Why Baseline Choice Matters
In a competing-risks joint model, each event cause has its own cause-specific hazard. The baseline hazard describes how that cause’s hazard changes over follow-up when the modeled covariates and longitudinal association terms are at their reference values. Choosing its form is not just a computational detail: if the baseline cannot represent the event-time pattern, the model may try to explain that pattern through other terms, including the association with the longitudinal marker.
The focus here is how to specify the baseline across causes. The association structure is treated as given. For illustration, let \(T_i\) be subject \(i\)’s event or censoring time, let \(k\) index event cause, and let \(b_i\) denote the subject-specific random effects. Write \(m_i(t)\) for the modeled longitudinal trajectory. A representative conditional cause-specific hazard is
Here \(\lambda_{0k}(t)\) is the baseline hazard for cause \(k\), \(w_i\) denotes other modeled covariates, and \(\gamma_k\) and \(\alpha_k\) are the corresponding parameters. This expression is an example of an association structure, not a requirement that every analysis use current value as its longitudinal association. The baseline is conditional on the specified reference values; with random effects in the model, it is not generally the marginal hazard obtained by averaging over the random-effects distribution.
The model assumes that each cause-specific hazard is well-defined among subjects still event-free at time \(t\), and that the chosen baseline form adequately represents its time pattern. Likelihood-based inference also requires an appropriate longitudinal model and a censoring assumption: censoring must be noninformative conditional on the quantities included in the model. These assumptions do not imply that the baseline must be identical across causes.
Three Baseline Representations
A parametric baseline uses a small number of parameters to describe the hazard over the full follow-up period. For example, an exponential baseline is constant, \(\lambda_{0k}(t)=\lambda_k\). A Weibull baseline can represent a monotone increase or decrease:
where \(\kappa_k>0\) is a shape parameter, \(\tau_k>0\) is a time-scale parameter, and \(\Lambda_{0k}(t)\) is the cumulative baseline hazard. The hazard is constant when \(\kappa_k=1\), increases when \(\kappa_k>1\), and decreases when \(\kappa_k<1\). Parametric forms can be efficient, but a strongly imposed shape can be problematic when the true hazard has features the form cannot express.
A piecewise-constant baseline divides follow-up into intervals and assigns a constant hazard within each interval. The intervals need not be the same for every cause. This representation allows step changes while keeping the baseline positive. Its flexibility depends on the number and placement of interval boundaries.
A spline baseline models a transformed hazard, commonly the log hazard, as a smooth combination of basis functions. For example, \(\log\lambda_{0k}(t)=\sum_{j=1}^{J}\theta_{kj}B_j(t)\), where \(B_j(t)\) are spline basis functions and \(\theta_{kj}\) are cause-specific coefficients. Exponentiating the result ensures a positive hazard. Knot placement and any smoothing or penalty determine how much local variation the fitted curve can represent.
Each cause can have a different baseline form. Alternatively, an analyst may impose a common form or share selected baseline parameters across causes. Such restrictions are assumptions about the time pattern, not consequences of competing-risks sampling. In particular, allowing cause-specific association parameters does not require identical baselines, and choosing identical baselines does not by itself impose equal association parameters.
Worked Baseline Calculations
Worked Example: Constant Baselines for Two Causes
Suppose a hypothetical model uses exponential baselines, with cause 1 having a baseline hazard of \(0.040\) per month and cause 2 having a baseline hazard of \(0.015\) per month. Consider the reference subject, so that the covariate and association multipliers equal 1. Calculate the probability of remaining free of either event through 12 months, conditional on this model.
The total hazard at each month is \(0.040+0.015=0.055\) per month. The cause-specific cumulative baseline hazards through 12 months are
The total cumulative hazard is \(0.480+0.180=0.660\). Under the specified constant hazards, the event-free probability is
Thus the model gives an event-free probability of about \(0.517\) through 12 months for the reference subject. This calculation depends on both baselines being constant over the interval; it is not a general consequence of having two causes.
Worked Example: Piecewise-Constant Baselines
For a hypothetical two-cause model, suppose the baseline hazards are constant within months 0–6 and 6–12. For cause 1, the hazards are \(0.020\) and \(0.050\) per month. For cause 2, they are \(0.010\) and \(0.030\) per month. Calculate the cumulative baseline hazard for each cause through month 12 and the corresponding event-free probability at the reference covariate values.
For cause 1, the cumulative baseline hazard is the sum of each interval’s hazard multiplied by its duration:
For cause 2, the corresponding calculation is
The total cumulative hazard is \(0.420+0.240=0.660\), so the event-free probability is \(\exp(-0.660)\approx0.5169\). The numerical result happens to match the previous example, but the time patterns do not: the piecewise hazards increase after month 6, whereas the exponential hazards were constant. Equal cumulative hazards at one horizon do not imply equal hazards at every time or equal cause-specific event probabilities.
Worked Example: Evaluating a Spline Baseline
Suppose a hypothetical spline model for cause 1 represents the log baseline hazard as \(\log\lambda_{01}(t)=\theta_0+\theta_1B_1(t)+\theta_2B_2(t)\). At two follow-up times, the basis values and fitted coefficients are \(\theta_0=-4.0\), \(\theta_1=-0.4\), \(\theta_2=0.2\). At month 6, let \(B_1(6)=0.5\) and \(B_2(6)=0\); at month 12, let \(B_1(12)=1\) and \(B_2(12)=0.5\). Calculate the fitted baseline hazards.
At month 6, the fitted log hazard is \(-4.0+(-0.4)(0.5)+(0.2)(0)=-4.2\). Therefore,
At month 12, the fitted log hazard is \(-4.0+(-0.4)(1)+(0.2)(0.5)=-4.3\), giving
These are fitted baseline hazards at the reference values, not hazards for every subject. A subject’s conditional hazard also includes the modeled covariate and association multiplier. The two evaluated times alone do not establish the entire curve; that depends on all the spline basis functions, knots, and fitted coefficients.
Why Baseline Flexibility Can Change an Association Estimate
The reason baseline choice matters can be seen from the conditional event-time likelihood contribution. Let \(\delta_{ik}=1\) if subject \(i\) experiences cause \(k\) at \(T_i\), and let \(\delta_{ik}=0\) otherwise. In a single-event competing-risks setting, at most one cause indicator is 1. Conditional on the random effects and the modeled histories, the event-time contribution is
The product supplies the hazard for the observed cause when an event occurs. The exponential factor is the probability contribution for remaining free of every event cause up to \(T_i\). It follows because the total hazard among event-free subjects is the sum of the cause-specific hazards, so the cumulative total hazard is the sum of their integrals.
For the illustrative current-value association, the part of the log likelihood involving \(\alpha_k\) has derivative
The first term uses the marker trajectory at an observed cause-\(k\) event. The second accumulates the marker-weighted hazard over the full time at risk. Both terms depend on the baseline through \(\lambda_{ik}\). If the baseline is misspecified, the model can misrepresent when events should occur and consequently change the association estimate that best balances these two terms. This is one reason an association estimate can move when the baseline form changes, even when the longitudinal model and association structure are held fixed.
Worked Example: A Time Pattern Mistaken for a Marker Association
Consider hypothetical person-time summaries in two periods. Assume the true event rate depends on period but not on the marker within a period. Early in follow-up, both marker groups have rate \(0.10\) events per person-year; later, both have rate \(0.02\). The high-marker group contributes 80 person-years early and 50 late, with 8 and 1 events, respectively. The low-marker group contributes 20 person-years early and 400 late, with 2 and 8 events. Calculate the crude rates and explain why a model with an inadequate time baseline could suggest a marker association.
The pooled high-marker rate is
The pooled low-marker rate is
The crude rate ratio is \(0.06923/0.02381\approx2.91\). Yet within each period the rates match: early, \(8/80=2/20=0.10\); late, \(1/50=8/400=0.02\). The pooled contrast arises because the high-marker group contributes more person-time in the high-rate early period. A model that forces a constant baseline may have difficulty representing that period pattern and may attribute some of it to the marker association. A sufficiently appropriate time baseline can separate this time pattern from the within-period marker contrast. This arithmetic is an illustration of the mechanism, not a fitted joint-model estimate or a claim about causal effects.
Choosing and Checking a Baseline
A practical choice balances shape assumptions against the amount of information available for each cause. A parametric baseline is economical and can stabilize estimation, but its shape may be too restrictive. A piecewise-constant baseline makes local changes easy to represent, but many short intervals can leave little information for estimating each hazard. A spline can produce a smooth, flexible curve, but its shape depends on knot placement and the degree of smoothing. Greater flexibility is not automatically better: it can increase uncertainty and make association estimates sensitive to sparse event information.
| Baseline form | Strength | Main risk |
|---|---|---|
| Parametric | Few parameters; smooth, compact representation | Bias if the imposed shape is inadequate |
| Piecewise-constant | Direct representation of interval-specific changes | Unstable estimates with too many or poorly supported intervals |
| Spline | Smooth flexible shape without a single global hazard trend | Sensitivity to knots or smoothing; uncertainty in sparsely supported regions |
Check baseline fit separately for each cause, because event counts and time patterns can differ. Useful checks include plotting fitted cause-specific hazard or cumulative-hazard patterns, inspecting event-time calibration, and comparing observed and model-predicted event patterns over follow-up. Such checks must respect the fitted covariates and longitudinal structure; a crude cause-specific curve is not automatically a diagnostic for the conditional baseline. Also compare association estimates and uncertainty across scientifically plausible baseline specifications. Large shifts are a warning that the association result is sensitive to assumptions about time.
When selecting among forms, consider whether the model reproduces the observed event-time distribution, whether each cause has enough events to support its baseline complexity, and whether the baseline and association estimates are stable under reasonable knot or interval changes. Information criteria or predictive comparisons can help, but they do not establish that a baseline is scientifically correct. Report the baseline specification clearly for each cause and treat a major change in the longitudinal association across plausible choices as a limitation.
Common Pitfall
Baseline choice is distinct from the choice of whether association parameters are shared across causes. It also does not remove other limitations: a flexible baseline cannot fix a misspecified longitudinal trajectory, informative censoring that is not handled by the model, or inadequate information for a rare cause. The goal is not maximal flexibility, but a defensible representation of each cause’s time pattern that does not force the longitudinal association to compensate for baseline error.
Check Your Understanding
Answer the questions, then compare your reasoning with the solutions.
- In a joint model, what reference condition defines a cause-specific baseline hazard?
- A piecewise baseline for one cause is \(0.03\) per month for 4 months and \(0.01\) per month for the next 6 months. What is its cumulative baseline hazard over those 10 months?
- Why can a baseline that is too restrictive affect an estimated longitudinal association?
- Give one advantage and one limitation of a spline baseline relative to an exponential baseline.
- If a marker association estimate changes substantially when reasonable baseline specifications are compared, what should the analyst conclude?
Solutions. (1) It is the hazard for that cause at the specified time when modeled covariates and association terms are at their reference values; it is conditional on the model’s conditioning quantities and is not generally a marginal hazard. (2) The cumulative hazard is \(0.03(4)+0.01(6)=0.12+0.06=0.18\). (3) Baseline misspecification can distort the modeled timing of events and hence the cumulative-hazard part of the likelihood; the association parameter may shift to compensate. (4) A spline can represent a wider range of smooth time patterns, while an exponential baseline has a simpler constant shape. The spline’s flexibility brings sensitivity to knot or smoothing choices and may be poorly supported when events are sparse. (5) The association result is sensitive to baseline assumptions and should be reported with that limitation; the analyst should inspect fit and uncertainty across plausible forms rather than selecting a form solely because it produces a preferred association estimate.