The Classical Two-Period Crossover ANOVA Model

Why the Crossover Model Has Several Terms

In a two-period crossover study, each participant receives both treatments, but in an assigned order. A participant in the TR sequence receives Test in period 1 and Reference in period 2; a participant in the RT sequence receives Reference first and Test second. Thus, an observed endpoint can reflect not only the treatment received, but also the period, the assigned sequence, and the participant’s own level of response.

For a pharmacokinetic endpoint such as AUC or Cmax, the classical analysis is commonly applied to its logarithm. The response \(Y\) in the model below is therefore a log-transformed endpoint, not the original endpoint on its arithmetic scale. The model describes how the design factors and subject-level variation contribute to that response. It does not, by itself, establish that its assumptions are satisfied or dictate every inferential calculation.

Model and Indexing

Let \(i\) index sequence, with \(i\in\{\mathrm{TR},\mathrm{RT}\}\); let \(q\) index subjects within sequence \(i\); and let \(j\in\{1,2\}\) index period. Write \(Y_{iqj}\) for the log endpoint measured on subject \(q\) in sequence \(i\), during period \(j\). Let \(d(i,j)\) denote the treatment received in that sequence and period. The treatment assignment is fixed by the design:

The model is

Here, \(\mu\) is the overall mean; \(S_i\) is the fixed sequence effect; \(B_{q(i)}\) is the random effect for subject \(q\) nested within sequence \(i\); \(P_j\) is the fixed period effect; and \(\tau_{d(i,j)}\) is the fixed treatment effect for the treatment assigned to that observation. The remaining term \(\varepsilon_{iqj}\) is within-subject error. The sequence effect compares groups assigned different orders; the period effect represents a systematic difference between periods; and the treatment effect represents a systematic difference between Test and Reference in this model.

A conventional set of constraints for the fixed effects is \(S_{\mathrm{TR}}+S_{\mathrm{RT}}=0\), \(P_1+P_2=0\), and \(\tau_{\mathrm{Test}}+\tau_{\mathrm{Reference}}=0\). These constraints identify the effects relative to the overall mean. Equivalently, software may use a reference category rather than sum-to-zero constraints; the parameter values then have a different reference-based interpretation, but the fitted cell means can represent the same model.

For the standard random-effects formulation, subjects are treated as a sample from a subject population, with \(B_{q(i)}\) having mean zero and variance \(\sigma_B^2\). The period-specific errors have mean zero and variance \(\sigma^2\), are independent across observations conditional on the subject effects, and are independent of the subject effects. This random-intercept structure allows the two observations from one subject to be correlated through \(B_{q(i)}\). Subjects are assumed independent of one another. The model also assumes a common residual variance and an additive structure: it has no sequence-by-period, treatment-by-period, or carryover term.

The phrase “fixed-effects ANOVA” is sometimes used broadly for this classical design analysis. More precisely, sequence, period, and treatment are fixed effects in the usual model, while subject within sequence is random. If subject indicators are instead entered as fixed nuisance effects, the resulting parameterization is not the same random-effects model. In particular, a subject-specific fixed effect for every participant absorbs all between-subject differences, so a separate sequence effect cannot be estimated independently without additional modeling choices.

Worked Examples: Reading the Design

What the Model Assumes—and How to Check It

The model is useful when the treatment and period effects can reasonably be represented as additive fixed effects, the subject-level differences are represented by a common random-intercept distribution, and the remaining within-subject errors have a common variance. It also presumes that the planned observations are correctly linked to subjects, periods, sequences, and treatments. Balance concerns the design allocation and completeness; it does not guarantee normality, independence between subjects, or absence of influential observations.

Before interpreting a fitted model, check the data structure. Each participant should have a sequence assignment, no more than one record per planned period, the correct treatment label for that sequence and period, and a response on the intended analysis scale. In a complete balanced design with \(n\) participants per sequence, there should be \(2n\) participants and \(4n\) period-level records. A treatment label inconsistent with the assigned sequence can change the model’s meaning even if the response values themselves are correct.

Residual plots can reveal large departures from constant variance, unusual observations, or systematic patterns left unexplained by the model. A normal quantile plot of residuals can help identify severe departures from the assumed error distribution. Because observations from the same participant are related in the random-intercept model, records should not be treated as independent subjects. A conspicuous period pattern not represented by a single period effect, or different treatment patterns across sequences, may signal that the additive model is inadequate. These checks support judgment; they do not prove that assumptions hold.

How This Model Relates to Other Analyses

The earlier material on subject-level log ratios describes a within-subject summary based on each participant’s two treatment measurements. That approach emphasizes the paired comparison directly. The ANOVA model instead represents the observations in their sequence and period structure, including a subject-within-sequence term. In a complete balanced two-period design, these views are closely connected, but the model presentation makes the design factors explicit.

A parallel-group analysis would compare participants receiving different treatments without each participant receiving both treatments. It therefore does not have the same within-subject structure or the same period and sequence terms. Conversely, the classical crossover model is not a general remedy for missing periods, imbalance, or more complicated repeated-measure patterns; those situations require attention to the data structure and an appropriate model. The present model should not be extended by adding terms mechanically: each additional term must be supported by the design and identifiable from the observations.

Common Mistakes

• Confusing sequence with treatment. Sequence is the assigned order, TR or RT; treatment identifies what was received in a particular period.
• Treating the two records from one participant as independent. Both records share the subject-within-sequence effect, so the model represents within-subject dependence.
• Assigning the wrong treatment to a period. TR receives Test first, whereas RT receives Reference first. Treatment labels must follow the sequence allocation.
• Calling a balanced allocation proof of model adequacy. Equal sequence sizes do not verify normality, equal residual variance, correct data linkage, or absence of carryover.
• Interpreting the subject term as another treatment effect. It represents subject-level variation within an assigned sequence; it is not a treatment administered to participants.