Penalized Fine-Gray Regression

Why Penalize a Fine-Gray Model?

Fine-Gray regression relates predictors to the subdistribution hazard for a specified event type, and thereby to its cumulative incidence function (CIF). When the number of candidate predictors is large relative to the number of observed events of that type, an unpenalized fit can be unstable: small changes in the data may produce large coefficient changes. Penalization adds a constraint that generally shrinks fitted coefficients toward zero. Depending on the penalty, it can also set some coefficients exactly to zero.

The target remains the Fine-Gray model for the chosen cause. Penalization changes how its coefficients are estimated; it does not turn the subdistribution hazard into a cause-specific hazard or directly estimate a risk ratio. The aim is often stable prediction or a more compact model, not an automatic test of which predictors have causal effects.

Model and Penalized Objective

Definition: Let $T$ be the event time, let $J$ be the event type, and let $k$ be the event type of interest. The cause-$k$ subdistribution hazard is the instantaneous rate of a cause-$k$ event among the subdistribution risk set: subjects who have not yet had any event and subjects who have already had a competing event. The proportional subdistribution-hazards model is $$ \lambda_k^*(t\mid Z)=\lambda_{k0}^*(t)\exp(\beta^\mathsf{T}Z), $$ where $Z$ is the predictor vector, $\lambda_{k0}^*(t)$ is an unspecified baseline subdistribution hazard, and $\beta$ is the coefficient vector. For fixed covariates, this model implies $F_k(t\mid Z)=1-\exp\{-\Lambda_{k0}^*(t)\exp(\beta^\mathsf{T}Z)\}$, where $F_k$ is the cause-$k$ CIF and $\Lambda_{k0}^*$ is the baseline cumulative subdistribution hazard.

The Fine-Gray partial likelihood uses a modified risk set. A subject with a competing event remains represented after that event, with a weight that accounts for censoring. A subject censored before a later event time cannot simply be treated as if still observed; inverse-probability-of-censoring weighting is used to adjust the risk-set contribution. Let $Y_j^*(t)$ indicate that subject $j$ contributes to the modified risk set at time $t$, and let $w_j(t)$ be the corresponding censoring weight. A standard form of the partial log likelihood, suppressing the specific tie adjustment, is

\ell_{\mathrm{FG}}(\beta) = \sum_{i:\,J_i=k} \left[ \beta^\mathsf{T}Z_i - \log\left\{ \sum_j Y_j^*(T_i)w_j(T_i)\exp(\beta^\mathsf{T}Z_j) \right\} \right].

The exact construction of $w_j(t)$ depends on the censoring adjustment and the available covariates. For example, a subject who experienced a competing event earlier can remain in the risk set with a weight involving the estimated probability of remaining uncensored from that competing-event time to $t$. Tied cause-$k$ event times require a tie method; the chosen method must be used consistently in the fit. The formula above illustrates the no-tie contribution and is not a replacement for those implementation details.

For $p$ coefficients, a common scaled objective is

\widehat{\beta} = \mathop{\arg\min}_{\beta} \left[ -\frac{\ell_{\mathrm{FG}}(\beta)}{n} + \lambda \left\{ \alpha\sum_{r=1}^{p}|\beta_r| + \frac{1-\alpha}{2}\sum_{r=1}^{p}\beta_r^2 \right\} \right].

Here $n$ is the sample size, $\lambda\geq 0$ controls the overall penalty, and $0\leq\alpha\leq1$ controls its form. At $\alpha=1$, this is the lasso; at $\alpha=0$, it is ridge; intermediate values give the elastic net. The division by $n$ is a scaling convention: software that uses an unscaled partial log likelihood will generally report a differently scaled $\lambda$. The baseline subdistribution hazard is not penalized by this expression.

What Shrinkage Means on the Fine-Gray Scale

A coefficient $\beta_r$ describes the multiplicative change in the subdistribution hazard associated with a one-unit increase in predictor $Z_r$, holding the other included predictors fixed. Its subdistribution hazard ratio is $\exp(\beta_r)$. Penalization pulls the estimated coefficient toward zero, so its estimated ratio is pulled toward 1. In a Fine-Gray model, that ratio concerns the subdistribution hazard; it is not generally a ratio of CIFs at a particular time.

With the lasso, some coefficients can be exactly zero, which excludes those predictors from the fitted linear predictor. Ridge usually retains all predictors but can stabilize estimates when predictors are numerous or correlated. The elastic net combines the two behaviors and can be useful when correlated predictors are expected. None of these penalties makes a misspecified proportional subdistribution-hazards model correct.

Worked Example: A One-Coefficient Shrinkage Update

Suppose an optimization step for one standardized predictor gives the quadratic approximation $Q(\beta)=\frac{1}{2}H(\beta-z)^2+\lambda|\beta|$, with curvature $H=4$, unpenalized local estimate $z=0.30$, and lasso penalty $\lambda=0.80$. This is an illustrative optimization calculation, not a complete dataset fit.

For a positive $z$, the minimizer is the soft-thresholded value $\max(0,z-\lambda/H)$. Substituting,

\widehat{\beta} = \max\left(0,\ 0.30-\frac{0.80}{4}\right) = \max(0,0.10) = 0.10.

The corresponding subdistribution hazard ratio is $\exp(0.10)=1.105$, compared with $\exp(0.30)=1.350$ at the unpenalized local estimate. In this approximation, the penalty reduces the estimated association toward a ratio of 1; it does not establish that the predictor has no effect. A larger penalty, $\lambda=1.20$, would give $\max(0,0.30-1.20/4)=0$, setting the coefficient to zero.

Why the Lasso Gives Exact Zeros

The thresholding in the example follows from the corner in the absolute-value penalty at zero. For the stated quadratic approximation, consider first a nonnegative coefficient. Differentiating $Q$ for $\beta>0$ gives $H(\beta-z)+\lambda=0$, so the positive solution is $\beta=z-\lambda/H$, provided $z>\lambda/H$. For a negative coefficient, differentiating gives $H(\beta-z)-\lambda=0$, yielding $\beta=z+\lambda/H$, provided $z<-\lambda/H$. If $-\lambda/H\leq z\leq\lambda/H$, neither nonzero solution lies in its required region, and the minimum occurs at zero. Combining these cases gives

\widehat{\beta} = \operatorname{sign}(z)\max\left(|z|-\frac{\lambda}{H},0\right).

This derivation explains a central practical distinction: lasso can produce a sparse fitted linear predictor, whereas the smooth ridge penalty shrinks coefficients without this exact-zero threshold. In a full Fine-Gray fit, optimization is typically multivariate and uses an iterative algorithm, but the same penalty shapes the solution.

Worked Risk-Set Contribution

Worked Example: A Censoring-Weighted Event Contribution

Consider a hypothetical cause-$k$ event at time 6. The event subject A has $Z_A=1$. Subject B has $Z_B=0$ and has not had an event by time 6. Subject C, with $Z_C=1$, had a competing event at time 4 and remains in the modified risk set with weight $0.60$ at time 6. Subject D was censored at time 5 and does not contribute at time 6. Suppose the fitted coefficient is $\beta=0.40$, and there are no tied cause-$k$ events at time 6.

The weighted denominator is

\exp(0.40\cdot 1) + \exp(0.40\cdot 0) + 0.60\exp(0.40\cdot 1) = 1.4918+1+0.8951 = 3.3869.

The event subject's numerator is $\exp(0.40)=1.4918$. Thus this event contributes

0.40-\log(3.3869) \approx -0.820.

The weight reduces C's denominator contribution because C had a competing event before time 6. Omitting C altogether would not represent the Fine-Gray modified risk set; treating C's weight as 1 would discard the stated censoring adjustment. The contribution is one component of the partial log likelihood, not by itself a coefficient estimate.

Fitting and Checking a Penalized Model

The model and censoring assumptions must be considered before interpreting a penalized fit. The proportional subdistribution-hazards form assumes that each predictor's log subdistribution-hazard ratio is constant over time, conditional on the other included predictors. Censoring must be independent of the event process conditional on the information used in the censoring adjustment, and the censoring model must be adequate. There must also be sufficient probability of remaining uncensored over relevant follow-up times; extreme weights can make an estimate unstable. As in ordinary regression, the observations must have the independence structure assumed by the variance and fitting method, or appropriate clustering methods are needed.

A practical fit usually standardizes continuous predictors before penalization so that a one-unit scale difference does not determine how strongly coefficients are shrunk. Binary predictors may be left in their original coding, but the scaling and penalty treatment should be documented. Any unpenalized adjustment variables should be identified in advance and not accidentally included in the penalized set. After fitting, coefficients on standardized predictors must be converted back to the original scale for interpretation.

Specify the target and model.
Choose the event type of interest, predictors, time scale, and any unpenalized adjustment terms. State the proportional subdistribution-hazards and censoring assumptions.

Construct the Fine-Gray likelihood.
Use the modified risk sets and censoring weights, and specify a tie method if cause-$k$ events are tied.

Scale predictors and tune the penalty.
Standardize predictors as appropriate, then choose $\lambda$ (and, for an elastic net, $\alpha$) using a stated criterion such as cross-validation.

Check stability and interpretation.
Review convergence, the coefficient path, censoring weights, and sensitivity to reasonable tuning choices. Interpret nonzero coefficients as shrunken conditional subdistribution-hazard associations.

Diagnostics should include whether the fitting algorithm converged, whether coefficients change sharply over a small range of $\lambda$, and whether a few observations receive unusually large censoring weights. Check whether the proportionality assumption is plausible over time, and assess predictive performance using an evaluation appropriate to the CIF target. If the selected predictors or estimates vary greatly across folds or modest changes in the data, the fitted model may be too unstable for confident interpretation.

Cross-validation is a way to tune predictive fit, not a guarantee that the selected model is the uniquely correct scientific model. Ordinary standard errors and p-values from an unpenalized fit should not be attached automatically to penalized estimates: shrinkage and data-driven selection affect their sampling behavior. For inferential aims, the analysis needs methods designed for inference after penalization or a prespecified low-dimensional model.

Worked Example: Interpreting a Penalized Fit

Suppose a hypothetical analysis of 180 patients considers 45 candidate predictors for a specified cause-$k$ event. After standardization and cross-validation, a lasso Fine-Gray fit retains a treatment indicator with coefficient $-0.25$ and an age predictor with coefficient $0.12$ per standard deviation; the other fitted coefficients are zero. Assume the model and censoring diagnostics do not show an obvious failure.

For the treatment indicator, the fitted subdistribution hazard ratio is $\exp(-0.25)=0.7788$. Conditional on the other included predictors, the fitted cause-$k$ subdistribution hazard is about 22.1% lower for the coded treatment group than for the reference group. For age, the ratio is $\exp(0.12)=1.127$ per standard deviation, or about 12.7% higher. These are shrunken model associations on the subdistribution-hazard scale; they are not direct statements that the CIF is respectively 22.1% lower or 12.7% higher at every time.

The zero coefficients mean those predictors do not contribute to this particular lasso linear predictor at the chosen tuning value. They do not prove that the corresponding variables are unrelated to the outcome, especially if predictors are correlated or the fitted model is unstable across folds.

How This Approach Differs From Alternatives

Approach	What is penalized or modeled	Useful distinction
Unpenalized Fine-Gray regression	Fits the same subdistribution-hazards target without a shrinkage penalty.	Can be unstable when predictors are numerous relative to cause-$k$ events.
Penalized cause-specific Cox regression	Penalizes coefficients in a cause-specific hazard model.	It targets a different hazard, so its coefficients are not Fine-Gray coefficients.
Lasso Fine-Gray regression	Adds an absolute-value penalty to subdistribution-hazard coefficients.	Can set coefficients exactly to zero.
Ridge Fine-Gray regression	Adds a squared-coefficient penalty to subdistribution-hazard coefficients.	Shrinks coefficients, generally retaining all predictors.

These methods answer different modeling needs. A cause-specific Cox model and a Fine-Gray model do not become interchangeable merely because both use a Cox-type partial-likelihood construction. Likewise, a penalty can improve numerical stability without resolving the choice of estimand or correcting inappropriate covariates, censoring assumptions, or model form.

Common Mistakes and Limits

Calling a subdistribution hazard ratio a risk ratio. The coefficient is on the subdistribution-hazard scale. The CIF is obtained through the fitted baseline subdistribution hazard as well as the covariates.
Assuming penalization removes the model assumptions. Shrinkage does not fix informative censoring, extreme weights, nonproportional effects, or an unsuitable predictor set.
Comparing coefficient sizes before accounting for predictor scale. Penalties act on the coefficient parameterization, so scale predictors appropriately and report units clearly.
Treating a lasso zero as proof of no association. It means the selected penalized fit assigns a zero coefficient at that tuning value, not that the population association is known to be zero.
Using ordinary p-values as if selection had not occurred. Penalization and tuning alter the usual inferential problem; use inference methods appropriate to the analysis goal.

The central limitation is that penalized Fine-Gray regression trades some bias for potential stability and, with lasso, sparsity. The resulting coefficient vector depends on the penalty, predictor scaling, censoring adjustment, and tuning procedure. Report those choices alongside the target event type, the fitted coefficients, and the diagnostics.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In a Fine-Gray model, what does $\exp(\beta_r)$ describe, and what does it not directly describe?
A standardized predictor has local unpenalized estimate $z=-0.50$, curvature $H=5$, and lasso penalty $\lambda=1.5$. Find its soft-thresholded estimate.
Why can a subject with an earlier competing event remain represented in a Fine-Gray risk set?
Name two assumptions or diagnostics that matter even when a Fine-Gray model is penalized.

Solutions: (1) It is the multiplicative subdistribution-hazard change per one-unit increase in the predictor, conditional on the other included predictors; it is not directly a CIF ratio. (2) $\operatorname{sign}(-0.50)\max(0.50-1.5/5,0)=-0.20$. (3) Fine-Gray models the subdistribution hazard for the chosen cause using a modified risk set that retains prior competing-event subjects with appropriate censoring adjustment. (4) Examples include proportional subdistribution hazards, adequate conditional independent-censoring adjustment, convergence, positivity of censoring survival, and stability of weights or coefficients.

What You'll Learn