Landmarking and Joint Models: Distinct Conditioning Sets

Why the Conditioning Set Matters

A predicted risk is meaningful only after specifying for whom it is predicted and what information is being conditioned on. At a landmark time, the risk set consists of people who are still event-free at that time. A landmark model predicts from that survivor population using the predictors included in its model. A joint model can also predict for an event-free person, but it uses the longitudinal record and the event-free status to update information about that person’s latent trajectory. These are related prediction strategies, but the conditioning information need not be the same.

Let $T_i$ be the event time for person $i$, let $s$ be a landmark time, and let $u>0$ be a prediction horizon measured from $s$. Write $Y_i(t)$ for the longitudinal measurement at time $t$, $H_i(s)$ for the person’s observed longitudinal history up to $s$, and $Z_i$ for baseline covariates. The event-free indicator at the landmark is $R_i(s)=I(T_i>s)$. The use of strict inequality means that a person who experiences the event at or before $s$ is not in the event-free risk set at $s$.

Definition: A landmark risk at time $s$ for horizon $u$ is a risk conditional on being event-free at $s$ and on the landmark predictors $X_i(s)$ used by the fitted model: $$ P(s<T_i\le s+u\mid T_i>s,\ X_i(s)). $$ A joint-model prediction for the same horizon is typically conditional on being event-free at $s$, baseline covariates, and the observed longitudinal history through $s$: $$ P(s<T_i\le s+u\mid T_i>s,\ Z_i,\ H_i(s)). $$ The two conditioning sets may overlap, but they are not automatically identical.

Here $X_i(s)$ denotes the landmark covariate representation supplied to the landmark model. It may summarize part of the history, such as a value measured at or before $s$, but it need not preserve every observed value or the same information as $H_i(s)$. If a model uses a different set of predictors, its conditional risk refers to a different information set. This distinction is about the target of prediction, not a claim that one method is universally better.

What the Landmark Risk Set Represents

A landmark analysis starts with people known to be event-free at $s$, then predicts events occurring after $s$ within the chosen horizon. Its target population is therefore the population of eligible survivors at $s$, possibly narrowed further by the predictor values under consideration. It is not the original baseline population: events occurring by $s$ are already in the past and are not part of the landmark prediction.

For example, a model using only the most recent biomarker value conditions on event-free status at $s$ and that value. Even if a person has a long measurement record, the fitted prediction does not condition on every detail in that record unless those details are included in $X_i(s)$. At a given value of the included predictors, the estimated risk describes the relevant survivor group represented by those predictors. In general, it is not a claim that all members of that group have identical underlying health trajectories.

The event-free condition must also be established correctly. Someone censored before $s$ is not known to be event-free at $s$ merely because no event was recorded before censoring. Such records require appropriate handling under the analysis design and censoring assumptions; they cannot simply be treated as confirmed survivors. The same basic care is needed when defining the people eligible for joint-model prediction.

Worked Example: Counting the Landmark Target Population

Consider a hypothetical cohort of 100 people with complete event ascertainment through $s+u$. By landmark time $s$, 20 have experienced the event, leaving 80 event-free. Of those 80 survivors, 30 have a specified landmark biomarker category, and 9 experience the event during $(s,s+u]$.

The risk for that landmark category is calculated among the 30 eligible survivors in the category: $$ \frac{9}{30}=0.30. $$ Thus the observed horizon risk is 30% for people in this category who were event-free at $s$. It is not $9/100=0.09$: that denominator includes 20 people who were not eligible for a post-$s$ prediction because their events had already occurred.

The calculation assumes that event status is completely ascertained for this hypothetical example and that category membership is defined using information available by $s$. If the question instead concerned cumulative event occurrence from baseline through $s+u$, the earlier 20 events would matter, but that would be a different target from the landmark risk.

How Joint-Model Conditioning Uses History

A joint model represents the longitudinal measurements and event process together, commonly by linking the event hazard to a person-specific latent trajectory. For a basic shared-random-effects formulation, one may write a longitudinal model $Y_i(t)=m_i(t)+\epsilon_i(t)$, where $m_i(t)$ is the modeled trajectory and $\epsilon_i(t)$ is measurement error. The event hazard conditional on the latent effects $b_i$ and baseline covariates can be written $h_i(t\mid b_i,Z_i)$. These model components, together with their distributional and censoring assumptions, must be appropriate for the data for the resulting predictions to be useful.

The observed history $H_i(s)$ informs the latent effects, while survival without an event through $s$ also carries information about them when the event hazard depends on those effects. The prediction averages over uncertainty about the latent effects rather than treating them as directly observed. One useful expression makes both parts of the conditioning explicit:

P(T_i>s+u\mid T_i>s,\ H_i(s),Z_i) = \int \frac{S_i(s+u\mid b_i,Z_i)}{S_i(s\mid b_i,Z_i)} \,p(b_i\mid H_i(s),T_i>s,Z_i)\,db_i.

Here $S_i(t\mid b_i,Z_i)=P(T_i>t\mid b_i,Z_i)$ is the event-free probability through time $t$ conditional on the latent effects and baseline covariates. The ratio is residual survival from $s$ to $s+u$ for a specified $b_i$; the density $p(b_i\mid H_i(s),T_i>s,Z_i)$ describes uncertainty about those effects after observing both the longitudinal history and survival through $s$. The event risk is one minus this conditional survival probability.

Equivalently, if the posterior for $b_i$ is first formed conditional on $H_i(s)$ and $Z_i$, survival through $s$ can be included by weighting that posterior by $S_i(s\mid b_i,Z_i)$ and normalizing. This is important when survival through $s$ is informative about the latent effects. The expression assumes the joint model is correctly specified and that the observed history and event status are represented as stated; it does not make the risk unconditional from baseline.

Worked Example: Updating Latent-Effect Uncertainty with Survival

Suppose a hypothetical joint model has two possible latent trajectory classes, high and low. After observing a person’s longitudinal history through $s$, but before using survival status, the model assigns probability 0.75 to the high class and 0.25 to the low class. Suppose the probability of surviving to $s$ is 0.50 in the high class and 1.00 in the low class. These values are hypothetical and are used only to show the conditioning calculation.

Among people with this history who are also known to have survived to $s$, the probability of the high class is $$ P(\text{high}\mid H_i(s),T_i>s) = \frac{0.75(0.50)}{0.75(0.50)+0.25(1.00)} = \frac{0.375}{0.625} = 0.60. $$ The corresponding probability of the low class is $1-0.60=0.40$.

Now suppose the model’s residual event risks over $(s,s+u]$ are 0.40 for the high class and 0.10 for the low class. The prediction conditional on both history and survival to $s$ is $$ 0.60(0.40)+0.40(0.10)=0.24+0.04=0.28. $$ The resulting risk is 28%. If survival to $s$ were ignored and the probabilities 0.75 and 0.25 were used instead, the calculation would be $0.75(0.40)+0.25(0.10)=0.325$, or 32.5%. That is a different conditioning calculation, not the requested prediction for a person known to have survived through $s$.

This example assumes the stated class probabilities and risks are valid under the hypothetical model. In an actual analysis, they would come from the fitted joint model and its posterior calculations, not from manually assigning two classes.

Deriving the Conditional Prediction

The residual-survival expression follows from conditioning on the latent effects and then averaging over their distribution among people with the observed information. Given $b_i$, $Z_i$, and survival through $s$, the probability of surviving an additional $u$ time units is

P(T_i>s+u\mid T_i>s,b_i,Z_i) = \frac{P(T_i>s+u\mid b_i,Z_i)} {P(T_i>s\mid b_i,Z_i)} = \frac{S_i(s+u\mid b_i,Z_i)}{S_i(s\mid b_i,Z_i)}.

The first equality uses the fact that $T_i>s+u$ implies $T_i>s$. To obtain the prediction given the observed history, apply the law of total probability over the posterior distribution of $b_i$ conditional on $H_i(s)$, $T_i>s$, and $Z_i$. This gives the integral displayed above. Finally, because the event by $s+u$ is the complement of surviving beyond $s+u$, conditional event risk is

P(s<T_i\le s+u\mid T_i>s,H_i(s),Z_i) = 1- \int \frac{S_i(s+u\mid b_i,Z_i)}{S_i(s\mid b_i,Z_i)} \,p(b_i\mid H_i(s),T_i>s,Z_i)\,db_i.

This derivation clarifies two separate sources of conditioning: the risk is residual from $s$, and the distribution used to average over latent effects reflects the person’s history and survival to $s$. A landmark model also targets residual risk among survivors, but it generally estimates that risk as a function of its chosen $X_i(s)$, rather than integrating over a joint-model posterior for latent effects.

Worked Example: Comparing Two Conditioning Sets

Suppose two hypothetical people are both event-free at $s$ and have the same most recent biomarker value. Person A has a history that, under a fitted joint model, gives posterior probability 0.60 for a high-risk latent class after conditioning on survival to $s$. Person B has a different earlier measurement pattern and has posterior probability 0.20 for that class. Suppose the joint model assigns residual risks of 0.40 and 0.10 to the high- and low-risk classes, respectively.

For person A, the joint-model prediction is $$ 0.60(0.40)+0.40(0.10)=0.28. $$ For person B, it is $$ 0.20(0.40)+0.80(0.10)=0.16. $$ The predictions are 28% and 16%, respectively, despite the same last observed value. The difference arises because the joint predictions condition on the full histories as represented by the model, as well as survival to $s$.

A landmark model whose only predictor is that last value assigns the same fitted risk to both people with the same value, subject to any other included predictors. That risk is estimated for the corresponding survivor group; it does not claim that the two people’s complete histories are identical. The example assumes the stated posterior probabilities and class-specific risks are supplied by a valid fitted model. It illustrates the conditioning distinction, not a general guarantee that joint-model predictions differ or are more accurate.

Interpreting Differences and Checking the Analysis

A landmark prediction and a joint-model prediction can both be conditional on $T_i>s$, yet differ because one conditions on $X_i(s)$ while the other conditions on $H_i(s)$ and $Z_i$, with latent effects updated under the joint model. They may also differ in their model assumptions and in how they summarize information. A numerical difference alone does not establish that either prediction is biased or that one method performs better. A fair comparison requires the same eligible population, landmark time, prediction horizon, event definition, and predictor information—or an explicit acknowledgment that the questions differ.

Several checks help keep the interpretation aligned with the calculation:

Verify eligibility at $s$. Confirm that each predicted person is known to be event-free at the landmark. Do not count people with an earlier event as members of the landmark risk set.
Make the time scale explicit. A risk over $(s,s+u]$ is a residual risk over $u$ time units, not a baseline-to-$s+u$ cumulative risk.
Check when predictors were observed. Landmark predictors must be available by $s$. Information recorded after $s$ cannot be used as if it were known at prediction time.
Keep the conditioning statement with the number. State whether the estimate is conditional on a landmark summary or on the longitudinal history and survival status used by the joint model.
Assess modeling assumptions. Landmark risk estimation requires appropriate handling of censoring and a suitable model for the target risk. Joint prediction additionally depends on the longitudinal model, event model, their association structure, and the assumptions used to integrate over latent effects.

Landmarking can be attractive when a direct prediction among survivors using a prespecified summary is the practical target. Joint models can use the longitudinal history through a shared latent structure, but this reliance makes their predictions sensitive to model specification and the adequacy of the assumed association between the longitudinal and event processes. Neither approach removes the need to define the target population and prediction time carefully.

Common Mistake

Common pitfall: “Both methods predict risk after $s$, so their predictions have the same conditioning set and should be numerically equal.” The correction is that both may condition on being event-free at $s$, while conditioning on different predictor information. A landmark risk conditional on $X_i(s)$ and a joint-model risk conditional on $H_i(s)$ and $Z_i$ are not automatically the same estimand. Agreement or disagreement must be interpreted in light of those conditioning sets and the fitted models.

A related error is to interpret a landmark risk as the risk for everyone enrolled at baseline. It is specifically a post-landmark risk among eligible survivors. Naming the population, time origin, horizon, and information set prevents this shift in interpretation.

Check Your Understanding

Answer each question using the conditioning information and target population explicitly. Brief solutions follow.

At time $s$, 50 people are event-free, and 10 experience the event by $s+u$. What is the observed landmark risk, and which people are its denominator?
In one sentence, distinguish the conditioning sets in a landmark risk using $X_i(s)$ and a joint-model prediction using $H_i(s)$ and $Z_i$.
A person was censored before $s$, with no event recorded before censoring. Is that person known to be event-free at $s$? Explain.
A joint model assigns posterior probability 0.30 to a high-risk class and 0.70 to a low-risk class among people with the observed history who survived to $s$. Their residual event risks are 0.50 and 0.10. Calculate the joint-model risk.

Solutions. (1) The observed risk is $10/50=0.20$, or 20%, among the 50 people known to be event-free at $s$. (2) The landmark risk conditions on survival to $s$ and the model’s landmark predictors $X_i(s)$; the joint prediction conditions on survival to $s$, the observed history $H_i(s)$, and baseline covariates $Z_i$. (3) No. Censoring before $s$ leaves later event status unknown, so event-free status at $s$ cannot be assumed without appropriate handling. (4) The risk is $0.30(0.50)+0.70(0.10)=0.15+0.07=0.22$, or 22%, conditional on the stated history and survival to $s$.