Plotting Post-Dropout Trajectories Across Candidate Models

Why Plot Post-Dropout Trajectories?

When participants stop providing outcomes, a trajectory plot can make the consequences of different missingness assumptions visible over time. Up to dropout, the data constrain the outcome history; after dropout, a selection model supplies predictions conditional on the observed history and the dropout information represented in that model. Candidate models can therefore agree on observed means yet imply different post-dropout paths.

A useful plot makes that distinction visible rather than drawing one apparently continuous line as though every point were measured. It aligns the time scale, identifies the group being summarized, marks when outcomes first become unavailable, and distinguishes observed estimates from each model’s predictions. The plot is a descriptive comparison of trajectories under stated models; it does not establish which unobserved trajectory is correct.

Definition: Let \(D_i\) denote participant \(i\)'s dropout time, defined as the first scheduled visit at which the outcome is missing. For a group whose members share dropout time \(d\), the observed mean at a visit before \(d\) is the arithmetic mean of their observed outcomes at that visit. The model-specific post-dropout mean at visit \(t \ge d\) is the average of their predicted outcomes at \(t\), conditional on their observed histories and dropout information under candidate model \(m\). These are different kinds of quantities: one summarizes observed values, while the other summarizes model-based predictions.

This definition uses the first missed visit as the dropout time. If a study uses a different convention, such as the last completed visit, state it explicitly and make the axis labels and vertical marker consistent with it. Otherwise, the plotted “dropout time” may be shifted by one visit.

Specify What Each Curve Represents

Before drawing lines, specify the outcome scale, visit times, and target group. A particularly direct display follows one dropout-time group: show its observed means at visits before dropout, then show a separate predicted series for each candidate model from the first missed visit onward. The predictions should be conditional on the observed histories and dropout status of that group, not predictions for a different population silently substituted at the missing visits.

For group \(G_d=\{i:D_i=d\}\), let \(Y_{it}\) be the outcome when observed and let \(\widehat{Y}^{(m)}_{it}\) be the conditional mean predicted by candidate model \(m\). Then the plotted observed and predicted means are:

\bar{Y}_{d,t}^{\,\mathrm{obs}} = \frac{1}{|G_d|}\sum_{i\in G_d}Y_{it} \quad (t<d), \qquad \bar{Y}_{d,t}^{(m)} = \frac{1}{|G_d|}\sum_{i\in G_d}\widehat{Y}^{(m)}_{it} \quad (t\ge d).

The observed mean is defined only when the group has observed outcomes at that visit. The predicted mean is defined only after the model and its conditioning information have been specified. Use different symbols or line styles for these two quantities—for example, filled circles for observed means and dashed lines with open circles for predictions. Label the candidate models directly in a legend.

A plot of the observed mean among everyone observed at each visit is also possible, but its composition can change over time as participants drop out. It is not automatically the observed part of a trajectory for one fixed dropout group. Label its denominator or group clearly, and do not connect it to predictions for another target group without explaining the change in estimand.

Uncertainty Intervals for Predicted Means

An interval around a model-specific predicted mean should represent uncertainty about the quantity actually plotted. If the target is the group’s conditional mean trajectory, use uncertainty in that estimated mean, including parameter uncertainty as appropriate. A predictive interval for an individual outcome answers a different question and is generally wider because it also includes individual outcome variation. The legend or caption should identify which kind of interval is shown.

For a large-sample, approximately normal estimate \(\widehat{\theta}\) of a plotted mean with standard error \(SE\), a pointwise 95% uncertainty interval is often calculated as \(\widehat{\theta}\pm1.96SE\). This approximation requires a suitable standard-error estimate and an approximately normal sampling distribution for the estimated mean. It is not a simultaneous interval covering the whole trajectory with 95% probability, and it is not a 95% probability statement about a fixed parameter.

For example, if model predictions for several participants are averaged, their prediction-estimate errors may be correlated because the same model parameters are used for everyone. If \(V\) is the covariance matrix of the vector of predicted conditional means and \(n_d=|G_d|\), then:

\operatorname{Var}\!\left(\bar{Y}_{d,t}^{(m)}\right) = \frac{\mathbf{1}^{\mathsf T}V\mathbf{1}}{n_d^2}, \qquad SE\!\left(\bar{Y}_{d,t}^{(m)}\right) = \sqrt{\frac{\mathbf{1}^{\mathsf T}V\mathbf{1}}{n_d^2}}.

This follows because the average is a linear combination with weights \(1/n_d\): its variance is the sum of all entries of \(V\), divided by \(n_d^2\). Ignoring positive covariance can make the interval too narrow. The uncertainty calculation must match how the model’s predictions were obtained.

Worked Example: Plotting Two Candidate Mean Trajectories

Suppose a hypothetical trial measures a symptom score at weeks 0, 4, 8, 12, and 16; higher scores indicate more symptoms. Ten participants first miss the outcome at week 12. Their observed group means and standard errors before dropout are 10.0 (0.40), 9.4 (0.45), and 8.8 (0.50) points at weeks 0, 4, and 8. Two candidate selection models give the following conditional mean predictions and standard errors for this same group:

Visit	Model A mean (SE)	Model B mean (SE)
Week 12	8.3 (0.60)	9.1 (0.65)
Week 16	7.9 (0.70)	9.0 (0.80)

Assume the reported standard errors quantify uncertainty in the respective group means, and use the approximate pointwise 95% interval \(\widehat{\theta}\pm1.96SE\). For Model A at week 12:

8.3\pm1.96(0.60) = 8.3\pm1.176 = [7.12,\ 9.48]\text{ points}.

For Model B at week 12, the interval is \(9.1\pm1.96(0.65)=9.1\pm1.274=[7.83,10.37]\) points. At week 16, Model A gives \(7.9\pm1.96(0.70)=[6.53,9.27]\) points, and Model B gives \(9.0\pm1.96(0.80)=[7.43,10.57]\) points, with endpoints rounded to two decimal places.

Plot the observed means only at weeks 0, 4, and 8, using filled symbols. At week 12, draw a vertical dropout marker and begin both model-specific predicted series with distinct dashed lines and open symbols. Add the calculated intervals to the predicted means. The two predicted means differ by 0.8 points at week 12 and 1.1 points at week 16; their intervals overlap. This visual comparison alone does not test equality of the model predictions, because it does not use the uncertainty of their difference.

Construct and Annotate the Display

Fix the target group and time convention.
State who is summarized and whether dropout time means the first missed visit or the last observed visit.

Calculate the observed summaries.
At each pre-dropout visit, calculate the mean from the group’s observed outcomes and report the number contributing to it.

Obtain predictions under each candidate.
At post-dropout visits, average model-based conditional means for the same group and calculate uncertainty for the plotted mean.

Align time and distinguish sources.
Use a shared visit axis, mark dropout at the first unavailable visit, and use separate symbols or line styles for observed and predicted values.

Explain intervals and missingness.
Identify the interval type, name the candidate models, and state plainly that post-dropout points are predictions rather than observed outcomes.

If dropout times vary across participants, a small rug or count annotation can show how many first dropouts occur at each visit. Keep that annotation distinct from the trajectory itself. If the plotted trajectory is for one dropout-time group, mark that group’s dropout time on its panel; a study-wide rug is context, not a replacement for the group-specific marker.

Worked Example: Averaging Predictions With Correlated Uncertainty

For four hypothetical participants who first miss the outcome at week 12, Model A predicts conditional means of 8, 10, 11, and 13 points at that visit. Their average is:

\bar{Y}^{(A)}_{12} = \frac{8+10+11+13}{4} = 10.5\text{ points}.

Suppose the estimated variance of each participant’s predicted mean is \(0.36\) points squared, and the covariance between every pair is \(0.09\) points squared. There are four variances and twelve off-diagonal covariance entries in the sum. Thus:

\operatorname{Var}(\bar{Y}^{(A)}_{12}) = \frac{4(0.36)+12(0.09)}{4^2} = \frac{2.52}{16} = 0.1575, \qquad SE=\sqrt{0.1575}=0.397\text{ points}.

The approximate 95% interval for the predicted group mean is \(10.5\pm1.96(0.397)=[9.72,11.28]\) points. It represents uncertainty in the predicted mean under the stated covariance assumptions, not the range in which 95% of individual outcomes are expected to fall. In a plot, draw this interval around Model A’s week-12 mean and label it as an interval for the mean.

Worked Example: Marking Dropout Times Without Shifting the Visit

In a hypothetical six-person panel, two participants first miss the outcome at week 8, three first miss it at week 12, and one first misses it at week 16. Suppose the time axis is week 0, 4, 8, 12, and 16, and the convention is that \(D_i\) is the first missed visit.

A rug below the horizontal axis can therefore place two marks at week 8, three at week 12, and one at week 16. For the subgroup with \(D_i=12\), draw its observed means only through week 8 and put its vertical dropout marker at week 12, where its model-based predictions begin. Do not put the marker at week 8 merely because that is the last visit with observed outcomes: that would change the stated convention.

If a plot instead defines dropout time as the last observed visit, it must say so and label the marker at week 8 for this subgroup. The two conventions can both be used, but mixing them shifts predictions by one scheduled visit and misrepresents when outcomes became unavailable.

Reading the Plot and Recognizing Failure Modes

A useful comparison asks where candidate predictions diverge, how their uncertainty changes over time, and how far the predicted continuation lies from the group’s observed history. A widening separation between model curves indicates sensitivity of the extrapolated trajectory to model specification. It is not evidence that the model with the more appealing line is better supported by observed post-dropout outcomes: those outcomes are unavailable.

Common plotting choices have different strengths. An overlay makes differences in predicted means easy to see, provided line styles and labels remain legible. Small multiples, one panel per candidate model with identical axes, can be clearer when there are many candidates or overlapping intervals. An observed-only plot avoids model-based extrapolation but cannot display the proposed post-dropout paths. Whichever display is chosen, keep the outcome scale and time axis comparable across panels.

Connecting observations to predictions without a boundary. A continuous solid line can imply that post-dropout values were measured. Use a visible change in symbol or line style at dropout.
Comparing unlike groups. A predicted mean for a dropout subgroup is not automatically comparable with an observed mean among all participants still observed at that visit. Name the group for every series.
Changing denominators without notice. If observed means use whoever remains observed at each visit, show the contributing sample size or explain that group membership changes over time.
Confusing intervals. A confidence or uncertainty interval for a mean is not an individual prediction interval. State which one is plotted and what uncertainty it includes.
Overinterpreting overlap. Overlapping pointwise intervals do not prove that two models have equal predictions; non-overlap is not a substitute for a properly calculated interval or test for their difference.
Treating the predicted path as observed evidence. A smooth line is still a consequence of model assumptions. Its visual continuity does not validate the extrapolation.

These plots complement, rather than replace, comparisons of observed-data fit and the substantive assessment of extrapolation assumptions. Their central limitation is unavoidable: observed data do not reveal the actual post-dropout outcomes for participants who dropped out. Even narrow intervals may reflect strong modeling or distributional assumptions, not direct information about those missing outcomes.

Key takeaway: Align candidate trajectories on one time scale, define the group behind every mean, mark the first unavailable visit, and visually separate observed outcomes from model-based predictions. Uncertainty intervals describe a specified estimated quantity; they do not turn extrapolations into observations.

Check Your Understanding

Answer each question, then compare with the solutions.

A group first misses its outcome at week 10. Under the first-missed-visit convention, where should the dropout marker go, and where may its observed means be plotted?
Two participants have predicted means of 6 and 10 points at a visit. What is their average predicted mean?
A plotted mean is 12 points with \(SE=0.5\). Calculate an approximate 95% interval using 1.96.
Why should a predicted mean for dropouts not be joined to an observed mean among all participants still observed without identifying the groups?

Solutions. (1) Mark week 10, the first missed visit; plot observed means only at scheduled visits before week 10 where their outcomes are available. (2) \((6+10)/2=8\) points. (3) \(12\pm1.96(0.5)=12\pm0.98=[11.02,12.98]\) points. (4) They summarize different groups and may have different composition, so a continuous line could conceal a change in the estimand and suggest unsupported continuity.