Risk Predictions at Short and Long Horizons

Why the Prediction Horizon Matters

A dynamic prediction is tied to both a landmark time and a future horizon. For a patient who is event-free at landmark $s$, the question might be the chance of an event within the next three months, or the chance within the next year. These are different predictions: the longer horizon accumulates risk over more time and may include portions of the patient’s projected hazard path that are irrelevant to a short-horizon prediction.

In a joint model, the hazard path depends on the patient’s longitudinal trajectory, possibly through a current marker value, slope, or cumulative exposure. The earlier material on conditional survival and integration over shared random effects establishes how history and latent trajectory uncertainty enter a prediction. Here the focus is on comparing horizons: exactly what is accumulated, what changes as the horizon grows, and why two patients’ risk rankings can depend on the horizon selected.

Definition: Let $T_i$ be patient $i$’s event time, $s$ a landmark at which the patient is known to be event-free, and $h>0$ a future horizon. Given the patient’s observed history through $s$, denoted $H_i(s)$, the conditional risk by horizon $h$ is $$ R_i(s,h)=P(T_i\leq s+h\mid T_i>s,H_i(s)). $$ The corresponding conditional survival probability is $$ S_i(s+h\mid s,H_i(s))=P(T_i>s+h\mid T_i>s,H_i(s)), $$ so $R_i(s,h)=1-S_i(s+h\mid s,H_i(s))$.

The horizon $h$ is a duration after the landmark, not necessarily the time $s+h$ measured from the study origin. For example, a 6-month risk at landmark month 10 concerns the interval from month 10 to month 16. A risk must therefore be reported with its landmark and horizon, along with the event and the population or patient to which it refers.

From the Hazard Path to Horizon-Specific Risk

Write $\lambda_i(u\mid b_i,\theta)$ for patient $i$’s hazard at time $u$, conditional on latent random effects $b_i$ and model parameters $\theta$. A hazard has units of inverse time. It describes the instantaneous event rate among individuals still event-free at time $u$; it is not itself a probability. For a specified trajectory and parameter value, assume the hazard is nonnegative and integrable over the prediction interval.

The cumulative hazard between the landmark and the horizon is the area under the hazard path. Conditional survival is the exponential of the negative cumulative hazard, and conditional risk is one minus that survival:

\Lambda_i(s,s+h\mid b_i,\theta) =\int_s^{s+h}\lambda_i(u\mid b_i,\theta)\,du, \qquad S_i(s+h\mid s,b_i,\theta) =\exp\{-\Lambda_i(s,s+h\mid b_i,\theta)\}, \qquad R_i(s,h\mid b_i,\theta)=1-\exp\{-\Lambda_i(s,s+h\mid b_i,\theta)\}.

A brief justification follows from the defining relationship between hazard and survival. If $S(u)$ denotes survival conditional on being event-free at $s$ and on the specified trajectory, then $dS(u)/du=-\lambda_i(u)S(u)$, with $S(s)=1$. Dividing by $S(u)$, integrating from $s$ to $s+h$, and exponentiating gives

\log S(s+h)-\log S(s) =-\int_s^{s+h}\lambda_i(u)\,du, \qquad S(s+h)=\exp\left\{-\int_s^{s+h}\lambda_i(u)\,du\right\}.

In an individual prediction, $b_i$ is usually not known. Conditional on fitted parameters, let $p(b_i\mid H_i(s),T_i>s,\theta)$ represent the distribution of the latent effects after using the patient’s history and survival to the landmark. The prediction averages survival over this distribution:

S_i(s+h\mid s,H_i(s),\theta) = \int \exp\left\{-\int_s^{s+h}\lambda_i(u\mid b_i,\theta)\,du\right\} p(b_i\mid H_i(s),T_i>s,\theta)\,db_i, \qquad R_i(s,h\mid H_i(s),\theta)=1-S_i(s+h\mid s,H_i(s),\theta).

This is an average of trajectory-specific survival probabilities, followed by subtraction from one. In general it is not equivalent to inserting an average latent effect into the hazard, or to exponentiating the negative cumulative hazard computed from an average hazard path. The exponential transformation is nonlinear.

Worked Example: A Rising Hazard at Short and Long Horizons

Suppose a hypothetical patient is event-free at landmark $s$. Conditional on the estimated trajectory, the hazard is $0.020$ events per month during months 0–3 after the landmark and $0.060$ events per month during months 3–12. Treat these piecewise-constant hazards as the specified path and hold the fitted parameters fixed.

For a 3-month horizon, the cumulative hazard is $0.020\times3=0.060$. Thus $$ R_i(s,3)=1-\exp(-0.060)\approx0.05824. $$ The estimated conditional risk is about 5.82% over the first three months.

For a 6-month horizon, the first 3 months contribute $0.020\times3=0.060$, and the next 3 contribute $0.060\times3=0.180$. The cumulative hazard is $0.240$, so $$ R_i(s,6)=1-\exp(-0.240)\approx 1-0.7866=0.2134. $$ The conditional risk over six months is about 21.3%.

For a 12-month horizon, the cumulative hazard is $0.020\times3+0.060\times9=0.600$, giving $$ R_i(s,12)=1-\exp(-0.600)\approx 1-0.5488=0.4512. $$ The 12-month risk is about 45.1%. It is larger because it includes both more time at risk and the later interval with the higher projected hazard. These numbers are conditional on the stated trajectory and fitted parameters; they do not account for uncertainty in those quantities.

Why Latent-Trajectory Averaging Can Change the Answer

When a patient’s history is compatible with several latent trajectories, each trajectory implies a different cumulative hazard at a given horizon. The correct conditional prediction averages their survival probabilities using their history-updated weights. As the horizon lengthens, the trajectories may diverge more, so the resulting risk can become more sensitive to the shape and weighting of the projected hazard paths.

Worked Example: Averaging Survival Over Two Possible Trajectories

For a hypothetical patient, suppose the history-updated latent-trajectory distribution assigns probability $0.70$ to a lower-hazard path and $0.30$ to a higher-hazard path. Assume fitted parameters are fixed. The lower path has hazards $0.010$ per month for the first 6 months and $0.040$ for the next 6; the higher path has hazards $0.050$ and $0.120$ per month over those intervals. Each path is nonnegative and piecewise constant, so its cumulative hazard gives survival by the formula above.

At 6 months, the lower-path survival is $\exp(-0.010\times6)=\exp(-0.060)$, and the higher-path survival is $\exp(-0.050\times6)=\exp(-0.300)$. Average these survivals using the stated weights: $$ S_i(s+6\mid s,H_i(s),\theta) =0.70\exp(-0.060)+0.30\exp(-0.300) \approx 0.70(0.9418)+0.30(0.7408) =0.8815. $$ Therefore $R_i(s,6)\approx1-0.8815=0.1185$, or 11.9%.

At 12 months, the lower path has cumulative hazard $0.010\times6+0.040\times6=0.300$. The higher path has cumulative hazard $0.050\times6+0.120\times6=1.020$. Hence $$ S_i(s+12\mid s,H_i(s),\theta) =0.70\exp(-0.300)+0.30\exp(-1.020) \approx 0.70(0.7408)+0.30(0.3606) =0.6268, $$ and $R_i(s,12)\approx1-0.6268=0.3732$, or 37.3%. The greater separation between the two paths by 12 months makes the posterior mixture especially consequential.

For contrast, the weighted average hazard is $0.70(0.010)+0.30(0.050)=0.022$ per month in the first interval and $0.70(0.040)+0.30(0.120)=0.064$ in the second. Using this average path as if it were the patient’s hazard would give 12-month cumulative hazard $0.022\times6+0.064\times6=0.516$, and risk $1-\exp(-0.516)\approx0.4031$, not 0.3732. The difference arises because averaging the exponential survival over trajectories is not the same operation as exponentiating the cumulative hazard of the average path.

Horizon-Dependent Comparisons and Interpretation

A longer horizon cannot reduce a patient’s conditional risk when the prediction conditions and model are held fixed. For $h_2>h_1$, the additional cumulative hazard is nonnegative, so survival cannot increase:

\Lambda_i(s,s+h_2)\geq \Lambda_i(s,s+h_1) \quad\Longrightarrow\quad \exp\{-\Lambda_i(s,s+h_2)\}\leq\exp\{-\Lambda_i(s,s+h_1)\} \quad\Longrightarrow\quad R_i(s,h_2)\geq R_i(s,h_1).

The amount of increase depends on the hazard path over the added interval, not merely on the horizon’s length. A long-horizon risk therefore gives greater weight to later projected marker values and their association with the hazard. It is also more exposed to extrapolation: if future marker behavior is poorly supported by observed data, a seemingly precise long-horizon number can rest heavily on the model’s assumptions.

Worked Example: Patient Risk Ranking Can Depend on the Horizon

Consider two hypothetical patients with fixed, model-based hazard paths, in events per month. Patient A has hazard $0.040$ during the first 3 months after the landmark and $0.010$ during the next 9 months. Patient B has hazard $0.020$ during the first 3 months and $0.030$ during the next 9 months. Assume both are event-free at the same landmark and use the same prediction assumptions.

At 3 months, patient A’s cumulative hazard is $0.040\times3=0.120$, so the risk is $1-\exp(-0.120)\approx0.1131$, or 11.3%. Patient B’s cumulative hazard is $0.020\times3=0.060$, giving risk $1-\exp(-0.060)\approx0.05824$, or 5.82%. At this short horizon, A has the higher predicted risk.

At 12 months, patient A’s cumulative hazard is $0.040\times3+0.010\times9=0.210$, giving risk $1-\exp(-0.210)\approx0.1894$, or 18.9%. Patient B’s cumulative hazard is $0.020\times3+0.030\times9=0.330$, giving risk $1-\exp(-0.330)\approx0.2811$, or 28.11%. At the longer horizon, B has the higher predicted risk because B’s hazard remains higher during the later interval. The ranking changes because the hazard paths cross; a single risk estimate cannot summarize the comparison for every time window.

A fixed-horizon risk is useful when a decision concerns a specific time window, such as whether to intensify monitoring over the next 6 months. Reporting a conditional survival curve over a range of horizons provides a broader time profile, but still relies on the same predicted hazard path and assumptions. Neither presentation replaces careful selection of a clinically relevant landmark and horizon. Comparisons are meaningful only when the landmark, event definition, conditioning information, and model settings are aligned.

Assumptions, Diagnostics, and Common Pitfalls

The calculations require a fitted hazard model that is appropriate for the event and the patient population, a nonnegative hazard that can be integrated over the interval, and a defensible projection of the longitudinal trajectory over the chosen horizon. The posterior distribution of latent effects must reflect the available history and survival to the landmark. If predictions hold fitted parameters fixed, the reported risks are conditional on those estimates and do not include parameter uncertainty.

Check the interval definition, hazard units, and trajectory carefully. If hazard is expressed per month, each interval length in the cumulative-hazard calculation must be in months. Confirm that the hazard is evaluated over the full interval from $s$ to $s+h$, and that survival contributions from different latent trajectories are weighted by their posterior probabilities. For long horizons, examine whether the marker trajectory is being extrapolated far beyond the observed follow-up and whether small changes in its late behavior substantially change the risk.

Mistake: treating hazard as risk. A hazard of $0.04$ per month does not mean a 4% risk over any arbitrary period. Integrate the hazard over the specified interval, then transform cumulative hazard to risk.
Mistake: comparing numbers without naming horizons. A 6-month risk and a 12-month risk answer different questions. State the landmark and duration in context.
Mistake: using an average hazard in place of trajectory averaging. If latent trajectories are uncertain, average their survival probabilities with the appropriate weights. Applying the survival formula to the average hazard generally gives a different result.
Mistake: assuming patient rankings are fixed across time. Crossing hazard paths can produce one risk ordering at a short horizon and another at a long horizon.

These predictions are model-based, not guarantees of future outcomes. Misspecified hazard associations, poorly estimated late trajectories, or unrepresented changes in treatment can make long-horizon risks unreliable. A risk at a longer horizon is mathematically at least as large as one at a shorter horizon under the same conditioning, but its accuracy does not automatically improve with the additional time considered.

Check Your Understanding

Use the conditional-risk definition and the cumulative-hazard calculation. Solutions are included below each question.

A patient’s hazard is $0.030$ per month for the first 2 months and $0.050$ per month for the next 3 months. What is the conditional risk by month 5 after the landmark?
A latent trajectory has posterior weight $0.40$ and constant hazard $0.010$ per month for 10 months; another has weight $0.60$ and constant hazard $0.040$ per month. Calculate the 10-month risk by averaging survival.
Under a fixed nonnegative hazard path and unchanged conditioning information, can conditional risk at 12 months be lower than risk at 6 months? Explain.
Why can a patient with higher risk at 3 months have lower risk at 12 months than another patient?

Solutions. (1) The cumulative hazard is $0.030(2)+0.050(3)=0.210$. Thus the risk is $1-\exp(-0.210)\approx0.1894$, or 18.9% by 5 months. (2) The averaged survival is $0.40\exp(-0.100)+0.60\exp(-0.400)\approx0.40(0.9048)+0.60(0.6703)=0.7641$. The risk is $1-0.7641=0.2359$, or 23.6%. (3) No. The added interval contributes nonnegative cumulative hazard, so survival cannot increase and risk cannot decrease. (4) The patients’ hazard paths can cross. One may have a larger early hazard, while the other has a larger hazard later, changing the accumulated risk ordering as the horizon extends.