What a Sequence Effect Describes
In a crossover study, participants are assigned to an ordered sequence of treatments. In a two-period study, for example, one sequence might receive treatment A in period 1 and treatment B in period 2 (AB), while the other receives B followed by A (BA). A sequence effect is a difference associated with these assigned orders: it compares outcomes for participants assigned to one sequence with outcomes for participants assigned to another.
That comparison is between participants, because each participant is assigned to only one sequence. It is not the same as a direct treatment effect, which concerns the difference in outcome attributable to receiving one treatment rather than another, with the design factors accounted for. The distinction matters: the fact that AB and BA participants have different average outcomes does not, by itself, show that treatment A has a different direct effect from treatment B.
A sequence effect can be useful as a model term: it represents a systematic difference between the groups assigned different orders, apart from the treatment and period terms specified in that model. But it is not a biological explanation for such a difference. Baseline participant characteristics, random variation, or features of the study and its response pattern may contribute. Random allocation helps make sequence groups comparable on average; it does not make their observed outcomes identical.
A Model for Separating the Terms
For a simple two-period, two-sequence design, consider a model for a continuous outcome. Let \(Y_{ij}\) be the outcome for participant \(i\) in period \(j\). Let \(S_i=+1\) for AB and \(S_i=-1\) for BA; let \(P_j=+1\) in period 1 and \(P_j=-1\) in period 2; and let \(T_{ij}=+1\) when the participant receives A and \(T_{ij}=-1\) when the participant receives B. One additive model is:
Here, \(\mu\) is the overall mean, \(\delta\) is the sequence coefficient, \(\pi\) is the period coefficient, and \(\tau\) is the direct treatment coefficient under this coding. The participant-specific term \(b_i\) represents a participant’s persistent departure from the overall mean. For this model, assume participants are independently sampled within their assigned sequence, \(b_i\) has mean zero, and the within-participant errors \(\varepsilon_{ij}\) have mean zero and are independent of \(b_i\). A random-intercept analysis commonly also assumes normally distributed random effects and errors, with a specified covariance structure. The additive mean model assumes the sequence, period, and treatment contributions are adequately represented by the stated terms.
With \(S_i\) coded as \(+1\) or \(-1\), the sequence-adjusted mean difference between AB and BA is \(2\delta\), not \(\delta\). The coefficient is half the difference because the two sequence codes differ by 2 units. The direct treatment contrast under the same coding is \(2\tau\). The terms are distinct even though treatment and period assignments are linked to sequence by the design.
Why the Sequence Contrast Has Its Own Meaning
The model’s distinction can be seen from the four population cell means. Write \(m_{A1}\) for the mean outcome under A in period 1, for example. Under the additive model above, the four means are:
Add the two means belonging to AB, then subtract the two belonging to BA. The overall mean, period contribution, and treatment contribution each cancel:
Each sequence average is a sum of two means divided by 2. Therefore, the difference between the AB and BA sequence averages is \(2\delta\). This derivation depends on the additive model: the treatment and period contributions must cancel in the stated way. If that model does not represent the outcome pattern adequately, this contrast may not have the intended interpretation as a single sequence term.
Worked Example: Distinguishing Sequence and Treatment Differences
Suppose hypothetical mean outcomes, measured in concentration units, are \(m_{A1}=105\), \(m_{B2}=97\), \(m_{B1}=95\), and \(m_{A2}=103\). Assume these means are adequately described by the additive model above. Calculate the sequence contrast and compare it with the across-period treatment averages.
The AB sequence average is \((105+97)/2=101\) concentration units. The BA sequence average is \((95+103)/2=99\) concentration units. Thus the AB-minus-BA sequence-average contrast is \(101-99=2\) concentration units. Equivalently, the sequence coefficient is:
The treatment A average across periods is \((105+103)/2=104\) concentration units; the treatment B average is \((95+97)/2=96\) concentration units. Their difference is 8 concentration units. The sequence contrast is 2 concentration units, while the treatment contrast under this simple additive setup is 8 concentration units. They are not interchangeable. These calculations describe the supplied means; without participant-level variability, they do not provide a standard error or an inferential conclusion.
Worked Examples in Interpretation
Worked Example: Reading a Sequence Assignment Correctly
In a hypothetical study, 24 participants are randomly allocated to AB and 24 to BA. An analyst reports that the mean outcome averaged over both periods is 12.4 units in AB and 11.9 units in BA. What does the difference of 0.5 units establish?
The arithmetic is \(12.4-11.9=0.5\) outcome units. This is an observed between-sequence difference in the reported averages. It is not, on its own, an estimate of the direct treatment effect: participants in each sequence receive both treatments, but in opposite periods. Nor does the observed difference alone establish that treatment order caused a difference. Interpretation depends on how the average was defined, participant-level variation, randomization, and whether the chosen model represents the data.
If the analysis defines and estimates a sequence coefficient while adjusting for treatment and period, the coefficient has a model-based sequence interpretation. If it merely compares unadjusted averages, it should be reported as an unadjusted sequence-group difference. In either case, the difference is not a substitute for the direct treatment comparison.
Worked Example: Randomization Does Not Guarantee Equal Observed Means
Consider a hypothetical study in which each participant has a complete two-period record. For each participant, calculate the average of the two outcomes. The AB participants’ averages are 100, 102, and 104 units; the BA participants’ averages are 95, 100, and 102 units. Calculate the observed sequence-group contrast.
The AB mean is \((100+102+104)/3=102\) units. The BA mean is \((95+100+102)/3=99\) units. The observed AB-minus-BA contrast is \(102-99=3\) units.
This is a descriptive difference between the averages of participants assigned to the two orders. It is not automatically a direct treatment effect or evidence that order has a reproducible effect. With only three participants per sequence in this illustration, participant-to-participant variability could have a large influence. Random assignment supports a causal interpretation of assigned sequence under appropriate design and analysis conditions, but a nonzero sample contrast can occur by chance. A precision assessment would require participant-level variability and an appropriate analysis.
Assumptions, Diagnostics, and Limits
A sequence coefficient is meaningful only relative to its model and data. Before interpreting it, check that sequence labels match the actual allocation, period and treatment are coded consistently, and the analysis uses the intended participant records. In a complete two-period design, each participant contributes one outcome under each assigned period-treatment combination. Missing records or departures from the planned schedule can change what the available averages represent.
For a model-based estimate, examine whether residual patterns suggest that the additive mean structure is inadequate, and whether a few participants dominate the between-sequence difference. Compare sequence-group participant characteristics and outcome distributions as descriptive diagnostics, while remembering that randomization does not require exact balance in a finite sample. A large observed difference may prompt careful review; it does not identify its cause. Conversely, a small estimated sequence contrast does not prove that the groups or all relevant aspects of the design are equivalent.
The interpretation also has limits. A sequence contrast is a between-participant comparison and is therefore exposed to participant variability. Its precision may be lower than that of within-participant treatment comparisons. The model’s sequence term summarizes a particular pattern of mean differences; it cannot explain why those differences arose. If the additive assumptions fail, one number may conceal differing patterns across periods or treatments. The sequence estimate should not be used as a standalone test of bioequivalence or as a replacement for the treatment estimand.
Choosing the Right Comparison
| Quantity | Comparison | Question answered |
|---|---|---|
| Unadjusted sequence-group difference | Observed averages for AB participants versus BA participants | How different were the reported group averages? |
| Model-based sequence contrast | Sequence term after accounting for specified treatment and period terms | Is there a sequence-group difference under the stated model? |
| Direct treatment contrast | Outcomes under A versus B, accounting for the crossover design | What is the treatment comparison of interest? |
These approaches are not competing answers to a single question. The unadjusted difference is simple but may not isolate the model’s sequence term. The model-based sequence contrast makes its assumptions explicit, but depends on those assumptions and does not reveal a mechanism. The direct treatment contrast targets a different estimand. Select and interpret each quantity according to the question it answers, rather than treating any sequence difference as evidence about the direct treatment effect.
Common Mistake
Check Your Understanding
Answer each question using the distinction between assigned order and treatment received.
- In an AB-versus-BA study, what kind of comparison is a difference between the mean outcomes of the two assigned groups?
- With sequence coded \(+1\) for AB and \(-1\) for BA, what is the AB-minus-BA contrast if the model’s sequence coefficient is \(\delta=0.7\) units?
- Why does random allocation not guarantee that the observed sequence-group means will be equal?
- Under the additive model, hypothetical cell means are \(m_{A1}=20\), \(m_{B2}=18\), \(m_{B1}=17\), and \(m_{A2}=19\) units. Calculate the sequence contrast.
Solutions. (1) It is a between-participant sequence-group comparison, not automatically a direct treatment contrast. (2) The contrast is \(2\delta=2(0.7)=1.4\) units. (3) Random allocation balances groups in expectation, but chance participant variation can produce unequal observed means in a finite sample. (4) The AB average is \((20+18)/2=19\) units; the BA average is \((17+19)/2=18\) units. The AB-minus-BA sequence contrast is \(19-18=1\) unit. Equivalently, \(\delta=[(20+18)-(17+19)]/4=0.5\) units, so \(2\delta=1\) unit.