Predicting With a Nonlinear Marker Trajectory

Why the Trajectory Shape Matters for Prediction

A dynamic prediction at landmark time \(s\) uses a patient’s marker history and event-free status through \(s\) to estimate risk after \(s\). The joint model links that risk to the patient’s latent marker trajectory, not simply to the last recorded measurement. When the trajectory bends over time, predicting with a straight line or a single marker value can misrepresent the hazard over the prediction interval.

The central task here is to evaluate the fitted, subject-specific trajectory at the times needed for prediction. Those times may include future times at which no marker has been measured. The resulting latent values enter the chosen hazard association, and the hazard is integrated over time. The introductory material on conditional survival from a joint model gives the broader prediction target; the focus here is how a nonlinear trajectory supplies its time-varying marker values.

Trajectory Representation and Assumptions

A convenient general representation is \(m_i(t)=\boldsymbol{B}(t)^\top\boldsymbol{\beta}+\boldsymbol{Z}(t)^\top\boldsymbol{b}_i\). The vectors \(\boldsymbol{B}(t)\) and \(\boldsymbol{Z}(t)\) contain fixed-effect and subject-specific time-basis functions, \(\boldsymbol{\beta}\) is a vector of fixed coefficients, and \(\boldsymbol{b}_i\) contains the subject’s random effects. For a quadratic trajectory, basis functions might include \(1,t,t^2\); for a spline, they include functions defined by its knots and boundary choices. The longitudinal observations fluctuate around \(m_i(t)\) according to the specified measurement-error model.

For clarity, suppose the event hazard conditional on the random effects has a current-value association:

Here \(h_0(t)\) is the baseline hazard, \(\boldsymbol{w}_i\) denotes baseline covariates, and \(\alpha\) is the association parameter. The hazard is nonnegative and has units of inverse time. With a slope association, the trajectory derivative \(m_i'(t)\) replaces \(m_i(t)\) in the association term; with a cumulative-exposure association, the relevant quantity is an integral of the trajectory. Thus a nonlinear trajectory may be evaluated, differentiated, or integrated according to the specified association structure.

The prediction relies on several modeling conditions: the fitted trajectory and hazard association must be appropriate over the interval; the time scale and marker units must agree with those used when fitting the model; and the subject-specific random effects must be averaged over their distribution conditional on the available marker history and survival to \(s\). If parameter uncertainty is being propagated, parameter draws must also be handled consistently. In the numerical examples below, model parameters are treated as fixed, while the displayed random-effect draws and weights represent the relevant conditional uncertainty in the trajectory.

From a Nonlinear Trajectory to Conditional Risk

For a given trajectory draw, conditional survival from landmark \(s\) to time \(u>s\) is obtained from the hazard along that draw. Define the cumulative hazard \(A_i(s,u\mid\boldsymbol{b}_i)\) as the integral of the conditional hazard. Conditional survival is its negative exponential. Because the random effects are uncertain after observing a patient’s history, the prediction averages this survival over their conditional distribution.

Here \(\mathcal{H}_i(s)\) is the information available through \(s\), and \(T_i>s\) denotes being event-free at the landmark. The corresponding conditional event risk by \(u\) is \(1-S_i(u\mid s,\mathcal{H}_i(s))\). This expectation is over the random-effect distribution updated by the patient’s marker history and event-free follow-up; it is not generally the survival calculated by substituting the mean trajectory into the hazard.

A numerical calculation commonly has two distinct averaging or integration tasks. First, for each trajectory draw, integrate its hazard over calendar time, using a sufficiently fine grid or a suitable quadrature rule. Second, average the resulting conditional survival values over the trajectory draws with their conditional weights. Keeping these tasks separate helps identify whether an inaccurate prediction comes from an inadequate time grid or an inadequate representation of trajectory uncertainty.

Evaluating a Spline at Prediction Times

A spline represents the trajectory with locally flexible basis functions. For example, a linear spline with a knot at month 3 can be written \(m_i(t)=b_{0i}+b_{1i}(t-2)+b_{2i}(t-3)_+\), where \((x)_+=\max(x,0)\). This curve is continuous at the knot, while its slope may change there. More flexible splines can be used when smoother curvature is needed. Whatever the basis, evaluate the complete fitted function at each integration time; do not use a basis value from one time point as if it were the marker itself.

Derivation, Diagnostics, and Limitations

The averaging step follows from conditioning on the latent random effects. Given \(\boldsymbol{b}_i\), the joint model specifies the conditional survival as \(\exp\{-A_i(s,u\mid\boldsymbol{b}_i)\}\). Removing the unobserved \(\boldsymbol{b}_i\) while retaining the observed history requires the law of total probability: take the conditional expectation of that survival over the distribution of \(\boldsymbol{b}_i\) given \(\mathcal{H}_i(s)\) and \(T_i>s\). This gives the prediction formula above. In general, the exponential is nonlinear, so averaging and substituting do not commute:

The right-hand side uses one average random effect; it is not, in general, the required average survival. The same issue applies if one averages marker trajectories first and then calculates risk. The trajectory draws must be carried through the hazard and survival calculation before their survival values are averaged.

Useful checks include plotting fitted trajectories against observed measurements, examining whether curvature is supported by the observation times, and checking that the time basis and knot locations are coded consistently at fitting and prediction. Compare numerical integration on successively finer grids; material changes in cumulative hazards or risks indicate that the original grid was inadequate. Check that computed hazards are nonnegative and in inverse-time units, and that predicted survival is between zero and one and does not increase as the prediction horizon advances.

A polynomial can behave implausibly when extrapolated far beyond the observed time range: even a modest quadratic term can eventually dominate. Splines allow more local flexibility but still depend on knot placement and boundary behavior, and sparse measurements near a boundary may leave the future curve weakly determined. A better numerical integral cannot correct a misspecified trajectory or hazard association. Nor does a close fit to recorded measurements alone establish that the latent curve is reliable for future prediction.