Why a Period Effect Is Not Necessarily an Interaction
In a crossover study, outcomes may change from one period to another for reasons that affect both treatments: changes in disease activity, study conditions, or the participant’s response over time. This is a period effect. A different question is whether the treatment contrast itself changes across periods. That is treatment-by-period interaction.
The distinction matters when a single treatment effect is used to summarize a study. If the treatment contrast is similar in every period, a common effect may be a reasonable summary. If it changes, averaging period-specific effects can hide a clinically relevant pattern. On a log exposure scale, for example, the treatment contrast describes a log ratio; a change in that contrast means the treatment ratio differs by period.
A Model and a Short Justification
A simple model makes the distinction explicit. Let \(Y_{ij}\) be the response for participant \(i\) in period \(j\), let \(b_i\) represent a participant-specific level, and let \(x_{ij}=1\) when the participant receives T and \(0\) when the participant receives R. A period-specific direct-effect model is
Here \(\pi_j\) is the common period effect, \(\tau\) is a treatment component, and \(\gamma_j\) allows the treatment contrast to vary by period. A constraint such as \(\sum_j\gamma_j=0\) is used to define the overall treatment component and period-specific deviations uniquely. The implied contrast in period \(j\) is \(\delta_j=\tau+\gamma_j\). If every \(\gamma_j=0\), the treatment contrast is common across periods; the \(\pi_j\) values may still differ.
This model is a starting point, not a guarantee that the interaction can be estimated from every schedule. Its interpretation as a direct treatment-by-period pattern requires comparable outcome measurement and a suitable model for repeated responses. In particular, residual carryover must be negligible or represented appropriately: if the previous treatment still affects the current response, an apparent period-specific direct effect may partly reflect treatment history. The design must also provide enough independent treatment-period information for the model terms to be estimable.
The contrast that directly expresses interaction between two periods is the difference between their treatment contrasts:
To see why a common period effect cancels, suppose both treatments rise by the same amount \(c\) from period 1 to period 2. Then \(\mu_{T2}=\mu_{T1}+c\) and \(\mu_{R2}=\mu_{R1}+c\). Substitution gives
Thus, a shift in both treatment means is not by itself evidence of interaction. What matters is whether the shift differs between treatments.
Worked Example: A Common Period Shift
Suppose hypothetical mean log-exposure responses are as follows. The log scale is used so that a treatment contrast corresponds to a log ratio.
| Period | Mean under R | Mean under T |
|---|---|---|
| 1 | 5.00 | 5.20 |
| 2 | 5.50 | 5.70 |
In period 1, the treatment contrast is \(\delta_1=5.20-5.00=0.20\) log units. In period 2, it is \(\delta_2=5.70-5.50=0.20\) log units. Therefore,
Both treatments rose by \(0.50\) log units between periods. The treatment contrast did not change, so these means show a common period shift, not treatment-by-period interaction. On the original exposure-ratio scale, the T-to-R ratio in either period is \(\exp(0.20)\approx1.221\). This is a descriptive calculation from hypothetical means, not an inferential test.
Worked Example: A Changing Treatment Contrast
Now consider different hypothetical means on the same log-exposure scale.
| Period | Mean under R | Mean under T |
|---|---|---|
| 1 | 5.00 | 5.20 |
| 2 | 5.50 | 5.40 |
The period-specific contrasts are \(\delta_1=5.20-5.00=0.20\) log units and \(\delta_2=5.40-5.50=-0.10\) log units. Their difference is
The T-to-R ratios corresponding to the two contrasts are \(\exp(0.20)\approx1.221\) in period 1 and \(\exp(-0.10)\approx0.9048\) in period 2. The contrast has changed, so these means display a treatment-by-period pattern. R increased by \(0.50\) log units across periods, whereas T increased by \(0.20\) log units. The difference between those increases, \(0.20-0.50=-0.30\), matches the interaction contrast. The pattern alone does not establish its cause or statistical uncertainty.
What Schedule Patterns Can Reveal
To investigate interaction, a design needs information about both treatments in the periods of interest. If T is observed only in one period and R only in another, changes in treatment and changes in period cannot be separated by simply comparing those observations. Having each treatment represented in each period makes period-specific contrasts descriptively possible, but representation is not the same as independent estimability in a crossover model.
For instance, consider the standard two-period AB/BA schedule, where A and B denote the two treatments. Every participant receives both treatments, and across the two sequences each treatment appears in each period. That balance is useful for estimating a common treatment contrast under the usual model assumptions. However, with participant-specific effects, the within-participant changes supply only two sequence-specific patterns. A model that also allows unrestricted period effects, treatment effects, and treatment-by-period interaction cannot generally separate all those components from this schedule alone. A fitted interaction may therefore depend on additional assumptions or constraints. This is a design limitation, not evidence that the interaction is zero.
Schedules that vary treatment allocation across periods and provide additional independent comparisons can help reveal whether contrasts change over time. Replicated or longer schedules may offer such comparisons, but more periods also mean more treatment history to consider. A useful schedule review asks not only whether each treatment occurs in each period, but also whether the planned model’s treatment, period, and interaction terms can be distinguished given the actual sequence patterns.
Worked Example: Reading Treatment-Period Coverage
Suppose a hypothetical three-period study randomizes 24 participants equally to two schedules, T-R-T and R-T-R. The table shows how many participants receive each treatment in each period.
| Period | Receiving T | Receiving R |
|---|---|---|
| 1 | 12 | 12 |
| 2 | 12 | 12 |
| 3 | 12 | 12 |
For example, in period 1 the T-R-T sequence contributes 12 T observations and the R-T-R sequence contributes 12 R observations. In period 2 the assignments reverse, giving 12 observations under each treatment again; period 3 returns to the period 1 pattern. Thus the design has treatment coverage in every period, a useful prerequisite for examining period-specific contrasts.
Coverage alone does not establish that an interaction estimate is unbiased or separately estimable under a chosen model. The T observation in period 3 follows T for one sequence but R for the other; period-specific direct effects could be entangled with previous-treatment effects if carryover remains. The schedule is therefore a prompt for a model-and-assumption check, not a standalone proof of interaction.
Assumptions, Diagnostics, and Alternatives
Before interpreting period-specific contrasts, specify the response scale and the treatment history represented by the model. On a log scale, differences correspond to log ratios; on the original scale, differences and ratios have different meanings. The direct-effect interpretation also relies on appropriate randomization and on handling repeated measurements correctly. If carryover is plausible, it must be addressed through the design, an appropriate model, or cautious interpretation rather than silently absorbed into a direct interaction term.
Useful checks include plotting the treatment contrast by period with uncertainty intervals, inspecting period-specific treatment means, and examining whether results depend heavily on a small number of sequences or participants. A conspicuous contrast change may be clinically important, but a noisy estimate may also fluctuate by chance. Conversely, a wide interval that includes zero interaction does not prove the contrasts are identical. Model diagnostics should also look for residual patterns by period and treatment history, and for estimates that change substantially when plausible carryover terms or covariance assumptions are considered.
The main alternatives answer different questions. A model with treatment and period main effects but no interaction assumes one common treatment contrast and is more parsimonious. A model with treatment-by-period terms allows the contrast to vary, but requires enough design information and may estimate the contrasts less precisely. Reporting period-specific contrasts can make the pattern transparent; if the design cannot support their separate estimation, that limitation should be stated instead of presenting constrained estimates as if they were unrestricted. A preliminary test for interaction should not be used as an automatic switch between models: limited power can fail to detect meaningful variation, while a small study can produce unstable apparent differences.
Common Mistakes and Practical Importance
- Calling any period change an interaction. If both treatment means rise by the same amount, their difference is unchanged. Compare treatment contrasts, not just period means.
- Assuming balance guarantees estimability. Seeing both treatments in each period is helpful, but the sequence structure and repeated-measures model determine whether interaction is separately estimable.
- Treating a point estimate as proof. A nonzero interaction contrast describes the estimated pattern. Its uncertainty and the adequacy of the model still matter.
- Interpreting a direct interaction despite carryover. If previous treatment affects the current response, the apparent treatment-by-period pattern may partly reflect treatment history.
The practical question is not simply whether outcomes differ across periods. It is whether the treatment contrast changes across periods, on a stated scale and under a design and model capable of supporting that comparison. This distinction helps prevent a common period shift from being mistaken for treatment-specific variation—and prevents a real, potentially important variation from being hidden inside one average contrast.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In period 1, the T and R means are 8.4 and 8.0; in period 2, they are 9.1 and 8.7. What are the two treatment contrasts, and is their difference evidence of a changing point-estimate pattern?
- Suppose \(\delta_1=0.15\) and \(\delta_2=0.35\) log units. Calculate \(D_{2,1}\), and state whether the T-to-R ratio is larger in period 1 or period 2.
- Why is the statement “both treatments increased between periods, so there is interaction” incorrect?
- A schedule places T in period 1 only and R in period 2 only. What makes a simple treatment comparison problematic?
Solutions. For question 1, \(\delta_1=8.4-8.0=0.4\) and \(\delta_2=9.1-8.7=0.4\); their difference is \(0.4-0.4=0\), so the point estimates show a common shift, not a changing contrast. For question 2, \(D_{2,1}=0.35-0.15=0.20\) log units. Since \(\exp(0.35)>\exp(0.15)\), the estimated T-to-R ratio is larger in period 2. For question 3, interaction concerns whether the treatment contrast changes. Equal increases in both treatments leave that contrast unchanged. For question 4, treatment and period are aligned: the observed difference could reflect a treatment effect, a period effect, or both, so the schedule does not provide a direct separation through that comparison.