Two Risk Questions at the Same Landmark
At a landmark time \(s\), a prediction over the next \(u\) time units might ask: “What is the chance of an event by \(s+u\) among people who are event-free at \(s\)?” That is a population-level question. A clinician or researcher may instead ask: “Given this particular patient’s biomarker measurements and their being event-free at \(s\), what is the patient’s chance of an event by \(s+u\)?” That is a patient-specific question.
The two answers can differ even when they come from the same joint model. The population prediction averages over a distribution of latent patient profiles among people at risk at \(s\). The individual prediction uses that patient’s data to update the distribution of their latent profile before averaging their risk. The distinction is about which information is conditioned on and which distribution is averaged over; it is not a distinction between two unrelated models.
Both risks are conditional on being event-free at the landmark. Unless stated otherwise, the model parameters are treated as known here so that the focus stays on the distinction between the two targets. Uncertainty in estimated parameters is a separate source of predictive uncertainty.
Model and Risk Definitions
Suppose the joint model specifies a hazard \(h_i(t\mid b_i,X_i)\) conditional on random effects and baseline covariates. The corresponding conditional survival function is \(S_i(t\mid b_i,X_i)=P(T_i>t\mid b_i,X_i)\). Under the usual hazard-to-survival relationship, \(S_i(t\mid b_i,X_i)=\exp\{-\int_0^t h_i(v\mid b_i,X_i)\,dv\}\). The patient’s conditional landmark risk, if \(b_i\) were known, is \(r_i(b_i;s,u)=1-S_i(s+u\mid b_i,X_i)/S_i(s\mid b_i,X_i)\).
The ratio conditions on survival to \(s\). It is the probability of an event by \(s+u\), given no event by \(s\), for a specified random-effects value and covariates. In a joint model, the hazard may depend on a latent longitudinal trajectory, such as its current value or slope; that dependence is carried into the survival function through \(b_i\).
Let \(p(b\mid X_i=x)\) be the model’s random-effects distribution for patients with covariates \(x\). Among those patients, being event-free at \(s\) changes the distribution of random effects: profiles with better survival to \(s\) are more prevalent in the risk set. Thus the population-average risk at \(s\), for covariates \(x\), is
Equivalently, define the random-effects distribution among patients with \(T>s\) by \(p(b\mid T>s,x)=S(s\mid b,x)p(b\mid x)/\int S(s\mid b',x)p(b'\mid x)\,db'\). Then \(R_{\mathrm{pop}}(s,u\mid x)\) is the average of \(r(b;s,u,x)\) over this event-free distribution. If the target is a population with varied baseline covariates, one must also specify how that covariate distribution is handled: for example, report risks within covariate strata or average appropriately over a stated target distribution.
Why the Two Random-Effects Distributions Differ
A patient-specific prediction uses the observed longitudinal history as well as survival to the landmark. Suppose, as in a standard joint-model formulation, the longitudinal measurements and event time are conditionally independent given \(b_i\) and \(X_i\). Write \(L_i(b)=p(Y_i(s)\mid b,X_i)\) for the longitudinal-data likelihood contribution. For a patient known to be event-free through \(s\), their posterior random-effects density is proportional to
The factor \(S_i(s\mid b_i,X_i)\) is present because the patient’s event-free status is informative about their latent profile whenever survival depends on that profile. The patient-specific risk is then
These definitions give a short derivation of the population formula. Conditional on \(b\), the probability of being event-free at \(s\) is \(S(s\mid b,x)\), and the probability of an event in \((s,s+u]\) is \(S(s\mid b,x)-S(s+u\mid b,x)\). Averaging each probability over the baseline distribution \(p(b\mid x)\) gives, respectively, the denominator and numerator in the displayed ratio. Dividing the joint probability of surviving to \(s\) and then having the event in the window by the probability of surviving to \(s\) yields the conditional risk. This also shows why simply averaging conditional landmark risks over the baseline distribution \(p(b\mid x)\) can target the wrong population: it does not account for selection into the event-free risk set.
Worked Example: Population Risk After Selection Into the Risk Set
Hypothetical data. Suppose there are two latent profiles, high-risk \(H\) and low-risk \(L\), each with probability \(0.50\) at baseline. At landmark \(s\), survival probabilities are \(S(s\mid H)=0.60\) and \(S(s\mid L)=0.90\). Over the next horizon, the conditional risks are \(0.50\) for \(H\) and \(0.10\) for \(L\).
The corresponding survival probabilities at \(s+u\) are \(0.60(1-0.50)=0.30\) for \(H\) and \(0.90(1-0.10)=0.81\) for \(L\). The probability of surviving to \(s\), averaged over the baseline profiles, is \((0.50)(0.60)+(0.50)(0.90)=0.75\).
The risk-set weights are therefore \(P(H\mid T>s)=(0.50)(0.60)/0.75=0.40\) and \(P(L\mid T>s)=(0.50)(0.90)/0.75=0.60\). The population-average landmark risk is \((0.40)(0.50)+(0.60)(0.10)=0.20+0.06=\mathbf{0.2600}\).
For a check using the ratio definition, the probability of being in the risk set and having the event in the window is \((0.50)(0.60-0.30)+(0.50)(0.90-0.81)=0.1500+0.0450=0.1950\). Dividing by \(0.75\) gives \(0.1950/0.75=0.2600\). Averaging the conditional risks using the baseline weights instead would give \(0.50(0.50)+0.50(0.10)=0.3000\), which is not the risk among those who have survived to the landmark.
Conditioning on One Patient’s Measurements
The patient-specific prediction additionally updates the random-effects distribution using that person’s longitudinal history. The next example uses a simple two-profile model so every posterior weight is visible. The profile probabilities and likelihoods below are hypothetical and are assumed to come from a correctly specified model.
Worked Example: Updating Risk With a Patient’s History
Hypothetical data. At baseline, \(P(H)=0.25\) and \(P(L)=0.75\). For a patient with covariates \(X_i=x\), suppose the longitudinal history has likelihoods \(p(Y_i(s)\mid H,x)=0.80\) and \(p(Y_i(s)\mid L,x)=0.20\). Survival to \(s\) is \(0.70\) for \(H\) and \(0.90\) for \(L\). Given each profile, the horizon risks are \(0.40\) for \(H\) and \(0.10\) for \(L\).
The unnormalized posterior weights, including both the history and event-free status, are \(0.25(0.80)(0.70)=0.1400\) for \(H\) and \(0.75(0.20)(0.90)=0.1350\) for \(L\). Their sum is \(0.2750\). Thus \(P(H\mid Y_i(s),T_i>s,x)=0.1400/0.2750=0.5091\), and the low-profile probability is \(0.1350/0.2750=0.4909\).
The patient-specific horizon risk is \((0.5091)(0.40)+(0.4909)(0.10)=0.2036+0.04909\approx\mathbf{0.2527}\). Using the unrounded weights, this is \(\{(0.1400)(0.40)+(0.1350)(0.10)\}/0.2750=0.06950/0.2750=0.2527\). The prediction is the patient’s event probability over the stated horizon conditional on their history and being event-free at \(s\), under the assumed model. It is not a probability that the model or its parameters are correct.
The longitudinal measurements shift the patient’s posterior toward profile \(H\), while survival to \(s\) shifts it toward profile \(L\), which has better survival to the landmark. The final posterior incorporates both pieces of information. Omitting either one changes the conditioning set and can change the prediction.
Comparing the Targets in Practice
Worked Example: Two Patients, One Population Risk
Hypothetical data. Return to the first example: among people event-free at \(s\), the high- and low-profile weights are \(0.40\) and \(0.60\), and their horizon risks are \(0.50\) and \(0.10\). Suppose patient A’s measurements and event-free status produce posterior profile probabilities \(P(H\mid Y_A(s),T_A>s)=0.80\) and \(P(L\mid Y_A(s),T_A>s)=0.20\). Suppose patient B’s corresponding probabilities are \(0.20\) and \(0.80\).
Patient A’s risk is \((0.80)(0.50)+(0.20)(0.10)=0.4000+0.0200=\mathbf{0.4200}\). Patient B’s risk is \((0.20)(0.50)+(0.80)(0.10)=0.1000+0.0800=\mathbf{0.1800}\). The population-average risk remains \(0.2600\), calculated using the risk-set profile weights \(0.40\) and \(0.60\). The different predictions answer different questions: the population value describes the at-risk population, whereas each individual value incorporates that patient’s observed history.
A population-average prediction can be useful for describing expected burden in a target population or comparing standardized risks across groups. A patient-specific prediction supports decisions for a particular person, provided their measurements, covariates, and event-free status are appropriately represented in the model. A patient-specific prediction is not automatically more accurate just because it is more tailored; it depends on adequate data and a suitable model.
| Prediction target | Random-effects distribution used | Question answered |
|---|---|---|
| Conditional risk for a fixed \(b\) | A specified random-effects value | What is the risk if this latent profile were known? |
| Population-average landmark risk | Distribution among patients with \(T>s\) | What is the risk over the horizon in the event-free population? |
| Patient-specific landmark risk | Posterior given \(Y_i(s)\), \(T_i>s\), and \(X_i\) | What is the risk for this patient given their observed information? |
Assumptions, Diagnostics, and Limitations
The formulas rely on a coherent joint model for the longitudinal and event processes. In particular, the longitudinal-data likelihood and survival contribution must be combined using the model’s conditional dependence assumptions; the displayed posterior factorization assumes conditional independence given \(b_i\) and \(X_i\). The hazard and longitudinal trajectory models must be adequate for the relevant time range, covariates must be handled consistently with the target population, and the patient’s event-free status and measurement times must be represented correctly. If censoring occurs before \(s\), the patient is not known to be event-free through \(s\); treating censoring as survival past the censoring time without an appropriate censoring assumption is not justified.
Useful checks include comparing predicted and observed event frequencies over the same landmark and horizon in a validation sample, checking calibration separately across risk groups and landmark times, and examining whether predictions change implausibly when a measurement is added or removed. A mismatch between predicted and observed risk may reflect misspecified longitudinal trajectories, an inadequate association between the marker and event hazard, incorrect handling of measurement times, or a change in the target population. Calibration for a population does not by itself establish accuracy for each patient.
- Wrong target weights. Averaging conditional risks over \(p(b\mid x)\) ignores that survival to \(s\) selects the risk set. Use weights proportional to \(S(s\mid b,x)p(b\mid x)\) for the population risk at the landmark.
- Incomplete patient conditioning. Using only the longitudinal likelihood, while ignoring known event-free status through \(s\), omits survival information when that status depends on \(b_i\). Include the survival contribution under the joint model.
- Unclear population target. A risk conditional on one covariate value is not automatically the average risk across a population with varying covariates. State the covariate stratum or standardization distribution.
These risks also have limitations. Both depend on model specification and available information; patient-specific estimates may be imprecise when the history is sparse, and population averages can conceal meaningful heterogeneity. The central distinction remains useful: identify the conditioning set, identify the random-effects distribution being averaged over, and interpret the result for precisely that target.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why does the population-average landmark risk generally not average conditional risks using the baseline random-effects distribution?
- In a two-profile model, risk-set weights are \(0.30\) for \(H\) and \(0.70\) for \(L\), with horizon risks \(0.60\) and \(0.20\). Calculate the population-average risk.
- A patient’s posterior profile probabilities, given their history and survival to \(s\), are \(0.25\) for \(H\) and \(0.75\) for \(L\). Using the risks in Question 2, calculate the patient-specific risk.
- What additional distribution must be specified to turn a risk conditional on fixed baseline covariates into a population-average risk over a population with varying covariates?
Solutions. (1) Survival to \(s\) changes the random-effects distribution in the risk set, so the relevant weights are proportional to survival to \(s\) times the baseline random-effects density. (2) The risk is \(0.30(0.60)+0.70(0.20)=0.18+0.14=\mathbf{0.32}\), or 32% over the stated horizon. (3) The patient-specific risk is \(0.25(0.60)+0.75(0.20)=0.15+0.15=\mathbf{0.30}\), or 30% over the horizon conditional on that patient’s information. (4) Specify the target population’s baseline-covariate distribution, then average or standardize conditional risks over that distribution.