Worked Model Selection for a Competing-Risk Dataset

Start With the Clinical Question

When a patient can experience one of several mutually exclusive first events, the choice of regression model should follow the question the analysis is meant to answer. A model for the instantaneous rate of a particular event among people still free of every event does not directly answer the same question as a model for the probability of that event by a specified time. Cause-specific Cox regression and Fine-Gray regression are both useful, but they target different hazard quantities.

Consider a hypothetical randomized clinical trial comparing two transplant strategies. The first event after transplant is either relapse or nonrelapse mortality (NRM); a patient who has one first event cannot subsequently have the other as a first event. The analysis team asks whether the strategies differ in relapse. That question needs clarification: does “differ in relapse” mean a difference in the relapse rate among patients who remain alive and relapse-free, or a difference in the probability of relapse by two years, accounting for NRM? The first points to a cause-specific hazard; the second points directly to cumulative incidence.

Definition: Let $T$ be time to the first event, $J$ its cause, and $X$ a vector of covariates. For cause $k$, the cause-specific hazard is the instantaneous rate of cause $k$ at time $t$, conditional on having had no event before $t$: $$ \lambda_k(t\mid X)=\lim_{\Delta t\to0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t,\ X)}{\Delta t}. $$ The cumulative incidence function (CIF) for cause $k$ is $F_k(t\mid X)=P(T\leq t,\ J=k\mid X)$, the probability of experiencing cause $k$ by time $t$ as the first event. The subdistribution hazard for cause $k$ is the hazard associated with that CIF, defined by $\tilde{\lambda}_k(t\mid X)=f_k(t\mid X)/[1-F_k(t\mid X)]$, where $f_k$ is the cause-$k$ event-time density. Its risk-set construction retains people who have already experienced a competing event, with appropriate weighting in the presence of censoring.

The distinction in the risk sets matters. A person who has had NRM is no longer at risk of relapse as a first event, so that person leaves the cause-specific risk set. In Fine-Gray regression, a person with a competing event remains in the constructed subdistribution risk set; this links the model to the CIF rather than to the event rate among people still event-free.

Two Models, Two Targets

A cause-specific Cox model for cause $k$ specifies

\lambda_k(t\mid X)=\lambda_{k0}(t)\exp(\beta_k^\mathsf{T}X).

Here $\lambda_{k0}(t)$ is an unspecified baseline cause-specific hazard. A coefficient $\beta_{kj}$ describes the log cause-specific hazard ratio for a one-unit increase in covariate $X_j$, holding the other modeled covariates fixed. A cause-specific Cox model treats competing events as ending observation for the cause-specific risk set; they are not ordinary losses to follow-up, but they do preclude the event of interest as the first event.

Fine-Gray regression instead specifies a proportional subdistribution hazards model:

\tilde{\lambda}_k(t\mid X)=\tilde{\lambda}_{k0}(t)\exp(\gamma_k^\mathsf{T}X).

The exponentiated coefficient, $\exp(\gamma_{kj})$, is a subdistribution hazard ratio. It describes the relative subdistribution hazard under the model; it is not itself a risk ratio, a risk difference, or a cause-specific hazard ratio. The Fine-Gray model is commonly selected when the target is a covariate association with the CIF over time. A fitted model can be used to estimate covariate-specific CIFs, subject to its assumptions and the chosen prediction method.

Neither model is universally preferable. The estimand—the specific quantity the analysis is intended to describe—should be identified before fitting either model. If the target is a contrast in instantaneous event rates among those still event-free, use the cause-specific formulation. If the target is the observed probability of the event by a time horizon in the presence of competing events, use the CIF as the target and consider Fine-Gray regression. Cause-specific models for all causes can also be used to construct CIF predictions; the choice is not a rule that Fine-Gray is the only route to a CIF.

Worked Example: Translate a Clinical Question Into a Model Target

In the hypothetical transplant trial, clinicians ask: “Among patients who are alive and relapse-free at each time, is the instantaneous relapse rate different between strategies?” This is a cause-specific hazard question. A cause-specific Cox model for relapse can estimate the treatment association in that rate. NRM removes a patient from the relapse risk set because relapse can no longer occur as that patient’s first event.

Now consider the question: “What is the probability of relapse by two years under each strategy, when NRM can occur first?” This is a CIF question. A Fine-Gray model is a natural regression choice for modeling that target. Alternatively, cause-specific models for relapse and NRM can be combined to obtain CIF predictions. In either case, the reported two-year quantity should be a CIF, not a hazard ratio.

The questions differ even though they use the same patients, causes, and treatment variable. They also need not produce effects of the same size or direction: a treatment can alter the competing-event rate, which changes the chance of remaining available to experience relapse. The model choice follows the target, not which coefficient appears more favorable.

Worked Analysis of a Hypothetical Dataset

Suppose the trial randomized 600 patients, 300 to each transplant strategy, and followed each patient through two years unless a first event occurred. The hypothetical event counts are:

Strategy	Relapse by two years	NRM by two years	Event-free at two years	Total
A	75	45	180	300
B	55	60	185	300

Assume for this illustration that there is no loss to follow-up before two years; patients without either first event are administratively censored at two years. The crude two-year relapse CIFs are therefore the observed proportions with relapse: $75/300=0.250$ for A and $55/300=0.1833$ for B. These proportions are descriptive; regression uses the event times, not only the counts in this table.

Worked Example: Fit and Interpret a Cause-Specific Cox Model

Suppose the analysis target is the treatment association with the instantaneous relapse rate among patients still alive and relapse-free. Fit a cause-specific Cox model with treatment as the predictor. The model assumes proportional cause-specific hazards for treatment over follow-up. For this hypothetical illustration, assume the individual relapse event times show no material evidence against that assumption and that follow-up is complete to two years for patients without an event. A software fit to those event times reports $\hat{\beta}=0.55$, $SE(\hat{\beta})=0.20$.

The estimated cause-specific hazard ratio for A versus B is

\widehat{HR}_{CS}=\exp(\hat{\beta})=\exp(0.55)=1.733.

The estimated log hazard ratio’s approximate 95% Wald interval is

0.55\pm1.96(0.20)=0.55\pm0.392=(0.158,\ 0.942).

Exponentiating the endpoints gives an approximate 95% confidence interval for the cause-specific hazard ratio:

\big(\exp(0.158),\ \exp(0.942)\big)=(1.171,\ 2.565).

Under the fitted model, the estimated cause-specific relapse hazard for strategy A is 1.73 times that for strategy B among patients who remain event-free, with the model’s proportional-hazards interpretation applying over the modeled follow-up. The approximate interval is compatible with cause-specific hazard ratios from about 1.17 to 2.57. It does not say that the relapse probability is 1.73 times as large.

Worked Example: Fit Fine-Gray Regression for the Two-Year CIF Question

Suppose the prespecified target is the association between strategy and the probability of relapse by two years, with NRM treated as a competing event. Fit a Fine-Gray proportional subdistribution hazards model. Assume its treatment proportionality is an adequate working description over the two-year period, and that censoring is independent of the event process conditional on modeled covariates (here, treatment), or is handled by a valid censoring-weight procedure. There is no pre-two-year censoring in this illustration.

Suppose the fitted log subdistribution hazard ratio for A versus B is $0.32$, with standard error $0.19$. Then

\widehat{SHR}=\exp(0.32)=1.377.

An approximate 95% Wald interval on the log scale is $0.32\pm1.96(0.19)=0.32\pm0.3724=(-0.0524,\ 0.6924)$. Exponentiating gives $(\exp(-0.0524),\exp(0.6924))=(0.949,\ 1.999)$. This interval includes 1, so these illustrative data do not provide strong evidence against a null subdistribution hazard ratio of 1 at the 5% two-sided level. The interval is not a 95% probability statement about the fixed parameter.

The estimated subdistribution hazard ratio is not the crude CIF ratio $0.250/0.1833=1.364$. The two numbers happen to be close here, but they are different quantities and need not be numerically similar. For the clinical report, estimate and present the model-based CIFs at two years, with uncertainty intervals, if the target is absolute relapse probability. The subdistribution hazard ratio is a regression coefficient summary, not a substitute for those probabilities.

A Practical Selection and Checking Process

Specify the event and horizon.
Define the first event of interest, competing causes, population, and time period. Here, relapse is the event of interest, NRM competes, and the clinical horizon is two years.

Name the target quantity.
Decide whether the question concerns a cause-specific event rate among event-free patients or the CIF, the probability of the event by time $t$ in the presence of competing events.

Choose a compatible model.
Use cause-specific Cox regression for a proportional cause-specific hazard target; use Fine-Gray regression for a proportional subdistribution hazard model linked to the CIF. Do not select by comparing their likelihood values or information criteria as though they were competing fits to the same likelihood.

Check data and model assumptions.
Confirm that each first event has the correct cause, censoring is represented correctly, and any delayed entry and time-dependent covariates are handled as required. Examine proportionality and the adequacy of the covariate form.

Report the target in interpretable units.
For an absolute-risk question, report CIF estimates at relevant times. For a hazard association, name which hazard the ratio concerns and state the risk-set interpretation.

Proportional hazards should not be assumed merely because a model converges. For cause-specific Cox regression, inspect suitable residual diagnostics and plots for time patterns in the treatment effect; if proportionality is doubtful, consider a time-varying coefficient or report time-specific contrasts. For Fine-Gray regression, assess whether a constant subdistribution hazard ratio is credible over the period and whether predicted CIFs are plausible and calibrated. A coefficient that masks a changing association may be a poor summary even when its standard error is small.

Censoring also deserves a specific check. Cause-specific Cox regression requires censoring to be independent of the event process conditional on the modeled information, for the usual partial-likelihood interpretation. Fine-Gray methods require appropriate handling of censoring for the subdistribution risk set; when censoring depends on covariates or time, inverse-probability censoring weights may be needed, and their specification must be defensible. Event coding, time origin, and competing-event definitions must match the clinical protocol. Sparse events or few events in a covariate group can make either model unstable; penalization may be considered, but it does not repair a mismatched estimand or incorrect coding.

Why the Choice Matters

The CIF depends on the cause-specific hazards for the event of interest and for all competing causes. A treatment can therefore change relapse probability by changing relapse rates, NRM rates, or both. A cause-specific hazard ratio isolates the relapse rate among those who remain event-free; it does not summarize the treatment’s total association with the observed relapse probability. Fine-Gray regression links its coefficient to the CIF but does not turn the subdistribution hazard ratio into a probability ratio.

For the hypothetical table, the crude relapse CIFs are 25.0% and 18.3%, while NRM proportions are 15.0% and 20.0%. The treatment groups differ in both event types. A report centered only on the cause-specific relapse ratio would omit the probability scale that clinicians requested; a report centered only on the subdistribution ratio would still benefit from presenting the CIFs themselves. Reporting the competing-event CIF can also clarify how the observed probability of relapse relates to NRM.

A common mistake is to fit both models and choose whichever produces the smaller p-value, larger effect, or better numerical fit criterion. That reasoning makes the target depend on the observed results. Instead, define the scientific quantity first, choose a model that addresses it, and evaluate the model’s assumptions. A second mistake is to call a cause-specific hazard ratio a “risk of relapse” ratio. The correction is to name it precisely: it compares instantaneous relapse rates among people still free of either first event. For an absolute risk at a stated time, report a CIF.

Key takeaway: Model selection in competing-risk regression is primarily estimand selection. Cause-specific Cox and Fine-Gray regression describe different hazard processes; neither coefficient alone is a universal measure of a treatment’s effect on event probability.

Check Your Understanding

Answer each question using the distinction between event rates among event-free patients and cumulative incidence.

A clinician asks for the probability of relapse by 18 months when death before relapse can occur. Which target should be reported?
In a cause-specific Cox model, what happens to a patient’s relapse risk-set contribution after a competing first event?
For a cause-specific Cox coefficient of $0.40$, calculate the hazard ratio and state what it compares.
Why is it not appropriate to choose Fine-Gray over cause-specific Cox solely because its p-value is smaller?

Solutions. (1) Report the relapse CIF at 18 months, which accounts for death as a competing first event. (2) The patient leaves the cause-specific risk set at the competing event because relapse can no longer be the first event. (3) $\exp(0.40)=1.492$; under the model, this compares instantaneous cause-specific event rates among patients still free of all first events, for a one-unit covariate increase. (4) The models target different hazard quantities, so the p-values are not a model-selection contest; choose the model based on the prespecified scientific target and check its assumptions.