Mixed-Model Analysis with a Missing Crossover Period

Why a Missing Period Changes the Analysis

In a two-period crossover, each subject is intended to receive both formulations, one in each period. If a subject has an unusable measurement in one period, the paired log difference used in a complete-subject analysis cannot be calculated. That does not mean the remaining observation must be discarded. A mixed-effects model can use the available log outcome while accounting for the fact that outcomes from the same subject are correlated.

The key is to fit the model to the observations that exist, rather than to pretend that the missing outcome was observed. Likelihood-based estimation does this by using the probability distribution of each subject’s observed measurements. Its validity depends on the model and on an assumption about why measurements are missing.

The Crossover Mixed Model with Incomplete Records

Let \(Y_{sij}\) be the log-transformed pharmacokinetic endpoint for subject \(i\) in sequence \(s\) and period \(j\), when that endpoint is observed. For example, the endpoint might be log AUC. Let \(d(s,j)\) identify the formulation assigned to that sequence and period.

The treatment effect of interest is the contrast between the test and reference formulations, \(\tau_T-\tau_R\). Because the endpoint is on the log scale, exponentiating this contrast gives the estimated test-to-reference exposure ratio. The fixed sequence and period terms adjust the treatment comparison for systematic differences associated with the assigned sequence and study period.

For a subject with both periods observed, the two outcomes share the random subject effect. Their marginal covariance matrix is

For a subject with just one observed period, the observed response has variance \(\sigma_b^2+\sigma_e^2\). In the likelihood, the unobserved component is not included in that subject’s observed response vector. More generally, the covariance matrix used for a subject is the submatrix corresponding to the periods actually observed.

This model assumes a common residual variance across periods and formulations, normally distributed subject effects and residuals, and the specified mean structure. These assumptions concern the distribution of the observed outcomes; the missingness assumption needed for likelihood estimation is an additional condition, not a consequence of normality.

Why the Observed-Data Likelihood Works

For subject \(i\), collect the observed outcomes into a vector \(Y_{i,\mathrm{obs}}\), and let \(X_{i,\mathrm{obs}}\) be the corresponding fixed-effect design matrix. Integrating over the random subject effect gives the marginal model

Here \(\beta\) contains the fixed effects and \(\theta\) the variance parameters. The likelihood includes only the observed elements for each subject. ML estimates maximize this likelihood; REML estimates variance parameters using a likelihood adjusted for estimation of fixed effects. Both approaches can use subjects with incomplete records. The earlier material on REML covers its role in estimating variance components.

The justification for omitting a separate model for missingness is called ignorability. In practical terms, the missingness mechanism must be missing at random (MAR): conditional on the observed outcomes and included observed information, the chance that an outcome is missing does not depend on that unobserved outcome. The parameters governing the missingness process must also be distinct from the outcome-model parameters. Missing completely at random is a stronger special case; MAR allows missingness to depend on observed information, such as sequence or an already recorded period measurement.

Under these conditions, the observed-data likelihood provides valid likelihood-based estimation for the specified outcome model. It does not fill in the missing values. Rather, it uses the distribution of the outcomes that were observed. If missingness depends on an unobserved exposure even after conditioning on observed information, the mechanism is not MAR, and the usual likelihood analysis may be biased unless that dependence is addressed.

What Incomplete Subjects Contribute to the Treatment Contrast

A subject with both periods observed supplies within-subject information: the contrast between that subject’s two outcomes is informative about treatment and period effects. A subject with one period supplies less direct information about the within-subject contrast, but the observed outcome can still contribute to the likelihood and to fixed-effect estimation. How much it contributes depends on the design, the fitted covariance, and the fixed-effect adjustment.

This contribution is not guaranteed merely because a model can be specified. The fixed-effect design must have sufficient information to estimate the treatment contrast. If, for example, all observations for one formulation or one sequence are absent, treatment may be confounded with other fixed effects. The design matrix must have adequate rank for the contrast of interest. Similarly, an incomplete pattern cannot by itself identify every variance component; estimation relies on the full set of observed patterns.

Likelihood-based estimation is particularly useful when different subjects have different numbers of observed periods. It avoids treating the missing response as if it were observed and avoids discarding a subject solely because a complete pair is unavailable. However, it does not guarantee that an estimate is precise, that a chosen model is appropriate, or that MAR is true.

Missingness Checks and Common Failure Modes

A missingness mechanism cannot generally be proved to be MAR from the observed outcomes alone, because the missing outcomes are unavailable for comparison. The study team should document why each period is missing and whether that reason could depend on the unobserved exposure. Compare missingness patterns across sequence, period, treatment assignment, and available subject information. Check that the fixed-effect design remains estimable and that enough subjects have both periods to support estimation of the within-subject covariance.

Inspect model convergence, estimated variance components, and standard errors. A variance estimate at a boundary, a singular fit, or a very large standard error can signal that the observed patterns provide too little information for the chosen covariance model. Compare fitted and observed cell counts and verify that missing values were represented as missing, not entered as zero. If missingness may depend on an unobserved outcome, consider sensitivity analyses under plausible departures from MAR rather than presenting the standard likelihood result as definitive.

A common mistake is to delete every subject with one missing period and call the remaining paired analysis unbiased. The correction is that complete-case analysis is not automatically unbiased: it effectively relies on the selected complete subjects giving an appropriate basis for inference. Under MAR, likelihood-based mixed modelling can use partial records while accounting for the specified covariance. Conversely, a mixed model does not automatically solve informative missingness; the validity still depends on its assumptions.

The main alternatives answer the problem differently. A complete-case paired analysis is simple but discards partial records and may lose precision or introduce selection bias. Single-value imputation creates artificial certainty and generally fails to preserve the uncertainty and covariance structure. The mixed model uses each observed response under its mean, covariance, and missingness assumptions. Its limitations include sensitivity to misspecified covariance or mean structure, weak information about variance components when few subjects are complete, and potential bias when MAR is implausible.