Deriving Conditional Survival From a Joint Model

From a Joint Model to a Landmark Prediction

A dynamic prediction asks about a person who has remained event-free up to a specified time and whose longitudinal measurements have been observed up to that time. The prediction concerns a later time. Keeping those two times distinct is essential: survival from the start of follow-up to a future time is not the same quantity as survival from a landmark to that future time.

Let \(T_i\) be the event time for person \(i\), let \(s\) be the landmark time, and let \(w>0\) be the prediction horizon. The target is the probability of remaining event-free for a further \(w\) time units, conditional on being event-free at \(s\) and on the longitudinal history available at \(s\). Write that history as \(\mathcal{H}_i(s)\), and let \(x_i\) denote baseline covariates.

The prediction is conditional on survival to \(s\), not merely on having a measurement at \(s\). If the reported horizon is three years for a person at a two-year landmark, the target endpoint is year five. It is not survival through year three from baseline.

Joint-Model Contributions and the Ratio

Let \(b_i\) denote the person-specific shared random effects in the joint model. Conditional on \(b_i\) and \(x_i\), suppose the event process has hazard \(\lambda_i(t\mid b_i,x_i)\). Its conditional survival function is

Let \(L_i\{\mathcal{H}_i(s)\mid b_i,x_i\}\) be the longitudinal-data likelihood contribution for the observed measurements through \(s\), conditional on the shared effects and covariates. Under the joint-model factorization, the contribution for the history together with being event-free through time \(t\), for \(t\ge s\), is

This expression is a joint-model contribution: it combines the observed longitudinal history, survival through \(t\), and the model distribution of the shared effects. For a fixed \(b_i\), the longitudinal likelihood is the same in the contributions at \(s\) and \(s+w\). The difference between those contributions is the required additional survival.

The ratio follows directly from conditional probability. The event \(T_i>s+w\) implies \(T_i>s\), so

The numerator and denominator are precisely the joint-model contributions \(D_i(s+w;\mathcal{H}_i(s),x_i)\) and \(D_i(s;\mathcal{H}_i(s),x_i)\). This derivation also explains why the denominator must use survival through \(s\): the prediction is conditional on having reached the landmark without the event.

For a given value of the shared effects, conditioning has a simpler form:

This identity is useful for understanding the time interval involved. But a prediction conditional on observed history alone is generally not obtained by simply taking this ratio at one fixed \(b_i\). In the joint contribution ratio, the history likelihood and survival terms occur together inside each contribution. The ratio is the precise target; how the shared effects are integrated or updated is a separate computational and inferential issue.

Worked Ratios Under Specified Hazards

Assumptions, Interpretation, and Checks

The ratio is meaningful when the joint model represents the relevant data-generating structure. In particular, the longitudinal likelihood and event survival must be correctly specified conditional on the shared effects and covariates, and the shared-effects distribution must be appropriate. The measurement and censoring processes must either be ignorable for the stated likelihood or be represented in the model. The conditioning history must contain measurements available by \(s\), not measurements collected after \(s\).

A fitted model often uses estimated parameters in these expressions. A plug-in prediction evaluates the ratio at those estimates; it does not by itself account for parameter-estimation uncertainty. Also, \(D_i(t;\mathcal{H}_i(s),x_i)\) is a likelihood contribution, not generally a probability that can be interpreted on its own. It is the ratio that gives the conditional survival probability.

Several checks can expose mistakes:

• For \(w=0\), the ratio should be \(1\), since \(D_i(s)/D_i(s)=1\).
• For \(w>0\), the prediction must lie between 0 and 1. A value above 1 usually signals a reversed ratio or inconsistent contributions.
• For fixed \(s\), increasing \(w\) must not increase survival: \(D_i(s+w)\) cannot represent a less restrictive event-free requirement than \(D_i(s+w')\) when \(w>w'\), under a valid model.
• The endpoint must be \(s+w\), and the survival condition in the denominator must be through \(s\). Substituting \(w\) directly for the endpoint answers a baseline-time question instead.

A common mistake is to report \(S_i(s+w)\) as the conditional prediction. That is survival from baseline to the endpoint, not survival beyond \(s\) among people known to be event-free at \(s\). The correction is to divide the joint contribution for survival through \(s+w\) by the corresponding contribution for survival through \(s\). Another mistake is to divide two marginal survival probabilities while ignoring the observed biomarker history. That may define a population-level conditional survival quantity, but it is not generally the history-specific prediction from a joint model.

The conditional ratio for a fixed shared-effect value offers a direct alternative description: it is survival over only the interval \((s,s+w]\). The joint-contribution ratio is the relevant formulation when the prediction conditions on observed longitudinal history and the shared effects are latent. A landmark analysis based on a separate event model is another possible approach, but it does not automatically use the joint longitudinal and event structure. The joint-model prediction depends on its assumptions, and model misspecification, sparse measurements, or extrapolation to long horizons can make it unreliable.