Why Update the Baseline Hazard?
A joint model combines repeated longitudinal measurements with an event-time model. When it is transported to a new cohort, its predicted survival can be systematically too high or too low even if the longitudinal trajectory and its association with the event remain useful. Updating the baseline hazard offers a targeted correction: it changes the cohort’s underlying event rate over time while retaining the model’s estimated longitudinal association structure.
The key distinction is between how much baseline event risk accumulates over time and how the model’s predictors modify that risk. A baseline-only update addresses the first. It does not, by itself, correct an incorrect association parameter, a misspecified longitudinal trajectory, or a changed relationship between predictors and events.
Model and Assumptions
For clarity, write a standard shared-parameter joint model as
Here \(y_{ij}\) is subject \(i\)’s longitudinal measurement at time \(t_{ij}\), \(b_i\) is the subject-specific random effect, and \(m_i(t;b_i)\) is the model-implied longitudinal value at time \(t\). The vector \(w_i\) contains event-model covariates. The parameters \(\beta\), \(\gamma\), and \(\alpha\), along with the random-effects distribution and other longitudinal parameters, are treated as fixed at their original estimates during a baseline-only update. The new cohort supplies information for estimating \(h_{0,\mathrm{new}}(t)\).
This update is appropriate when the endpoint definition, time origin, and prediction horizon are compatible across cohorts, and when the original longitudinal and association structure remains adequate. The new cohort must provide enough event information across the time range of interest to estimate its baseline hazard. Censoring must be independent of the event time conditional on the modeled information, or handled by a valid method that accounts for the censoring process. There must also be adequate representation of the predictor patterns for which predictions will be used.
These assumptions are substantive, not just computational. For example, if the association between the latent marker and event risk differs in the new cohort, holding \(\alpha\) fixed may leave systematic errors that no baseline update can remove. Likewise, a different time origin or outcome definition can make the estimated baseline hazards incomparable.
Estimating the New Baseline Hazard
If each subject’s event linear predictor \(\eta_i(t)\) is treated as known, where \(\eta_i(t)=\gamma^{\mathsf T}w_i+\alpha m_i(t)\), a familiar estimator is the Breslow estimator. At an event time \(t_k\), let \(d_k\) be the number of events and \(R(t_k)\) the subjects still at risk just before that time. The estimated increment in cumulative baseline hazard is
The denominator is the total relative risk in the risk set. Subjects with larger \(\eta_i(t_k)\) contribute more expected events per unit baseline hazard, so a given observed event count implies a smaller baseline-hazard increment when the risk set has greater total relative risk. Summing the increments up to time \(t\) gives \(\widehat H_{0,\mathrm{new}}(t)\), and survival conditional on a specified predictor path is
This form makes clear why the update preserves the association structure: the relative-risk multiplier remains \(\exp\{\eta_i(t)\}\), while the cumulative baseline hazard changes. For a constant \(\eta_i\), the expression simplifies to \(\widehat S_i(t)=\exp\{-\exp(\eta_i)\widehat H_{0,\mathrm{new}}(t)\}\).
In a joint model, however, the random effects and hence \(m_i(t;b_i)\) are not known exactly. A plug-in approach substitutes estimated subject-specific trajectories into \(\eta_i(t)\) and then uses the Breslow formula. This is computationally convenient, but it treats uncertain latent trajectories as if they were known. A more principled update estimates the baseline hazard using the new cohort’s joint likelihood, integrating over the random effects while fixing the original model parameters other than the baseline hazard.
For an individual with observed longitudinal data \(y_i\), event or censoring time \(T_i\), and event indicator \(\delta_i\), the relevant likelihood contribution has the form
The product of these contributions can be used to estimate a flexible baseline hazard, for example by representing it with increments at event times. Because the integration occurs after combining the longitudinal and event contributions, the exact estimator need not equal a Breslow calculation based on a single estimated trajectory per subject. The choice between a full integrated-likelihood update and a plug-in approximation should be reported, along with how latent-trajectory uncertainty was handled.
Worked Calculations
Worked Example: Estimating Two Baseline-Hazard Increments
Suppose a hypothetical validation cohort has three subjects at risk at time \(t_1\), with known event-model relative risks \(1\), \(2\), and \(0.5\). One event occurs at \(t_1\). Just before \(t_2\), the first subject has experienced the event, leaving the subjects with relative risks \(2\) and \(0.5\); one event occurs at \(t_2\). Treat these predictors as known for this illustration and use the Breslow estimator.
At \(t_1\), the risk-set sum is \(1+2+0.5=3.5\), and \(d_1=1\). Therefore,
At \(t_2\), the risk-set sum is \(2+0.5=2.5\), and \(d_2=1\), so
Thus the cumulative baseline hazard after \(t_2\) is \(0.2857+0.4000=0.6857\). The arithmetic assumes the stated risk sets and known predictors; in an actual joint-model update, uncertainty in latent trajectories would need to be handled by the chosen estimation method.
Worked Example: Converting the Updated Baseline to Survival
Continue with the hypothetical increments above. For a subject whose event linear predictor is \(\eta=0\) throughout both intervals, the relative risk is \(\exp(0)=1\). The estimated survival through \(t_2\) is
The corresponding predicted event risk through \(t_2\) is \(1-0.5038=0.4962\), or about 49.62%. For a subject with constant \(\eta=\log(1.5)\), the relative risk is \(1.5\), giving
The second subject has lower predicted survival because the retained event-model structure assigns a 1.5-fold hazard multiplier. These are model-based predictions conditional on the stated predictor values, not estimates of an individual’s known future outcome.
Worked Example: Why the Shape of the Baseline Update Matters
Suppose, hypothetically, the original model’s baseline-hazard increments in two intervals were \(0.1000\) and \(0.3000\), while the new cohort’s estimated increments are \(0.2857\) and \(0.4000\). The interval-specific ratios are
The ratios differ substantially. Multiplying the old baseline by one constant cannot reproduce both new increments: matching the first interval requires a multiplier of \(2.857\), whereas matching the second requires \(1.333\). This illustrates why an update that estimates the baseline hazard over time can correct a changed time pattern that a single overall risk adjustment would miss. The example is a comparison of hypothetical increments, not evidence that either cohort has a particular clinical risk.
Justification and Practical Interpretation
The Breslow increment follows by matching observed events to their modeled hazard contribution. With fixed predictors and a baseline increment \(\Delta H_0(t_k)\), each subject \(i\) at risk contributes approximately \(\exp\{\eta_i(t_k)\}\Delta H_0(t_k)\) to expected events at that time. Summing over the risk set gives the expected event count
Equating this expected count to the observed count \(d_k\) and solving for the increment yields
This justification explains both the calculation and its conditions: the risk predictors must be fixed for the calculation, the risk set must be correctly defined, and the event model’s relative-risk form must be suitable. With uncertain random effects, the plug-in formula uses estimated rather than known predictors, which is why an integrated joint-likelihood approach may be preferable.
Diagnostics, Alternatives, and Limitations
After updating, assess whether predicted and observed event experience agree over time in the new cohort, using methods that account for censoring. Compare observed event counts with model-based expected counts across time intervals, and examine calibration summaries with their uncertainty. Large discrepancies concentrated in particular periods can indicate a poorly estimated baseline shape, sparse events, an incorrect time origin, or a time-varying association that the fixed event structure cannot represent. Good overall agreement alone does not guarantee that predictions are appropriate across all predictor values.
A baseline-only update is most defensible when the original relative-risk structure remains plausible and the principal transport problem is a changed event rate over time. It can be unstable in periods with few events or little person-time, especially if the baseline is estimated very flexibly. Predictions beyond the event-time support of the validation cohort require extrapolation and are not justified by the nonparametric increments alone. Uncertainty in the updated baseline should also be reflected in uncertainty intervals for predictions, rather than treating the estimated curve as known.
| Approach | What changes | When it may help | Main limitation |
|---|---|---|---|
| Baseline-hazard update | Baseline hazard over time; predictor effects are fixed | New cohort has a different underlying event rate or time pattern | Cannot repair a wrong relative-risk structure |
| Prediction-intercept recalibration | Overall prediction level through an intercept-type adjustment | A broad level shift is plausible | May not represent a changing baseline pattern over time |
| Prediction-slope recalibration | Strength of the predictor signal | Predictions are too extreme or not extreme enough | Changes relative-risk strength rather than only the baseline |
| Full model refitting | Longitudinal and event-model parameters | There is adequate new-cohort data and structural change is suspected | Can lose the stability and transportability of the original model |
A common mistake is to interpret a lower updated survival curve as evidence that the longitudinal marker has become more strongly associated with the event. That conclusion does not follow from a baseline-only update: the association parameter was held fixed, and the change can arise entirely from a higher estimated baseline hazard. Conversely, if errors vary systematically with the event linear predictor, the baseline update may not be enough; changing only the baseline leaves relative-risk contrasts unchanged.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A risk set has relative risks \(1\), \(1.5\), and \(0.5\), with two events at a tied event time. Using the Breslow formula, what is the baseline-hazard increment?
- If the updated cumulative baseline hazard is \(0.4\) and a subject has constant \(\eta=\log(2)\), what is the predicted survival probability?
- Why does a baseline-only update not correct a changed association between the longitudinal marker and event risk?
- What is a key limitation of using plug-in subject-specific trajectories in the Breslow calculation for a joint model?
Solutions. (1) The risk-set sum is \(1+1.5+0.5=3\), so the increment is \(2/3=0.6667\). (2) The relative risk is \(\exp\{\log(2)\}=2\), so survival is \(\exp(-2\times0.4)=\exp(-0.8)\approx0.4493\). (3) The update holds the association parameter fixed, so it changes the baseline event rate but not the relative-risk contrast associated with the marker. (4) A plug-in trajectory treats an estimated latent trajectory as known and can ignore uncertainty in the random effects; an integrated joint-likelihood update accounts for that uncertainty more directly.