A report says eight participants completed both study conditions, producing sixteen measurements. Before judging the result, identify what was randomized and what was compared. Sixteen measurements do not become sixteen independent participants.
The learning task is to reconstruct a within-person comparison, then ask whether time, previous exposure or incomplete follow-up could change its meaning. This article concerns the simple two-condition, two-period design; it does not provide a treatment schedule.
Start with the sequence
In an AB/BA crossover, participants are randomized to receive A then B, or B then A. Each person contributes a comparison between conditions. This can remove stable between-person differences from that contrast. The design needs a setting where the condition is sufficiently stable and the intervention effect sufficiently reversible. Cochrane Handbook, section 23.2
Write two labels on your reading notes:
- Sequence: the assigned order, AB or BA.
- Contrast: the outcome under B minus the outcome under A.
The contrast keeps the same direction even when the order changes. “Second minus first” answers a different question in the BA sequence.
Fictional worked example: two temporary display settings
An original fictional rehabilitation research exercise compares display settings A and B during a simulated task. Higher scores indicate more successful task performance. The score is invented for arithmetic and has no established clinical interpretation.
Eight participants are randomized equally between sequences. Everyone completes both periods. For this first calculation, assume independent participants, negligible carryover, negligible period effects and comparable measurement conditions. Those assumptions are supplied for the exercise; the table cannot verify them.
| Participant | Sequence | Score under A | Score under B | B minus A |
|---|---|---|---|---|
| 1 | AB | 40 | 43 | 3 |
| 2 | AB | 50 | 55 | 5 |
| 3 | AB | 60 | 64 | 4 |
| 4 | AB | 70 | 76 | 6 |
| 5 | BA | 42 | 45 | 3 |
| 6 | BA | 52 | 57 | 5 |
| 7 | BA | 62 | 66 | 4 |
| 8 | BA | 72 | 78 | 6 |
The differences sum to 36 points. The mean difference is therefore 36 ÷ 8 = 4.5 points. Participant 5 contributes +3 even though B occurred first: 45 minus 42, not 42 minus 45.
The mean A score is 56 and the mean B score is 60.5, giving the same point estimate. However, those two means alone do not supply the uncertainty of the paired comparison. That depends on how the two measurements travel together within people. Under the stated conditions, a paired analysis uses the participant-level differences. Cochrane Handbook, section 23.2.5
Here, the sample standard deviation of the eight differences is approximately 1.195 points. The standard error is:
1.195 ÷ √8 = 0.423 points.
For illustration, also assuming approximately normally distributed differences, a paired t interval uses seven degrees of freedom. Its approximate 95% confidence interval is 4.5 ± 2.365 × 0.423, or 3.50 to 5.50 points.
This small, constructed dataset demonstrates a calculation. It does not establish a useful clinical effect, validate the invented score or justify using eight participants in a real trial. A narrow interval under a model cannot establish that the model's assumptions hold.
Separate a period effect from carryover
A period effect is a systematic difference associated with when measurement occurs. Carryover is an effect of the earlier intervention that persists into the later period. Sequence balance can address a common additive period effect; it does not establish absence of carryover. Cochrane Handbook, section 23.2.3
Consider a separate fictional arithmetic model. Suppose B adds four points relative to A, while familiarity adds two points in period two regardless of the display:
- AB produces B minus A = 4 + 2 = 6.
- BA produces B minus A = 4 − 2 = 2.
- Equal sequence weighting gives (6 + 2) ÷ 2 = 4.
The average recovers the stipulated condition effect. If everyone followed AB, the six-point difference would combine the display effect with familiarity. If completion became unequal between sequences, simply pooling the available contrasts could also lose the cancellation.
This algebra assumes the same additive period effect and no residual effect of the earlier display. Actual analyses may need participant, period and intervention terms rather than the simple paired calculation above. The crossover CONSORT extension asks authors to make their within-person analysis explicit. Dwan and colleagues, 2019
A washout is an assumption to justify
Imagine replacing temporary display settings with teaching a lasting task strategy. After learning it under B, a participant may use it during A. Removing the display does not remove the learned strategy. A longer gap is not automatically an adequate solution when the intended intervention effect is durable.
The relevant question is whether the first exposure can still influence the outcome measured in period two. The 2019 crossover reporting extension calls for a rationale for negligible carryover and cautions against testing for carryover and then choosing only first-period data according to that test. Dwan and colleagues
For the display example, ask what supports reversibility: prior task familiarization, an explanation of the mechanism, timing of outcome measurement and evidence about lingering effects. These are appraisal questions, not a proposed clinical protocol. A nonsignificant carryover test would not prove that the residual effect is zero.
Follow incomplete periods through the analysis
Now change the fictional case: participant 8 completes B first but leaves before A. There is no observed B-minus-A contrast for that person. Entering zero would invent equal outcomes; carrying the B score into the A column would invent a measurement under the other condition.
A complete-pair calculation now describes seven completers. Whether it estimates the intended population effect depends on why the pair is missing. Leaving because the task was unusually difficult raises a different concern from an unrelated scheduling conflict, although either explanation needs evidence.
Models can use partially observed records, but their validity depends on the model and missing-data assumptions. The crossover reporting extension discusses mixed models under a missing-at-random assumption and sensitivity analysis, alongside participant flow by sequence and period. Dwan and colleagues
For a practical reading note, record how many people entered each sequence, completed each period and contributed to the reported estimate. Then describe which missing outcomes could plausibly change the conclusion.
Transfer exercise: interpret two sequence contrasts
In another original fictional study, equal numbers complete AB and BA. The mean B-minus-A contrast is seven points in AB and three points in BA. Assume no carryover, a common additive condition effect and a common additive period-two effect.
Which explanation fits those stipulated conditions?
A. B adds seven points; BA results should be discarded.
B. B adds five points, and period two adds two points.
C. B adds three points; AB results reflect only bias.
D. The four-point sequence difference proves carryover.
Feedback: B. Let the condition effect be δ and the period effect be π. The two equations are δ + π = 7 and δ − π = 3. Adding them gives 2δ = 10, so δ = 5. Subtracting gives 2π = 4, so π = 2.
A and C each mistake one sequence's contrast for the isolated condition effect. D contradicts the exercise's assumptions and treats a difference that period effects can explain as proof of carryover. In real data, uncertainty and alternative mechanisms still require evaluation; this algebra cannot diagnose the mechanism by itself.
Take this to your next paper
Write one sentence identifying the randomized sequence, one defining the signed within-person contrast, and one stating why later measurements can reasonably be interpreted after earlier exposure. Add the completion counts. If any element is missing, keep the effect estimate provisional while you examine the methods.
Sources and further reading
- Cochrane Handbook, Chapter 23: Including variants on randomized trials, section 23.2.
- Dwan et al. (2019): CONSORT 2010 statement—extension to randomised crossover trials. This is the dated crossover extension, not a claim about the latest general CONSORT statement.