Landmark Time, Prediction Time, and Time Origin

Why the Time Origin Matters

A prediction such as “the probability of occupying a particular state six months from now” is incomplete until “now” and the time scale are clear. In a multi-state study, records may be indexed by calendar date, time since a participant entered the study, or time since that participant entered a state. These are different coordinates for describing time. Confusing them can move the target date, so that an analysis answers a different prediction question than intended.

The practical rule is to identify the target date first. Then express that date on any desired time scale by subtracting that scale’s origin. A landmark is a reference time at which the prediction is defined; a prediction horizon is the amount of time from that landmark to the target. Neither is automatically the time origin used to index a model.

Definition: Let \(t\) denote a calendar time on a common time axis. For individual \(i\), let \(E_i\) be the calendar time of study entry, \(S_i\) the calendar time of entry into the state currently occupied at the landmark, and \(L\) the calendar landmark time. A prediction horizon \(h\) is the elapsed time from \(L\) to the target time. Thus the target calendar time is \(t^*=L+h\). The elapsed time since study entry at time \(t\) is \(u_i(t)=t-E_i\), and the elapsed time since state entry is \(v_i(t)=t-S_i\).

These definitions require a consistent time unit and a shared calendar scale for the dates being compared. For example, if dates are recorded to the day, the conversion should preserve day-level precision; if time is recorded in months, the study’s convention for measuring months should be applied consistently. In the examples below, all dates fall on the first of a month, so calendar-month arithmetic is exact.

At the landmark, the elapsed times are \(L-E_i\) since study entry and \(L-S_i\) since state entry. At the target, they are \(L-E_i+h\) and \(L-S_i+h\), respectively. These quantities are not competing definitions of the target: they are different coordinates for the same date.

Converting the Landmark and Horizon

Start with a calendar landmark and a horizon measured from that landmark. The calendar target is \(L+h\). To express the landmark relative to study entry, subtract \(E_i\); to express the target relative to study entry, subtract \(E_i\) from the target as well. The same operation applies to state-entry time, using \(S_i\) as the origin.

\begin{aligned} \text{Landmark, since study entry} &= L-E_i, \\ \text{Target, since study entry} &= (L+h)-E_i=(L-E_i)+h, \\ \text{Landmark, since state entry} &= L-S_i, \\ \text{Target, since state entry} &= (L+h)-S_i=(L-S_i)+h. \end{aligned}

The same conversion works in reverse. If the landmark is recorded as \(u_L\) units after study entry, its calendar time is \(E_i+u_L\). A horizon \(h\) then gives target calendar time \(E_i+u_L+h\). If instead the landmark is \(v_L\) units after state entry, the corresponding target is \(S_i+v_L+h\). These expressions describe the same target when the input dates and elapsed times are consistent.

Key relation: At any time \(t\), the two elapsed-time coordinates differ by a fixed offset for an individual: \(v_i(t)=u_i(t)+E_i-S_i\). This follows because \(t-S_i=(t-E_i)+(E_i-S_i)\). At a fixed landmark, changing from one elapsed-time coordinate to the other changes the origin, not the calendar date.

Worked Example: One Target, Two Different Horizons

Suppose a hypothetical participant entered a state on July 15, 2021, and the landmark is January 15, 2022. The prediction target is September 15, 2022. Express that target as a horizon from the landmark and as elapsed time since state entry.

The time from state entry to the landmark is six months: July 15, 2021 to January 15, 2022. The time from the landmark to the target is eight months: January 15 to September 15, 2022. The time from state entry to the target is fourteen months: July 15, 2021 to September 15, 2022.

Thus the same target is eight months after the landmark and fourteen months after state entry. The conversion is \(14=6+8\): elapsed time since state entry at the landmark plus the landmark horizon equals elapsed time since state entry at the target.

The target can therefore be represented as \(14\) months after state entry or \(8\) months after the landmark. Those are adjusted horizons for one target date. By contrast, a horizon of \(14\) months after the landmark would target March 15, 2023, not September 15, 2022. Using the same numerical horizon from different origins does not generally preserve the target date.

Worked Conversions Across All Three Time Scales

Worked Example: From Study Entry to a Calendar Target

In a hypothetical study, an individual enters on March 1, 2020. The landmark is December 1, 2021, and the requested prediction horizon is six months. The individual entered the current state on October 1, 2021. Find the landmark and target in calendar time, time since study entry, and time since state entry.

Calendar time. The landmark is December 1, 2021. Adding six months gives a target of June 1, 2022.

Time since study entry. The landmark is \(21\) months after March 1, 2020. The target is \(21+6=27\) months after study entry. Checking by date, March 1, 2020 to June 1, 2022 is 27 months.

Time since state entry. The state began two months before the landmark, so the landmark is \(2\) months after state entry. The target is \(2+6=8\) months after state entry. Checking directly, October 1, 2021 to June 1, 2022 is 8 months.

All three descriptions identify June 1, 2022 as the target. The study-entry and state-entry elapsed times differ because their origins differ, not because the prediction horizons have changed.

Worked Example: A Shared Calendar Landmark for Two Participants

Consider two hypothetical participants at a calendar landmark of January 1, 2021, with a prediction horizon of six months. Participant A entered the study on January 1, 2020, and the current state on March 1, 2020. Participant B entered the study on June 1, 2020, and the current state on May 1, 2020. Find the target date and each participant’s elapsed times at the landmark and target.

The calendar target for both participants is \(L+h\): January 1, 2021 plus six months is July 1, 2021.

For participant A, the landmark is 12 months after study entry and 10 months after state entry. At the target, the elapsed times are \(12+6=18\) months since study entry and \(10+6=16\) months since state entry. Direct date checks give January 1, 2020 to July 1, 2021 as 18 months, and March 1, 2020 to July 1, 2021 as 16 months.

For participant B, the landmark is seven months after study entry and eight months after state entry. At the target, the elapsed times are \(7+6=13\) months since study entry and \(8+6=14\) months since state entry. Direct date checks give June 1, 2020 to July 1, 2021 as 13 months, and May 1, 2020 to July 1, 2021 as 14 months.

The calendar target is shared, but the elapsed times from each participant’s entry dates are not. This distinction matters whenever a study reports elapsed times from individual-specific origins while defining predictions for a common calendar landmark and horizon.

Why the Conversion Is Valid

The conversion is a consequence of how elapsed time is defined, rather than a special property of a particular statistical model. Suppose \(t^*=L+h\). Elapsed time since a chosen origin \(O_i\) at the target is \(t^*-O_i\). Substituting the target definition gives \((L+h)-O_i=(L-O_i)+h\). The first term is elapsed time from that origin to the landmark; the second is the horizon. This proves the conversion for any fixed origin, including study entry or state entry.

Equivalently, adding the same horizon to two elapsed-time coordinates preserves their offset. At the landmark, the difference between state-entry time and study-entry time is \((L-S_i)-(L-E_i)=E_i-S_i\). At the target, the difference is \(((L+h)-S_i)-((L+h)-E_i)=E_i-S_i\). The offset does not depend on the horizon. A conversion that fails this check usually signals a date, unit, or origin error.

Fix the target definition.
Write the calendar landmark \(L\) and landmark horizon \(h\). Calculate the target date \(t^*=L+h\) before changing time scales.

Name the origin.
For study-entry time use \(E_i\); for state-entry time use \(S_i\). Do not use “time zero” without stating which origin it means.

Subtract the origin from both dates.
Calculate \(L-E_i\) and \(t^*-E_i\), or \(L-S_i\) and \(t^*-S_i\). The target elapsed time should equal the landmark elapsed time plus \(h\).

Check by adding back.
Add each elapsed time to its origin. The reconstructed landmark and target dates must match the original calendar dates.

When the Time Scale Is Part of the Model

Changing coordinates to describe a fixed target is different from fitting a model whose time variable is elapsed time. A model indexed by calendar time can represent calendar-period patterns; one indexed by time since study entry represents duration after enrollment; one indexed by time since state entry represents duration in the state. Those choices can imply different descriptions of how event rates vary over time. Merely translating the dates does not establish that a model using one time scale is appropriate on another.

The distinction is especially important if individuals enter at different calendar dates or reach a state at different times. At one shared calendar landmark, people may have different durations since study entry and different sojourn times in their current states. Conversely, equal sojourn times need not correspond to equal calendar dates. The time scale used in a model should match the scientific time pattern being represented, while the prediction target should remain explicitly defined.

This tutorial concerns the bookkeeping that defines the time point of a prediction. It does not determine which model is best, or how to calculate conditional state probabilities. Those questions require additional assumptions about the process and its history. Whatever method is later used, the time origin and target date must be clear enough that the modeled quantity corresponds to the intended prediction.

Diagnostics, Pitfalls, and Limitations

A practical diagnostic is to reconstruct the calendar target independently from each representation. If the target is \(q\) months after study entry, add \(q\) months to \(E_i\). If it is \(r\) months after state entry, add \(r\) months to \(S_i\). Both calculations should return \(L+h\). Also check that the landmark is not inadvertently replaced by the study-entry date or state-entry date when defining the requested horizon.

Using one number of months from two origins. A target 14 months after state entry is not generally the target 14 months after the landmark. Translate the target date and then calculate the adjusted horizon from the new origin.
Confusing the landmark with an origin. The landmark is the reference point for the prediction horizon. It does not have to be time zero for the model or the data.
Ignoring individual-specific entry dates. The same calendar landmark can represent different durations since study entry or state entry for different individuals.
Mixing units or date conventions. Adding 180 days is not always identical to adding six calendar months. Use the convention specified for the analysis and apply it consistently.
Using a state-entry origin without a known state-entry date. If \(S_i\) is unavailable or ambiguous, elapsed time since that state entry cannot be determined from the available information without additional assumptions.

A conversion also does not resolve missing or uncertain dates, delayed recording, or ambiguity about which state-entry episode is relevant. Those issues require a clear data definition before the arithmetic can be trusted. A correct calculation cannot compensate for an incorrectly recorded origin.

Common mistake: “The landmark is six months after state entry, so a 14-month prediction horizon from the landmark is 14 months after state entry.” This adds durations that begin at different dates without converting the target. If the intended target is 14 months after state entry and the landmark is six months after state entry, the horizon from the landmark is \(14-6=8\) months. Conversely, an eight-month landmark horizon reaches 14 months after state entry. Always calculate the target date, then express it relative to the requested origin.

Key takeaway: Define the landmark and prediction horizon on a stated time scale, calculate the target date, and only then translate to time since study entry or state entry. A change of origin changes the numerical elapsed time; it need not change the target or the estimand when the conversion is done correctly.

Check Your Understanding

Answer each question by naming the origin and showing how the target time is obtained.

A hypothetical participant entered a state on April 1, 2022. The landmark is October 1, 2022, and the target is February 1, 2023. What are the horizon from the landmark and elapsed time since state entry?
A participant entered a study on January 1, 2021. The landmark is January 1, 2023, with a 9-month horizon. Give the target date and the target elapsed time since study entry.
At a landmark, a participant is 5 months after state entry. A target is 12 months after state entry. What is the horizon from the landmark? Explain why it is not 12 months.
Two participants share the same calendar landmark and horizon but entered the study on different dates. Must their target calendar dates match? Must their target times since study entry match? Explain.

Solutions. (1) October 1 to February 1 is a 4-month landmark horizon. April 1, 2022 to February 1, 2023 is 10 months since state entry, and \(6+4=10\). (2) The target is October 1, 2023. It is \(24+9=33\) months after study entry: 24 months to the landmark plus 9 months to the target. (3) The horizon is \(12-5=7\) months. Twelve months is measured from state entry; the landmark is already 5 months after that origin. (4) Their target calendar dates must match because they share \(L\) and \(h\), so both targets are \(L+h\). Their target elapsed times since study entry need not match because each is calculated by subtracting that participant’s own \(E_i\).