Conditional Survival Given a Patient’s History

Why the Landmark Changes the Prediction

A survival prediction made at diagnosis answers a different question from one made after a patient has remained event-free and provided longitudinal measurements. At a later landmark, the prediction should describe the patient’s future from that point onward, using the information available then. It should not include measurements collected later or treat a patient who has already had the event as eligible for prediction at that landmark.

The target is conditional on two pieces of information: the patient is event-free through the landmark, and the patient’s observed history up to that time is known. The result is a probability for that patient’s future, not a statement about the probability that the patient has an event at the landmark.

Definition: Let $T_i$ be patient $i$’s event time, $t$ a landmark time, and $u\geq 0$ a future prediction horizon. Let $\mathcal Y_i(t)=\{(t_{ij},y_{ij}):t_{ij}\leq t\}$ denote that patient’s longitudinal measurements and their observation times through $t$, and let $X_i$ denote baseline covariates. The conditional survival function at horizon $u$ is $$ S_i(u\mid t)=P\!\left(T_i>t+u\mid T_i>t,\mathcal Y_i(t),X_i\right). $$ It is the probability that the patient remains event-free for at least the next $u$ time units, conditional on being event-free at $t$ and on the specified information available by $t$.

The landmark time $t$ is measured on the study’s event-time scale, while $u$ measures time forward from that landmark. For example, if time is measured in years since enrollment, a prediction at $t=3$ for $u=2$ concerns being event-free beyond year 5. The corresponding absolute-time version is $P(T_i>s\mid T_i>t,\mathcal Y_i(t),X_i)$, where $s=t+u$. Stating both the landmark and the horizon prevents ambiguity.

At $u=0$, the patient is known to be event-free at the landmark, so $S_i(0\mid t)=1$. As $u$ increases, conditional survival cannot increase: surviving beyond a longer horizon implies surviving beyond every shorter horizon. The conditional event risk by horizon $u$ is $1-S_i(u\mid t)$. It is a risk over the interval after the landmark, not the risk accumulated from the beginning of follow-up.

What Information Is Being Conditioned On?

The history $\mathcal Y_i(t)$ includes the recorded marker values and when they were measured. The times matter: the same pair of values can describe different histories if one patient’s later value was measured much closer to the landmark. Baseline covariates $X_i$ may also be included, but the target must state what information is used.

The condition $T_i>t$ is essential. A patient who had the event before $t$ is not part of the population for whom a prediction at $t$ is defined. Thus, this is a prediction for an event-free patient at the landmark, rather than a baseline prediction for every person enrolled in the study. In observed data, a patient censored before $t$ is not known to be event-free through $t$; censoring is not itself evidence that the patient remained event-free.

A prediction at $t$ uses measurements available by $t$, not the patient’s future marker values. Using later values would change the information set and would not represent a prediction that could actually have been made at the landmark. The conditional target is defined by its probability statement; choosing a model to estimate that probability is a separate step.

Two Equivalent Ways to Express the Target

One can define the target directly as a conditional probability, as above. Alternatively, suppose $G_i(s\mid\mathcal Y_i(t),X_i)=P(T_i>s\mid\mathcal Y_i(t),X_i)$ denotes survival to absolute time $s$, conditional on the history and baseline covariates but not yet conditioning on survival to $t$. Provided the denominator is positive, the rule for conditional probabilities gives:

S_i(u\mid t) = \frac{P(T_i>t+u\mid\mathcal Y_i(t),X_i)} {P(T_i>t\mid\mathcal Y_i(t),X_i)} = \frac{G_i(t+u\mid\mathcal Y_i(t),X_i)} {G_i(t\mid\mathcal Y_i(t),X_i)}.

The numerator describes survival past the future absolute time; the denominator restricts attention to survival through the landmark. This ratio makes clear why an unconditional survival probability from time zero is not the target for a patient already known to be event-free at $t$. The history is held fixed at the information observed by $t$ in both terms.

A conditional hazard offers another expression. If $\lambda_i(t+v\mid t)$ is the event hazard $v$ time units after the landmark, conditional on the patient’s landmark information and continued survival up to that point, then under a continuous event-time model:

S_i(u\mid t) = \exp\!\left\{-\int_0^u \lambda_i(t+v\mid t)\,dv\right\}.

The integral is cumulative hazard over the future interval, not from the study’s time origin. This expression requires a valid nonnegative, integrable hazard on the interval. It provides a calculation once a conditional hazard is specified; it does not by itself say how a joint model estimates that hazard from longitudinal history.

Worked Calculations

Worked Example: Constant Conditional Hazard

Suppose a hypothetical patient is event-free at year $t=4$. Their recorded marker history through year 4 is used to specify a constant conditional hazard of $0.08$ per year over the next two years. Under this assumed constant hazard, calculate the probability of remaining event-free for two more years.

The hazard is nonnegative and constant over the stated future interval, and the target is conditional on event-free status at year 4 and the observed history through year 4. The cumulative hazard over $u=2$ years is $0.08\text{ per year}\times2\text{ years}=0.16$. Therefore,

S_i(2\mid 4)=\exp(-0.08\times2)=\exp(-0.16)=0.8521.

The conditional probability of an event by year 6 is $1-0.8521=0.1479$. In context, under the stated hazard assumption, the patient has an estimated 85.21% probability of remaining event-free through the next two years, given that they were event-free at year 4 and given the specified history. The prediction concerns survival beyond year 6, not survival from enrollment to year 6 without conditioning.

Worked Example: A Piecewise Conditional Hazard

For a second hypothetical patient, suppose the conditional hazard after landmark $t=2$ is $0.04$ per year during the first year after the landmark and $0.10$ per year during the following two years. Assume these hazards apply conditional on the patient’s history through year 2 and on remaining event-free up to each point. Calculate conditional survival at horizons of two and three years.

For a two-year horizon, one year is spent at each hazard level. The cumulative hazard is $0.04\times1+0.10\times1=0.14$, so

S_i(2\mid 2)=\exp[-(0.04\times1+0.10\times1)] =\exp(-0.14)=0.8694.

For a three-year horizon, the second hazard applies for two years. Thus, $0.04\times1+0.10\times2=0.24$, and

S_i(3\mid 2)=\exp[-(0.04\times1+0.10\times2)] =\exp(-0.24)=0.7866.

The corresponding conditional event risks are $1-0.8694=0.1306$ by two years and $1-0.7866=0.2134$ by three years. Survival at the longer horizon is lower, as it must be. The calculation uses time intervals after the landmark; the first one-year interval is years 2 to 3, and the next two-year interval is years 3 to 5.

Worked Example: Conditioning on Event-Free Survival to the Landmark

Suppose a hypothetical model, conditional on a patient’s marker history through $t=3$ and baseline covariates, gives $G_i(3\mid\mathcal Y_i(3),X_i)=0.72$ and $G_i(4\mid\mathcal Y_i(3),X_i)=0.60$. Find conditional survival for the next year among patients with this information who are event-free at year 3.

The denominator is positive, so the conditional-probability ratio applies:

S_i(1\mid3)=\frac{G_i(4\mid\mathcal Y_i(3),X_i)} {G_i(3\mid\mathcal Y_i(3),X_i)} =\frac{0.60}{0.72}=0.8333.

Thus, the conditional probability of remaining event-free from year 3 through year 4 is 83.33%; the conditional event risk over that year is $1-0.8333=0.1667$, or 16.67%. Reporting $0.60$ as the patient’s one-year conditional survival would be wrong: it is survival to year 4 under the stated conditioning before restricting to those event-free at year 3. The denominator performs that restriction.

Assumptions, Diagnostics, and Alternatives

The probability target itself requires clear definitions of the event, time origin, landmark, horizon, and available history. Estimating it requires additional assumptions from the chosen statistical model. For example, the conditional hazard calculation assumes that the specified hazard adequately represents risk over the future interval. If event times are censored, estimation also depends on how censoring is handled; a prediction cannot simply treat a censoring time as an event or as proof of survival thereafter.

Basic checks can reveal errors before a prediction is interpreted. The estimated survival should equal 1 at horizon zero, stay between 0 and 1, and be nonincreasing as the horizon grows. The conditional risk should be $1-S_i(u\mid t)$, also between 0 and 1, and nondecreasing in $u$. Check that rates have units inverse to time and that each hazard interval has the stated duration. With observed data, assess predictive calibration and discrimination at the intended landmark and horizons, using methods appropriate for censoring. A plausible-looking curve alone does not establish that the model is well calibrated.

Approach or quantity	What it answers	Key distinction
Landmark-conditional survival	Probability of surviving a future horizon given event-free status and information available at the landmark	Patient’s risk set and information are updated at the landmark
Unconditional survival from the time origin	Probability of surviving from the start of follow-up to an absolute time	Does not, by itself, condition on having survived to the landmark
Landmark prediction using a separate model	Conditional risk estimated from patients’ information at a selected landmark	Can be tailored to that landmark; its assumptions depend on the chosen model
Prediction from a joint model	Conditional survival estimated using a model for longitudinal and event data	Uses a joint modeling framework; the conditional target remains the probability defined above

These approaches can target the same conditional probability if they use the same event definition, landmark, horizon, and conditioning information. They differ in how they estimate it and in their assumptions. Unconditional survival is useful for population-level summaries, but it is not interchangeable with a prediction for a patient known to have survived to $t$. Landmark-based approaches can be easier to tailor to a chosen prediction time, while a joint model offers a structured way to use longitudinal and event information together. Neither approach removes the need to specify the target carefully or examine predictive performance.

Common Mistakes and Limitations

Mixing up the time scales. A horizon $u=2$ at landmark $t=4$ means prediction through absolute year 6, not through year 2.
Forgetting the survival condition. A prediction among patients event-free at $t$ is not an unconditional probability from baseline. When using absolute-time survival probabilities, divide by survival to $t$, if the stated conditioning makes the ratio valid.
Using future marker values. A prediction made at $t$ must use information available by $t$. Later measurements cannot be used as though they were already observed.
Treating censoring as event-free follow-up. Censoring before $t$ leaves event status after censoring unknown; it does not establish $T_i>t$.

Finally, a conditional survival probability is not automatically a causal effect, and it does not guarantee an accurate forecast for an individual. It is a model-based probability for patients with the stated information, under the model and data conditions used for estimation. Predictions can be unreliable outside the observed covariate and history patterns, at very long horizons with little follow-up, or when the model’s assumptions fail.

Check Your Understanding

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

At landmark $t=5$, what absolute time does horizon $u=1.5$ refer to?
If a patient’s conditional survival for the next two years is $0.78$, what is the conditional event risk over those two years?
Suppose $G_i(t\mid\mathcal Y_i(t),X_i)=0.80$ and $G_i(t+1\mid\mathcal Y_i(t),X_i)=0.68$. Calculate one-year conditional survival.
Why is a patient censored at year 4 not automatically known to be event-free at landmark $t=6$?
A model gives survival $0.91$ at a one-year horizon and $0.94$ at a two-year horizon for the same target. What check has failed?

Solutions. (1) It refers to survival beyond year $5+1.5=6.5$, conditional on event-free status and the specified history at year 5. (2) The conditional event risk is $1-0.78=0.22$, or 22% over the next two years. (3) The ratio is $0.68/0.80=0.85$, so conditional survival is 85%. (4) Censoring means event status after year 4 is unobserved; it does not establish $T_i>6$. (5) Survival has increased with the horizon, violating the required nonincreasing behavior of a survival function; check the time intervals, calculations, and prediction procedure.