Pooled Landmark Models Across Multiple Times

Why Pool Landmark Times?

A landmark analysis predicts risk among patients who are still eligible at a specified time, using information available by that time. When predictions are needed at several landmark times, fitting a separate regression at every time is one option. Pooling the eligible records into one regression is another: it can use information across times and provide a single framework for estimating landmark-specific risks. The crucial detail is that one patient may contribute more than one record, so the stacked rows are generally not independent observations.

The pooled model must also preserve the meaning of the prediction. A risk prediction for a patient eligible at time \(t_k\) concerns the event over a stated future horizon, conditional on eligibility and predictor information at \(t_k\). It is not an unconditional risk from the start of follow-up.

Eligibility must be determined separately at each landmark, typically requiring that the patient is alive and has not yet experienced the event of interest at \(t_k\). The event indicator must refer to the same specified horizon and event definition for every row. If follow-up is censored before the horizon, the binary outcome may not be fully observed; simply coding such a row as no event is generally incorrect. The analysis then needs an appropriate approach to incomplete outcome ascertainment, such as a justified censoring adjustment.

Building the Pooled Regression

A basic pooled logistic regression models the probability of the horizon event using landmark time and the covariates recorded at that time. With a separate intercept for each landmark, a common-slope model is

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\), \(\alpha_k\) is the intercept for landmark \(t_k\), and \(\beta\) is a vector of covariate effects constrained to be the same across landmarks. The landmark-specific intercepts allow the baseline event odds to differ by time. The common-slope assumption is substantive: a one-unit covariate contrast has the same conditional log-odds association at each landmark.

If effects may differ by landmark, include interactions between landmark indicators and covariates. Equivalently, write

This more flexible model estimates separate covariate slopes at each landmark. A pooled regression is therefore not automatically a model with identical effects over time: the chosen parameterization determines which features are shared. With enough data, interactions can be assessed and retained when needed. Sparse eligible risk sets, however, may make fully landmark-specific effects unstable.

Dependence Within a Patient

Rows from different patients can often be treated as independent, conditional on the model and sampling design. Rows from the same patient generally cannot. They may share unmeasured characteristics, and a patient’s later eligibility and covariates are related to that patient’s earlier history. Consequently, the ordinary logistic-regression standard error that treats every stacked row as independent can be too small or otherwise misleading.

A common approach is to fit the pooled regression while using a patient-clustered sandwich variance estimator. Let \(z_{ik}\) be the design vector for a row, including its landmark terms and covariates, and let \(\hat p_{ik}\) be its fitted probability. In an unweighted logistic model, the row score at the fitted coefficients is \(u_{ik}=z_{ik}(Y_{ik}-\hat p_{ik})\). Sum these scores within patient to obtain \(U_i=\sum_{k:R_{ik}=1}u_{ik}\). The sandwich variance has the form

where \(\hat\theta\) collects the fitted regression coefficients. The middle term uses each patient’s total score, so it allows arbitrary dependence among that patient’s rows under the working model. It does not require rows within a patient to be independent. The usual large-sample justification relies on having a sufficiently large number of independent patient clusters and a correctly specified mean model; a small number of clusters can make the sandwich approximation unreliable.

A short justification

The pooled logistic fit solves the estimating equation \(\sum_i U_i(\theta)=0\). If patients are independent, the variability of this total score is estimated by summing the outer products of the patient-level scores, \(\sum_i U_iU_i^\mathsf{T}\). Linearizing the estimating equation around the true parameter gives an approximate coefficient error equal to the inverse sensitivity matrix times the total score. Taking the variance of that approximation yields the sandwich form above. If instead each row’s score were treated as a separate independent contribution, cross-products between rows from the same patient would be omitted, which is why row-wise independence-based uncertainty estimates need not be valid.

Prediction at a Specified Landmark

To report an absolute risk, select a landmark \(t_k\), the corresponding covariate values \(x\), and the horizon \(h\), then calculate the linear predictor for that landmark and transform it with the inverse-logit function. In a model with landmark-specific slopes, the prediction is

This is the estimated probability of the defined event within the horizon among patients eligible at \(t_k\) with covariates \(x\), under the fitted model. It is not the probability of being event-free from the original study baseline. For a common-slope model, replace \(\hat\beta_k\) with \(\hat\beta\). Clustered standard errors address uncertainty in estimation; they do not change the fitted probability formula.

Assumptions, Diagnostics, and Alternatives

Pooling is useful only when the data structure and model match the prediction target. Check that eligibility is applied at each landmark; covariates use information available by that landmark; the horizon and event definition are clear; and censoring has not been mistaken for absence of an event. The logistic mean model should also be adequate. Assess calibration by landmark, compare observed and predicted event frequencies across meaningful risk groups, and examine whether residual patterns or estimates suggest that a common covariate slope is untenable. Sparse events at a particular landmark, extreme fitted probabilities, and unstable interaction estimates are warning signs.

The principal alternatives have different trade-offs. Separate regressions at each landmark allow all coefficients to vary, but may be inefficient and difficult to compare when some risk sets are small. A pooled model with common slopes is more parsimonious, but imposes the same covariate associations across times. Landmark-by-covariate interactions offer a middle ground by allowing selected effects to vary. A model with patient-specific random effects is another possible way to represent dependence, but its assumptions and interpretation differ from a cluster-robust marginal analysis; it should not be treated as a routine substitute without considering the target prediction and the model’s assumptions.

Cluster-robust standard errors do not repair a misspecified risk model, incorrect eligibility rules, informative missingness, or unaddressed censoring. Nor do they make predictions at different landmarks interchangeable. They adjust uncertainty for within-patient dependence under appropriate large-sample conditions. A pooled analysis also remains limited by the information in each landmark risk set: a large total number of stacked rows can conceal few independent patients or few events at a particular time.