Predicting Under a Current-Value Association

Why the Current Value Matters for Prediction

In a joint model, a patient’s biomarker is not necessarily treated as a directly observed, error-free value at every time. Instead, the longitudinal submodel describes an underlying trajectory, and the event submodel can link the hazard to that trajectory. Under a current-value association, the hazard at time $t$ depends on the latent biomarker value at that same time. A prediction over a future interval must therefore account for how the patient’s latent value may evolve during that interval, as well as uncertainty about the trajectory given the available data.

The target here is conditional survival beyond a landmark $s$, for a patient who has remained event-free through $s$ and has longitudinal history $H_i(s)$ observed by then. The fitted model parameters are treated as fixed in the calculations below. The integration is over uncertainty in that patient’s latent trajectory, represented by their random effects; it does not include uncertainty in the estimated model parameters.

Current-Value Joint Model and Prediction Target

Definition: Let $m_i(t;b_i)$ denote patient $i$’s latent biomarker value at time $t$, as specified by the longitudinal model, with random effects $b_i$. A current-value association specifies the event hazard, conditional on $b_i$ and baseline covariates $w_i$, as $$ h_i(t\mid b_i,w_i)=h_0(t)\exp\{\gamma^\top w_i+\alpha m_i(t;b_i)\}. $$ Here $h_0(t)$ is the baseline hazard, $\gamma$ is the vector of baseline-covariate coefficients, and $\alpha$ is the association parameter. Conditional survival from time 0 through time $t$ is $$ S_i(t\mid b_i,w_i)=\exp\left\{-\int_0^t h_i(v\mid b_i,w_i)\,dv\right\}. $$

The hazard is instantaneous, whereas survival over an interval depends on the integrated hazard throughout that interval. Consequently, knowing the latent value at the landmark alone is generally insufficient: the current-value association uses $m_i(v;b_i)$ at every time $v$ in the prediction interval. The sign of $\alpha$ describes the direction of association, conditional on the other model terms. For example, if the biomarker is measured in units $u$, then $\alpha$ is a log-hazard change per one-unit increase in the latent biomarker, holding time and baseline covariates fixed. It does not by itself establish a causal effect.

Write $H_i(s)$ for the biomarker measurements and their observation times available through $s$. The patient must also be known to be event-free at $s$. The desired prediction at a future time $u>s$ is $P(T_i>u\mid T_i>s,H_i(s),w_i)$. The history and event-free follow-up both inform which latent trajectories remain plausible.

Integrating Over the Patient’s Latent Trajectory

Let $p(b_i)$ be the random-effects distribution and let $p\{H_i(s)\mid b_i\}$ be the longitudinal-history likelihood conditional on the random effects. Under conditional independence of the longitudinal and event processes given the shared random effects and model covariates, the posterior density of $b_i$ for a patient event-free at $s$ is proportional to $p\{H_i(s)\mid b_i\}S_i(s\mid b_i,w_i)p(b_i)$. The survival factor matters: among trajectories that fit the biomarker history, those implying substantially greater hazard are less compatible with having survived to the landmark.

q_i(b_i\mid H_i(s),T_i>s,w_i) = \frac{p\{H_i(s)\mid b_i\}S_i(s\mid b_i,w_i)p(b_i)} {\int p\{H_i(s)\mid b_i'\}S_i(s\mid b_i',w_i)p(b_i')\,db_i'}.

For a given trajectory $b_i$, the probability of surviving from $s$ to $u$ is the ratio $S_i(u\mid b_i,w_i)/S_i(s\mid b_i,w_i)$. Averaging this conditional probability over the posterior trajectory distribution gives the prediction. Equivalently, the numerator and denominator below are the joint probabilities of biomarker history and survival to $u$, and of biomarker history and survival to $s$, respectively.

\begin{aligned} P(T_i>u\mid T_i>s,H_i(s),w_i) &=\int \frac{S_i(u\mid b_i,w_i)}{S_i(s\mid b_i,w_i)} q_i(b_i\mid H_i(s),T_i>s,w_i)\,db_i\\ &= \frac{\int p\{H_i(s)\mid b_i\}S_i(u\mid b_i,w_i)p(b_i)\,db_i} {\int p\{H_i(s)\mid b_i\}S_i(s\mid b_i,w_i)p(b_i)\,db_i}. \end{aligned}

This identity is the central calculation. It follows by substituting the posterior density into the first integral: the factor $S_i(s\mid b_i,w_i)$ in the posterior cancels the denominator in the conditional survival ratio, leaving the numerator integral divided by the posterior normalizing constant. That constant is exactly the denominator shown in the second line.

In practice, the integral can be approximated with draws or weighted points representing the posterior distribution of $b_i$. For each point, calculate the latent trajectory, integrate the corresponding hazard over the relevant time interval, convert that integrated hazard to conditional survival, and average using the posterior weights. The points must represent the distribution conditional on both the observed history and survival to the landmark, not merely the population random-effects distribution.

Worked Example: Conditional Survival for a Known Trajectory

Consider a hypothetical patient with no baseline covariate contribution, baseline hazard $h_0(t)=0.04$ per year, association parameter $\alpha=0.5$ per biomarker unit, and latent trajectory $m_i(t)=1+0.2t$ biomarker units, where time is measured in years. First suppose this trajectory is known without uncertainty. Find the probability of surviving from $s=2$ years to $u=3$ years.

The hazard is $0.04\exp\{0.5(1+0.2t)\}$ per year. The conditional survival is the exponential of the negative integrated hazard from year 2 to year 3:

\begin{aligned} P(T_i>3\mid T_i>2,b_i) &=\exp\left\{-\int_2^3 0.04e^{0.5(1+0.2t)}\,dt\right\}\\ &=\exp\left\{-0.04e^{0.5}\frac{e^{0.3}-e^{0.2}}{0.1}\right\}\\ &=\exp(-0.08473)\\ &\approx 0.9188. \end{aligned}

Thus, under this specified trajectory and hazard model, the conditional probability of surviving the next year is about 91.88%. This is a trajectory-conditional result; it is not yet averaged over uncertainty about the patient’s trajectory.

Worked Example: Updating Two Trajectory Weights

Suppose a hypothetical fitted model represents the patient’s uncertain trajectory by two possibilities, low and high. Before using this patient’s history, each has probability 0.5. Their longitudinal-history likelihoods are 0.8 and 0.3, respectively. The model gives survival probabilities to landmark $s$ of 0.9 and 0.6. Calculate the posterior trajectory weights after incorporating both the history and survival to $s$, then suppose their conditional survival probabilities from $s$ to $u$ are 0.95 and 0.70.

The unnormalized posterior weights are prior probability multiplied by history likelihood and landmark survival:

\begin{aligned} a_{\mathrm{low}}&=0.5(0.8)(0.9)=0.36,\\ a_{\mathrm{high}}&=0.5(0.3)(0.6)=0.09. \end{aligned}

Their sum is $0.36+0.09=0.45$, so the normalized weights are $0.36/0.45=0.8$ and $0.09/0.45=0.2$. The dynamic prediction is therefore

P(T_i>u\mid T_i>s,H_i(s))=(0.8)(0.95)+(0.2)(0.70)=0.76+0.14=0.90.

The predicted conditional survival is 0.90. The two survival probabilities used here are hypothetical model outputs for the two trajectories; each corresponds to a nonnegative integrated hazard on the interval. The history and event-free follow-up jointly favor the low trajectory in this example, so it receives four times the posterior weight of the high trajectory.

Why Averaging the Predictions Matters

The exponential hazard link makes survival a nonlinear function of the latent biomarker path. In general, evaluating survival at an average trajectory is not the same as averaging survival over trajectories. The following example shows the difference while keeping the fitted hazard parameters fixed.

Worked Example: Posterior Averaging Versus an Average Trajectory

Suppose a hypothetical posterior assigns probability 0.5 to each of two trajectories. Across the one-year prediction interval, the first has constant latent biomarker value $m=0$, and the second has constant value $m=\log(4)$. Let $h_0=0.1$ per year, $\alpha=1$ per biomarker unit, and omit baseline covariates. For a constant trajectory over one year, conditional survival is $\exp\{-0.1e^m\}$.

For the first trajectory, survival is $e^{-0.1}=0.9048$. For the second, it is $e^{-0.4}=0.6703$. Integrating over the two equally weighted trajectories gives

$$ (0.5)(0.9048)+(0.5)(0.6703)=0.7876. $$

The posterior mean latent value is $0.5(0)+0.5\log(4)=\log(2)$. If that mean value is substituted into the hazard first, the resulting survival is $e^{-0.1e^{\log(2)}}=e^{-0.2}=0.8187$, which differs from 0.7876. The integrated prediction averages survival probabilities, not biomarker values before applying the nonlinear hazard and survival calculations.

Assumptions, Diagnostics, and Alternatives

The calculation relies on a correctly specified joint model for the longitudinal measurements and event time, including the random-effects distribution, the latent trajectory form, and the current-value hazard association. The conditional-independence structure must be appropriate: after conditioning on the shared random effects and included covariates, the model must adequately describe how the longitudinal and event processes are connected. The prediction also assumes that the fitted relationship is useful over the future interval being considered. In particular, a trajectory model that fits observed measurements poorly or extrapolates implausibly can yield a precise-looking but unreliable survival prediction.

Useful checks include examining whether the fitted longitudinal model captures the observed measurements and trajectory shapes, whether the event model gives plausible hazards over the prediction interval, and whether predictions change materially with reasonable choices for numerical integration accuracy. Check that posterior weights are nonnegative and sum to one, that predicted survival lies between 0 and 1, and that conditional survival does not increase as the prediction horizon moves later. A near-zero denominator in the ratio form can signal that the observed history and survival to the landmark have very low probability under the fitted model; this warrants investigation rather than uncritical reporting.

A plug-in alternative calculates a prediction using a single estimated random-effects value or a single estimated trajectory. It is simpler, but it omits the averaging over trajectory uncertainty shown above. A baseline-marker model is another possible approach, but it does not represent the time-varying current-value link in the hazard. Current-value association is appropriate when the scientific model links instantaneous risk to the marker’s latent level at the same time; it should not be treated as interchangeable with other association structures. The later material on slope and cumulative-exposure associations addresses different hazard links.

This integration handles uncertainty in the patient’s latent trajectory conditional on the fitted parameters. It does not, by itself, include uncertainty in those parameter estimates, nor does it correct misspecification, measurement processes that the model omits, or changes in clinical care. Predictions should therefore be reported with the landmark, prediction horizon, conditioning history, and model assumptions made clear. Their usefulness depends on both the model and the information available for the individual patient.

Common pitfall: Using the population random-effects distribution, or a single estimated trajectory, as though it were the patient’s prediction distribution ignores information from that patient’s biomarker history and survival to the landmark. Instead, weight trajectories by their compatibility with the observed history and event-free follow-up, then average their conditional survival probabilities.

Check Your Understanding

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

Under a current-value association, what feature of the latent trajectory determines the hazard at time $t$?
Why does the posterior distribution of random effects for a patient event-free at $s$ include a factor $S_i(s\mid b_i,w_i)$?
A model gives posterior weights 0.7 and 0.3 to two trajectories. Their conditional survival probabilities to $u$ are 0.9 and 0.6. What is the integrated prediction?
For the two trajectories in question 3, suppose the biomarker-history likelihoods were already included in the weights, but survival to the landmark was not. Is the prediction fully conditioned on being event-free at the landmark?

Solutions. 1. The hazard at $t$ depends on the latent current value $m_i(t;b_i)$, evaluated at that same time, through the term $\exp\{\alpha m_i(t;b_i)\}$. 2. A trajectory implying low survival to $s$ is less compatible with observing that the patient survived to $s$; multiplying by $S_i(s\mid b_i,w_i)$ incorporates that information before predicting beyond the landmark. 3. The prediction is $(0.7)(0.9)+(0.3)(0.6)=0.63+0.18=0.81$, or 81% conditional survival. 4. No. The trajectory weights must also account for survival to $s$, as well as the biomarker-history likelihood, before they are used for a prediction conditional on being event-free at the landmark.