The Baseline Hazard in a Joint Model

Why the Baseline Hazard Matters

A joint model links a longitudinal marker to an event process, but the marker association does not by itself describe how event risk changes over time. That time pattern is supplied by the baseline hazard. Together, the baseline hazard and the marker-linked component determine the subject-specific hazard at each time. Understanding their distinct roles is essential for interpreting conditional survival and for checking that a fitted model represents plausible event risk.

Use \(i\) to index subjects, \(t\) for time, and \(b_i\) for subject \(i\)'s shared random effects. Write \(m_i(t)\) for the subject-specific latent marker value at time \(t\), as defined by the longitudinal submodel. The event hazard conditional on \(b_i\) is

Here \(h_0(t)\) is the baseline hazard and \(\alpha\) is the current-value association parameter. This form assumes that, conditional on the shared random effects and the latent marker history represented by \(m_i(t)\), the marker modifies the hazard multiplicatively. Other terms may be included in a fuller model, but this tutorial focuses on the baseline and current-value components.

The baseline is therefore a reference hazard curve, not a constant event probability and not necessarily a constant hazard. If time is measured in years, for example, \(h_0(t)\) has units of inverse years. The exponent must be dimensionless, so \(\alpha\) has units reciprocal to those of the marker. When \(m_i(t)=0\), the multiplier is one and the conditional hazard equals \(h_0(t)\). When \(m_i(t)\) is positive, the hazard is multiplied by \(\exp\{\alpha m_i(t)\}\); a negative value of \(\alpha\) reverses the direction of that association.

From Hazard to Conditional Survival

A hazard describes an instantaneous rate among subjects who have remained event-free up to the time in question. It is not itself a probability. To obtain a probability of surviving through an interval, the hazard must be accumulated over time. Conditional on \(b_i\), define the cumulative hazard through time \(t\) by

The integration variable is \(u\), because the hazard and latent marker can change throughout the interval. The conditional survival function is

This relationship follows from the definition of hazard. Over a short interval of length \(dt\), conditional survival changes approximately by a factor \(1-h_i(t\mid b_i)dt\). Thus \(dS_i(t\mid b_i)/dt=-h_i(t\mid b_i)S_i(t\mid b_i)\). Dividing by survival and integrating from 0 to \(t\), with \(S_i(0\mid b_i)=1\), gives \(\log S_i(t\mid b_i)=-\int_0^t h_i(u\mid b_i)\,du\), and exponentiating gives the stated result.

The cumulative hazard is dimensionless, as required by the exponential. The conditional probability of an event by \(t\), for a subject event-free at time 0, is \(1-S_i(t\mid b_i)\). For an interval beginning at \(s\), the event probability conditional on being event-free at \(s\) is \(1-\exp\{-[H_i(t\mid b_i)-H_i(s\mid b_i)]\}\), for \(t>s\). Both quantities depend on the hazard over the full relevant interval, not just its value at the endpoint.

A Changing Baseline and a Changing Marker

When either component varies with time, the cumulative hazard generally requires integration of their product. In particular, it is usually incorrect to multiply the cumulative baseline hazard by the marker multiplier evaluated at the final time. That shortcut would ignore the marker values and baseline levels at earlier times. Numerical integration is one way to evaluate the required integral when no convenient closed form is available.

The integral also explains why the baseline and marker components cannot usually be separated into two independent cumulative terms. In general,

The right-hand side is valid only in special cases, such as when \(m_i(u)\) is constant throughout the interval. Otherwise, the multiplier changes inside the integral and must be evaluated along the latent trajectory.

Instantaneous Association Is Not a Risk Ratio

At a fixed time, comparing two subjects under the same baseline hazard isolates the marker-linked hazard ratio. For subjects \(i\) and \(j\), conditional on their respective random effects,

This is a ratio of instantaneous hazards at time \(t\), not a ratio of event probabilities over a follow-up period. Cumulative probabilities also depend on how the baseline and each subject's marker value evolve over the entire interval.

Conditional and Population-Level Interpretation

The hazard formula above is conditional on the shared random effects. Those effects connect the longitudinal and event submodels: they help determine the latent marker trajectory and therefore the marker-linked hazard. A subject-specific conditional survival is consequently not the same as survival averaged across the population. If \(G(b)\) denotes the random-effects distribution, the population survival at \(t\) is

In general, this is not \(\exp\{-\int h_i(t\mid b)\,dG(b)\,dt\}\), nor is it obtained by simply inserting an average marker trajectory into the conditional formula. The exponential and averaging operation do not generally commute. Interpretation should therefore specify whether a reported quantity is conditional on random effects or averaged over their distribution.

Assumptions, Diagnostics, and Alternatives

The baseline-hazard representation relies on several conditions. The baseline must be nonnegative and integrable over the time intervals used, so that cumulative hazards are finite there. The multiplicative association assumes that the marker changes the hazard through the stated exponential factor; in particular, a one-unit marker difference gives the same hazard ratio \(\exp(\alpha)\) at each time, conditional on other modeled quantities. The latent marker trajectory must be defined on the times at which its value enters the event model. The shared-random-effects formulation also requires that the conditional event model be interpreted given those random effects and the modeled latent history.

Useful checks follow directly from the structure of the model:

• Check the baseline scale and units. A hazard stated per month cannot be integrated against time measured in years without converting units.
• Check nonnegativity and cumulative monotonicity. A baseline hazard should not be negative, and a computed cumulative hazard should not decrease as time advances.
• Inspect the implied time pattern. A baseline that is too restrictive may fail to represent changing event risk when the marker is at its reference value. Compare fitted and observed event patterns over time using appropriate model diagnostics.
• Check the marker contribution. Confirm the marker scale and the direction and size of \(\alpha\). A unit change in a rescaled marker changes the numerical value and interpretation of \(\alpha\).
• Check integration accuracy. When the integrand changes quickly, a coarse grid can distort cumulative hazards and survival probabilities. Recalculate with a finer grid and compare results.

A common mistake is to call \(h_0(t)\) the subject's event probability at time \(t\), or to report \(\exp(\alpha)\) as the event-probability ratio. The correction is that \(h_0(t)\) is a reference rate, and \(\exp(\alpha)\) is the conditional hazard ratio for a one-unit difference in latent marker value at a fixed time. Probabilities require integrating the subject-specific hazard over an interval and converting cumulative hazard to survival or event probability.

Baseline-hazard choices trade assumptions against flexibility. A fully specified parametric baseline gives a compact form but commits to a particular time pattern. A semiparametric approach leaves the baseline less restricted while estimating the association, and flexible estimated curves can represent more varied patterns but may require careful smoothing or numerical calculations. These approaches do not change the basic relationship \(h_i(t\mid b_i)=h_0(t)\exp\{\alpha m_i(t)\}\); they differ in how \(h_0(t)\) is represented and estimated. No baseline choice compensates for a misspecified marker association, an unsuitable time scale, or inadequate event information.