Why Visit Timing Matters at a Landmark
A landmark analysis predicts future event risk among people who have survived to a chosen time, using information available by that time. But a biomarker is not necessarily measured on a common schedule for everyone. Patients with worsening symptoms may be seen more often, while those doing well may have no recent visit. Consequently, a biomarker recorded near the landmark can be available more often for people whose future risk differs from that of people without a measurement.
This is a problem of selective availability, distinct from measurement error. Even if every recorded biomarker value were measured perfectly, the subset with a usable value could differ systematically from the landmark-eligible population. A landmark analysis handles the conditioning on survival to the landmark; that conditioning alone does not remove selection caused by who is seen and measured.
The definition is conditional on eligibility: the target population for a landmark prediction is not everyone enrolled at baseline, but those known to be event-free and uncensored at \(t_L\). A usable-measurement subset is smaller still. A model fit only to that subset naturally describes risk among eligible people with an available measurement, unless additional assumptions and methods justify inference for a broader target.
Selection Changes the Composition of the Measured Group
Suppose \(U\) denotes a prognostic characteristic, such as a high-risk disease state. It may be measured, partially observed, or latent. Among eligible subjects, Bayes' rule gives the proportion with that characteristic among those with a measurement:
Both the availability probabilities and the underlying proportions are conditional on both surviving past the landmark and remaining uncensored through it. Conditioning only on \(T>t_L\) would not in general describe the eligible cohort if censoring before \(t_L\) were associated with \(U\) or visit availability.
If people with higher risk are more likely to have a recent visit, the measured subset will tend to contain more high-risk people. A prediction model can then be accurate for that selected subset while its risk estimates or predictor relationships do not transport to all landmark-eligible people. In the other direction, if lower-risk people receive more routine monitoring, the measured subset could be lower-risk. The direction is determined by the visit process, not by landmarking itself.
Worked Example: Risk Composition Among Patients With a Visit
Consider a hypothetical landmark-eligible cohort of 500 people: 200 have a high-risk state \(U=1\), and 300 have a low-risk state \(U=0\). A usable measurement is available for 80% of the high-risk group and 30% of the low-risk group. Suppose the probability of an event by a fixed prediction horizon is 0.40 for high-risk people and 0.10 for low-risk people, for both \(V=1\) and \(V=0\) within each \(U\) group. This last condition allows the group-specific risks to be used for the measured subset as well as the full cohort.
The expected measured counts are \(200(0.80)=160\) high-risk people and \(300(0.30)=90\) low-risk people, or 250 measured people altogether. The high-risk fraction among those measured is \(160/250=0.640\), compared with \(200/500=0.400\) among all eligible people.
The expected event count among the measured people is \(160(0.40)+90(0.10)=64+9=73\). Thus their expected event risk is \(73/250=0.292\), or 29.2%. For all 500 eligible people, the expected event count is \(200(0.40)+300(0.10)=80+30=110\), giving risk \(110/500=0.220\), or 22.0%. In this hypothetical example, selective availability raises the measured subset's event risk by 7.20 percentage points. That difference follows from the specified within-group risk assumption; the availability rates by themselves would not determine the event count.
Availability Is Not the Same as Measurement Error
The earlier material on Measurement Error in Last-Observed Biomarkers concerns how a recorded value may differ from the underlying value it is intended to represent. Informative visit timing raises a separate question: which people have a value recorded at all, and when? Both problems can occur together. A biomarker can be measured precisely at each visit, but the most recent value may be much older for some patients, and patients with frequent visits may be disproportionately represented in analyses of last-observed values.
For example, if symptoms trigger a visit, a recent measurement may be especially likely during a period of deterioration. In that setting, the visit process carries information about health status. The association need not mean that a future event caused an earlier visit; worsening health or another prognostic factor may influence both the visit schedule and future outcome. Merely adding the number of visits as a predictor also does not guarantee correction: it may help describe risk, but it does not necessarily recover the biomarker distribution or prediction target for people without measurements.
Worked Example: A Measured Mean Can Differ From the Eligible-Cohort Mean
Suppose a hypothetical eligible cohort has 100 people. At the landmark, 50 have a latent biomarker value of 10 and 50 have a value of 4. Assume these values are measured without error when a visit occurs. In a particular realized dataset, a usable measurement is available for 40 of the 50 people with value 10 and for 10 of the 50 people with value 4.
The mean among all eligible people is \(\{50(10)+50(4)\}/100=700/100=7.00\) units. The measured subset contains 50 people, so its mean is \(\{40(10)+10(4)\}/50=(400+40)/50=8.80\) units. The 1.80-unit difference arises because the measured subset has 40/50, or 80%, with the higher value, whereas that group is 50/100, or 50%, of the eligible cohort.
This calculation demonstrates selection, not a biased assay: each observed value was stipulated to be exact. It also does not establish that any particular real-world visit process will raise the measured mean. If visit availability favored people with lower values, the direction could reverse.
When Weighting Can Address Availability
One possible adjustment is inverse-probability weighting. Let \(H_i\) denote information available for every eligible person that is used to model measurement availability, and let \(\pi(H_i)=P(V_i=1\mid H_i,E_i=1)\). To use measured outcomes to represent the eligible population, the key condition is that, within \(H_i\), availability is independent of the relevant outcome or biomarker quantity \(Y_i\). There must also be a positive probability of observation for every relevant history: \(\pi(H_i)>0\). If availability still depends on unmeasured prognosis after conditioning on \(H_i\), weighting by the modeled probabilities need not remove the selection.
The basic justification is that, under these conditions, weighting an observed value by \(1/\pi(H_i)\) corrects for its chance of being observed. Conditional on \(H_i\), independence gives \(E(V_iY_i\mid H_i,E_i=1)=E(V_i\mid H_i,E_i=1)E(Y_i\mid H_i,E_i=1)\). Since \(E(V_i\mid H_i,E_i=1)=\pi(H_i)\), it follows that:
Averaging over eligible subjects' histories then targets the eligible-population mean, provided the assumptions hold and the availability probabilities are correctly specified. In practice, very small \(\pi(H_i)\) values produce large weights and unstable estimates. Weight truncation can reduce instability but changes the estimator and may introduce bias; it is not a substitute for positivity or a credible availability model.
Worked Example: Weighting a Measured-Subset Risk
Return to the hypothetical cohort of 200 high-risk and 300 low-risk eligible people. Suppose the high-risk measurement probability is 0.80 and the low-risk probability is 0.30, and the event risks are 0.40 and 0.10 in each risk group regardless of measurement availability. The expected measured counts and event counts are 160 and 64 for the high-risk group, and 90 and 9 for the low-risk group.
Give each measured high-risk person weight \(1/0.80=1.25\), and each measured low-risk person weight \(1/0.30=3.333\). The weighted event total is \(64(1.25)+9(3.333)=80+30=110\). The weighted population total is \(160(1.25)+90(3.333)=200+300=500\). The weighted risk is therefore \(110/500=0.220\), matching the stipulated risk in the full eligible cohort.
This result is justified here because the availability probabilities are known, positive, and constant within the two \(U\) groups, and event risk is assumed equal for observed and unobserved people within each group. If \(U\) were unmeasured, or if visit availability still depended on outcome risk within the modeled groups, this calculation would not establish that weighting recovers the full-cohort risk.
Diagnostics, Alternatives, and Limits
The first diagnostic is descriptive: compare visit counts, time since last visit, and the proportion with a usable measurement across available prognostic groups. If measured history is recorded, examine whether visit frequency varies with prior biomarker values or clinical status. These comparisons can reveal patterns, but they cannot prove that availability is independent of unobserved prognosis. An apparently balanced measured subset can still be selectively observed for unrecorded reasons.
For a weighting approach, inspect the estimated availability probabilities and resulting weights. Probabilities near zero, very large weights, or strong dependence of availability on histories omitted from the model warn that the estimate may be unstable or biased. Also check whether the availability model includes the relevant observed predictors and timing information. A model for the probability of any measurement may be inadequate if the age of the last measurement is important to the prediction.
Different strategies answer different problems:
| Approach | What it can address | Main limitation |
|---|---|---|
| Scheduled measurement | Reduces variation in availability and timing by using a common measurement schedule. | May be impractical; missed scheduled visits can remain informative. |
| Inverse-probability weighting | Can reweight observed records toward the eligible population when availability is explained by measured history. | Requires positivity, a credible availability model, and conditional independence given modeled history. |
| Imputation | Can fill in missing covariates under an appropriate missing-data model. | Does not automatically solve informative missingness; results depend on the imputation assumptions. |
| Joint modeling of outcome and visit processes | Can represent dependence between visits, longitudinal measurements, and outcome through a specified model. | Requires additional modeling assumptions and may be sensitive to misspecification. |
Landmarking still has an important role: it defines prediction among people who have reached \(t_L\), using information available by then. But it does not make biomarker availability random, undo selective monitoring, or recover unmeasured values without assumptions. If the intended prediction target is all landmark-eligible people, an analysis restricted to \(V_i=1\) may not directly answer that target. If the intended target is specifically people with an observed measurement, that selected population should be stated clearly.
Common Pitfall
- Wrong reasoning: “The analysis starts at the landmark, so any selection problem has been handled.”
- Correction: Landmark eligibility handles conditioning on survival and observability through \(t_L\). Whether a biomarker is observed within the eligible group is a separate selection process, which may still depend on prognosis.
- Wrong reasoning: “The measured subset has higher risk, so the biomarker measurement caused the increase.”
- Correction: A difference between measured and eligible populations can arise because prognostic factors affect both visits and outcomes. The contrast is not, by itself, a causal effect of measurement or of attending a visit.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Among eligible subjects, 30 of 100 high-risk people and 20 of 100 low-risk people have a measurement. What fraction of measured people are high risk? What fraction of the eligible cohort is high risk?
- Why must the eligibility condition in an availability calculation include both \(T>t_L\) and \(C>t_L\), when eligibility is defined using both?
- What two main conditions allow inverse-probability weighting to correct for availability using observed history?
- A biomarker assay is exact, but people with worsening symptoms are more likely to be measured. Is selective availability still possible? Explain.
Solutions. (1) The measured group has \(30+20=50\) people, so \(30/50=0.600\), or 60.0%, are high risk. In the eligible cohort, \(100/200=0.500\), or 50.0%, are high risk. (2) If censoring before \(t_L\) is related to risk or availability, conditioning only on \(T>t_L\) does not describe the population satisfying both eligibility requirements. (3) Availability must be conditionally independent of the target outcome or biomarker given the modeled history, and each relevant history must have positive availability probability. The availability model must also be credible. (4) Yes. Exact measurement addresses error in observed values, not whether values are selectively observed. Symptom-related visits can make the measured subset differ from the eligible cohort.