Withdrawals and Incomplete Two-Period Records

What Withdrawal Changes

A two-period crossover assigns each participant to one of two sequences: test then reference (TR), or reference then test (RT). A participant who completes both periods contributes responses under both treatments, so the participant can contribute a within-person treatment comparison. A participant who withdraws after period 1 contributes only the response under the period 1 treatment. That observation is not useless, but it cannot form the participant’s two-treatment comparison.

This distinction matters because crossover designs use within-person comparisons to reduce variation between participants. Withdrawal can therefore reduce precision, change the composition of the complete-case group, and leave different amounts of information for different analyses. The key is to identify what was observed and what comparison it supports before choosing an analysis.

For clarity, let \(Y_{ij}\) denote the response for participant \(i\) in period \(j\). In a bioequivalence analysis, the response is often a log-transformed pharmacokinetic measure, so a treatment contrast on this scale corresponds to a ratio on the original scale. The example calculations below use hypothetical response units and focus on the treatment contrast, not on a bioequivalence decision.

Which Comparisons Remain Available?

For a complete participant, the treatment contrast can be constructed from the two observed responses. In the TR sequence, the contrast \(Y_{i1}-Y_{i2}\) is test minus reference; in the RT sequence, \(Y_{i2}-Y_{i1}\) is test minus reference. Under a common additive period effect, the period contribution has opposite signs in the two sequence-specific contrasts. Averaging equally across the two sequences cancels that contribution.

Here \(\tau\) is the test-minus-reference treatment effect under the additive model. The cancellation relies on the model and the sequence weighting; a simple unadjusted average of observed differences need not have this property if sequence representation is unbalanced or other relevant effects are present. The earlier material on deriving the treatment contrast and confounding among sequence, period, and treatment provides the design context for this comparison.

A participant with only a period 1 response cannot contribute \(C_i\), because the second treatment response is not observed. However, period 1 is a randomized comparison: participants assigned to TR receive test in period 1, while participants assigned to RT receive reference in period 1. If period 1 responses are collected for all randomized participants, their difference in means estimates the period 1 treatment contrast in expectation under randomization, subject to the usual conditions for interpreting that comparison. It does not use the within-person comparison and is generally less precise.

Assumptions and a Short Derivation

The period 1 comparison has a simple justification. Let \(Z_i=1\) for a participant assigned test in period 1 and \(Z_i=0\) for reference in period 1. Under a model with a common period 1 effect and an additive treatment effect, write \(Y_{i1}=\mu+\pi_1+\tau Z_i+\varepsilon_{i1}\). If sequence assignment is randomized, the two groups are comparable in expectation at baseline. If period 1 outcomes are observed regardless of later withdrawal, taking expectations over the randomized groups gives:

This derivation depends on using the randomized period 1 groups, not just participants who later complete both periods. It also depends on recording the response before withdrawal, or otherwise having a defensible approach to any missing period 1 responses. If withdrawal happens before the period 1 measurement, or if some responses are absent for other reasons, randomization by itself does not guarantee that the observed period 1 groups remain comparable.

A complete-case crossover analysis makes a different assumption: the participants with both responses provide a suitable basis for estimating the treatment contrast of interest. That can be reasonable when completion is not related to the relevant outcomes after accounting for factors in the analysis. But if completion depends on treatment response, tolerability, or other outcome-related factors, the complete cases may not represent all randomized participants. Different withdrawal rates by sequence are a warning to investigate, not proof of bias by themselves.

Choosing an Analysis for Incomplete Records

There is no single analysis that makes every incomplete record equivalent to a complete crossover. The available approaches use different information and depend on different assumptions.

A likelihood-based repeated-measures model can use available responses without requiring every participant to have two measurements. Its interpretation depends on the model for the responses and on the missing-data assumption. A common assumption is that the probability of a response being missing may depend on observed information included in the analysis, but not on the unobserved response after conditioning on that information. This is called missing at random (MAR). If missingness still depends on an unobserved response after conditioning on observed information, MAR does not hold, and a standard MAR analysis may be misleading.

Model-based use of incomplete observations is not the same as observing the missing treatment response. The result is conditional on the stated assumptions. The analysis plan should identify the primary approach, the population it targets, and how departures from the missingness assumptions will be examined. A first-period-only analysis may be a useful alternative or sensitivity analysis, but it generally sacrifices the efficiency expected from a crossover.

Design and Data-Collection Measures

The most useful response to anticipated withdrawal is to preserve interpretable observations and document why records become incomplete. Planning can improve both the amount of usable information and the ability to assess whether missingness may be informative.

• Collect period 1 outcomes before asking participants to proceed. Ensure that the required samples and measurements are scheduled and recorded even when a participant is unlikely to enter period 2.
• Continue permitted follow-up after treatment discontinuation. Where appropriate and acceptable, obtaining planned outcome measurements after stopping study treatment can distinguish treatment discontinuation from loss of outcome data.
• Record sequence, period, treatment exposure, and timing. Document the last dose, sample times, predose observations, reason for withdrawal, and whether the event preceded or followed a measurement.
• Make withdrawal reasons specific. Distinguish adverse events, participant choice, protocol deviations, administrative loss, and failure to meet a period-specific requirement. Reasons help assess plausibility of missing-data assumptions.
• Protect follow-up without compromising safety or consent. Participants must remain free to withdraw; data collection plans should respect consent and applicable procedures.

Diagnostics should start with a disposition summary: randomized counts by sequence, numbers completing each period, and reasons and timing of withdrawal. Compare withdrawal proportions by sequence and by period 1 treatment, and summarize observed baseline and period 1 characteristics for completers and noncompleters. These comparisons can reveal patterns needing explanation, but a nonsignificant difference does not prove that missingness is harmless. Examine whether withdrawals cluster around adverse events, extreme responses, protocol deviations, or particular sample times. Where the analysis depends on MAR, use sensitivity analyses that make the missingness assumption more explicit rather than treating the primary model as assumption-free.

Common Mistakes and Limitations

• Wrong reasoning: “A participant who withdrew did not complete the crossover, so none of that participant’s data are useful.” Correction: The period 1 response can still inform the randomized period 1 comparison and may contribute to an appropriate model-based analysis. It cannot, by itself, provide that participant’s within-person treatment contrast.
• Wrong reasoning: “Randomization means the complete cases are still comparable.” Correction: Randomization balances treatment assignment at entry. Conditioning on later completion can disrupt that balance if completion is related to sequence, treatment experience, or outcomes.
• Wrong reasoning: “Imputing the last observed response as the missing period 2 response restores the crossover pair.” Correction: Carrying a period 1 value forward assumes that it stands in for an unobserved later response. That assumption is generally not justified by the design and does not create a measured response under the missing treatment.
• Limitation: A period 1 comparison avoids dependence on period 2 completion when period 1 data are complete, but it discards the within-person treatment comparison and can have substantially lower precision.

A practical analysis should therefore report how many participants contribute to each comparison, not just the number randomized. The complete-case crossover estimate and a period 1 estimate may differ because they use different subsets and design information; their disagreement deserves investigation rather than automatic selection of whichever result is more favorable.