Plotting Observed and Imputed Reference-Based Trajectories

Why the Plot Must Show Where Outcomes Come From

A reference-based trajectory plot is not just a picture of means over visits. It represents both outcomes recorded in the trial and outcomes generated under assumptions for participants who dropped out. If those sources look identical, readers can mistake an imputation-based prediction for an observed result. If they are shown as unrelated curves without context, readers may miss that the imputed portion depends on the dropout pattern and the selected reference rule.

This tutorial concerns continuous outcomes measured at scheduled visits, with monotone dropout for ease of presentation: after a participant stops providing outcomes, later outcomes are missing. The plotting principles also apply when there are more complicated missingness patterns, provided the source of each plotted value is identified. J2R (jump to reference) and copy-reference (CR) imputations use the reference arm to model post-dropout outcomes for participants assigned to the treatment arm. Under J2R, the treatment benefit is assumed to be lost after dropout; under CR, post-dropout response behavior follows the reference-arm model, conditional on the specified history. The exact predictive distributions are determined by the reference-based imputation procedure.

Before plotting, fix the outcome scale, visit labels, treatment-arm definition, analysis population, and reference arm. The imputation draws must follow the chosen J2R or CR procedure, including its treatment of parameter uncertainty and observed history. For a multiple-imputation plot, use the same participants and pattern weights across the imputation datasets. A plot cannot establish that the imputation assumptions are correct; it can make their consequences easier to see.

Choose a Display That Separates Observation and Imputation

A useful figure often has two panels. The first shows observed-data summaries by randomized arm and visit, with the number observed at each visit shown in the caption or beneath the axis. The second shows the model-based completed-data trajectory, with observed and imputed portions distinguished by line style or symbols. For example, use filled circles for observed means and open circles or dashed segments for post-dropout imputed means. Include a legend that defines the symbols in words, not only by color.

When dropout time matters, show pattern-specific curves or facets for \(K_i\), and identify the number of participants in each pattern. Mark the last observed visit for each pattern. Do not connect an observed mean for one changing group of participants directly to an imputed mean for a different group without indicating the change in source. For a summary across patterns, show how the pattern means are combined, or make the pattern-specific display available alongside the pooled curve.

For multiple imputation, calculate the plotted mean in each completed dataset and average those means. If uncertainty bands are shown, calculate them using a multiple-imputation variance procedure appropriate to the estimand; do not use the standard error from a single completed dataset as if the imputations were observed data. A pointwise interval at each visit is not a simultaneous band for the whole trajectory. State whether the displayed bands are pointwise and what population contrast or mean they describe.

Pooling Imputation Results and Dropout Patterns

With \(M\) imputed datasets, let \(Q_m\) be the mean or other plotted estimand calculated from completed dataset \(m\), and let \(U_m\) be its estimated variance within that dataset. The pooled point estimate is \(\bar Q=M^{-1}\sum_{m=1}^{M}Q_m\). The between-imputation variance describes how the estimates vary across completed datasets; it contributes to uncertainty because the missing outcomes were not known. For a mean, the usual Rubin-rule quantities are

Here \(T\) is the estimated total variance, so \(\sqrt{T}\) is the pooled standard error. This pooling separates within-imputation sampling uncertainty from variation across imputations. It is not correct to draw an uncertainty band from \(\bar U\) alone when \(B\) is nonzero. Degrees of freedom and interval construction should follow the inference procedure used for the analysis.

A pooled overall mean can also be expressed as a weighted average of pattern-specific means. If \(w_k\) is the proportion of the analysis population in dropout pattern \(k\), and \(\mu_{kj}\) is that pattern’s mean at visit \(j\), then the overall mean is \(\sum_k w_k\mu_{kj}\), with weights summing to one. This identity follows by partitioning the population into mutually exclusive patterns: the total outcome sum is the sum of the pattern-specific outcome sums, and division by the total sample size gives the weighted average. For imputed outcomes, the pattern means are model-based and should be labelled accordingly.

Diagnostics, Alternatives, and Limitations

Read the graph for both statistical and display problems. Check that visit labels and outcome units are consistent, observed counts decline as dropout accumulates, and the legend identifies which segments are imputed. Confirm that pattern weights correspond to the stated analysis population and add to one. Where a plot has many patterns, verify that rare patterns are not hidden by a single pooled line. Compare J2R and CR using the same population, visit scale, and display conventions; otherwise visual differences may reflect inconsistent summaries rather than the imputation rule.

The available-case observed-data plot is a useful descriptive reference, but its mean at each visit may represent a different subset as participants drop out. It is not automatically an estimate for the original randomized population. A single pooled completed-data curve is compact, but can conceal how dropout patterns contribute. Pattern-specific plots reveal that structure but can become crowded and imprecise when patterns are small. Tables can report exact means, sample sizes, and intervals more clearly, but are less effective at showing the timing of changes. These displays answer complementary questions; none removes dependence on the missing-data assumptions.

A common mistake is to draw one uninterrupted line and label every point “observed mean.” The reasoning is that all points are means for the same randomized arm, so they seem comparable. The correction is to identify the source at every visit: some later values may be imputed, and the observed-data means may use changing subsets. Use distinct symbols or panels, state the analysis population, and explain whether the plotted trajectory is observed-only, pattern-specific, or pooled across completed datasets.

Plots are descriptive summaries of the data and imputation model, not tests of the assumptions. A smooth curve does not validate a reference-based procedure, and a visible difference between J2R and CR does not by itself establish which assumption is more plausible. Averages can also conceal skewness, extreme imputed values, or substantial variation within a pattern. Pair trajectory plots with numerical summaries and distributional checks suited to the outcome and analysis.