Time-Varying Effects in Fine-Gray Regression

Why Let a Subdistribution Effect Change Over Time?

A proportional Fine-Gray model assigns a covariate a constant subdistribution hazard ratio. That assumption can be too restrictive: a treatment or prognostic factor may have a stronger association with the target event early in follow-up and a weaker association later. A time-varying coefficient allows the association to change while retaining the Fine-Gray model’s focus on the cumulative incidence of a particular cause.

The model’s risk set is a subdistribution risk set, not the ordinary cause-specific risk set. In particular, individuals who experience a competing event are retained in the Fine-Gray risk-set construction. The earlier material on Fine-Gray regression establishes this construction and its weighted partial-likelihood estimation. Here the central question is how to represent and interpret a coefficient that depends on time.

Definition: For target cause $k$, a time-varying-coefficient Fine-Gray model is $$ \tilde{\lambda}_k(t\mid x)=\tilde{\lambda}_{k0}(t)\exp\{x\beta(t)\}, $$ for a scalar covariate $x$, where $\tilde{\lambda}_k(t\mid x)$ is the cause-$k$ subdistribution hazard, $\tilde{\lambda}_{k0}(t)$ is an unspecified baseline subdistribution hazard, and $\beta(t)$ is a coefficient function. For $x=1$ versus $x=0$, the time-specific subdistribution hazard ratio is $\exp\{\beta(t)\}$. With several covariates, $x\beta(t)$ is replaced by the sum of their covariate-coefficient terms.

A common specification uses a prespecified function $g(t)$:

\beta(t)=\beta_0+\beta_1g(t).

Then $\exp\{\beta_0+\beta_1g(t)\}$ is the subdistribution hazard ratio at time $t$. The proportional subdistribution hazards model is the special case $\beta_1=0$. The function $g(t)$ should be chosen for scientific interpretability and adequate numerical behavior over the observed follow-up; for example, $g(t)=\log(t/t_*)$ compares effects on a log-time scale relative to a chosen reference time $t_*>0$.

Assumptions and Interpretation

The model requires a well-defined target cause and time origin, appropriate handling of right censoring in Fine-Gray estimation, and a correctly specified form for the coefficient’s time dependence. In particular, a linear effect in $g(t)$ is an assumption about how the log subdistribution hazard ratio changes; it is not a consequence of fitting a time interaction. Covariates in this formulation are fixed baseline covariates. The method also relies on sufficient information in the weighted subdistribution risk sets across the time range where effects are reported.

A time-specific subdistribution hazard ratio compares the subdistribution hazards at that time for covariate values differing by one unit, conditional on the model. It is not a ratio of cause-specific hazards, and it is not generally a ratio of cumulative incidences. It describes an association on the subdistribution-hazard scale; by itself it does not establish a causal effect.

Worked Example: Evaluating a Log-Time Effect

Suppose a hypothetical Fine-Gray model for a binary exposure $x$ and a specified target cause is $\beta(t)=\beta_0+\beta_1\log(t/2)$, with time $t$ measured in years. The fitted coefficients are $\hat\beta_0=-0.20$ and $\hat\beta_1=0.35$. Estimate the exposure subdistribution hazard ratio at years 1, 2, and 4.

At year 1, $g(1)=\log(1/2)=-0.693147$. Thus $\hat\beta(1)=-0.20+0.35(-0.693147)=-0.442601$, and $\exp\{\hat\beta(1)\}=\exp(-0.442601)\approx0.6424$.

At year 2, $g(2)=0$, so $\hat\beta(2)=-0.20$ and the estimated ratio is $\exp(-0.20)\approx0.8187$.

At year 4, $g(4)=\log 2=0.693147$. Therefore $\hat\beta(4)=-0.20+0.35(0.693147)=0.042601$, giving $\exp(0.042601)\approx1.0435$.

The fitted association changes over time: the estimated ratio is below 1 at year 1 and slightly above 1 at year 4. These are time-specific subdistribution hazard ratios, not statements that the exposure reduces or increases the cumulative incidence by those same percentages.

Estimation and Uncertainty at a Given Time

In practice, $\beta_0$ and $\beta_1$ are estimated through Fine-Gray weighted partial-likelihood methods, with the time interaction incorporated into the covariate term. For a binary exposure, the time-specific linear predictor contribution is $x\{\beta_0+\beta_1g(t)\}$. The Fine-Gray risk-set construction and censoring weights still matter; introducing a time interaction does not turn the analysis into an ordinary Cox fit.

Uncertainty in a time-specific coefficient must include the covariance between its component estimates. If the estimated covariance matrix for $(\hat\beta_0,\hat\beta_1)$ has variances $V_{00}$, $V_{11}$ and covariance $V_{01}$, then:

\widehat{\operatorname{Var}}\{\hat\beta(t)\} =V_{00}+g(t)^2V_{11}+2g(t)V_{01}.

This follows by applying the variance rule for a linear combination: $\hat\beta(t)=\hat\beta_0+g(t)\hat\beta_1$, so its variance is the variance of the first term plus the variance of the second term plus twice their covariance. A pointwise confidence interval for the log ratio is $\hat\beta(t)\pm z_{1-\alpha/2}SE\{\hat\beta(t)\}$; exponentiating both endpoints gives a pointwise interval for the ratio.

Worked Example: Confidence Intervals at Two Times

Continue the hypothetical fit above. Suppose $\widehat{\operatorname{Var}}(\hat\beta_0)=0.0100$, $\widehat{\operatorname{Var}}(\hat\beta_1)=0.0144$, and $\widehat{\operatorname{Cov}}(\hat\beta_0,\hat\beta_1)=-0.0040$. Calculate pointwise 95% confidence intervals at years 1 and 4.

At year 1, $g=-0.693147$. The estimated variance is $0.0100+(-0.693147)^2(0.0144)+2(-0.693147)(-0.0040)\approx0.02246$. Thus $SE\approx\sqrt{0.02246}=0.1499$. Using $\hat\beta(1)=-0.442601$, the log-scale interval is $-0.442601\pm1.96(0.1499)$, approximately $[-0.7364,-0.1488]$. Exponentiating gives an interval of approximately $[0.4788,0.8617]$ for the year-1 subdistribution hazard ratio.

At year 4, $g=0.693147$. The variance is $0.0100+(0.693147)^2(0.0144)+2(0.693147)(-0.0040)\approx0.01137$, so $SE\approx0.1067$. The log-scale interval is $0.042601\pm1.96(0.1067)$, approximately $[-0.1664,0.2516]$, giving a ratio interval of approximately $[0.8467,1.286]$.

Each interval is pointwise: it describes uncertainty for the ratio at that selected time under the fitted model. It is not a simultaneous confidence band covering all times with 95% confidence.

Testing Whether the Effect Changes

For the linear time function above, the proportional subdistribution hazards hypothesis is $H_0:\beta_1=0$. A Wald test divides $\hat\beta_1$ by its standard error and compares the resulting statistic with its null reference distribution. A likelihood-ratio or score test can instead compare the time-varying model with its proportional counterpart. These tests address the selected time function: failure to reject does not prove proportionality or rule out departures that the chosen function cannot represent.

Worked Example: Testing the Time Interaction

Using $\hat\beta_1=0.35$ and $SE(\hat\beta_1)=\sqrt{0.0144}=0.12$, test $H_0:\beta_1=0$ against a two-sided alternative at $\alpha=0.05$.

The Wald statistic is $z=0.35/0.12=2.917$. Under $H_0$, this is compared with a standard normal distribution; the two-sided p-value is approximately $0.0035$. Since $0.0035<0.05$, reject $H_0$. Under the specified log-time model, the data provide evidence that the log subdistribution hazard ratio changes with time.

For a direct contrast, the fitted log ratio at year 4 minus that at year 1 is $\hat\beta(4)-\hat\beta(1)=2\log(2)\hat\beta_1\approx1.386294(0.35)=0.4852$. The ratio of the year-4 to year-1 subdistribution hazard ratios is $\exp(0.4852)\approx1.625$. Its log-scale standard error is $1.386294(0.12)=0.1664$, giving a 95% interval $0.4852\pm1.96(0.1664)\approx[0.1591,0.8113]$, or approximately $[1.173,2.251]$ after exponentiation. This contrast quantifies the model’s estimated change between those two times.

Why a Hazard Ratio Is Not a Cumulative Incidence Ratio

The subdistribution hazard is linked to the cumulative incidence of the target cause, but its ratio does not equal the ratio of cumulative incidences. For an event-free starting population under the usual subdistribution-hazard definition, $F_k(t)=1-\exp\{-\tilde\Lambda_k(t)\}$, where $\tilde\Lambda_k(t)$ is the cumulative subdistribution hazard. Consequently, cumulative incidence depends on the accumulated subdistribution hazard over time, not only on the ratio at one instant.

Worked Example: A Changing Ratio and Cumulative Incidence

Consider hypothetical binary groups and a model in which the baseline subdistribution hazard for the target cause is $0.10$ per year in each of two one-year intervals. Suppose the exposure subdistribution hazard ratio is 2 during the first year and 0.5 during the second year. This piecewise example is used only to show how accumulated hazards relate to incidence.

At year 1, the reference group has cumulative subdistribution hazard $0.10$, so its cumulative incidence is $1-\exp(-0.10)\approx0.0952$. The exposed group has cumulative subdistribution hazard $0.20$, giving incidence $1-\exp(-0.20)\approx0.1813$. The incidence ratio is about $0.1813/0.0952=1.905$, not 2.

At year 2, the reference group’s cumulative subdistribution hazard is $0.10+0.10=0.20$, so its incidence is approximately $0.1813$. For the exposed group it is $0.20+0.05=0.25$, giving incidence $1-\exp(-0.25)\approx0.2212$. The incidence ratio is approximately $1.220$, even though the current interval’s subdistribution hazard ratio is 0.5. Earlier accumulation remains part of the year-2 cumulative incidence.

Diagnostics, Alternatives, and Limitations

A useful diagnostic sequence is to inspect a plot of the estimated time-specific coefficient or ratio, examine residual-based or other proportionality diagnostics suitable for Fine-Gray fits, and compare the prespecified time interaction with the proportional model. Also inspect the number of target events and the effective weighted risk-set information over time: few late events or highly variable censoring weights can make late estimates unstable. Diagnostics should be interpreted together rather than treated as a single definitive test.

Check the chosen time function. A straight-line trend in $\log(t)$ may miss a threshold, plateau, or reversal. Sensitivity to reasonable alternative time functions is informative.
Check time support and scale. The reference time $t_*$ in $\log(t/t_*)$ should be positive and interpretable, and estimates should not be extrapolated far beyond observed follow-up.
Check uncertainty and stability. Very wide intervals, large changes after modest model changes, or sparse event information at late times signal that a detailed time-specific interpretation may be unreliable.
Check the censoring and weighting setup. The time interaction does not repair inappropriate censoring assumptions or poorly estimated censoring weights.

The main alternative is a proportional Fine-Gray model, which is simpler and yields one constant ratio when that summary is adequate. If proportionality is not plausible, a prespecified piecewise coefficient can summarize distinct time periods; a flexible time function can describe more complicated patterns but may require substantially more information and careful uncertainty reporting. Cause-specific Cox models answer a different question about cause-specific hazards and should not be substituted as though they estimate the same effect.

Common mistake: “The year-4 subdistribution hazard ratio is 1.04, so exposure raises the year-4 cumulative incidence by 4%.” This is incorrect because a subdistribution hazard ratio is not a cumulative incidence ratio, and cumulative incidence reflects the entire prior hazard history. Report the model-scale ratio as such, and use model-based cumulative incidence estimates when the scientific question concerns absolute risk.

Key takeaway: In a time-varying Fine-Gray model, $\exp\{\beta(t)\}$ is a time-specific subdistribution hazard ratio for the target cause. Its interpretation depends on the chosen coefficient-time function, the weighted Fine-Gray estimation assumptions, and the time point; it is not itself a cumulative incidence contrast.

Check Your Understanding

Try each question, then compare with the solutions below.

If $\beta(t)=\beta_0+\beta_1\log(t/t_*)$, what hypothesis corresponds to proportional subdistribution hazards?
Why must the covariance between $\hat\beta_0$ and $\hat\beta_1$ be included when calculating a time-specific standard error?
At a time when the estimated subdistribution hazard ratio is 0.70, what does that ratio compare, and what does it not directly tell you?
Name two reasons a late-follow-up time-varying estimate might be unreliable.

Solutions. (1) $H_0:\beta_1=0$; under this specified model the coefficient, and hence the ratio, is constant over time. (2) The estimate is the linear combination $\hat\beta_0+g(t)\hat\beta_1$, whose variance includes $2g(t)\widehat{\operatorname{Cov}}(\hat\beta_0,\hat\beta_1)$. Omitting it generally gives the wrong standard error. (3) It compares subdistribution hazards at that time for the covariate values in the contrast. It does not directly give a cumulative incidence ratio or a 30-day risk difference. (4) Examples include few late target events, unstable or highly variable censoring weights, and extrapolation beyond the well-supported follow-up period.