Repeated Measures Before Event Readout

Why Summarize Repeated Measurements?

In a time-to-event MAMS trial, the final event outcome may not yet be sufficiently observed when a treatment arm must be selected. By then, however, participants may have provided several measurements of an intermediate outcome, such as a symptom score or laboratory measure. The selection rule needs a way to reduce those measurements to a prespecified score for each arm. This tutorial focuses on three options: a fitted slope, a time-weighted average, and a value at a fixed landmark.

The summary is not a generic search for whichever feature makes an arm look best. Its time window, calculation, treatment of missing visits, direction of benefit, and rule for comparing arms should be specified before the interim data are examined. As in the earlier material on choosing an intermediate outcome, the summary must also be scientifically relevant to the selection objective. A numerical difference on an intermediate measure is not, by itself, proof of benefit on the final event endpoint.

For clarity, suppose larger outcome values are better; if lower values are better, reverse the direction of the comparison in advance. A simple selection score for arm \(a\) relative to control is the difference between their arm-level summaries. The summary’s units and time window matter: a slope has units of outcome per unit time, whereas an average or landmark value has the outcome’s original units.

Three Ways to Summarize the Trajectory

Fitted slope

A fitted slope condenses the direction and rate of change across several scheduled visits. With complete measurements at the same \(m\) times for a participant, define the ordinary least-squares slope as follows:

This is the fitted change in outcome per unit of time. It is defined by the chosen linear summary even if the participant’s trajectory is not truly linear; in that case, it is an overall linear rate, not a claim that the outcome changes linearly at every moment. The calculation requires at least two distinct measurement times. If participants have different sets of observed visits, a rule for calculating or handling incomplete trajectories must be prespecified.

Time-weighted average

A time-weighted average gives measurements over the specified interval appropriate weight when visits are unequally spaced. If the observed times satisfy \(t_1<t_2<\cdots<t_m=\tau\), the trapezoidal rule approximates the area under the trajectory from \(t_1\) through the cutoff. Dividing by the interval length gives an average outcome over that window:

The trapezoidal rule treats the outcome as changing linearly between adjacent measurements for purposes of approximating the area. The condition \(t_m=\tau\) is essential for this expression to average over the full stated window \([t_1,\tau]\). If the last observed time is earlier than \(\tau\), dividing the area only through that last visit by \(\tau-t_1\) does not produce the full-window average. One may instead define an average over the observed span, dividing by \(t_m-t_1\), or prespecify how to handle the unobserved interval.

Landmark value

A landmark summary is the outcome at a fixed scheduled time \(t_L\leq\tau\), such as \(Y_{ia}(t_L)\). It is straightforward to interpret and does not require summarizing the shape between visits. It does, however, use only the measurement at that time. The landmark and the rule for participants without a recorded value there must be chosen before selection; substituting each participant’s last available value can create a different, follow-up-dependent summary.

From Participant Summaries to Arm Comparisons

For any prespecified participant-level summary \(S_{ia}\), a basic arm-level summary is its mean among the participants included under the prespecified rule:

where arm \(0\) denotes control, \(n_a\) is the number of participants contributing to the summary in arm \(a\), and \(\widehat{\Delta}_a\) is the intermediate-outcome contrast. When larger values are favorable, a larger contrast is directionally favorable for selection. The contrast is in the units of \(S\), not automatically in units of the final event outcome.

For slopes, there is an important distinction between averaging participant-level slopes and fitting a slope to arm-level means. If the same participants contribute at every visit and all have the same scheduled times, the two calculations agree. To see why, let \(\bar{Y}_a(t_j)\) be the mean at visit \(j\) among that same set of participants. Then:

The denominator and visit times are common to every participant, and the numerator is linear in the outcomes. Thus the right-hand side is the slope fitted to the arm-mean trajectory. If different participants contribute at different visits, the visit means can reflect changing composition; the equality no longer follows. A slope of those displayed arm means may still be calculated as a descriptive quantity, but it should not be labelled the mean participant-level slope.

Choosing a Summary and Checking It

These summaries answer different questions. A slope uses the pattern across visits and can be efficient when change over time is relevant, but it can conceal nonlinearity and is sensitive to which visits are available. A time-weighted average reflects outcomes across an interval and handles unequal spacing explicitly, but relies on interpolation between observations and needs a clear rule at both ends of the window. A landmark is easy to communicate and avoids averaging across a trajectory, but ignores information at other visits and can be vulnerable to missing data at the chosen time.

The following checks are useful before interpreting an interim score:

• Verify the time window. Confirm that every measurement used was available by \(\tau\), that visit times and units are correct, and that the time-weighted average’s final measurement is at \(\tau\) when it is described as covering the full interval.
• Check visit availability and composition. Tabulate the number contributing at each visit by arm. If this changes, an arm-mean trajectory can mix outcome change with changes in who is observed. Do not treat its fitted slope as the mean individual slope without the common-participant condition.
• Inspect trajectories, not only their summaries. Plot arm means over scheduled times and examine participant-level patterns. A single slope can hide a plateau, a reversal, or substantial variation in individual changes.
• Assess missingness and measurement consistency. Compare completion patterns across arms and visits, and check that measurements were taken and processed comparably. A summary based on observed values need not represent all randomized participants when missingness is informative.
• Check direction and scale. Make sure that higher or lower values really are favorable under the prespecified rule, and report contrasts with their units and time window.

These checks diagnose whether the summary is being calculated and interpreted as intended; they do not establish that the intermediate outcome is a valid substitute for the final endpoint. Selection based on an intermediate summary is also distinct from a confirmatory test of the final event outcome. A practical alternative to one summary is to select on a different prespecified summary, such as a landmark rather than a slope. Choosing among them after seeing which produces the preferred arm comparison can undermine the intended selection rule.

Common Pitfalls and Limits

• Using a full-window denominator with an early last visit. If measurements stop at month 4 but the cutoff is month 5, an area calculated only through month 4 divided by 5 months is not the average over months 0 to 5. Define an observed-span average, or specify how the interval from month 4 to month 5 is handled.
• Confusing the arm-mean slope with the mean individual slope. Their equality requires common participants at each visit and common scheduled times. With changing contributors, a shift in arm means can reflect composition rather than within-participant change.
• Treating a fitted slope as a complete description. Two arms can have similar slopes but different outcome levels, or similar averages but different trajectories. The selected summary discards information, by design.
• Assuming the intermediate contrast predicts the final effect. An arm that looks favorable on an intermediate score is not thereby shown to benefit the event endpoint. The summary is a selection input, not a guarantee about the final result.

The central design choice is therefore not simply which formula is convenient. It is which prespecified summary represents the relevant aspect of the longitudinal intermediate outcome at the time selection must occur, with a transparent rule for incomplete data and a defensible interpretation of the resulting arm comparison.