Landmark Analysis

Landmark Analysis in Survival Studies and Economic Models

Landmark analysis answers a question that ordinary survival analysis gets wrong, namely whether something that happens after follow-up begins, such as responding to treatment, predicts what happens next. Grouping patients from the start by a status they only reach later hands one group a stretch of follow-up during which it cannot fail, and the comparison then flatters that group. Health economists meet the problem when survival or response data are split to populate a model, for example survival after response, after progression or after a switch of treatment, and a biased split carries straight into the life years and QALYs a model produces. The method starts the clock at a fixed time and compares only patients still at risk then, which removes that bias at the cost of some data and generalisability.

Why grouping by a later event biases survival comparisons

Anderson, Cain and Gelber described the problem in 1983 for cancer trials that compared survival of responders and non-responders. A patient counted as a responder must have lived long enough to respond, so the time from entry to response is immortal time for that group, while patients who die early are counted only among non-responders. Giobbie-Hurder, Gelber and Regan call the same error guarantee-time bias and show that it can make an ineffective factor look strongly protective.

The bias does not need a real difference in prognosis. Even if response changed nothing, responders would appear to live longer because the classification itself selects the survivors. Mantel and Byar made the same point for heart transplant data in 1974, where patients who received a transplant had by definition survived the wait for a donor heart.

How the landmark method works

The analysis fixes a landmark time before looking at the outcome data, chosen so that the classifying status is known for most patients by then. Patients who have had the event or been censored before the landmark are excluded, the rest are grouped by their status at the landmark, and follow-up restarts at the landmark for everyone. A Kaplan-Meier estimator or a Cox proportional hazards model is then applied to time after the landmark.

$$S(t \mid T > s) = \frac{S(t)}{S(s)}, \qquad t \geq s$$

where $s$ is the landmark time, $T$ is the time to the event, $S(t)$ is the survival function from the start of follow-up and $S(t \mid T > s)$ is survival to time $t$ for patients still event-free at the landmark. A Cox model fitted from the landmark uses only information available at that time.

$$h(t \mid x(s), T > s) = h_{0s}(t)\exp\left(\beta_s\, x(s)\right), \qquad s \leq t \leq s + w$$

where $x(s)$ is the patient's status at the landmark, such as responder or non-responder, $h_{0s}(t)$ is a baseline hazard for patients at risk at the landmark, $\beta_s$ is the log hazard ratio for that status, and $w$ is an optional horizon after which follow-up is cut off. Van Houwelingen extended this to a series of landmarks combined in one model, known as a landmark supermodel, for dynamic prediction of survival as a patient's history unfolds.

Choosing the landmark time

The landmark time is the main analytic choice and changes the answer. Dafni sets out the trade-off. An early landmark keeps more patients and events, but many patients have not yet reached the classifying status, so some later responders are counted as non-responders. A late landmark classifies status more completely, but excludes more patients and describes a smaller, more selected group.

Good practice is to fix the landmark from clinical reasoning before analysis, such as the time by which most responses occur, to report how many patients were excluded at the landmark, and to repeat the analysis at other landmarks as a sensitivity analysis. Results that change markedly with the landmark suggest that the classifying status is still forming or that its effect varies over time.

Worked example with tumour response at six months

The figures in this example are illustrative. A cohort of 200 patients starts treatment, and tumour response is assessed at six months. By six months 30 patients have died, all before response could be assessed. Of the 170 still alive, 70 are responders and 100 are non-responders. Two years after the six-month assessment, 49 responders and 50 non-responders are alive.

A naive analysis groups patients from the start, so the 30 early deaths fall among non-responders.

$$\text{Responders } \frac{49}{70} = 0.70, \qquad \text{non-responders } \frac{50}{100 + 30} = 0.385$$

where the proportions are survival at 30 months from the start of treatment. A landmark analysis at six months uses only the 170 patients alive at the landmark.

$$\text{Responders } \frac{49}{70} = 0.70, \qquad \text{non-responders } \frac{50}{100} = 0.50$$

where the proportions are survival over the two years after the landmark. The naive comparison shows a gap of 31.5 percentage points and the landmark comparison a gap of 20 points, because the naive analysis charged the 30 early deaths to non-responders while responders were guaranteed six months of survival. For the whole cohort, the conditional survival formula gives $S(30 \mid T > 6) = 0.495 / 0.85 = 0.582$, which equals the 99 survivors out of the 170 patients alive at six months. Even the landmark gap describes association, not the effect of causing a response, because responders may differ from non-responders in prognosis.

Use in economic evaluation and HTA

Decision models often need survival split by a status reached during follow-up. Response-based state transition models, models that separate survival before and after progression, and analyses of survival after a treatment switch all face the same risk of crediting immortal time to one group. A landmark analysis gives estimates of survival from a common starting point for patients in each state, which a model can apply from the cycle at which the status is assigned. A partitioned survival model, which works from overall and progression-free survival curves for the whole arm, avoids conditioning on response altogether.

Landmark results describe only patients alive at the landmark. A model that applies them must still account for patients who died or progressed before it, usually from the unconditional curves up to the landmark, or it will overstate survival in the cohort as a whole. Conditional survival estimates from a landmark are also useful for checking the plausibility of extrapolated curves, since survival beyond the end of a trial for patients who have already lived a year can be compared with registry data on the same conditional basis.

Landmark analysis and other approaches

A Cox model with the status treated as a time-dependent covariate is the main alternative. It keeps every patient, moves each from one group to the other at the time the status changes, and uses all events, so it is usually more efficient, but its estimate is harder to turn into a survival curve for a model state. Landmark analysis gives a simple conditional curve that is easy to explain and to use in a model, and with Kaplan-Meier curves it needs no model of how the hazards of the groups relate after the landmark. When death before the landmark is itself an outcome of interest, methods for competing risks or multi-state models describe the whole pathway rather than one conditional slice of it.

What a landmark analysis cannot show

Landmark analysis removes immortal time bias but does not remove confounding. Responders may have had less advanced disease, so a difference after the landmark can reflect prognosis rather than any effect of responding. The method discards patients and events before the landmark, which lowers precision, and patients whose status changes after the landmark remain in the group they were in at the landmark, which can dilute the comparison. Choosing the landmark after seeing the data invites selective reporting. A frequent error in models is to apply landmark survival from time zero, which ignores the deaths that occurred before the landmark and overstates life expectancy.

Sources

• Anderson JR, Cain KC, Gelber RD. Analysis of survival by tumor response. Journal of Clinical Oncology. 1983;1(11):710-719.
• Mantel N, Byar DP. Evaluation of response-time data involving transient states: an illustration using heart-transplant data. Journal of the American Statistical Association. 1974;69(345):81-86.
• Dafni U. Landmark analysis at the 25-year landmark point. Circulation: Cardiovascular Quality and Outcomes. 2011;4(3):363-371.
• Giobbie-Hurder A, Gelber RD, Regan MM. Challenges of guarantee-time bias. Journal of Clinical Oncology. 2013;31(23):2963-2969.
• van Houwelingen HC. Dynamic prediction by landmarking in event history analysis. Scandinavian Journal of Statistics. 2007;34(1):70-85.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.