Fixed-Horizon Landmark Logistic Models

Risk Over a Fixed Horizon After a Landmark

A landmark prediction answers a specific question: among patients who are event-free at a chosen time, what is the probability of an event during a specified period after that time? A logistic model is a direct way to represent this probability when the endpoint is binary, such as event by 6 months versus no event by 6 months. It is not a model of the exact timing of the event within the period.

Let $s$ be the landmark time and $h>0$ the prediction horizon. For patient $i$, let $T_i$ be the event time, $C_i$ the censoring time, and $Z_i(s)$ the covariate vector formed using information available at the landmark. The target population consists of patients with $T_i>s$ who are observed and eligible at $s$. The outcome is whether the event occurs in $(s,s+h]$.

Definition: The fixed-horizon landmark risk at covariate value $z$ is $$ p_h(z)=P\{T\leq s+h\mid T>s,\ Z(s)=z\}. $$ A fixed-horizon landmark logistic model specifies $$ \operatorname{logit}\{p_h(z)\} =\log\left\{\frac{p_h(z)}{1-p_h(z)}\right\} =\beta_0+\beta^\mathsf{T}z. $$ Here $p_h(z)$ is the conditional probability of an event by the horizon, among patients event-free at the landmark. The covariates must be available at $s$; information recorded later cannot be used as a predictor for a prediction made at $s$.

This definition assumes a single event endpoint. If another event prevents the event of interest from occurring, how that competing event is handled must be specified as part of the estimand; it should not silently be treated as ordinary censoring. The logistic model also assumes independent patients, a correctly specified conditional mean on the log-odds scale, and enough outcome variation to estimate the coefficients. In particular, a continuous predictor is assumed to have a linear association with the log odds unless nonlinear terms are included.

Fitting and Interpreting the Model

If every patient’s event status at $s+h$ is known, define $D_i=1$ when the event occurs by $s+h$, and $D_i=0$ otherwise. Standard logistic regression estimates the coefficients by maximizing the Bernoulli log-likelihood

\ell(\beta_0,\beta) =\sum_i\left[D_i\log\{p_h(Z_i(s))\} +(1-D_i)\log\{1-p_h(Z_i(s))\}\right].

A coefficient $\beta_j$ is a conditional log odds ratio for a one-unit increase in covariate $Z_j$, holding the other included covariates fixed. Thus $\exp(\beta_j)$ is the corresponding conditional odds ratio for event by the specified horizon. It is not a risk ratio, and it is not a hazard ratio. To obtain a predicted probability, transform the fitted linear predictor $\eta$ using $p=1/(1+\exp(-\eta))$.

Worked Example: A Binary Landmark Predictor

Suppose 200 hypothetical patients are eligible at a landmark, and every patient’s 6-month event status is observed. Among 100 patients in a lower-marker group, 20 have an event by 6 months; among 100 in a higher-marker group, 40 have an event. Let $Z=1$ denote the higher-marker group and $Z=0$ the lower-marker group.

The observed risks are $20/100=0.20$ and $40/100=0.40$. Their odds are $0.20/(1-0.20)=0.25$ and $0.40/(1-0.40)=0.6667$, respectively. In the model $\operatorname{logit}(p)=\beta_0+\beta_1Z$, the fitted intercept and slope are therefore

\hat\beta_0=\log(0.25)=-1.386,\qquad \hat\beta_1=\log\left(\frac{0.6667}{0.25}\right) =\log(2.6668)=0.9809.

For the lower-marker group, the fitted risk is $1/\{1+\exp(1.386)\}=0.200$. For the higher-marker group, the fitted log odds are $-1.386+0.9809=-0.4051$, giving risk $1/\{1+\exp(0.4051)\}=0.400$. The estimated odds ratio is $\exp(0.9809)=2.667$: in these hypothetical data, the higher-marker group has 2.667 times the odds of an event by 6 months, compared with the lower-marker group. This is an odds comparison, not a claim that its event probability is 2.667 times as large.

When Censoring Hides the Horizon Outcome

A patient who has the event before the horizon has a known positive outcome if the event is observed. A patient observed event-free through $s+h$ has a known negative outcome. But a patient censored before $s+h$ without an event has an unknown horizon outcome: the patient might have had an event later in the period. Coding such a patient as event-free is generally incorrect.

Write $V_i=T_i-s$ for time from the landmark to event and $W_i=C_i-s$ for time from the landmark to censoring. Let $R_i=1$ when the horizon outcome is ascertainable: an event is observed by $h$, or the patient is observed event-free through $h$. For an ascertainable outcome, $D_i=1$ if $V_i\leq h$ and $D_i=0$ if the patient remains event-free through $h$.

When some outcomes are not ascertainable, inverse-probability-of-censoring weighting (IPCW) can retain information without pretending that early-censored patients are non-events. Let $G(v\mid z)=P(W\geq v\mid Z(s)=z)$ denote the probability of remaining uncensored through $v$, conditional on landmark covariates. For an observed event, ascertainment requires remaining uncensored until its event time; for an observed non-event, it requires remaining uncensored through the horizon. The corresponding weights are $1/G(V_i\mid Z_i(s))$ for events and $1/G(h\mid Z_i(s))$ for non-events.

Assumptions for IPCW: Conditional on the included covariates, censoring is independent of the event time; the censoring survival probabilities are correctly estimated or otherwise adequately modeled; and $G(v\mid z)>0$ for the event times and horizon relevant to the target population. The last condition is positivity: patients with the covariate patterns of interest must have a nonzero chance of remaining observed long enough for the outcome to be ascertained.

For ascertainable records, IPCW fits the weighted logistic likelihood

\ell_w(\beta_0,\beta) =\sum_{i:R_i=1} w_i \left[D_i\log\{p_h(Z_i(s))\} +(1-D_i)\log\{1-p_h(Z_i(s))\}\right], \qquad w_i=\frac{1}{G(q_i\mid Z_i(s))},

where $q_i=V_i$ for an observed event and $q_i=h$ for an observed non-event. Under the stated assumptions, weighting compensates for the chance that a record with a particular outcome and follow-up time is observable. An early-censored patient without an event is not assigned $D_i=0$; that record contributes no observed binary outcome to this likelihood.

Worked Example: Assigning Censoring Weights

Consider three hypothetical patients eligible at a landmark, with a 6-month horizon. A fitted censoring model gives the following probabilities of remaining uncensored: $G(2\mid z)=0.90$, $G(5\mid z)=0.80$, and $G(6\mid z)=0.75$ for the relevant patients.

Patient A has an observed event 2 months after the landmark. The outcome is known to be positive, and the weight is $1/0.90=1.111$. Patient B is observed event-free through 6 months. The outcome is known to be negative, and the weight is $1/0.75=1.333$. Patient C is censored event-free at month 4. The patient’s status at month 6 is unknown, so this record is not assigned a zero outcome or a horizon-outcome weight in this binary likelihood.

These weights do not mean that Patient A had a 1.111 probability of an event. They weight the patient’s likelihood contribution to represent comparable patients who may have been censored before an outcome could be ascertained. If the estimated probability of remaining observed is very small, the corresponding weight becomes large and can make the fit unstable.

Why the Weighting Argument Works

The key justification is that inverse-probability weighting removes the expected loss of outcome information due to censoring. Let $Q_i=\min(V_i,h)$ for patients whose horizon outcome is ascertainable. Under conditional independent censoring, the probability that the outcome is ascertainable given the event process and covariates is $G(Q_i\mid Z_i(s))$. Consequently, for a contribution $a_i$ to the logistic score or likelihood that is defined from the full outcome,

E\left[ \left. \frac{R_i}{G(Q_i\mid Z_i(s))}a_i \right|\,\text{event process},Z_i(s) \right] = \frac{a_i}{G(Q_i\mid Z_i(s))} E\{R_i\mid \text{event process},Z_i(s)\} =a_i.

The weighted contribution has, conditionally, the same expectation as the contribution that would have been available with complete follow-up. Summing these contributions gives an estimating basis for the full landmark cohort. This argument depends on the censoring model being appropriate and on the event process being conditionally independent of censoring; weighting cannot repair unmeasured dependence between censoring and outcome after conditioning on the covariates.

Worked Example: Interpreting an Adjusted Coefficient

Suppose a hypothetical IPCW logistic model for 12-month event risk includes a biomarker measured at the landmark in units of 10 mg/L and adjusts for age and treatment group. The estimated biomarker coefficient is $\hat\beta=-0.55$, with standard error $SE=0.22$.

The odds ratio for a 10 mg/L increase, conditional on the other model covariates, is $\exp(-0.55)=0.577$. Thus the fitted odds of an event by 12 months are about 0.577 times as large per 10 mg/L increase, under the model. This does not establish a causal effect and does not say that risk falls by 42.3 percentage points.

A Wald statistic is $z=-0.55/0.22=-2.50$, giving a two-sided normal-approximation $p$-value of about 0.0124. A 95% Wald interval for the log odds ratio is $-0.55\pm1.96(0.22)=[-0.9812,-0.1188]$. Exponentiating the endpoints gives an odds-ratio interval of $[0.375,0.888]$. Under the model and its uncertainty approximation, the data are inconsistent with a zero coefficient at the 0.05 level. The interval concerns the conditional odds ratio for event by 12 months, not a hazard ratio.

Diagnostics, Alternatives, and Limitations

Check the outcome construction first: eligibility must be determined at the landmark, the horizon must be fixed, and early censoring without an event must remain unknown. Then examine whether the logistic mean model is plausible. Useful checks include calibration of predicted risks, the functional form of continuous predictors on the log-odds scale, influential observations, sparse event counts, and signs of complete or near-complete separation. Since IPCW adds a censoring model, inspect its fit and the distribution of weights as well. A few very large weights indicate weak positivity or a poorly supported censoring model; report this problem rather than treating a numerically converged fit as sufficient evidence of reliability.

Approach	When it can be useful	Main concern
Ordinary logistic regression	Horizon status is known for everyone, for example because follow-up is complete through the horizon.	It is not valid to code early-censored, event-free patients as non-events.
Complete-case logistic regression	Only when outcome ascertainment does not select patients differently by outcome after conditioning on modeled covariates, or when follow-up is complete through the horizon.	With censoring, ascertainability can depend on event status and event time, so restricting to known outcomes can distort risk.
IPCW logistic regression	Some horizon outcomes are unknown, and conditional independent censoring and positivity are credible.	Results can be unstable with extreme weights or misspecification of the censoring model.

A fixed-horizon model is attractive when the clinical question concerns a particular period, but it compresses event timing into a yes-or-no endpoint: events at different times within the horizon receive the same outcome value. Its odds ratios depend on the chosen horizon, and odds ratios are non-collapsible, so adjusted and unadjusted estimates can differ even without confounding. If the horizon changes, the estimand and generally the fitted coefficients change. A landmark Cox model is an alternative when modeling residual event-time information is the goal; it answers a different modeling question rather than repairing a misspecified fixed-horizon logistic model.

Common Mistakes

Calling an early-censored patient a non-event. Follow-up ending before the horizon does not show that the patient remained event-free for the rest of the period.
Interpreting an odds ratio as a risk ratio. Logistic coefficients describe odds at the stated horizon; calculate predicted risks separately when absolute probabilities are needed.
Using information measured after the landmark. Such information is not available for a prediction made at the landmark and can introduce information from after the prediction time.
Assuming IPCW solves every censoring problem. It relies on conditional independence, a suitable censoring model, and positivity; weighting does not fix violations of these assumptions.

Key takeaway: Fixed-horizon landmark logistic regression models event probability by a prespecified time among landmark-eligible patients. Correctly defining the eligible cohort and horizon outcome is essential; when censoring makes some outcomes unknown, IPCW is justified only under explicit censoring and positivity assumptions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

At a 3-month landmark, a patient is event-free but censored 2 months later, before a 6-month prediction horizon. Is the 6-month outcome zero, one, or unknown?
A fitted coefficient for a one-unit covariate increase is $0.40$. What is the conditional odds ratio, and what does it describe?
An observed event occurs 4 months after the landmark. If the estimated probability of remaining uncensored through month 4 is $0.80$, what is its IPCW weight?
Name two assumptions needed for IPCW and explain one consequence of violating positivity.

Solutions. (1) Unknown: the patient could have had an event between censoring and the 6-month horizon. (2) $\exp(0.40)=1.492$, approximately 1.49. It is the conditional odds ratio for event by the specified horizon per one-unit increase, holding the other modeled covariates fixed. (3) The weight is $1/0.80=1.25$. (4) Examples are conditional independence of censoring and event time given modeled covariates, an adequately specified censoring model, and positivity. If positivity fails, some patients have effectively no chance of having the outcome observed; weights become extreme or undefined, so the target risk is not reliably identified from the observed data.