Why the Representation Matters
At a landmark time, a patient may have a record of several biomarker measurements. A prediction dataset needs a specified way to turn that history into covariate information. The choice changes what aspects of the history are retained: the most recent observed level, the level at baseline, the net difference between those levels, or a summary of measurements in a defined period. These are alternative representations of observed history, not interchangeable labels for the same quantity.
The representation should be chosen to match the question and the measurement process. A latest value can describe the current recorded level; a baseline value can retain starting status; a change can describe net movement between two defined times; and a window summary can reduce reliance on one visit. Each also discards information. In particular, none of these summaries, by itself, reconstructs the full trajectory.
Four Common Representations
Write the observed measurement times for patient \(i\) as \(t_{ij}\), with corresponding values \(Y_{ij}=Y_i(t_{ij})\). Only records available by \(t_L\) can contribute. A measurement taken after the landmark is future information relative to the prediction and must not be used.
- Latest observed value: select the value at the greatest observed time not exceeding \(t_L\). Its measurement time is \(t_{i,\mathrm{last}}=\max\{t_{ij}:t_{ij}\leq t_L\}\), and the covariate is \(Y_i(t_{i,\mathrm{last}})\). The value should be considered together with how long before the landmark it was measured.
- Baseline value: select a value according to a prespecified baseline definition, such as the measurement at a scheduled study baseline. Baseline must be defined consistently; it need not be the earliest record in every patient’s file if measurement schedules differ.
- Change from baseline: subtract the defined baseline value from the latest value, giving \(Y_i(t_{i,\mathrm{last}})-Y_i(t_{i,\mathrm{base}})\). This is a difference in biomarker units. It is a net difference between two observations, not a description of every intervening measurement.
- Summary over a look-back window: specify a window \([t_L-w,t_L]\) of length \(w\), then summarize measurements in that window. A common choice is the arithmetic mean of the observed values in the window. Other summaries are possible, but the choice and the window must be stated explicitly.
These definitions require a decision about patients without a required measurement. For example, a latest-value covariate is unavailable if there is no recorded value by the landmark; a change is unavailable if either its defined baseline or latest value is missing. A window mean is undefined if the window contains no measurements. Such cases should be handled according to a prespecified missing-data strategy, not by silently substituting a value from after the landmark.
Worked Example: Latest Value, Baseline, and Change
Suppose a hypothetical patient has biomarker measurements of 4.2 mg/L at month 0, 5.0 mg/L at month 6, and 6.1 mg/L at month 10. The landmark is month 12, and month 0 is the defined baseline.
The latest observed measurement by month 12 is the month-10 value, 6.1 mg/L. It is not a month-12 value: no measurement was recorded at that time.
The baseline representation is 4.2 mg/L. The change from baseline to the latest observed measurement is
Thus the three possible covariates are 6.1 mg/L, 4.2 mg/L, and 1.9 mg/L, respectively. The latest value was recorded two months before the landmark. Reporting that timing is useful because a value observed earlier may be less representative of the biomarker level at the landmark than a value recorded more recently.
Summaries Depend on the Measurement Schedule
A window summary can use information from more than one visit while excluding older measurements outside the chosen period. For the arithmetic mean, each observed value receives equal weight, regardless of the time between measurements. Consequently, this mean is an average of recorded values, not necessarily an average biomarker level across calendar time.
Worked Example: Mean in a Prespecified Look-Back Window
Suppose the landmark is month 12 and the prespecified look-back window is the six months from month 6 through month 12. A hypothetical patient has values of 5.0 mg/L at month 6, 5.6 mg/L at month 9, and 6.2 mg/L at month 11. A month-3 value of 4.7 mg/L is outside the window.
The window mean uses the three in-window values:
The month-3 measurement is excluded because it does not fall in the specified window. This calculation gives equal weight to the month-6, month-9, and month-11 observations. The intervals between those visits are not equal, so the result should not be described as a time-weighted average over months 6 through 12. If the clinical question calls for a time-weighted summary, that is a different representation and requires its own explicit definition and assumptions.
Window choice and summary choice are linked. A short window emphasizes recent measurements but may contain few or no values for some patients. A longer window is more likely to include observations, but it can mix older and newer biomarker levels. With irregular visits, patients with more frequent measurements contribute more recorded values to an unweighted mean. That may make the mean reflect visit frequency as well as biomarker history.
What Each Representation Retains
Consider two hypothetical patients measured at months 0, 6, and 12. Patient A has values 4.0, 5.0, and 6.0 mg/L; patient B has values 5.5, 5.8, and 6.0 mg/L. At a month-12 landmark, both have the same latest value, but their earlier histories differ.
Worked Example: Same Latest Value, Different Histories
For patient A, the baseline is 4.0 mg/L and the latest value is 6.0 mg/L, so the change is \(6.0-4.0=2.0\) mg/L. For patient B, the baseline is 5.5 mg/L and the latest value is 6.0 mg/L, so the change is \(6.0-5.5=0.5\) mg/L.
For a window from month 0 through month 12, the unweighted means are
The latest-value representation gives both patients 6.0 mg/L. The baseline, change, and window-mean representations distinguish aspects of their earlier observed levels. None establishes why the histories differ, and none alone determines a patient’s future outcome.
This example illustrates why choosing a representation is a scientific decision rather than a formatting decision. If current recorded level is central, the latest value may be a natural candidate. If initial status is important, baseline may be useful. If net movement from a defined starting point matters, change may be informative. If a single visit is thought to be unusually noisy, a window summary may be preferable, provided the window contains an adequate and interpretable set of observations.
A Redundancy to Watch For
Latest value, baseline, and change are algebraically linked when they use the same baseline and latest measurement. Let \(L\) denote the latest value, \(B\) the baseline value, and \(D=L-B\) the change. Including all three as separate covariates in a model with an intercept creates exact linear dependence: \(L-B-D=0\). The model cannot estimate three unrestricted coefficients for these three quantities as though they varied independently.
This algebra does not imply that every pair of representations is redundant in every analysis. It does show that change is defined using its two component values and that interpretation depends on the chosen parameterization. It also helps explain why a model using latest value and change answers a different question from one using latest and baseline, even though the covariate information is algebraically equivalent when both are included consistently.
Conditions, Diagnostics, and Alternatives
The covariate definitions above are descriptive rules, but their use still depends on sound data construction. Check that recorded times and values are correctly aligned to each patient, that the baseline rule is applied consistently, and that every selected measurement was available by the landmark. For a window summary, verify the window boundaries, the number of included observations, and how a patient with no in-window values is handled. Also examine the time from the latest measurement to the landmark and the distribution of visit counts across patients.
A long gap between the last measurement and the landmark is a warning that “latest” may not mean “current.” Very different visit counts across patients can make unweighted window means difficult to compare. Missing baselines can make changes unavailable for a selected group of patients. These patterns may reflect clinical practice or patient status, so the observed covariate distribution may differ systematically between patients with complete and incomplete histories. The representation itself does not correct such differences; they need to be considered in the analysis plan.
| Representation | Emphasizes | Key limitation to check |
|---|---|---|
| Latest observed value | Most recent recorded level | Measurement may be distant from the landmark |
| Baseline value | Starting level under a defined baseline rule | Does not summarize later measurements |
| Change from baseline | Net difference between two defined measurements | Requires both measurements and can hide intervening variation |
| Window summary | Multiple measurements within a specified period | Depends on window, visit frequency, and summary definition |
The main alternatives are therefore not simply different calculations of one universally best covariate. They trade off recency, starting status, net change, and use of repeated observations. A full longitudinal model can represent the measurement trajectory more directly, whereas a landmark representation compresses the observed history into covariates for a prediction at a chosen time. The compression is often convenient, but it necessarily omits some detail. The later landmark model should use the covariates as defined, without implying that their associations are causal.
Common Pitfall
A common mistake is to call the arithmetic mean of the last few recorded values a “recent average” without defining which values qualify or accounting for when they were measured. For example, the last three values could span three weeks for one patient and a year for another. The correction is to specify a calendar-time look-back window, state the summary rule, report or inspect the number and timing of included observations, and acknowledge that an unweighted mean gives equal weight to each recorded visit rather than each unit of time.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At a month-9 landmark, a patient has values 3.0 mg/L at month 0, 4.0 mg/L at month 5, and 5.0 mg/L at month 10. Which measurements may be used for the latest observed value?
- Using the month-0 value as baseline, calculate the change for the patient in question 1.
- A look-back window contains values 2.0, 3.0, and 7.0 mg/L. Calculate the unweighted mean and state one reason it may not be a time-average.
- Why should latest value, baseline, and change not be entered as three unrestricted covariates when change is defined as latest minus baseline?
- Give one diagnostic check that is especially relevant to a latest-value representation and one relevant to a window mean.
Solutions. 1. The month-0 and month-5 measurements are eligible; month 10 is after the landmark and must be excluded. The latest eligible value is 4.0 mg/L. 2. The change is \(4.0-3.0=1.0\) mg/L. 3. The mean is \((2.0+3.0+7.0)/3=4.00\) mg/L. It is not necessarily a time-average because the values receive equal weight even if their measurement times are unequally spaced. 4. The identity \(D=L-B\) means the three covariates are exactly linearly dependent, so their separate unrestricted coefficients are not identifiable. 5. For a latest value, examine the gap between its measurement time and the landmark. For a window mean, examine the number and timing of measurements per patient, including whether some patients have no observations in the window.