Baseline Hazard Choices for Transition Regression

Why the Baseline Hazard Is a Modeling Choice

A transition regression model separates two questions: how the transition hazard changes with covariates, and how it changes with time when covariates are at their reference values. The second part is the baseline hazard. Its specification can affect estimation, predictions, and the assumptions a model makes about time. It does not, by itself, change the definition of a covariate hazard ratio when the proportional-hazards model is correctly specified.

For a transition \(r\), let \(Y_{ir}(t)\) indicate whether individual \(i\) is at risk for that transition at time \(t\), and let \(N_{ir}(t)\) count that individual’s observed transitions of type \(r\) by time \(t\). Let \(Z_i(t)\) denote the covariate vector used for this transition. A transition-specific proportional-hazards regression model is

Here \(\lambda_{0r}(t)\) is the baseline hazard for transition \(r\), and \(\beta_r\) is its regression coefficient vector. The hazard has units of inverse time; the hazard ratio for a one-unit increase in a covariate is dimensionless. The baseline is transition-specific unless the model explicitly imposes a shared structure.

The baseline choice is made within a model whose other assumptions must also be credible: the transition risk sets must be correctly defined; measured covariates must be suitable for the model; censoring must be conditionally independent of the event process for the likelihood or partial likelihood being used; and the proportional-hazards form must be appropriate unless time-varying effects are included. These conditions are distinct from the question of whether the baseline is unspecified, piecewise constant, or parametric.

Three Ways to Represent the Baseline

All three choices can be paired with the same proportional-hazards covariate term. The main difference is how they describe the baseline time pattern and how much information about that pattern is assumed in estimation. A misspecified baseline can harm fitted hazards and predictions even if the covariate effect is of primary interest.

Unspecified Baseline: What Partial Likelihood Removes

With an unspecified baseline, the Cox partial likelihood estimates \(\beta_r\) by comparing the covariates of the individual who experiences a transition with those of individuals in the risk set for that transition. Suppose one transition occurs at time \(t\), there are no tied events at that time, and the risk set is \(R_r(t)\). Conditional on one event occurring among those at risk, the model assigns probability

to individual \(i\) being the one who experiences the transition. The common baseline hazard cancels in this conditional probability. Multiplying such contributions across event times gives the partial likelihood. This is why the regression coefficients can be estimated without first choosing a parametric form for \(\lambda_{0r}(t)\). Tied events require a specified tie method, such as an exact method or a standard approximation; simply applying the no-tie expression to tied events is not generally justified.

The cancellation does not mean the baseline is irrelevant. After estimating \(\beta_r\), one can estimate the baseline cumulative hazard. For example, the Breslow estimator adds, at event time \(t\), the number of transition events divided by the covariate-weighted risk set at that time. In a simple case with one event, its increment is \(1/\sum_{j\in R_r(t)}\exp\{\hat\beta_r^\mathsf{T}Z_j(t)\}\). The resulting estimate supports fitted cumulative hazards within the observed time range, but does not impose a smooth baseline shape.

Piecewise-Constant Baseline: Local Rates Over Time

A piecewise-constant baseline divides follow-up into intervals and assumes a constant baseline hazard within each interval. For cut points \(0=\tau_0<\tau_1<\cdots<\tau_K\), one representation is \(\lambda_{0r}(t)=\lambda_{rk}\) when \(\tau_{k-1}\leq t<\tau_k\). The baseline can change between intervals, while the covariate hazard ratio remains constant over time if \(\beta_r\) does not vary with time.

This choice makes interval boundaries part of the model. The intervals should be selected with the timing of the process and the amount of event information in mind. Very wide intervals may obscure meaningful baseline changes; very narrow intervals may leave unstable baseline estimates, especially where few events occur. A piecewise model is not automatically a good approximation merely because it uses many intervals.

Parametric Baseline: A Distributional Time Pattern

A parametric baseline specifies a particular function of time with a finite number of parameters. For example, a Weibull baseline can be written

where \(\kappa>0\) is a dimensionless shape parameter and \(\eta>0\) is a scale parameter measured in units of time. Under this parameterization, the baseline hazard is increasing when \(\kappa>1\), constant when \(\kappa=1\), and decreasing when \(0<\kappa<1\). The model estimates baseline parameters along with the regression coefficients. A smooth and compact description can be useful, but it assumes that the chosen form describes the baseline adequately over the time range of interest.

What Changes in Estimation and Interpretation?

Under a correctly specified proportional-hazards model, the transition-specific hazard ratio for a one-unit increase in covariate \(Z_q\), holding the other modeled covariates fixed, is \(\exp(\beta_{rq})\). It compares instantaneous transition hazards among individuals who are at risk for transition \(r\) at the same time and with the same values of the other modeled covariates. It is not a ratio of transition probabilities or a causal effect by itself.

The baseline choice affects how the time pattern is estimated and how the likelihood uses information. An unspecified baseline leaves its shape unrestricted at event times and estimates coefficients through partial likelihood. A piecewise-constant baseline estimates a rate for each interval under a constant-within-interval assumption. A parametric baseline estimates a small set of parameters under a specified curve. In finite samples these choices can produce different coefficient estimates and standard errors, because the estimators use different modeling restrictions and likelihood information. If all assumptions hold and there is sufficient information, the substantive coefficient target is the same proportional-hazards effect; the numerical estimates need not be identical.

Baseline assumptions matter especially when the goal includes absolute transition hazards or predictions over time. A hazard ratio alone does not give an absolute hazard: the baseline is needed. Extrapolation is particularly consequential for parametric models, because the assumed curve continues beyond observed event times. A flexible unspecified baseline avoids committing to a curve but offers less direct support for long-range extrapolation.

Diagnostics, Failure Modes, and Alternatives

Check the fit by transition, rather than relying only on an overall impression. Compare the observed timing of events with fitted baseline patterns; look for intervals with very few events or sharp changes that a chosen baseline cannot represent; and examine whether predictions are sensitive to reasonable changes in piecewise cut points or parametric form. A parametric model that fits poorly near the end of follow-up should not be trusted for distant extrapolation merely because its curve is smooth.

The earlier material on Schoenfeld residual diagnostics by transition and Martingale residual diagnostics for transition covariate forms addresses other important questions: whether covariate effects vary over time and whether covariate functional forms are adequate. A baseline choice does not repair a violation of proportional hazards, an unsuitable covariate form, or an incorrectly constructed risk set. Also check whether sparse transition events make estimates unstable. With an unspecified baseline, event times provide the baseline’s support; with a piecewise baseline, sparse intervals can yield imprecise rates; and with a parametric baseline, apparent numerical stability can conceal a poor functional-form assumption.

These choices are not the only possible modeling strategies. A stratified transition model can allow separate baseline hazards across groups while estimating common covariate effects, whereas covariate adjustment represents group differences through coefficients. Those approaches answer a different structural question about how groups or transitions share information; the baseline specification still needs to be considered within the chosen model.

• Common mistake: “The baseline is unspecified, so no assumptions are needed.” The correction is that the baseline shape is left flexible, but assumptions about risk sets, censoring, covariate effects, and the transition model remain.
• Common mistake: “A piecewise baseline makes the exposure hazard ratio different in each interval.” Not if the model uses a constant \(\beta_r\). The baseline changes by interval; the hazard ratio changes over time only if the covariate effect is modeled as time-varying.
• Common mistake: “A parametric baseline gives a more precise answer because it is smooth.” Smoothness can reduce variability when the form is appropriate, but misspecification can bias fitted hazards and predictions. Precision does not establish adequacy.
• Common mistake: “A hazard ratio of 1.8 means an 80% higher probability of transition.” It means an 80% higher instantaneous hazard among those currently at risk, under the model; it is not a probability ratio.