Current Biomarker Value and Recurrent-Event Intensity

Why Link Intensity to the Current Biomarker Value?

A recurrent-event process may be associated with a subject’s evolving biomarker level. For example, a higher underlying marker value might coincide with a higher instantaneous rate of repeated disease events. The statistical question is not simply whether recorded biomarker measurements and event counts tend to vary together. Measurements are noisy and usually occur only at selected visits, whereas recurrent events can occur between visits. A joint model addresses this by linking event intensity to the subject’s latent current biomarker value: the underlying value at the event-process time, as represented by the longitudinal model.

The link specifies a conditional association, not a causal effect. It describes how the model’s instantaneous event rate changes with the latent marker while other model terms are held fixed. Its meaning depends on the biomarker scale, the recurrent-event intensity model, and the conditioning variables.

Model Definition and Assumptions

Let \(N_i(t)\) count recurrent events for subject \(i\) by time \(t\), and let \(Y_i(t)\) equal 1 while that subject is under observation and at risk for another event, and 0 otherwise. The conditional intensity is the instantaneous event rate given the modeled history just before \(t\). A standard current-value specification uses a latent longitudinal trajectory \(m_i(t)\):

X_{ij}=m_i(t_{ij})+\varepsilon_{ij}, \qquad m_i(t)=\mathbf{x}_i(t)^\top\boldsymbol{\beta} +\mathbf{z}_i(t)^\top\mathbf{b}_i,

Here \(X_{ij}\) is the biomarker measured for subject \(i\) at visit time \(t_{ij}\); \(\varepsilon_{ij}\) is measurement error; and \(\mathbf{b}_i\) represents subject-specific random effects. The function \(m_i(t)\) is the latent mean biomarker trajectory, not the individual noisy measurement \(X_{ij}\).

\lambda_i(t\mid\mathcal{F}_{i,t-},\mathbf{b}_i,u_i) = Y_i(t)\lambda_0(t) \exp\!\left\{ \boldsymbol{\gamma}^{\top}\mathbf{w}_i(t) +u_i +\alpha q_i(t) \right\}, \qquad q_i(t)=\frac{m_i(t)-m_{\mathrm{ref}}}{c}.

In this expression, \(\mathcal{F}_{i,t-}\) denotes the modeled event and covariate history immediately before \(t\); \(\lambda_0(t)\) is a baseline intensity; \(\mathbf{w}_i(t)\) contains other covariates; and \(u_i\) is a subject-level frailty term if included. The constant \(m_{\mathrm{ref}}\) is a reference biomarker value and \(c\) is a chosen positive scaling increment, such as 10 biomarker units. Thus \(q_i(t)\) expresses the current latent value relative to the reference, in increments of size \(c\). The intensity is zero when \(Y_i(t)=0\).

Definition: In a current-value recurrent-event model, the event intensity at time \(t\) depends on the latent biomarker trajectory through its value \(m_i(t)\) at that same time. With the scaling above, \(\alpha\) is the log conditional rate ratio for a \(c\)-unit increase in the latent current biomarker value, holding the other model terms fixed.

The intensity has the usual small-interval interpretation: conditional on the modeled history and subject-specific effects, the probability of one event in \([t,t+dt)\) is approximately \(\lambda_i(t)\,dt\), with the probability of two or more events of smaller order than \(dt\). This formulation requires a correctly specified recurrent-event risk process and an appropriate at-risk indicator. It also relies on a suitable longitudinal model for \(m_i(t)\), including an adequate description of measurement error and subject variation. If the random effects in the longitudinal and event models are associated, their joint distribution must be represented adequately; treating them as unrelated when they are not can distort the estimated association.

The model further assumes that, conditional on the included history and effects, the current latent value and other specified terms adequately describe the event intensity. The covariates used in the intensity should be known just before the event time, and the chosen functional form for the current-value association should be plausible. These are modeling assumptions, not properties guaranteed by fitting a joint model.

Deriving the Conditional Rate Ratio

Compare two otherwise identical at-risk situations at the same time \(t\), with the same covariates, frailty, baseline intensity, and modeled history, but with latent current biomarker values \(m\) and \(m+\Delta\). Their scaled values differ by \(\Delta/c\). Dividing the two intensities cancels every shared term:

\frac{\lambda_i(t\mid m+\Delta,\ \text{other terms})} {\lambda_i(t\mid m,\ \text{other terms})} = \frac{\lambda_0(t)\exp\{\boldsymbol{\gamma}^{\top}\mathbf{w}_i(t)+u_i+ \alpha[(m+\Delta-m_{\mathrm{ref}})/c]\}} {\lambda_0(t)\exp\{\boldsymbol{\gamma}^{\top}\mathbf{w}_i(t)+u_i+ \alpha[(m-m_{\mathrm{ref}})/c]\}} = \exp\!\left(\alpha\frac{\Delta}{c}\right).

In particular, for an increase of \(c\) biomarker units, the ratio is \(\exp(\alpha)\). If \(\alpha>0\), the conditional intensity is higher at higher current values; if \(\alpha<0\), it is lower; and if \(\alpha=0\), the model has no current-value association after conditioning on its other terms. This is a comparison of instantaneous rates, not a comparison of cumulative event counts or of probabilities of ever having an event.

Worked Example: Comparing Intensities at Two Current Values

Suppose a hypothetical fitted model uses \(m_{\mathrm{ref}}=100\) units, \(c=10\) units, and \(\alpha=0.35\). For illustration, take \(\lambda_0(t)=0.040\) events per person-year at a particular time, and suppose the shared covariate and frailty contribution is \(\boldsymbol{\gamma}^{\top}\mathbf{w}_i(t)+u_i=0.10\). The subject is at risk in both comparisons.

At \(m=100\), \(q=(100-100)/10=0\), so the intensity is \(\lambda=0.040\exp(0.10+0.35\times0)=0.04421\) events per person-year.

At \(m=120\), \(q=(120-100)/10=2\), so the intensity is \(\lambda=0.040\exp(0.10+0.35\times2)=0.08903\) events per person-year.

The ratio is \(0.08903/0.04421=2.014\), which agrees with the direct calculation \(\exp\{0.35(120-100)/10\}=\exp(0.70)=2.014\). Under the model, a subject with a latent current value 20 units higher has about 2.01 times the instantaneous recurrence rate, conditional on the same time, risk status, covariates, and frailty. The comparison does not say that raising a person’s biomarker would cause their rate to change by this amount.

Estimating and Reporting the Association

In practice, \(\alpha\) is estimated jointly with parameters describing the longitudinal trajectory and recurrent-event process. The event contribution depends on the latent trajectory, while the biomarker measurements inform that trajectory. This joint estimation is important: substituting each noisy observed biomarker measurement directly into the event model can treat measurement error as if it were part of the true current value, and measurements may not even be available at event times.

A report should state the biomarker scaling and the conditioning implicit in the rate ratio. For example, \(\exp(\hat{\alpha})\) is the estimated conditional rate ratio per \(c\)-unit increase, not necessarily per one-unit increase. If an approximate Wald interval is used, it is computed on the coefficient scale and then exponentiated:

\hat{\alpha}\ \pm\ 1.96\,SE(\hat{\alpha}), \qquad \left[ \exp\{\hat{\alpha}-1.96\,SE(\hat{\alpha})\}, \exp\{\hat{\alpha}+1.96\,SE(\hat{\alpha})\} \right].

Worked Example: Interpreting an Estimated Association

Suppose a hypothetical joint-model fit gives \(\hat{\alpha}=0.22\) and \(SE(\hat{\alpha})=0.08\), with \(c=10\) biomarker units. The estimated conditional rate ratio per 10-unit increase is \(\exp(0.22)=1.246\), or about 1.25.

The approximate 95% Wald interval on the log scale is \(0.22\pm1.96(0.08)=0.22\pm0.1568\), giving \((0.0632,0.3768)\). Exponentiating the endpoints gives \((1.065,1.458)\). Under the model and its assumptions, this interval is a confidence interval for the conditional rate ratio per 10-unit increase; it is not a 95% probability statement about the fixed parameter.

For a two-sided test of \(H_0:\alpha=0\), the Wald statistic is \(z=0.22/0.08=2.75\), with an approximate two-sided \(p\)-value of 0.0060. At a 0.05 significance level, this is evidence against no current-value association in the specified model. It does not establish causation, nor does it test whether the model’s assumptions are correct.

Latent-Value Uncertainty and Diagnostics

The conditional intensity is defined given the latent trajectory and subject-specific effects. In applications those quantities are not directly observed. A subject-specific intensity averaged over uncertainty in the latent value involves averaging the exponential term, not simply inserting the posterior mean value. Because exponentiation is nonlinear, these calculations generally differ.

Worked Example: Why Plugging in a Mean Can Differ

Suppose, at a particular time, a hypothetical subject’s posterior distribution for the scaled current value \(q_i(t)\) assigns probability 0.5 to \(q=0\) and probability 0.5 to \(q=2\). Let the remaining intensity terms combine to 0.050 events per person-year and let \(\alpha=0.40\). Averaging the conditional intensity over these two possibilities gives \(0.5(0.050e^{0})+0.5(0.050e^{0.80})=0.050(1+2.2255)/2=0.08064\) events per person-year.

The posterior mean scaled value is \(0.5(0)+0.5(2)=1\). Plugging in that mean instead gives \(0.050e^{0.40}=0.07459\) events per person-year, which is not the same. The first calculation averages the model intensity over the stated uncertainty; the second calculates the intensity at the posterior mean. Which quantity is needed depends on the inferential target, but they should not be described as interchangeable.

Diagnostics should examine both parts of the joint model. Check whether the longitudinal model describes the observed biomarker patterns and whether its residual behavior and random-effects structure are plausible. For the event component, inspect recurrent-event residuals or other model-based checks for systematic departures over time and across subjects. Compare observed and model-predicted recurrence patterns in relevant covariate groups. A poor fit in either component can undermine the interpretation of \(\hat{\alpha}\).

A constant \(\alpha\) assumes the log intensity changes linearly with the scaled current value and that the association does not itself vary with time. A visibly nonlinear relationship or a changing association may therefore call for a different specification. Sparse biomarker measurements, misspecified measurement-error behavior, informative measurement timing not accounted for in the model, and an incorrect joint random-effects distribution can all make the reconstructed current value unreliable. Results should also be checked for sensitivity to reasonable modeling choices rather than judged by a single coefficient or significance test.

How This Link Differs From Alternatives

Different event models encode different sources of association. The following distinctions help keep the current-value coefficient’s meaning specific.

Specification	What the event intensity uses	Interpretive distinction
Current-value link	The latent biomarker value at time \(t\)	\(\exp(\alpha)\) is the conditional rate ratio for a \(c\)-unit current-value increase.
Frailty-only association	A subject-level latent event propensity, without a direct current-value term	Accounts for persistent subject differences but does not assign a rate ratio to a change in current biomarker value.
Observed-value substitution	A recorded biomarker measurement used as though it were the true value	Does not by itself separate measurement error from the underlying trajectory and may not provide a value at event times.

A current-value term can be included alongside frailty, as in the model above. In that case, \(\alpha\) describes the current-value association conditional on the frailty term as well as the other included terms. The alternatives answer different modeling questions; model choice should follow the scientific target and the adequacy of the assumptions, not merely which specification yields a more striking association.

Common Mistake and Practical Limits

Common mistake: “The estimated coefficient is 0.35, so every one-unit increase in the biomarker multiplies the recurrence rate by \(\exp(0.35)\).” This is incorrect when the model defines \(q_i(t)=(m_i(t)-m_{\mathrm{ref}})/10\). The coefficient is for a 10-unit increase; the one-unit rate ratio is \(\exp(0.35/10)\). Always read the scaling inside the model before interpreting the coefficient.

Even with a carefully specified joint model, the rate ratio is conditional and model-based. It does not establish that intervening on the biomarker would alter recurrence rates. It also does not remove the need to define the event process, at-risk periods, and relevant covariates correctly. If subjects leave the risk process for reasons related to their event history or latent trajectory, that feature must be handled by an appropriate design and model; the current-value term alone does not solve it.

The main benefit of this specification is a direct, interpretable link between an estimated underlying biomarker value and the instantaneous recurrence rate. Its main limitation is that this interpretation is only as credible as the latent trajectory, event-intensity, and joint-distribution assumptions supporting it.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A model uses \(q=(m-50)/5\) and estimates \(\alpha=-0.20\). What is the conditional rate ratio for a 5-unit increase in current latent biomarker value?
With the same model, what is the rate ratio for a 10-unit increase?
Why can an observed biomarker measurement at the last visit be an unsuitable substitute for the latent current value at a later event time?
In one sentence, state what is held fixed when interpreting the current-value rate ratio.

Solutions. (1) A 5-unit increase changes \(q\) by 1, so the ratio is \(\exp(-0.20)=0.8187\), conditional on the other model terms. (2) A 10-unit increase changes \(q\) by 2, so the ratio is \(\exp(-0.20\times2)=\exp(-0.40)=0.6703\). (3) The recorded value may contain measurement error and may differ from the underlying trajectory at the later event time. (4) Hold fixed the time and risk status, covariates, frailty, and other included model terms; the comparison changes only the latent current biomarker value.