Sequence, Period, and Treatment Effects in Crossover ANOVA

Three Terms, Three Comparisons

In a two-period crossover, each participant receives both treatments, but the order differs by randomized sequence. For example, sequence TR means test treatment in period 1 and reference treatment in period 2; sequence RT means reference in period 1 and test in period 2. The ANOVA terms for sequence, period, and treatment therefore do not represent interchangeable versions of one treatment comparison. They refer to different features of the design.

This distinction matters especially when the endpoint is a pharmacokinetic measure analyzed on the log scale. A treatment contrast on that scale represents a difference in mean log exposure and, after exponentiation, a ratio of geometric means. A sequence contrast instead compares groups of participants assigned different orders. A period contrast compares measurements collected at different study times. Neither is, by itself, the treatment effect.

Model and Meaning of the Terms

Let \(y_{ijk}\) be the log-transformed endpoint for participant \(j\) in sequence \(i\) and period \(k\). A conventional complete-data crossover ANOVA includes sequence, participant within sequence, period, treatment, and residual variation. To describe its fixed-effect means, write the expected log endpoint as

Here \(i\) identifies sequence, \(k\) identifies period, and \(l\) identifies the treatment received in that period. The terms \(\alpha_i\), \(\pi_k\), and \(\tau_l\) are the sequence, period, and treatment effects. A sum-to-zero convention, such as \(\alpha_{TR}+\alpha_{RT}=0\), \(\pi_1+\pi_2=0\), and \(\tau_T+\tau_R=0\), makes the parameters identifiable alongside the overall mean \(\mu\). The treatment contrast of interest is \(\delta=\tau_T-\tau_R\); the period contrast is \(\Delta_P=\pi_2-\pi_1\).

This mean structure is appropriate when treatment and period effects are additive: the treatment contrast is not assumed to change between periods. Its usual interpretation also relies on a properly randomized sequence assignment, complete measurements for the standard two-period analysis, independent participants, and no residual effect of a treatment from one period that changes the outcome in the next period. For statistical inference, the model’s error assumptions must also be reasonable: the relevant errors are centered around zero, have a suitable common variance, and are independent across participants. A log transformation can help make variability more stable, but it does not guarantee these assumptions.

The participant term is important conceptually: the two observations from one person are paired, not two independent observations. The sequence comparison is between groups of participants, whereas the treatment contrast uses the within-person treatment differences. The earlier material on Expected Mean Squares in the Two-Period Crossover ANOVA establishes why these comparisons belong to different error strata. Interpreting an effect estimate does not itself determine its test or error term.

Deriving the Treatment Contrast

The four sequence-by-period means make the distinction concrete. Denote by \(\bar y_{TR,1}\) and \(\bar y_{TR,2}\) the mean log outcomes in periods 1 and 2 for sequence TR, with corresponding notation for RT. Under the additive model, the mean difference oriented as T minus R is

In sequence TR, T is measured in period 1 and R in period 2, so the treatment difference includes the period-1 versus period-2 difference. In sequence RT, orient the difference the same way—T minus R—even though T occurs second:

Adding these two equations cancels the period contribution. Subtracting them cancels the treatment contribution. Thus, in a balanced design,

This is a useful justification for the crossover treatment contrast: each sequence supplies a within-person comparison, and averaging the oppositely ordered comparisons removes an additive period effect. It also explains why a treatment comparison based only on one sequence, or on one period, need not isolate treatment.

Interpreting Each Effect Carefully

Treatment. The treatment term estimates the adjusted mean difference between log outcomes under T and R. In a balanced two-period design, it can be understood through the average of the two sequence-specific T-minus-R differences. When the endpoint is log exposure, exponentiating the estimated contrast gives the ratio of geometric means. For example, a contrast of 0.20 log units corresponds to a ratio of \(\exp(0.20)\), not an increase of 0.20 in the original exposure units.

Period. The period term describes a systematic difference between period 2 and period 1, averaging over the treatment allocation under the model. It may reflect study-time changes that affect outcomes independently of the treatment contrast. The term is not a treatment effect and does not, by itself, identify why the measurements differ by period.

Sequence. The sequence term compares participants assigned to TR with participants assigned to RT. It is a between-subject comparison, even though every participant receives both treatments. It can reflect differences between the randomized groups as well as design-related patterns represented by the model. It does not directly estimate the within-subject T-versus-R contrast. In particular, a sequence difference alone does not establish that carryover occurred, and a nonsignificant sequence result does not prove that carryover is absent.

Assumptions, Diagnostics, and Alternatives

The additive mean structure is a central condition for the simple decomposition above. If treatment behaves differently by period, the two sequence-specific treatment differences need not combine into a single common treatment contrast in the intended way. The complete, balanced example also relies on comparable numbers of participants and observations in the two sequences. With missing periods or unequal sequence sizes, the simple averages shown here may not equal the model-based estimates; the chosen sums-of-squares or model approach then matters.

Useful checks include reviewing the sequence-by-period cell means, plotting participant-level paired outcomes or differences, and inspecting residuals from the fitted model. Large discrepancies between the two sequence-specific treatment contrasts can signal a possible treatment-by-period pattern, influential observations, or chance variation. Marked changes in spread across groups, skewed residuals, or a few extreme log outcomes may make the usual error model questionable. These checks flag concerns; they do not by themselves prove which assumption failed.

The main alternative summaries answer different questions. Comparing the two treatment means after pooling all observations is straightforward in a balanced design, but it should be understood as the model-adjusted treatment contrast only when the design and model justify that pooling. Comparing T and R within one period uses different sequence groups and can mix treatment with sequence differences. Comparing sequence averages answers a between-group question, not a within-person treatment question. A parallel-group comparison, in turn, has no within-person treatment contrast because each participant receives only one treatment.

The classical two-period ANOVA is also limited by its assumptions and design. It does not make an additive model true, repair measurements affected by prior treatment, or supply information about participants who lack a required period. A sequence result should not be used as a shortcut for diagnosing carryover or deciding that the treatment comparison is valid. Conclusions about treatment should be based on the treatment contrast, its appropriate uncertainty, and the credibility of the model and study conduct.

Common Pitfalls and Practical Use

• Calling every ANOVA term a treatment effect. Sequence compares assigned-order groups, period compares study times, and treatment compares T with R.
• Comparing treatments in just one period. This uses different sequence groups and can mix treatment and between-group differences.
• Reversing the contrast orientation. In RT, T is in period 2; calculate T minus R as period 2 minus period 1 so both sequence contrasts have the same sign convention.
• Interpreting a sequence p-value as a carryover test. A sequence comparison does not by itself identify carryover, and failure to detect a sequence difference is not evidence that carryover is impossible.
• Reporting a log contrast as a raw-unit difference. A log contrast is in log units; exponentiate it to report a geometric-mean ratio on the original exposure scale.

In practice, first state which term is being interpreted and what units its contrast uses. Then identify whether its comparison is within participants or between sequences. For the treatment term, report the log-scale estimate and, when relevant, its exponentiated ratio with uncertainty. Keep the period and sequence terms as distinct model comparisons rather than treating either as a substitute for the treatment result.