Baseline Hazard Choices For Absolute Risk

Why the Baseline Choice Matters

A relative hazard model describes how covariates multiply a hazard, but an absolute-risk prediction also needs its baseline level. In a cause-specific Cox model, the regression coefficients describe relative differences between covariate profiles, while the estimated baseline cumulative hazards set the scale of risk over time. Consequently, two implementations can give similar relative hazard estimates yet different cumulative incidence function (CIF) predictions.

The baseline estimate is not a cosmetic detail. It depends on how tied event times are handled, on the reference point used to report the baseline, and on how estimated cumulative hazards are converted to event-free probabilities and CIFs. These choices must be compatible with one another. The earlier material on cause-specific Cox fits and absolute risk provides the model context; here the focus is the baseline component of those predictions.

The baseline cumulative hazard is a nondecreasing function, usually estimated as a step function with jumps at observed event times. Given an estimated coefficient vector, define the risk score for subject \(i\) as \(r_{ik}=\exp(\hat\beta_k^\mathsf{T}z_i)\). The risk set \(R(t_j)\) contains subjects still under observation and event-free just before time \(t_j\).

Breslow and Efron Baseline Estimates

For a cause-specific Cox model, the Breslow estimator assigns a jump at each cause-\(k\) event time. If \(d_{kj}\) events of cause \(k\) occur at time \(t_j\), its estimated jump is

The denominator is the sum of the fitted relative hazards among those at risk. The cumulative estimate is the sum of these jumps up to the time of interest. The formula is the baseline counterpart of the Breslow treatment of tied event times in the partial likelihood.

When there are tied events, another common choice is the Efron approach. At a tied time, let \(D_{kj}\) be the subjects with cause-\(k\) events, and let \(E_{kj}=\sum_{i\in D_{kj}}r_{ik}\). The Efron baseline jump is

For a single event at a time, the two formulas coincide. With tied events, the Efron denominators remove an increasing fraction of the tied subjects’ risk scores across the tied failures. Its jump will generally differ from the Breslow jump. The coefficient estimation and baseline estimation should use a compatible tie convention; it is not sensible to treat the resulting predictions as if the convention had no effect.

Reference Points and Centering

Software may report a baseline cumulative hazard for covariates set to zero, or for covariates set to a reference vector such as their sample means. These are different parameterizations of the same fitted hazard model when the linear predictor is adjusted consistently. Let \(z_{\mathrm{ref}}\) be the reference vector and let \(\widehat H_{k,\mathrm{ref}}(t)\) be the cumulative hazard for that reference. Then the hazard for profile \(z\) is represented by

This is algebraically equivalent to using a zero-covariate baseline, provided that \(\widehat H_{k0}(t)=\widehat H_{k,\mathrm{ref}}(t)\exp(-\hat\beta_k^\mathsf{T}z_{\mathrm{ref}})\). The reference point changes the reported baseline level, not the fitted hazard for a given profile. Errors arise when a centered baseline is combined with an uncentered linear predictor, or vice versa.

From Cause-Specific Baselines to CIFs

For competing risks, each cause has its own baseline cumulative hazard and coefficient vector. At a profile \(z\), the fitted cause-specific cumulative hazard is \(\widehat H_k(t\mid z)=\widehat H_{k0}(t)\exp(\hat\beta_k^\mathsf{T}z)\), with the reference adjustment used if the baseline is centered. The cause-specific cumulative hazards must be combined: event-free probability depends on all causes, not just the cause whose CIF is being predicted.

One useful derivation applies when the cause-specific hazards are constant over an interval, or when their cumulative hazards over a horizon have a constant cause proportion. Write \(A_k\) for cause \(k\)'s cumulative hazard over that horizon and \(A=\sum_k A_k\). Event-free probability is \(\exp(-A)\), so total failure probability is \(1-\exp(-A)\). The fraction of the total hazard assigned to cause \(k\) is \(A_k/A\). Therefore,

This formula is not a general replacement for integrating CIFs when cause proportions change over time. In general, the CIF accumulates each cause’s hazard weighted by the probability of having had no earlier event. But it gives a transparent calculation for a single interval with constant hazards and shows why changing one baseline can affect more than one cause’s CIF: it changes both the cause’s contribution and overall event-free probability.

Probability Conversion, Diagnostics, and Alternatives

A cumulative hazard is not itself an event probability. For one cause, \(1-\exp\{-H(t)\}\) converts cumulative hazard to risk under the corresponding survival model. With competing causes, the correct calculation must also account for the other causes. For example, applying \(1-\exp\{-H_k(t\mid z)\}\) to each cause separately generally produces net risks, not competing-risk CIFs, and those values need not sum to the overall probability of failure.

A practical implementation must also choose how to handle event-time jumps. A product-integral calculation based on conditional event probabilities updates survival and CIFs by multiplying the event-free probability by the probability of no event at each step. When model-based cumulative-hazard increments are used, some prediction procedures instead use an exponential conversion over each interval; under constant cause-specific hazards within an interval, the conditional event probability is \(1-\exp(-\sum_k\Delta H_k)\), allocated across causes in proportion to their increments. These approaches agree to first order for small increments, but need not agree for large jumps. State the convention and use it consistently; do not treat a hazard increment as a probability without justification.

The Aalen-Johansen estimator is a nonparametric alternative based on observed risk sets and event counts. It does not require fitting cause-specific Cox relative effects, but it does not provide the same covariate-adjusted model predictions. Fine-Gray models use a subdistribution-hazard formulation rather than combining cause-specific hazard predictions in this way. These approaches answer related prediction questions, but their baseline quantities should not be substituted into a cause-specific Cox calculation.

Several checks help reveal implementation problems:

• Confirm that each cause’s denominator uses the appropriate risk set and fitted risk scores, and that the chosen tie method matches the model fit.
• Check that the baseline reference point and the profile’s linear predictor use the same centering convention.
• Inspect jumps and risk-set sizes. Large jumps or very small late risk sets can make absolute predictions sensitive to individual event times and modeling choices.
• Verify that predicted event-free probability and all CIFs are nonnegative and that their sum is consistent with the total failure probability under the chosen calculation.
• Compare predictions across reasonable baseline and tie-handling choices, especially when ties are common or the prediction horizon is late in follow-up.

These diagnostics do not establish that the model is well calibrated or that its assumptions hold. The Cox model still relies on proportional cause-specific hazards and suitable handling of censoring; baseline estimation cannot repair violations. Nor does a close match between two baseline methods guarantee accurate predictions outside the observed support of follow-up.