Diagnosing Proportionality in Fine-Gray Models

Why Diagnose Proportionality?

A Fine-Gray model summarizes a covariate’s association with the cumulative incidence of a specified event type through its subdistribution hazard ratio. That summary assumes the ratio is constant over follow-up. If the association strengthens, weakens, or reverses with time, a single estimated ratio can obscure an important feature of the data. Diagnostics assess this assumption; they do not establish that a covariate causes an event.

Fine-Gray diagnostics need to respect the model’s pseudo risk sets: individuals who experience a competing event remain represented in the subdistribution risk set, with appropriate weighting. Thus a diagnostic borrowed from cause-specific Cox regression cannot simply be applied to an unweighted ordinary risk set. The earlier material on proportional hazards and the Fine-Gray weighted partial likelihood provides the model context; here the focus is on checking whether its covariate effects remain stable over time.

Definition: For event type 1, the proportional subdistribution hazards model is $$ \lambda_1^*(t\mid Z)=\lambda_{10}^*(t)\exp(\beta^\top Z), $$ where $\lambda_1^*(t\mid Z)$ is the subdistribution hazard, $\lambda_{10}^*(t)$ is an unspecified baseline subdistribution hazard, and $Z$ is a covariate vector. Proportionality means each component of $\beta$ is constant over follow-up. For a one-unit increase in a covariate $Z_k$, the subdistribution hazard ratio is $\exp(\beta_k)$, conditional on the other modeled covariates.

Diagnostics are meaningful only if the other ingredients of the fit are credible. The event type and follow-up origin must be defined consistently, the covariate specification must be adequate, and censoring must be independent of the event process conditional on the variables used to model censoring. When inverse-probability-of-censoring weights are used, their model must be appropriate and the estimated probability of remaining uncensored must not approach zero over the time range being interpreted. Otherwise, apparent changes in effects can reflect misspecification or extreme weights rather than a genuine time pattern.

Time-Interaction Tests

A direct diagnostic augments the proportional model with a covariate-by-time interaction. For one covariate $Z$, choose a time function $g(t)$, such as $g(t)=\log(t/t_0)$, where $t_0$ is a positive reference time. The resulting linear predictor is $\beta Z+\gamma Zg(t)$. The proportionality null hypothesis is $H_0:\gamma=0$; under the alternative, the covariate effect changes according to the chosen function of time.

\log\{\text{subdistribution hazard ratio for a one-unit increase in } Z \text{ at } t\} =\beta+\gamma g(t).

The interaction test is specific to the selected time function. A log-time interaction can detect a smooth trend on the log-time scale, but it may miss a nonmonotone pattern or a sharp change at a particular time. For a categorical covariate with several levels, proportionality requires all relevant time-interaction coefficients to be zero, so the test is joint. Use a Wald or score test with variance estimation appropriate to the weighted Fine-Gray fit, including robust variance when required by the fitting procedure. An ordinary likelihood-ratio reference distribution should not be assumed automatically for a weighted partial likelihood.

Worked Example: Testing a Log-Time Interaction

Suppose a hypothetical Fine-Gray model for a binary treatment indicator $Z$ includes $Z\log(t/12)$, with time measured in months. The fitted coefficients are $\hat{\beta}=0.42$ and $\hat{\gamma}=-0.18$. The robust standard error for $\hat{\gamma}$ is $0.07$. Assume the fit and censoring-weight model are adequate for this illustration.

For the proportionality test, $H_0:\gamma=0$. The Wald statistic is $$ z=\frac{\hat{\gamma}}{SE(\hat{\gamma})} =\frac{-0.18}{0.07}=-2.571. $$ A two-sided standard normal reference gives $p\approx0.0101$. At a 0.05 significance level, this is evidence against a constant treatment subdistribution hazard ratio under this specified log-time interaction.

At 12 months, $g(12)=\log(1)=0$, so the estimated log ratio is $0.42$, and the ratio is $\exp(0.42)=1.522$. At 3 months, $$ \hat{\beta}+\hat{\gamma}\log(3/12) =0.42+(-0.18)(-1.386)=0.6695, $$ giving $\exp(0.6695)=1.953$. At 24 months, $$ 0.42+(-0.18)\log(2)=0.2953, $$ giving $\exp(0.2953)=1.343$. These are estimated subdistribution hazard ratios at the stated times, not probabilities of the event. The negative interaction indicates a decreasing log ratio as time increases on this model’s log-time scale. The test does not show that this particular functional form is the only possible departure from proportionality.

Residual-Based Diagnostics

A residual diagnostic examines whether covariate contributions at event times systematically depart from what the fitted proportional model predicts. In a weighted Fine-Gray partial likelihood, a Schoenfeld-type residual at an event time compares the event subject’s covariate value with a model-weighted covariate average over the pseudo risk set. Write $R^*(t_j)$ for that risk set, $a_i(t_j)$ for its effective censoring and risk-set weight, and $\hat{\eta}_i=\hat{\beta}^{\top}Z_i$. For a scalar covariate, the basic residual contribution is

r_j=Z_{(j)}- \frac{\sum_{i\in R^*(t_j)} a_i(t_j)\exp(\hat{\eta}_i)Z_i} {\sum_{i\in R^*(t_j)} a_i(t_j)\exp(\hat{\eta}_i)}.

Here $Z_{(j)}$ is the covariate value for the individual with the event of interest at $t_j$. Implementations can incorporate event contributions and censoring weights in additional ways, so residuals should be obtained from a method designed for the fitted Fine-Gray procedure rather than reconstructed from an ordinary Cox fit. Under a correctly specified proportional model, residuals should not show a systematic time trend; random scatter is expected. A smooth curve through residuals against time, often with uncertainty bands, can reveal a trend that is difficult to see in individual points.

Worked Example: Calculating a Weighted Residual

At a hypothetical event time of 8 months, suppose the effective pseudo-risk-set contributions are as follows. The fitted relative risk is $\exp(\hat{\eta})=1$ for $Z=0$ and $1.5$ for $Z=1$. The three risk-set records have $(Z,a)$ values $(0,1.0)$, $(1,0.8)$, and $(1,0.5)$. The event of interest occurs for a subject with $Z_{(j)}=1$. Treat these as the complete effective contributions for this illustrative residual calculation.

The weighted denominator is $$ 1.0(1)+0.8(1.5)+0.5(1.5)=1.0+1.2+0.75=2.95. $$ The weighted numerator for $Z$ is $$ 0(1.0)(1)+1(0.8)(1.5)+1(0.5)(1.5)=0+1.2+0.75=1.95. $$ Thus the risk-set expected covariate value is $1.95/2.95=0.6610$, and the residual is $r_j=1-0.6610=0.3390$. This positive residual says that this event subject’s covariate value exceeds the weighted risk-set expectation. One residual is not evidence of nonproportionality; the diagnostic question is whether residuals show a systematic pattern across event times.

For multiple covariates, inspect residuals separately and account for their dependence when assessing patterns. A cumulative score process or a formal time-interaction score test can complement a smoothed residual plot. Pointwise confidence bands around a smooth are not the same as simultaneous bands over the entire follow-up period: a curve crossing a pointwise band somewhere can occur by chance when many times are inspected. Treat plots as diagnostic evidence and use an appropriate formal test when a specific hypothesis is required.

Plots of Effects Over Follow-Up

A plot can make the practical size and direction of a time pattern clearer than a single test statistic. One option is to plot a smoothed estimate of the covariate effect over time, with uncertainty intervals. Another is to plot time-specific estimates from a small number of prespecified follow-up intervals. Under proportionality, the estimated log subdistribution hazard ratio should be approximately horizontal. Estimates at late times may be much less precise because few individuals remain informative and censoring weights can be large.

Worked Example: Reading a Residual Trend Plot

In a hypothetical analysis, a smoothed curve through weighted residuals for a binary covariate has approximate values $0.30$, $0.18$, $-0.12$, and $-0.28$ at 2, 6, 18, and 36 months, respectively. Assume the curve uses the model’s pseudo-risk-set residuals and the censoring weights are stable over this range.

The curve moves from positive to negative as follow-up increases. Early events therefore tend to have covariate values above their weighted fitted expectations, while later events tend to have values below them. That pattern is compatible with a changing covariate effect and motivates checking a time interaction or other prespecified time pattern. It does not by itself quantify the subdistribution hazard ratio at any time, and the four smooth values are not four independent tests. The conclusion should be tempered if the late portion has few events or wide uncertainty bands.

Diagnostics, Comparisons, and Failure Modes

No single diagnostic is decisive. A useful sequence is to inspect the fitted effect and residual patterns, test a scientifically interpretable time interaction, and assess whether the conclusion is sensitive to the time scale and follow-up region. Where a trend is suggested, report the time-specific effect or pattern that the model can support, rather than presenting one constant ratio as if proportionality were unquestioned.

Check the analysis inputs.
Confirm the event of interest, time origin, covariate coding, and censoring-weight construction. Investigate extreme weights and sparse late risk sets before interpreting a curve.

Inspect model-based patterns.
Plot weighted residuals against time and, where available, plot estimated effects with uncertainty. Look for sustained trends or reversals rather than isolated points.

Test a stated alternative.
Fit a prespecified time interaction and use inference appropriate to the weighted fit. For a multi-level covariate, test the time interactions jointly.

Describe the limits.
Report the time scale, the follow-up region with adequate information, and whether plots and formal tests agree. Do not extrapolate a late-time pattern supported by very few observations.

Cause-specific Cox diagnostics use the cause-specific risk set, whereas Fine-Gray diagnostics must reflect the subdistribution risk set and its censoring weights. Their residuals and estimated effects therefore need not show the same pattern. A Fine-Gray proportionality diagnostic addresses the time stability of the subdistribution hazard ratio; it does not diagnose proportionality for every cause-specific hazard, nor does it replace examination of the cumulative incidence curve. A time-interaction test is a focused formal test, while residual plots are broader exploratory checks. Piecewise time-specific estimates can be easier to communicate, but dividing follow-up into many intervals loses precision and can make results depend on cut points.

A common failure mode is to interpret a late residual trend without checking its information base. For example, suppose four late risk-set records have weights $1.1, 1.2, 4.0,$ and $5.0$. The effective sample size based on these weights is $$ n_{\mathrm{eff}}=\frac{(\sum_i a_i)^2}{\sum_i a_i^2} =\frac{(11.3)^2}{1.1^2+1.2^2+4.0^2+5.0^2} =\frac{127.69}{43.65}=2.925. $$ Although four records appear in the calculation, the highly unequal weights make their information comparable to only about 2.9 equally weighted records by this measure. A sharp late-time pattern could therefore be unstable. Check weight distributions, numbers at risk and events, and uncertainty bands; if positivity is poor, restrict interpretation to the supported time range and revisit the censoring model rather than treating the plot as definitive.

Common mistake: “The residual plot is flat, so proportionality is proven,” or “one point crosses a confidence band, so proportionality is false.” Neither conclusion follows. Diagnostics have limited power, and plotted points are noisy and dependent. Look for coherent patterns, use a formal test suited to the fitted weighted model, and report uncertainty. Failure to reject $H_0$ is not proof that the effect is constant.

Even a well-conducted diagnostic has limitations. A test can miss departures that differ from its selected time function; a flexible smoother can overreact to sparse data; and misspecified covariate form, omitted predictors, or an inadequate censoring model can mimic time variation. A significant interaction indicates evidence against the specified proportional model, not which alternative is correct. More elaborate time-varying Fine-Gray models require their own specification and interpretation; a diagnostic alone does not choose one automatically.

Key takeaway: Assess Fine-Gray proportionality with diagnostics that preserve the weighted subdistribution risk-set structure. Combine residual or effect plots with an appropriately tested time interaction, and interpret any apparent late-follow-up trend in light of censoring weights and available information.

Check Your Understanding

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

For a model with linear predictor $\beta Z+\gamma Z\log(t/6)$, what is the proportionality null hypothesis, and at what time does the covariate effect equal $\beta$?
A fitted time-interaction coefficient is $-0.12$ with robust standard error $0.06$. Calculate the Wald statistic and give the approximate two-sided p-value. What does it test?
Why should an ordinary cause-specific Cox residual plot not be substituted for a Fine-Gray residual diagnostic?
A smooth residual curve trends upward only in the final months, when censoring weights are extreme and few events occur. Name two checks before interpreting this as nonproportionality.

Solutions. (1) $H_0:\gamma=0$; at $t=6$, $\log(6/6)=0$, so the effect is $\beta$. (2) $z=-0.12/0.06=-2.00$, with two-sided $p\approx0.0455$. This is evidence against a constant effect under the specified log-time interaction, subject to the validity of the weighted fit and its variance estimate. (3) The Cox diagnostic uses a cause-specific risk set, while Fine-Gray uses a subdistribution risk set with censoring-related weighting; the diagnostic must match the fitted model. (4) Check the numbers of events and risk-set support late in follow-up, and inspect the censoring-weight distribution or effective information. Also assess uncertainty bands and the censoring-weight model before drawing a conclusion.