Why an Unavailable Landmark Value Needs a Rule
At a landmark time, patients may have measurements taken on different schedules. One patient’s last biomarker result might be from yesterday; another’s might be several months old. A third patient may have no usable result at all. A landmark model needs a reproducible way to turn these records into predictors, and the rule must use only information available at the landmark.
The earlier material on building a landmark dataset establishes the risk set and prediction time. Here the focus is the additional decision made when a biomarker is unavailable under the chosen measurement rule. This decision is not merely a data-cleaning detail: excluding patients, retaining them with a missingness indicator, and imputing values can produce different analysis samples and different populations to which model results apply.
“Usable” must also be defined in advance. For example, a record may be unusable because the assay failed or the value is outside a prespecified range. If two valid measurements share the latest timestamp, a protocol might choose one according to a fixed assay-priority rule or use their mean. Do not decide these details separately for individual patients after looking at their outcomes.
A look-back window prevents an arbitrarily old result from being treated as current, but it does not make measurements equally recent. The elapsed time from the selected visit to the landmark can be recorded as a separate predictor when that is scientifically appropriate. The window width, tie rule, and treatment of invalid results should be specified before evaluating predictions, rather than selected to give the most favorable apparent performance.
Define Availability Before Choosing a Missing-Data Strategy
Let \(A_i=1\) when patient \(i\) has a usable measurement under the look-back rule, and \(A_i=0\) otherwise. Then \(Y_i^*\) denotes the selected landmark value when it is available. This separates two facts: whether a value was obtained, and what value was obtained. A patient with \(A_i=0\) may have had no visit in the window, a visit without a recorded assay, or an unusable assay; these reasons can be recorded separately if they matter clinically.
The landmark risk set is defined by the landmark eligibility criteria, such as being alive and event-free at \(t_L\). Missing a biomarker does not by itself mean that a patient fails those criteria. However, a complete-case analysis additionally requires the predictor to be available, so its fitted model uses fewer patients. That distinction matters when describing the population represented by the analysis.
Three broad approaches are common. Complete-case analysis omits patients without the biomarker. A missingness-indicator approach retains them by supplying a fixed value for the missing biomarker and adding an indicator for its absence. Imputation replaces unavailable values with estimates, potentially using multiple imputed data sets to represent uncertainty. None is automatically correct in every application.
A missingness mechanism helps describe the concern. Missing completely at random (MCAR) means availability is unrelated to observed or unobserved data. Missing at random (MAR) means that, conditional on observed information, availability does not depend on the unobserved biomarker value. Missing not at random (MNAR) means that availability still depends on unobserved information after conditioning on observed data. These are assumptions about the data-generating process, not properties established just by adding an indicator or running an imputation procedure.
Worked Example: Apply a Prespecified Look-Back Window
Suppose the landmark is day 100 and the prespecified look-back window is 30 days, so only measurements from day 70 through day 100 are eligible. Four hypothetical patients have the following visit records.
| Patient | Recorded values (day, biomarker units) | Selected landmark value | Days before landmark |
|---|---|---|---|
| A | (65, 8.1), (92, 7.4) | 7.4 | 8 |
| B | (74, 6.8) | 6.8 | 26 |
| C | (60, 5.9) | Unavailable | — |
| D | (88, invalid), (96, 7.9) | 7.9 | 4 |
For patient A, day 92 lies in the window and is more recent than day 65, so the selected value is 7.4. Patient B’s only measurement is 100 − 74 = 26 days before the landmark and is therefore included. Patient C’s only measurement is 100 − 60 = 40 days before the landmark, outside the window, so \(A_C=0\). Patient D’s day-88 record is unusable, but the valid day-96 value is in the window; the selected value is 7.9.
Three of these four patients have an available biomarker under the rule. If all four otherwise belong to the landmark risk set, the look-back rule does not remove patient C from that risk set; it makes C’s biomarker predictor unavailable. The next analysis choice determines whether C is omitted or retained by another method.
What the Main Strategies Do
Complete-case analysis fits the model only to patients whose required predictors are available. It is straightforward and avoids inventing biomarker values, but it reduces the sample and may change the represented population. It is not generally justified merely because the remaining sample is large. Under some conditions, such as MCAR, complete-case estimates may avoid bias for certain model parameters, but they can still be less precise. Under other missingness patterns, selection into the complete cases can distort estimated relationships.
Missingness indicators retain patients by filling unavailable values with a chosen constant \(c\) and adding \(M_i=1-A_i\). For a biomarker term, the model can include both \(Z_i\) and \(M_i\), where \(Z_i=Y_i^*\) if available and \(Z_i=c\) otherwise. The indicator allows the fitted model to distinguish a filled value from an observed value equal to \(c\). The constant is a computational convention, not an estimate of the missing patient’s true biomarker.
The indicator can be useful when absence itself carries predictive information, for example when testing was ordered selectively. It is not a universal correction for missing-data bias, and it does not recover the patient’s unobserved biomarker. Its predictions for patients with missing values describe the missingness pattern represented in the development data. If that pattern changes, performance may change too.
Imputation supplies plausible values for unavailable biomarkers using a model based on available information. A simple mean imputation replaces each missing value with the observed mean, but treats the replacement as known and can understate uncertainty. Multiple imputation draws several plausible values from an imputation model, fits the analysis across the resulting data sets, and combines results to reflect imputation uncertainty. For prediction, the imputation procedure must be usable with information available at prediction time; it cannot rely on future measurements or future outcomes for a new patient.
Worked Example: Build a Missingness-Indicator Predictor
Consider six hypothetical patients in the landmark risk set. Their outcomes are shown only to illustrate how a data set is assembled; they are not used to decide biomarker availability. The landmark biomarker values are 4, 5, 6, 7, missing, and missing. Let \(c=0\) be the prespecified fill value.
The constructed predictors are \(Z_i=Y_i^*\) when observed and \(Z_i=0\) otherwise, with \(M_i=1\) for missing and \(M_i=0\) for observed. The six \((Z_i,M_i)\) pairs are \((4,0),(5,0),(6,0),(7,0),(0,1),(0,1)\). Thus all six patients remain in the analysis data, while the indicator distinguishes the last two from patients with an observed value of zero.
For a fixed-horizon binary outcome, one possible model form is
Here \(\beta_1\) describes the modeled change in log odds per one biomarker unit among records with \(M_i=0\), and \(\beta_2\) allows the missing group to differ from the observed group through the model’s specified terms. The equation does not assert that missingness is harmless or that missing patients truly have biomarker value zero. Those interpretations would be incorrect.
The risk set contains six patients and the indicator-model data retain six; a complete-case model would have four. The indicator approach therefore avoids losing these two records, but its estimated relationship depends on how missingness and outcome relate in the development sample. The coefficient estimates cannot be calculated from these values alone: they would also require the six outcomes and the fitted-model computation.
Worked Example: Compare Complete Cases and Mean Imputation
Suppose five hypothetical patients in the landmark risk set have biomarker values 2, 4, 6, missing, and missing. The observed mean is
A complete-case analysis has \(n=3\) patients for a model requiring this biomarker. If simple mean imputation is used instead, the constructed values are 2, 4, 6, 4, and 4, and the analysis data contain \(n=5\) patients. Their mean is
The equality of means is built into mean imputation; it does not show that the two missing values truly equal 4. The imputed data also contain less variation than the unobserved complete data could have had. For a model-based analysis, treating those two inserted values as observed can produce uncertainty estimates that are too small. Multiple imputation can represent uncertainty about the missing values, provided its model and assumptions are defensible.
Both approaches retain the same five patients in the original landmark risk set. But the complete-case estimate is based on the three observed cases, whereas the imputed analysis uses all five under an explicit imputation assumption. These analyses need not estimate the same quantity under every missingness process, so report the rule and the resulting sample counts rather than presenting the two as interchangeable.
Justification, Diagnostics, and Comparisons
The effect of complete-case selection can be seen directly. Let \(R_i=1\) mean that patient \(i\)'s biomarker is observed under the chosen rule. The complete-case analysis estimates relationships using the conditional distribution among \(R_i=1\), rather than automatically using the distribution among the whole landmark risk set. If availability is associated with the biomarker or outcome, then in general
The two conditional risks can differ because the selected patients are not representative of the risk set. This is why the eligible sample is part of the statistical interpretation, not just a bookkeeping quantity. Conversely, retaining everyone with an indicator changes the predictor representation, but does not by itself make the missingness process ignorable.
Before fitting a model, tabulate the number in the landmark risk set, the number with a usable value, the number missing by reason, and the number retained by the chosen strategy. Compare observed patient characteristics and outcomes across availability groups as a diagnostic. Such comparisons can reveal patterns, but cannot prove MCAR, MAR, or the absence of MNAR dependence. Also check the distribution of measurement ages: many values near the edge of a wide window may indicate that the nominal landmark predictor is often stale.
The main choices have different strengths and limitations:
| Approach | What happens to unavailable values? | Main concern |
|---|---|---|
| Complete case | Patients lacking required predictors are omitted | Smaller, potentially selected analysis sample |
| Missingness indicator | Use a fixed fill value plus an absence indicator | Does not reconstruct the biomarker or guarantee unbiased estimates |
| Imputation | Replace missing values using an imputation model | Results depend on the imputation model and its assumptions |
For every approach, use the same prespecified availability rule when constructing development and evaluation data. In any resampling or validation procedure, estimate preprocessing quantities such as an imputation mean using the training portion only, then apply that rule to held-out data. Otherwise, information from the evaluation data can leak into model construction and make performance appear better than it would be for new patients.
A look-back window also has limits: a narrow window can leave many values unavailable, while a wide window admits older values that may be less representative of the landmark state. These are competing design considerations, not a reason to select the window after inspecting evaluation outcomes. Report the window, the distribution of measurement ages, the handling of ties and unusable results, the missingness strategy, and the resulting sample counts.
Common Pitfall
A tempting argument is: “We filled every missing biomarker with the observed mean, so nobody was excluded and missingness is no longer a problem.” The correction is that imputation changes the data representation; it does not reveal the unobserved values or establish that the imputation model is appropriate. Similarly, a missingness indicator preserves records but does not automatically remove selection bias. State the assumptions and assess whether the chosen procedure is plausible for the intended prediction setting.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- The landmark is day 50 and the look-back window is 14 days. A patient has valid measurements on days 34 and 41. Which value is selected, and how old is it?
- If a patient is otherwise eligible for the landmark risk set but has no usable measurement in the window, must that patient be removed from the risk set? Explain.
- In a missingness-indicator approach, what does the fill value mean, and what does the indicator add?
- Why can mean imputation understate uncertainty if inserted values are treated as observed?
- Why must an imputation mean for validation be estimated from training data rather than the entire data set?
Solutions. (1) Day 41 is the latest value within the window; it is \(50-41=9\) days old. (2) No. The patient remains in the risk set if the landmark eligibility criteria are met; the analysis strategy determines whether the patient is retained for modeling. (3) The fill value is a computational placeholder, not a claim about the true biomarker; the indicator flags that the value was unavailable. (4) A fixed imputed value is treated as known even though the missing value is uncertain, so variation and uncertainty due to missingness are not represented. (5) Using held-out data to calculate the mean leaks information from evaluation records into preprocessing and can make validation performance optimistic.