differences
Stratified Fine-Gray model	Separate baseline for each stratum	Common across strata by default	Flexible baselines; no coefficient for the stratifying variable
Stratum-by-covariate interaction model	Depends on chosen baseline specification	Can vary by stratum	Answers a heterogeneity question but requires more information

Stratification does not repair dependent censoring, lack of covariate overlap, or an incorrect target-event definition. Nor does it turn a conditional regression association into a causal effect. It can also make absolute risk prediction less straightforward if stratum-specific baseline estimates are not retained and reported. The central limitation is that flexibility in baseline hazards is purchased by giving up a direct regression coefficient for the stratification variable, while retaining a common-effect assumption for the modeled covariates.

Common Mistakes

Adding strata to the denominator. The incorrect reasoning is that all subjects at the same follow-up time should be compared together. In a stratified model, each event contribution uses only its own stratum’s weighted Fine-Gray risk set.
Interpreting the stratum as an estimated hazard ratio. Stratification does not estimate a coefficient for the stratum. It allows a separate baseline subdistribution hazard in each stratum.
Assuming all covariate effects differ by stratum. Separate baselines do not imply separate regression coefficients. Effects are common unless interactions or stratum-specific coefficients are explicitly included.
Reading a subdistribution hazard ratio as a risk ratio. A shared subdistribution hazard ratio is not a fixed-time cumulative-incidence ratio or an absolute risk difference.
Treating competing events as ordinary censoring. Fine-Gray risk sets retain competing-event subjects through the weighted construction. Simply removing them changes the estimand and the method.

Key takeaway: Stratified Fine-Gray regression uses a separate baseline subdistribution hazard and a separate within-stratum weighted risk-set denominator for each stratum, while retaining common coefficients for the covariates unless interactions are specified.

Check Your Understanding

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

In a stratified Fine-Gray model, what is allowed to vary freely across strata, and what remains common by default?
Why should subjects from a different stratum not appear in an event-time denominator?
A fitted shared coefficient for treatment is $-0.2231$. Calculate and interpret its subdistribution hazard ratio.
What does it mean if treatment is constant within every stratum?
Which model feature would be needed to estimate different treatment coefficients across strata?

Solutions. (1) The baseline subdistribution hazard may differ freely by stratum; the modeled covariate coefficients are common. (2) The partial-likelihood contribution conditions on comparisons within the event subject’s stratum, where the same stratum-specific baseline cancels. (3) $\exp(-0.2231)=0.8000$; under the model, a one-unit increase in treatment coding corresponds to an estimated 20% lower cause-specific subdistribution hazard, conditional on modeled covariates, with the same ratio across strata. This is not a 20% reduction in cumulative incidence. (4) There is no within-stratum treatment contrast, so the common treatment effect is not identified from these comparisons. (5) Include treatment-by-stratum interactions or otherwise specify stratum-specific treatment effects; this requires adequate information within the strata.