Two Different Ways Treatment Can Affect an Outcome
In a crossover study, a participant receives treatments in more than one period. An outcome measured in a period can reflect the formulation given in that period, but it can also reflect an earlier treatment if that treatment has a residual influence. These are different scientific effects, even though both may contribute to the same observed measurement.
The distinction matters when interpreting a formulation comparison. If the outcome in a period is affected by the treatment received before it, an observed difference between current formulations need not be due solely to their direct effects. Conversely, the existence of a previous treatment does not by itself establish that carryover occurred: the earlier treatment must still influence the outcome being measured.
For a precise statement, let \(Y_i(t,h,p)\) denote the outcome that participant \(i\) would have in period \(p\) if the current formulation were \(t\) and the prior treatment history were \(h\). The history includes relevant earlier treatments and their timing. Let \(T\) and \(R\) denote test and reference formulations. A population-average direct contrast at a specified history and period can be written as
A carryover contrast for a specified current formulation can be written as
where \(h_T\) and \(h_R\) are the two histories being compared, such as prior exposure to \(T\) versus prior exposure to \(R\). These contrasts define different comparisons: the direct contrast changes the current formulation, while the carryover contrast changes the prior history.
These definitions are conditional on the conditions named in the contrast. A direct effect can differ across histories, and a carryover effect can differ across current formulations. Calling an effect “direct” does not imply that it is constant in every period or independent of prior exposure.
What Must Be Held Fixed
The comparisons are causal in meaning only when the other relevant conditions are held fixed, or are made comparable by the design and analysis. For a direct-effect comparison, the prior history, period, outcome definition, and relevant measurement conditions must be the same. For a carryover comparison, the current formulation and those same conditions must be the same while the prior history changes.
In practice, researchers do not observe both potential outcomes for a participant under the same conditions. Randomization of treatment sequences supports comparability of treatment groups, but it does not make every possible history-specific contrast directly observable. Interpreting an observed contrast as one of these effects requires appropriate design and assumptions. Among the standard causal assumptions are consistency (the observed outcome corresponds to the treatment history actually received), no interference between participants, and adequate randomization or adjustment for relevant differences.
A period effect is different from either treatment effect: it is a change associated with the period itself, such as a systematic shift in measurement conditions. A sequence effect is associated with the assigned order of treatments. Neither term is a synonym for carryover. A period or sequence difference may coexist with carryover, and observing one does not establish the other.
Worked Example: A Direct Effect on Log Exposure
Suppose a hypothetical, period-standardized comparison evaluates the log-transformed exposure endpoint for \(T\) and \(R\) under the same treatment-naive history. For 12 matched participant comparisons, the sum of the log differences, defined as log exposure on \(T\) minus log exposure on \(R\), is \(0.84\). Assume the comparisons represent the same period and measurement conditions, and that there is no prior treatment whose residual effect contributes to these outcomes.
Calculate the average contrast. The mean log difference is
This is a direct contrast for the specified treatment-naive history: current formulation changes from \(R\) to \(T\), while history and conditions are held fixed. Because the endpoint is on the natural-log scale, exponentiating the contrast gives the geometric-mean ratio:
Under the stated hypothetical conditions, average exposure on \(T\) is about \(1.073\) times that on \(R\), or about \(7.25\%\) higher. This example illustrates the scale conversion, not a bioequivalence decision: no acceptance criterion or uncertainty interval has been supplied. If prior treatment history differed between the two current-formulation groups, the contrast would not automatically isolate this direct effect.
How a Prior Treatment Can Leave a Residual Effect
Carryover can arise through more than one biological pathway. Drug remaining in the body can contribute to a later concentration measurement. A prior treatment can also produce a persistent pharmacodynamic response or another biological change that affects the next period’s endpoint. The key question is whether the earlier treatment changes the outcome now, under the same current treatment and measurement conditions.
Worked Example: A Persistent Pharmacodynamic Response
Consider a hypothetical study of a treatment that reduces systolic blood pressure. The endpoint is the reduction in mmHg after the current-period dose. Participants now receive \(R\), and the measurement schedule and current dose are the same in both groups. Suppose standardized average reductions are \(8.0\) mmHg when the previous treatment was \(T\), and \(5.0\) mmHg when the previous treatment was \(R\). Assume the groups are otherwise comparable and the difference is attributed to a persistent response to prior treatment.
Calculate the history contrast. With the current formulation held at \(R\),
The hypothetical carryover contrast is a \(3.0\) mmHg greater reduction after prior \(T\) than after prior \(R\). It is not a direct effect of current \(T\) versus \(R\), because the current formulation was \(R\) in both conditions. If the current formulation or period also differed between groups, that contrast would mix the history difference with other changes and would not isolate carryover.
An endpoint may also contain both effects. For instance, current-period exposure may depend on the formulation administered in that period while residual drug from the preceding period contributes to measured concentrations. In that situation, the measured endpoint is not a pure measure of the current dose’s direct effect unless the residual contribution is absent or appropriately handled.
Worked Example: Direct and Residual Effects Together
Worked Example: Why the Period-2 Contrast Can Mislead
Consider a hypothetical two-period, two-sequence crossover with sequences \(TR\) and \(RT\). For illustration only, suppose the endpoint is measured in arbitrary units and has a common baseline of 10. Assume a constant direct effect of \(+2\) units for current \(T\) relative to current \(R\), a period-2 shift of \(+1\) unit, and a residual effect of \(+3\) units in period 2 after prior \(T\). Assume no residual effect after prior \(R\), no interaction among these contributions, and no random error. These are stipulated values for demonstrating how the effects combine, not estimates from study data.
In period 1 there is no earlier study treatment. Thus the expected outcomes are 12 for current \(T\) and 10 for current \(R\). In period 2, the \(TR\) sequence receives \(R\) after \(T\), while the \(RT\) sequence receives \(T\) after \(R\). Applying the stated assumptions gives:
| Sequence and period | Current formulation | Prior treatment | Calculation | Expected outcome |
|---|---|---|---|---|
| TR, period 1 | T | None | 10 + 2 | 12 |
| RT, period 1 | R | None | 10 | 10 |
| TR, period 2 | R | T | 10 + 1 + 3 | 14 |
| RT, period 2 | T | R | 10 + 1 + 2 | 13 |
The period-2 contrast comparing current \(T\) with current \(R\) is \(13-14=-1\) unit. Yet the stipulated direct effect is \(+2\) units. The discrepancy occurs because the \(T\)-after-\(R\) observation and the \(R\)-after-\(T\) observation have different prior histories; the latter also has the stipulated residual \(+3\) units. In this example the period shift cancels in the comparison, but the carryover contribution does not.
This arithmetic is not a general formula for every crossover analysis. It demonstrates why a current-treatment contrast is not automatically a direct-effect contrast when the histories differ. In a two-period, two-sequence design, current treatment and prior treatment are linked in period 2: \(T\) follows \(R\), and \(R\) follows \(T\). Without additional restrictions or information, those contributions cannot be separated freely from the observed data. The earlier material on why carryover is not separately identifiable in a two-period design develops that design limitation.
Assumptions, Diagnostics, and Practical Choices
The examples rely on clearly stated conditions. A causal interpretation requires that the compared treatment histories be well defined, that outcomes be measured in comparable ways, and that relevant differences between histories not explain the contrast. In a randomized crossover, sequence randomization helps balance participant characteristics, but it does not guarantee that a residual effect is absent. Nor does a balanced schedule alone establish that all history-specific effects are equal.
Practical warning signs include unexpectedly different pre-dose measurements across periods, an outcome that plausibly persists beyond the dosing period, or a contrast that changes substantially when the analysis is restricted to outcomes measured before a subsequent dose. Such observations can prompt investigation, but none by itself proves a carryover effect. For example, a pre-dose difference could also reflect period conditions, measurement variation, or other changes. A diagnostic is evidence to interpret, not a substitute for defining the effect and comparison.
The main design alternatives address different needs. A sufficiently long washout is intended to reduce residual influence before the next period; it does not prove that the influence has vanished. A parallel-group study avoids giving the same participant different formulations in successive periods, but loses the within-subject comparison that motivates a crossover. Replicated crossover designs can provide more information about within-participant variation and treatment histories, at the cost of additional periods and burden. The suitable choice depends on the expected persistence of treatment effects, the endpoint, and the scientific question.
A common mistake is to call every difference between periods “carryover,” or to treat any observed difference after prior \(T\) as proof of a residual treatment effect. The correction is to specify the comparison: a direct effect changes the current formulation with history fixed; a carryover effect changes prior history with the current formulation fixed. If both change, the observed contrast may combine them.
Check Your Understanding
Answer each question by naming what changes and what is held fixed.
- A participant receives current formulation \(T\) in one comparison and current formulation \(R\) in another, with the same prior history and period conditions. Which effect is being contrasted?
- Participants all receive current \(R\). The outcome differs according to whether their prior treatment was \(T\) or \(R\), with other conditions comparable. Which effect is being contrasted?
- Why is a period-2 comparison of current \(T\) and current \(R\) potentially affected by carryover in a two-sequence \(TR/RT\) design?
- Does a difference in pre-dose measurements by itself establish carryover? Explain.
Solutions. (1) A direct effect, because the current formulation changes while prior history and period conditions are fixed. (2) A carryover effect, because prior history changes while current \(R\) is fixed. (3) In period 2, current \(T\) follows \(R\), whereas current \(R\) follows \(T\); therefore, current formulation and prior history differ together, so a contrast can include both direct and residual contributions. (4) No. A pre-dose difference is a possible warning sign, but period conditions, measurement variation, or other changes could also produce it; interpretation requires additional evidence and a specified comparison.