Diagnosing Treatment-by-Period Interaction

Why Compare Treatment Effects Across Periods?

A two-period crossover gives each participant both treatments, but the treatment received in a given period depends on sequence. Participants in the RT sequence receive reference in period 1 and test in period 2; participants in the TR sequence receive test in period 1 and reference in period 2. A treatment contrast calculated separately in each period therefore compares participants from different sequences. If the sequence groups differ in their mean outcome for reasons apart from treatment, the two period-specific contrasts can differ even when the treatment effect itself is constant.

On the log scale, such differences matter because the common-effect crossover model represents one treatment contrast across periods. Comparing period-specific contrasts is a useful diagnostic for whether that representation is plausible. In a standard two-sequence, two-period design, however, a contrast difference is not automatically a separately identifiable treatment-by-period interaction: sequence differences can produce the same pattern.

Model and Definitions

Let the endpoint be measured on the log scale. Denote the population mean for the RT sequence, apart from period and treatment effects, by \(\alpha_{RT}\), and the corresponding mean for the TR sequence by \(\alpha_{TR}\). Define the sequence difference as \(q=\alpha_{RT}-\alpha_{TR}\), so positive \(q\) means a higher mean for RT than TR. Let \(\pi_1\) and \(\pi_2\) be the period effects, and let \(\theta_1\) and \(\theta_2\) be the test-minus-reference treatment effects in periods 1 and 2, respectively.

For clarity, consider the population cell means under an additive model for sequence and period, with no carryover:

The sequence means here represent population-level sequence differences; individual participants also vary around them. This description assumes treatment assignment follows the stated sequences, outcomes are analyzed on the log scale, and the period and treatment effects are additive as written. It also assumes no carryover effect contaminates an outcome in the period after treatment. These assumptions define the interpretation of the contrasts; they do not ensure that every component can be estimated separately from this design.

Deriving the Period-Specific Contrasts

In period 1, the test mean comes from TR and the reference mean from RT. In period 2, the test mean comes from RT and the reference mean from TR. Define \(d_1\) and \(d_2\) as the observed population treatment-minus-reference contrasts in those periods. Subtracting the relevant cell means gives:

The signs follow directly from which sequence receives each treatment in each period. Averaging the contrasts cancels the sequence difference, while subtracting them retains it:

Thus, the average of the two period-specific contrasts corresponds to the average treatment effect under this model. But their difference combines the treatment-by-period interaction contrast with twice the sequence difference. Under a common treatment effect, \(\theta_1=\theta_2=\delta\), the relationships reduce to \(d_1=\delta-q\) and \(d_2=\delta+q\). Unequal period-specific contrasts can therefore occur even when there is no treatment-by-period interaction.

What Is and Is Not Identifiable

In the standard two-sequence, two-period design, the four sequence-by-period cell means do not provide separate information for all of sequence, period, treatment, and treatment-by-period effects. The treatment-by-period pattern is tied to the sequence allocation: RT receives R then T, whereas TR receives T then R. In the contrast equations, there are two observed contrasts, \(d_1\) and \(d_2\), but three quantities that could explain them: \(\theta_1\), \(\theta_2\), and \(q\). Without an additional restriction or information, the interaction contrast and sequence difference cannot be separated from these contrasts alone.

This is an identifiability limitation, not merely a problem of a small sample or a large standard error. A model may produce estimates after imposing a constraint, such as assuming no interaction or setting the sequence difference to zero. Those estimates then depend on that assumption. The standard common-effect crossover model imposes \(\theta_1=\theta_2\); it cannot also provide an independent test of whether that equality holds using the same two-period design.

Using the Contrasts as a Diagnostic

A useful assessment begins with estimates of the treatment-minus-reference contrast in each period, their average, and their difference. Report the scale and units: here, each contrast and their difference are in log units. Examine whether the direction and magnitude of the contrasts are similar, and consider their uncertainty. Large, opposing period-specific estimates may signal that a single common effect is an inadequate description, but they may also reflect sequence differences, sampling variability, missing observations, or departures from the model assumptions.

When a treatment effect on the original positive outcome scale is needed, exponentiating a log-scale contrast gives a ratio of geometric means. For example, a log contrast of \(0.20\) corresponds to \(\exp(0.20)\approx1.221\), or a test-to-reference geometric-mean ratio of about 1.221. This transformation does not resolve the identifiability issue: a difference between period-specific log contrasts remains ambiguous unless the model or design supplies information to separate its sources.

Assumptions, Alternatives, and Failure Modes

The contrast interpretation relies on correct treatment and period labels, comparable endpoint definitions across periods, and the absence of carryover in the model above. It also relies on the additive mean structure used to define the effects. A missing period or a substantial sequence imbalance can make the estimated contrasts less precise or change what a simple difference in means represents. Such issues should be addressed by an appropriate model rather than by treating the raw contrast difference as a definitive test.

There are three related approaches, each answering a different question:

• Common-effect model: imposes one treatment effect across periods and estimates that common effect using the crossover structure. It is efficient when the assumption is appropriate, but does not independently test it in the standard two-period design.
• Period-specific contrasts: describe the treatment-minus-reference comparisons in each period and reveal whether the estimates look similar. Their difference is a diagnostic summary, not a uniquely identified interaction estimate when sequence differences are unrestricted.
• Models or designs with additional identifying information: may allow treatment-by-period effects to be distinguished from sequence differences, subject to their own assumptions. Any formal interaction test must use a model and design in which the interaction contrast is estimable, with its uncertainty calculated using the relevant covariance structure.

A common mistake is to reason, “The period 1 and period 2 contrasts differ, so the treatment effect changes by period.” The correction is to first check the contrast definitions and the treatment-to-sequence mapping, then recognize that \(d_1-d_2=(\theta_1-\theta_2)-2q\). In the standard design, a difference may arise from interaction, sequence difference, or both. Do not report a formal interaction test or a p-value for an interaction unless the model identifies that effect and the test uses an appropriate error structure.

The practical value of the diagnostic is therefore not that it automatically decides whether the common-effect model is valid. It makes the model’s common-effect assumption visible, highlights patterns that deserve investigation, and prevents a pooled estimate from concealing substantial period-specific differences. Interpretation should distinguish a descriptive pattern from a conclusion supported by an identifiable model.