Attrition in a Vaccine Immunogenicity Trial

Why Attrition Is Especially Important for Vaccine Titers

A vaccine trial may measure antibody titers soon after vaccination and again months later, when waning can be assessed. If participants leave before the later visit, their later titers are unknown. This is consequential: early measurements do not reveal exactly how each participant’s antibody level subsequently changed. A comparison restricted to participants who returned may therefore describe a selected subset rather than the randomized groups.

The analysis below targets the mean log10 antibody titer at a specified later visit for all randomized participants in each arm. It compares responses across observed follow-up patterns, then adjusts predictions for missing titers using delta parameters. The delta adjustment makes an explicit sensitivity assumption about how the missing values differ from predictions under a missing-at-random (MAR) model. It does not make those values observed or identify their distribution from the trial alone.

Titers are commonly analyzed on the log scale because antibody concentrations are positive and often right-skewed. A difference in mean log10 titer can be converted to a ratio of geometric means by exponentiating base 10. This ratio compares the arms’ geometric mean titers; it is not, by itself, proof of protection from infection or clinical disease.

Define the Estimand and Follow-Up Patterns

For example, patterns may distinguish participants with a measured later titer from participants who discontinued before that visit. More detailed patterns can distinguish intermittent missed visits from dropout, or dropout at different times, if the data support those distinctions. Pattern membership is observed, but the later titer for a dropout pattern is not. Thus the pattern-specific mean for missing outcomes requires a model and, for departures from MAR, sensitivity assumptions.

Write \(q_{ag}\) for the proportion of randomized participants in arm \(a\) belonging to pattern \(g\), and \(\mu_{ag}\) for that pattern’s mean later titer. The arm-specific target mean is

For a pattern with observed later titers, its mean can be estimated from those measurements, possibly after adjustment or standardization for baseline and earlier antibody measurements. For a pattern with missing later titers, a model predicts the mean given the available history. Under MAR, the prediction assumes that, conditional on included observed information, missing later titers have the same distribution as comparable observed outcomes. Delta adjustment replaces that assumption with a stated departure:

A negative \(\delta_{ag}\) says that the missing participants’ mean log10 titer is lower than the MAR prediction, after conditioning on the information in the model. A shift of \(-0.30\) log10 units corresponds to multiplying the predicted titer by \(10^{-0.30}\), approximately 0.501. It does not mean subtracting 0.30 from the raw titer or from every observed participant’s measurement.

Assumptions Before Calculation

The calculations require clearly stated assumptions. Randomization supports an intention-to-treat comparison of the assigned arms, subject to the chosen estimand and appropriate handling of outcomes. It does not make missing later titers ignorable. The pattern proportions \(q_{ag}\) are estimated from the randomized participants in each arm, and the prediction model should use relevant observed information, such as baseline titer, early post-vaccination titer, age, and timing of dropout. Its form should be adequate for the data and should not extrapolate silently beyond the observed covariate range.

The MAR calculation relies on the conditional missingness assumption described above. A delta-adjusted calculation additionally assumes that the selected shifts are meaningful descriptions of departures from those predictions. The trial data generally cannot determine the correct delta for outcomes that are never observed. Clinical input may help define plausible shifts, for example by considering assay variability, known antibody kinetics, and the time between the last measurement and the missed visit. Such input motivates sensitivity values; it does not turn them into empirically identified parameters.

If some patterns have very few participants, their observed means and predictions may be unstable. If a pattern has no comparable observed later outcomes, its MAR prediction may rely heavily on extrapolation. In either case, the weighted mean can look precise while resting on weak information. Pattern definitions, model diagnostics, and uncertainty from estimating means and proportions should therefore be reported alongside the point estimates.

Delta Adjustment and Its Consequences

A useful way to see the effect of delta is to substitute the adjusted pattern means into the weighted total. If only one pattern is missing, with proportion \(q\), then adding \(\delta\) to that pattern’s MAR mean changes the arm mean by \(q\delta\). The weighting matters: a large shift in a small missing group may have less impact on the arm mean than a modest shift in a large missing group.

This follows directly by replacing the missing-pattern mean \(\mu_{a,\mathrm{mis}}^{(\mathrm{MAR})}\) with \(\mu_{a,\mathrm{mis}}^{(\mathrm{MAR})}+\delta_a\) in the weighted sum. The observed-pattern terms cancel, leaving \(q_{a,\mathrm{mis}}\delta_a\). With multiple missing patterns, the total change is the sum of each missing pattern’s proportion times its own delta.

Diagnostics, Alternatives, and Limitations

Check whether the pattern and prediction model are credible before interpreting a sensitivity result. Compare baseline and early titers across follow-up patterns within each arm; large differences can signal that a simple unadjusted pattern mean is inadequate. Examine distributions and residuals for the observed later titers, confirm that predictions are within a defensible range, and inspect whether missing participants have covariate histories similar to participants contributing observed later outcomes. Report the number of participants in every arm-pattern cell. Sparse cells and extreme predictions are warnings, not reasons to conceal the pattern.

A complete-case analysis is simpler but uses only participants with observed later titers. It estimates a survivor-pattern comparison and can be biased for the all-randomized estimand if response and retention are related. A MAR-based analysis uses observed histories to predict missing outcomes but depends on its conditional missingness assumption. Delta-adjusted pattern-mixture analysis keeps that model as a reference and makes departures explicit. Selection models are a different alternative: they model outcome distributions and the probability of observation jointly or conditionally. They can encode nonignorability differently, but do not eliminate the need for untestable assumptions about missing outcomes.

No single delta is certified by the observed data when later titers are missing. A defensible report explains the clinical rationale for the range examined, states whether shifts apply equally or differently by arm and dropout timing, and presents the resulting arm-specific means and contrasts. The analysis is limited by the quality of the prediction model, the plausibility of the delta values, assay properties, and the amount and timing of attrition. It estimates antibody response, not vaccine efficacy, unless a separate design and analysis justify that clinical claim.