Why Sequence and Period Patterns Need Separate Interpretation
A crossover study gives each subject more than one treatment, so the treatment comparison is usually informed by within-subject contrasts. Yet a results table also reports observations by sequence and period. Those summaries can look striking: one sequence may have higher overall exposure, or the second period may have higher values than the first. Neither pattern, by itself, answers whether the treatments differ. Interpretation depends on where the variation occurs and on how the model forms the treatment comparison.
The focus here is the common two-treatment, two-period design. Subjects are randomized to one of two sequences: test then reference (TR), or reference then test (RT). For a pharmacokinetic endpoint such as AUC or Cmax, analysis commonly uses the natural logarithm of the endpoint. A treatment difference on this scale corresponds to a ratio on the original scale.
The Model and Its Assumptions
A standard model for the log-transformed endpoint includes fixed effects for sequence, period, and treatment, together with a subject effect nested within sequence. One way to write it is:
Here \(Y_{ijk}\) is the log endpoint for subject \(j\) in sequence \(i\) during period \(k\), and \(\ell\) is the treatment administered in that period. The subject term represents stable between-subject differences; the residual \(\varepsilon_{ijk}\) represents remaining within-subject variation. In the usual model, subjects are independent, residuals are appropriately modeled (often as normal with common variance), and the treatment and period effects are additive. The model-based treatment interpretation also relies on the crossover design and conduct supporting a comparison of the assigned current treatments; a pattern in a summary table cannot establish that assumption.
The sequence contrast is between randomized groups, whereas period and treatment are observed within subjects under different combinations. Consequently, a sequence estimate can be imprecise and can reflect chance differences between groups. It is not a substitute for the within-subject treatment estimate. The period estimate describes an average shift from one period to another; it does not identify why that shift occurred.
In a standard 2×2 crossover, treatment-by-period interaction is not separately estimable from sequence. For example, the combination of test in period 1 and reference in period 2 identifies the TR sequence, while the opposite combination identifies RT. Adding a separate interaction term would therefore confound that term with sequence. Do not interpret a reported “interaction” from a model that does not have independent design information to estimate it.
Worked Example: Reading the Four Treatment-by-Period Means
Suppose a hypothetical balanced study reports the following mean log-AUC values. Assume the same number of evaluable subjects in each sequence, and that the entries are descriptive means rather than individual observations.
| Sequence | Period 1 | Period 2 | Sequence average |
|---|---|---|---|
| TR | T: 4.60 | R: 4.50 | 4.55 |
| RT | R: 4.30 | T: 4.55 | 4.425 |
The treatment means, averaging equally over the two sequence-period cells for each treatment, are \(T=(4.60+4.55)/2=4.575\) and \(R=(4.30+4.50)/2=4.400\). The descriptive treatment contrast is \(4.575-4.400=0.175\) log units.
The period means are \(P1=(4.60+4.30)/2=4.450\) and \(P2=(4.50+4.55)/2=4.525\). The descriptive period shift is \(4.525-4.450=0.075\) log units. The sequence averages are 4.550 for TR and 4.425 for RT, a between-sequence difference of 0.125 log units.
These three arithmetic contrasts describe different features of the table. In particular, the 0.125 sequence difference is not the treatment effect: it compares different groups of subjects. The treatment contrast is based on the test-reference comparisons within each sequence, combined across sequences. These summary means alone provide no standard error or confidence interval, so they do not establish statistical precision or a bioequivalence conclusion.
Why the Balanced Treatment Contrast Handles a Period Shift
The role of period can be seen directly from the two within-subject differences. Let \(\tau\) denote the test-minus-reference treatment effect on the log scale, and let \(\pi\) denote the period-2-minus-period-1 effect. Under an additive model, the expected contrast for a TR subject is \(\tau-\pi\): test is given in period 1 and reference in period 2. For an RT subject, the expected contrast is \(\tau+\pi\): test is given in period 2 and reference in period 1.
Thus, with balanced sequence information and the additive model, averaging the two sequence-specific treatment contrasts cancels the common period shift. The fitted model performs this comparison while also accounting for subject-level variability and the design. If sequence sizes are unequal, data are missing, or the model differs from this simple structure, the raw average need not equal the fitted treatment estimate; use the prespecified model output.
Worked Example: Opposite Sequence Contrasts Can Reflect Period
Consider another hypothetical balanced set of cell means, in log units: TR has T in period 1 at 5.00 and R in period 2 at 4.70; RT has R in period 1 at 4.40 and T in period 2 at 4.90.
For TR, the within-subject treatment contrast is \(5.00-4.70=0.30\). For RT, it is \(4.90-4.40=0.50\). Their balanced average is \((0.30+0.50)/2=0.40\) log units.
The period shifts point in opposite directions within the sequences: for TR, period 2 minus period 1 is \(4.70-5.00=-0.30\); for RT, it is \(4.90-4.40=0.50\). Their average is \((-0.30+0.50)/2=0.10\) log units. Under the additive model, the contrasts \(0.40-0.10=0.30\) for TR and \(0.40+0.10=0.50\) for RT match the observed sequence-specific treatment contrasts.
The differing contrasts are therefore not, on their own, evidence that the treatment effect truly changes by period. In this design, a separate treatment-by-period interaction cannot be disentangled from sequence. The pattern is a reason to inspect the summaries and model assumptions carefully, not a license to assign a unique cause.
Interpreting Model Output Alongside Summaries
A useful report puts the treatment estimate and its uncertainty first for the treatment comparison, then uses sequence and period terms to describe the data structure. Consider hypothetical model output with a test-minus-reference estimate of 0.035 log units, \(SE=0.025\), and \(df=38\). Suppose the 90% confidence interval is calculated using \(t_{0.95,38}\approx1.686\):
Exponentiating gives a test/reference geometric-mean ratio of \(\exp(0.035)=1.036\), with a 90% interval of approximately \((\exp(-0.00715),\exp(0.07715))=(0.993,\ 1.080)\). This interval describes uncertainty in the treatment ratio under the model. It is not an interval for the sequence or period effect.
Suppose the same output gives a period estimate of 0.060 log units with \(p=0.03\), and a sequence estimate of 0.040 log units with \(p=0.62\). The period estimate indicates an estimated average increase of 0.060 log units from period 1 to period 2, conditional on the model. The p-value indicates that, under the null hypothesis of no period effect and the model assumptions, a test statistic at least as extreme as the observed one would have probability 0.03. The sequence result does not show evidence against a zero sequence contrast at conventional levels, but it does not establish that the groups are identical. Neither p-value changes the treatment interval or determines the treatment conclusion by itself.
Worked Example: Translating a Sequence Estimate Without Overclaiming
Suppose a hypothetical model reports a TR-minus-RT sequence estimate of 0.18 log units, \(SE=0.08\), with \(p=0.04\). The sequence groups' estimated geometric-mean level ratio is \(\exp(0.18)=1.197\), or about 1.20. This describes a between-sequence difference in average level on the modeled scale.
It does not mean that test exposure is 20% higher than reference exposure. The sequence contrast compares subjects randomized to different orders; it is not the within-subject test-reference contrast. The appropriate treatment interpretation must come from the model's treatment estimate and its confidence interval. The sequence pattern may motivate a review of study conduct, baseline balance, and the data summaries, but it does not identify a cause or prove that the treatment comparison is biased.
Diagnostics, Alternatives, and Limitations
Before interpreting a pattern, confirm the labels and cell counts: each sequence should receive the intended order, and each reported mean should correspond to the correct treatment and period. Then inspect a table or plot with treatment-period cells separated by sequence. A period pattern that appears similarly across sequences is different descriptively from one driven by a single cell. Compare those displays with the fitted period and treatment estimates, their uncertainty, and the analysis population. This is a diagnostic description, not a replacement for the prespecified model.
A formal model-based analysis is generally preferable to treating cell means as independent, because the two observations from a subject are correlated. Conversely, descriptive tables are valuable for spotting patterns that a single model coefficient can hide. A raw treatment comparison that ignores period or subject structure is not automatically an acceptable alternative. If the design is unbalanced or has missing observations, simple equal-weight calculations may not reproduce the fitted estimate.
The sequence term is often less informative than its p-value might suggest: it compares randomized groups and can have limited precision. The period term is an average across the treatments and sequences under the model; it does not explain the mechanism behind a time trend. In the standard 2×2 design, sequence-specific contrast differences cannot be assigned uniquely to a separately estimated treatment-by-period interaction. These limitations make cautious wording essential.
Check sequence, treatment, period, and subject counts before interpreting any visible pattern.
Separate between-sequence level differences from within-subject period shifts and treatment contrasts.
Use the treatment estimate and interval for the treatment comparison; use sequence and period terms as supporting descriptions, not substitutes.
In a standard 2×2 design, do not claim a separately identified treatment-by-period interaction or a cause for a sequence pattern.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a balanced 2×2 crossover, what does a sequence effect compare, and why is it not the treatment effect?
- If the period-2 mean exceeds the period-1 mean, what can be concluded from that description alone?
- For hypothetical sequence-specific contrasts of 0.12 and 0.28 log units, calculate their balanced average treatment contrast.
- Why can a standard 2×2 design not separately estimate treatment-by-period interaction and sequence?
- A sequence test has \(p=0.70\). Does this establish that the sequence groups are identical?
Solutions. (1) It compares average outcomes between groups randomized to different treatment orders; it is between-subject, whereas the treatment contrast compares test with reference within the crossover structure. (2) It describes an average period shift in the summarized data; by itself it does not identify its cause or determine the treatment conclusion. (3) \((0.12+0.28)/2=0.20\) log units. (4) Each treatment-period pairing determines the assigned sequence, so a treatment-by-period contrast is confounded with sequence. (5) No. The result is not evidence against a zero sequence contrast at that test's resolution, but it does not prove equality.