Why Survival to the Landmark Changes the Weights
A dynamic prediction at time \(s\) concerns a patient who is still event-free at \(s\), as well as the longitudinal measurements available by that time. The observed biomarker history helps identify which latent trajectories are plausible. But it is not the only information about a patient’s latent random effects: remaining event-free also provides evidence. If some random-effects values imply a higher event hazard, patients with those values are less likely to have survived to the landmark. The distribution of random effects among patients who reach \(s\) is therefore generally different from the distribution among patients before follow-up.
The key operation is to include the probability of surviving to \(s\) in the weights assigned to each possible random-effects value. This is not a new survival model: it is conditioning under the joint model. The earlier material on integrating over shared random effects supplies the prediction framework; here the focus is the additional information contributed by the event-free landmark condition.
The Landmark Posterior Distribution
Write \(L_{Yi}(b_i)=f\{Y_i(s)\mid b_i\}\) for the longitudinal-data likelihood, with any fixed covariates suppressed. Let \(h_i(t\mid b_i)\) be the event hazard conditional on \(b_i\), and define the corresponding conditional survival function by
Assume the random-effects density before observing this patient’s history is \(f(b_i)\). Conditional on \(b_i\), suppose the longitudinal measurements and event process are independent, and suppose the longitudinal measurement process can be treated as ignorable for this likelihood. Then, for a patient known to be event-free through \(s\), Bayes’ rule gives
The normalizing denominator is the integral of the right-hand side’s unnormalized numerator over all possible \(b_i\); the proportionality multiplier is the reciprocal of that integral. The factor \(S_i(s\mid b_i)\) is the survival likelihood contributed by knowing that the patient has had no event by \(s\). Where the hazard increases with a component of \(b_i\), this factor tends to be smaller for values that imply higher risk. It does not have to reduce every component or every summary of the distribution: the direction and magnitude of reweighting depend on the hazard model and on the biomarker history.
This formula requires that event-free status through \(s\) be known. A patient censored before \(s\) cannot simply be treated as known to have survived event-free through \(s\). If censoring is involved, its mechanism must be handled under an appropriate censoring assumption; in the simple expression above, censoring is assumed not to invalidate the observed event-free status or the likelihood factorization.
Worked Example: Survival Alone Reweights Two Risk Levels
Suppose \(b_i\) takes two possible values, 0 and 1, each with prior probability 0.5. There is no biomarker information in this example. Over follow-up to the landmark, let the cumulative baseline hazard be 0.4, and let the hazard multiplier be \(\exp(b_i)\). Thus \(S_i(s\mid b_i)=\exp\{-0.4\exp(b_i)\}\). Calculate the random-effects distribution among patients who survive to \(s\).
For \(b_i=0\), the survival likelihood is \(S_i(s\mid 0)=\exp(-0.4)=0.6703\). For \(b_i=1\), it is \(S_i(s\mid 1)=\exp\{-0.4e\}=\exp(-1.0873)=0.3371\). The high-to-low survival likelihood ratio is
With equal prior probabilities, the posterior probability of \(b_i=1\) is
The high-risk value had prior probability 0.5, but it has probability about 0.3346 among those who reach the landmark. The likelihood ratio is about 0.5029, not 0.5084; using that ratio, rather than an incorrect numerical approximation, is essential to getting the posterior weight right.
Using the Landmark Distribution for Prediction
Once the landmark distribution is formed, future survival is averaged over it. For a prediction horizon \(u>0\), conditional survival from \(s\) to \(s+u\) is
The survival ratio is the probability of surviving the additional interval \((s,s+u]\), conditional on \(b_i\) and survival through \(s\). Equivalently, substitution of the landmark density gives a ratio of integrals:
This form makes an important bookkeeping point visible: survival to \(s\) is already included in the landmark distribution. One may average the future survival ratio using that distribution, or use the ratio of integrals above. One must not apply \(S_i(s\mid b_i)\) a second time after using the landmark distribution. The conditional event risk over the horizon is one minus the conditional survival probability.
Worked Example: Combining Biomarker and Survival Evidence
Continue with the two random-effects values and landmark survival probabilities from the previous example. Now suppose the observed biomarker history has likelihood 1 at \(b_i=0\) and likelihood 2 at \(b_i=1\), so the biomarker history alone favors \(b_i=1\) by a likelihood ratio of 2. Calculate the landmark weights and the predicted probability of surviving one additional interval for which the cumulative baseline hazard is 0.2.
The unnormalized weights, including the equal prior probabilities, are \(0.5(1)(0.6703)=0.3352\) for \(b_i=0\), and \(0.5(2)(0.3371)=0.3371\) for \(b_i=1\). Therefore,
Biomarker information alone would give probability \(2/(1+2)=0.6667\) to \(b_i=1\). Survival to the landmark reduces that weight, although the combined weight remains close to 0.5 because the biomarker likelihood favors the high-risk value.
For the additional interval, survival conditional on \(b_i=0\) is \(\exp(-0.2)=0.8187\), and survival conditional on \(b_i=1\) is \(\exp(-0.2e)=0.5806\). Averaging with the landmark weights gives
Thus the predicted event risk over this additional interval is \(1-0.6993=0.3007\), or about 30.07%. If survival to \(s\) were ignored, the biomarker-only weights would instead give survival \((1/3)(0.8187)+(2/3)(0.5806)\approx 0.6600\), and risk about 34.00%. The difference illustrates why the landmark condition belongs in the weights.
A Short Derivation and a Piecewise-Hazard Example
The landmark formula follows directly from conditional probability. Given a random-effects value, the joint contribution of the biomarker history and being event-free at \(s\) is \(L_{Yi}(b_i)S_i(s\mid b_i)\), by the conditional-independence assumption. Multiplying by the prior density gives the joint density of those observations and \(b_i\). Integrating over \(b_i\) gives the marginal probability density of the observed information, so division by that integral normalizes the conditional density:
The same conditioning also justifies the future prediction. Given \(b_i\) and survival through \(s\), the chance of surviving to \(s+u\) is \(S_i(s+u\mid b_i)/S_i(s\mid b_i)\). Averaging that conditional probability over the landmark distribution yields the prediction formula.
Worked Example: Calculating Weights With Piecewise Baseline Hazard
Suppose \(b_i\) is 0 or 1 with equal prior probabilities, the biomarker likelihood is equal at both values, and \(h_i(t\mid b_i)=h_0(t)\exp\{(\log 2)b_i\}\). The baseline cumulative hazard is 0.1 over the first interval and 0.2 over the second, with the landmark \(s\) at the end of both intervals. Over the next interval, the baseline cumulative hazard is 0.25. Find the landmark probability of \(b_i=1\) and the conditional event risk over the next interval.
The cumulative baseline hazard through \(s\) is \(0.1+0.2=0.3\). Since \(\exp\{(\log 2)b_i\}=2^{b_i}\), survival through the landmark is \(\exp(-0.3)=0.7408\) at \(b_i=0\) and \(\exp(-0.6)=0.5488\) at \(b_i=1\). With equal prior weights and no biomarker preference,
For the next interval, conditional survival is \(\exp(-0.25)=0.7788\) at \(b_i=0\) and \(\exp(-0.50)=0.6065\) at \(b_i=1\). The landmark-weighted conditional survival is
The conditional event risk over the next interval is therefore about \(1-0.7055=0.2945\), or 29.45%. This calculation uses the integrated hazard on each interval and conditions on survival through the landmark; it does not treat the hazard at a single time as the cumulative hazard.
Assumptions, Diagnostics, and Alternatives
The reweighting is valid only to the extent that the joint model and conditioning assumptions are appropriate. In particular, the hazard must represent the event process conditional on the random effects, the longitudinal likelihood must describe the observed history, and the stated conditional-independence or likelihood-factorization assumptions must be defensible. A shared-random-effects model provides the connection: the same \(b_i\) can help explain the marker history and the event hazard. If important dependence remains after conditioning on \(b_i\), the simple product of likelihood contributions may be inadequate.
In computation, the integrals are often approximated by quadrature or Monte Carlo draws. Evaluate the products on a log scale when likelihoods are small, normalize the resulting weights, and inspect whether only a few draws carry nearly all the weight. Very concentrated weights can reflect substantial survival selection, but can also indicate poor integration, an implausible model, or inadequate Monte Carlo sampling. Repeating calculations with more quadrature points or draws is a useful numerical check. Predictions should also be checked for calibration among patients who are actually event-free at comparable landmark times.
| Approach | What it uses | Main limitation |
|---|---|---|
| Biomarker-history weights alone | The marker likelihood and random-effects distribution | Leaves out the information in remaining event-free to the landmark |
| Landmark weights | Marker history, survival through the landmark, and the random-effects distribution | Depends on the joint model and its assumptions about the event and marker processes |
| Plug-in random effects | A single estimated value of \(b_i\) | Does not average over uncertainty in the patient’s latent random effects |
A landmark analysis that models risk using observed predictors can be a useful alternative, but it does not automatically perform the same latent random-effects integration. Conversely, the joint-model calculation can be sensitive to misspecification of the marker trajectory, hazard association, or random-effects distribution. Survival reweighting is therefore a principled consequence of the specified joint model, not a guarantee that the model captures all sources of patient risk.
Common Mistakes and Practical Interpretation
- Leaving out survival to the landmark. Biomarker-only weights do not represent patients known to be event-free at \(s\). Include \(S_i(s\mid b_i)\) in the landmark density.
- Counting landmark survival twice. After using \(f\{b_i\mid Y_i(s),T_i>s\}\), average the future survival ratio, not the future survival function without adjustment.
- Assuming every random-effects summary must decrease. Higher-risk values are generally downweighted when the hazard increases with them, but the posterior change also depends on biomarker evidence and the form of the hazard.
- Treating censoring before \(s\) as event-free survival to \(s\). A patient lost to follow-up before the landmark is not thereby known to have survived through it.
The interpretation is conditional: the predicted risk describes a patient with the specified observed history who is known to be event-free at the landmark, under the fitted joint model. It is not a causal effect of surviving to \(s\), nor does it imply that the patient’s random effects have been observed exactly.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Why does survival to a landmark usually reduce the weight on random-effects values associated with a higher hazard?
- Suppose the biomarker likelihood and prior density are fixed. What factor must be added to obtain the random-effects density conditional on being event-free through \(s\)?
- When the landmark posterior is used for prediction, what quantity should be averaged over it to obtain survival from \(s\) to \(s+u\)?
- Why is a patient censored before \(s\) not automatically eligible as known to be event-free at \(s\)?
- In the piecewise-hazard example, what would the event risk be if landmark survival were incorrectly ignored and the prior weights remained equal?
Solutions. (1) Such values imply a smaller \(S_i(s\mid b_i)\), so their posterior weight is reduced by the survival likelihood. (2) Multiply by \(S_i(s\mid b_i)\), then normalize by integrating over \(b_i\). (3) Average \(S_i(s+u\mid b_i)/S_i(s\mid b_i)\), the conditional survival over the additional interval. (4) Censoring before \(s\) means later event status is unobserved; it does not establish \(T_i>s\). (5) Equal prior weights would give survival \(\{0.7788+0.6065\}/2=0.6927\), hence event risk \(1-0.6927=0.3073\), or 30.73%. The difference from 29.45% shows the effect of landmark conditioning.