The Event Submodel Conditional on Latent History

Why Condition the Event Model on a Latent Trajectory?

In a joint model, the event process may depend on more than a subject’s baseline characteristics. A biomarker’s evolving trajectory can contain information about the subject’s changing clinical state, and the event submodel can represent that dependence by conditioning on a subject-specific latent trajectory. The key modeling decision is how the trajectory history is summarized for the hazard; the hazard does not automatically depend on every past or future biomarker value.

The longitudinal submodel represents the underlying trajectory, which can be written as \(m_i(t; b_i)\) for subject \(i\) at time \(t\), given latent subject-specific quantities \(b_i\). For example, in a linear mixed model this trajectory may be \(m_i(t; b_i)=x_i(t)^\mathsf{T}\beta+z_i(t)^\mathsf{T}b_i\). The event submodel uses a specified function of that trajectory history, alongside event-related covariates. Here the focus is the conditional event model; the full joint specification of random effects across submodels is a separate modeling question.

Definition: The conditional hazard \(h_i(t\mid w_i(t),b_i)\) is the instantaneous event rate at time \(t\) for subject \(i\), conditional on being event-free just before \(t\), the relevant covariates \(w_i(t)\), and the subject-specific latent quantities \(b_i\). Formally, it is the limit of the conditional probability of an event in \([t,t+\Delta t)\), divided by \(\Delta t\), as \(\Delta t\) approaches zero. Its units are inverse time.

A general proportional-hazards specification uses a nonnegative baseline hazard \(h_0(t)\), a covariate effect \(\gamma\), and a chosen summary \(q_i(t;b_i)\) of the latent trajectory history:

h_i(t\mid w_i(t),b_i) = h_0(t)\exp\left\{\gamma^\mathsf{T}w_i(t)+\alpha q_i(t;b_i)\right\}.

Here \(w_i(t)\) is a vector of event-model covariates, and \(\alpha\) describes the contribution of the selected trajectory summary on the log-hazard scale. The summary \(q_i(t;b_i)\) might encode a feature of the trajectory up to \(t\), such as its level, rate of change, or accumulated exposure. Its definition and units must be stated; these choices are substantive parts of the model. This general formulation does not require a particular summary or link.

Assumptions and Interpretation

The formula is meaningful under several conditions. The baseline hazard must be nonnegative, and the exponential term must be defined. Covariates and the trajectory summary used at time \(t\) must be available from information up to that time; using future trajectory values would not describe a hazard conditional on the subject’s history at \(t\). The time origin and units for event time and longitudinal time must align. Finally, the exponential form assumes that the specified covariate and trajectory-summary contributions are additive on the log-hazard scale. A different functional form may be needed if that relationship is implausible.

Conditional means important here: the hazard refers to subjects with the same included covariates and the same latent quantities, at the same time and while event-free. It is not automatically the hazard averaged over subjects’ latent quantities. Nor does a conditional association establish that changing the biomarker would cause a change in event risk.

Key distinction: The event submodel needs an explicit rule connecting latent trajectory history to the hazard. Writing “the hazard depends on the biomarker” is incomplete until the trajectory summary, its time reference, and its scale are specified.

Worked Conditional Hazards

Worked Example: A Standardized History Summary

Suppose a hypothetical event submodel has baseline hazard \(h_0(t)=0.020\) per month at a particular time, a binary covariate \(w_i=1\), coefficient \(\gamma=0.40\), and trajectory summary \(q_i=0.80\). The summary is cumulative exposure divided by 10 biomarker-months, so it is dimensionless. Let \(\alpha=0.30\). Calculate the subject’s conditional hazard.

The model is \(h_i(t)=h_0(t)\exp\{\gamma w_i+\alpha q_i\}\). Substituting the stated values gives

h_i(t) = 0.020\exp\{(0.40)(1)+(0.30)(0.80)\} = 0.020\exp(0.64).

Since \(\exp(0.64)\approx1.896\), the conditional hazard is \(0.020(1.896)=0.0379\) per month. This is an instantaneous rate conditional on the stated covariate and latent history, not a probability of an event during a month.

For comparison, holding the covariate and baseline hazard fixed but changing the summary from \(0.80\) to \(0.20\) changes the hazard ratio to \(\exp\{0.30(0.80-0.20)\}=\exp(0.18)\approx1.197\). Under this model, the first specified history has a conditional hazard about 1.20 times that of the second at this time.

Worked Example: A Trajectory-Slope Summary

Consider a hypothetical longitudinal trajectory \(m_i(s;b_i)=12+0.50s\), measured in biomarker units when \(s\) is in months. At the event time of interest, suppose the event submodel uses the slope \(q_i=0.50\) biomarker units per month. Let \(h_0(t)=0.010\) per month, \(\gamma w_i=0\), and \(\alpha=-0.40\) per (biomarker unit per month). Calculate the conditional hazard.

The slope is the coefficient of \(s\) in the stated trajectory. Substitution gives

h_i(t) = 0.010\exp\{(-0.40)(0.50)\} = 0.010\exp(-0.20) \approx 0.00819 \quad\text{per month}.

Thus the fitted conditional hazard for this specified slope is approximately \(0.00819\) per month. The calculation depends on the summary being the slope in biomarker units per month. If time were instead measured in years, the numerical slope and the coefficient’s units would need to change consistently; otherwise the exponent would no longer represent the same model.

Worked Example: An Accumulated-History Summary

Suppose a hypothetical trajectory is \(m_i(s)=2+0.20s\) biomarker units, with \(s\) measured in months. At \(t=5\) months, the event model uses accumulated exposure over \([0,t]\), scaled by 10 biomarker-months: \(q_i(t)=\int_0^t m_i(s)\,ds/10\). Let \(h_0(5)=0.015\) per month, \(\gamma w_i=0.30\), and \(\alpha=0.20\) per unit of the scaled summary.

First calculate the unscaled accumulation:

\int_0^5 (2+0.20s)\,ds = \left[2s+0.10s^2\right]_0^5 = 10+2.5 = 12.5 \quad\text{biomarker-months}.

Therefore \(q_i(5)=12.5/10=1.25\). The conditional hazard is

h_i(5) = 0.015\exp\{0.30+(0.20)(1.25)\} = 0.015\exp(0.55) \approx0.0260 \quad\text{per month}.

This calculation uses the trajectory only through month 5, as required by a history-based summary at that time. The units and scaling are explicit: \(\alpha\) multiplies a dimensionless accumulated-exposure summary.

Why the Conditional Form Has This Structure

The exponential form keeps the hazard nonnegative and makes the covariate contributions additive on the log-hazard scale. It also makes comparisons straightforward when the baseline hazard and other covariates are held fixed. For two specified latent histories \(q_a\) and \(q_b\) at the same time, with the same \(w_i(t)\), the conditional hazard ratio follows by division:

\frac{h_i(t\mid q_a,w_i(t))}{h_i(t\mid q_b,w_i(t))} = \frac{h_0(t)\exp\{\gamma^\mathsf{T}w_i(t)+\alpha q_a\}} {h_0(t)\exp\{\gamma^\mathsf{T}w_i(t)+\alpha q_b\}} = \exp\{\alpha(q_a-q_b)\}.

The common baseline hazard and covariate term cancel. This comparison is conditional on the specified histories and covariates. It does not compare marginal hazards for groups whose latent quantities vary, and it does not imply a causal effect.

The selected summary determines what aspect of history can be associated with the event hazard. A level summary, a slope summary, and accumulated exposure are different scientific models, not interchangeable algebraic descriptions. More complex summaries or nonlinear functions may be appropriate, but they must still use information available by time \(t\), be adequately supported by the data, and be defined with interpretable units.

Diagnostics, Alternatives, and Limitations

A conditional event submodel can be misspecified even when its algebra is correct. Check that the event and longitudinal time scales share a clear origin and units, that the trajectory summary is computed only up to the event time, and that its numerical scaling is sensible. Inspect whether the assumed log-linear contribution is plausible and whether conclusions are sensitive to reasonable alternative summaries. Survival-model diagnostics, including checks of proportional-hazards structure and overall fit, remain relevant; a latent-history term does not remove the need to assess the event model.

Common warning signs include implausibly large or small linear predictors caused by unscaled summaries, a hazard that appears to depend on future biomarker information, and a trajectory summary that changes meaning when time units change but whose coefficient is left unchanged. Sparse longitudinal measurements can also make a subject’s latent history uncertain. The fitted conditional hazard is then only as credible as the longitudinal trajectory representation and the event model that uses it.

The main alternatives differ in what information enters the event model. A survival model with only fixed covariates does not explicitly use the evolving latent trajectory. A model using a time-varying recorded biomarker instead uses observed measurements as covariates; it is not automatically the same as a model based on a latent trajectory summary. The conditional latent-history approach makes the chosen trajectory feature explicit, but adds modeling assumptions about that trajectory and how it relates to the event.

Event-model approach	Information used	Key limitation
Fixed-covariate survival model	Specified covariates, often measured at baseline	Does not represent evolving latent trajectory history
Time-varying observed covariate model	Recorded biomarker values over time	Uses observed values directly rather than an explicitly specified latent-history summary
Conditional latent-history model	Covariates and a defined function of subject-specific trajectory history	Depends on the trajectory model and the chosen history function

No single history summary is universally correct. A simple summary can be easier to estimate and explain but may omit relevant features; a flexible summary can be harder to identify and may be sensitive to modeling choices. These models describe conditional associations, and a fitted association should not be interpreted as evidence that an intervention on the biomarker would alter event risk.

Common Mistake

Wrong reasoning: “The event model includes a biomarker, so I can put the most recently recorded measurement into the hazard at every time, even if that measurement occurred after the event time.”
Correction: A hazard at time \(t\) must be conditioned on information available by \(t\). Define a time-aligned trajectory summary using history through \(t\), state its units, and use a model that respects that information boundary.

Key takeaway: A conditional event submodel combines a baseline hazard, event covariates, and an explicitly defined summary of latent trajectory history. The summary’s definition, time alignment, and scale are essential parts of the model.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

If \(h_0(t)=0.012\) per month, \(\gamma w_i=0.20\), \(\alpha=0.25\), and \(q_i(t)=0.40\), calculate the conditional hazard.
At the same time and covariate values, what is the conditional hazard ratio for \(q_a=1.0\) versus \(q_b=0.2\) when \(\alpha=0.30\)?
Why is a trajectory summary based on values after time \(t\) inappropriate for a hazard at time \(t\)?
If a slope is expressed per year rather than per month, what must be checked before using the same event-model coefficient?

Solutions. (1) \(h_i(t)=0.012\exp\{0.20+(0.25)(0.40)\}=0.012\exp(0.30)\approx0.0162\) per month. (2) The ratio is \(\exp\{0.30(1.0-0.2)\}=\exp(0.24)\approx1.271\). (3) The hazard at \(t\) is conditional on information available by \(t\); future values would introduce information not yet available at that time. (4) The slope’s numerical value and units change with the time scale, so the coefficient and summary must be scaled consistently to preserve the same dimensionless exponent.