Crossover Design

Crossover Design in Clinical Trials and Health Technology Assessment

In a crossover design the treatment comparison is made inside each person, because every participant is randomised to an order of treatments and is measured under each one in turn. Stable differences between people cancel out of that comparison, so a crossover trial can reach the same statistical power as a parallel-group trial with a fraction of the participants. The price is a set of conditions that many diseases fail, chiefly a chronic and stable condition and treatment effects that wear off before the next period. Health economists meet the design when its results enter a meta-analysis or a cost-effectiveness model, and they also meet the word "crossover" in cancer appraisals, where it often means something else, namely control patients switching onto the new drug.

Sequences, periods and the AB/BA design

The simplest form is the 2x2 or AB/BA design. ICH E9 describes it as each subject receiving each of two treatments in randomised order in two successive treatment periods, often separated by a washout period. Participants are randomised to a sequence rather than to a single treatment.

ICH E9 names the most common extension as comparing n treatments in n periods, with each subject receiving all n, and notes variations in which subjects receive a subset of treatments or in which treatments are repeated within a subject. The CONSORT extension for crossover trials lists bioequivalence studies, Balaam's design, extra period designs, n-of-1 designs and incomplete block designs among the less frequent versions. For bioequivalence of two immediate-release oral formulations, ICH M13A recommends a randomised, single-dose crossover design. An N-of-1 trial applies the same logic to a single patient over many periods. In a cluster crossover trial, whole clusters such as wards switch between interventions, and the Cochrane Handbook notes that the analysis must then respect both the pairing of periods and the similarity of people within clusters.

ICH E9 adds that block randomisation gives crossover trials the means of obtaining balanced designs, with greater efficiency and easier interpretation. Balance matters because the Cochrane Handbook explains that equal numbers in the two sequences let period effects cancel, while unequal allocation requires period effects in the statistical model. The Handbook also points out that the first period of a crossover trial is in effect a parallel-group trial.

Why each participant acting as their own control saves participants

The gain comes from removing between-person variance from the treatment contrast. A simple model splits each measurement into a participant effect and a within-person error.

$$Y_{ik} = \mu_k + s_i + e_{ik}$$

where $Y_{ik}$ is the outcome for participant $i$ under treatment $k$, $\mu_k$ is the mean outcome under treatment $k$, $s_i$ is the participant effect with between-person variance $\sigma_s^2$, and $e_{ik}$ is within-person error with variance $\sigma_e^2$. The participant effect appears in both of a person's measurements, so it drops out of their difference.

$$\frac{N_{parallel}}{N_{crossover}} = \frac{2(\sigma_s^2 + \sigma_e^2)}{\sigma_e^2} = \frac{2}{1 - \rho}$$

where $N_{parallel}$ and $N_{crossover}$ are the total participants each design needs for the same precision with no carryover, and $\rho = \sigma_s^2 / (\sigma_s^2 + \sigma_e^2)$ is the within-person correlation between measurements under the two treatments. With $N/2$ participants per arm, the parallel trial's difference in group means has variance four times $(\sigma_s^2 + \sigma_e^2)/N$, while the crossover's mean paired difference has variance twice $\sigma_e^2/N$.

Wellek and Blettner give the practical scale. When between-subject variance is twice the variance due to measurement error, a parallel-group study needs six times as many patients for the same power, which is what the ratio gives with $\rho = 2/3$. They add that each crossover patient supplies twice as many measurements and must complete two periods with a washout between them. At a correlation of zero the crossover still needs half the participants but the same number of measurements, which is why the Cochrane Handbook states that a correlation of zero or below gives no statistical benefit over a parallel-group design. The CONSORT extension accordingly asks for a sample size calculation that accounts for within-participant variability.

Worked example comparing sample size and trial cost

The figures are illustrative. A trial compares two inhaled treatments on a symptom score with a target difference of 5 points, two-sided significance of 5% and power of 80%, so $z_{0.975} + z_{0.80} = 1.96 + 0.8416 = 2.8016$, and its square is 7.849. Between-person variance is 60 and within-person variance is 40, so the total variance is 100 and $\rho$ is 0.6. The normal approximation is used throughout.

Step 1 (parallel-group trial). Each arm needs $n = 2 \times 100 \times 7.849 / 25 = 62.8$ patients, rounded up to 63, so 126 participants and 126 measurements in total.

Step 2 (crossover trial). The variance of a paired difference is $\sigma_d^2 = 2 \times 40 = 80$, so the trial needs $N = 80 \times 7.849 / 25 = 25.1$, rounded up to 26 participants, who supply 52 measurements. The ratio of 126 to 26 is about 4.8, close to the theoretical ratio of 5 given by the formula above with $\rho = 0.6$ once rounding is allowed for.

Step 3 (attrition). If 15% of participants are expected not to complete both periods, recruitment rises to $N = 26 / 0.85 = 30.6$, rounded up to 31. The parallel trial is assumed to lose no participants, which slightly favours it.

Step 4 (cost). Suppose a participant costs GBP 3,000 in the parallel trial and GBP 5,000 in the crossover trial, which runs two periods and a washout. The parallel trial costs 126 times GBP 3,000, or GBP 378,000, and the crossover costs 31 times GBP 5,000, or GBP 155,000.

The crossover is cheaper here because $\rho$ is high. If $\rho$ were 0.2, within-person variance would be 80, the paired variance 160 and the crossover would need $N = 160 \times 7.849 / 25 = 50.2$, rounded up to 51 participants, or 60 after attrition at a cost of GBP 300,000. That is still below the parallel trial's GBP 378,000, but the saving has shrunk and the timetable is longer.

Conditions the design depends on

ICH E9 states that the disease should be chronic and stable, that the relevant effects of the medication should develop fully within the treatment period, and that washout periods should be long enough for complete reversibility, with these conditions established in advance from prior information. The Cochrane Handbook gives asthma, rheumatoid arthritis and epilepsy as fields where the design is used. It rules the design out when the condition evolves over time, as in a degenerative or cyclic disorder, when a treatment such as a vaccine causes long-term change, when a drug's elimination half-life is very long, and when washout itself causes withdrawal or rebound.

The main threat is the carryover effect, in which an earlier treatment still influences outcomes in a later period. ICH E9 notes that in the 2x2 design carryover cannot be told apart from treatment-by-period interaction and that the test for it lacks power, which is why the decision has to rest on knowledge of the drug and disease rather than on the data. Outcomes that cannot recur, such as death, generally rule the design out.

Dropout is the other weak point. ICH E9 advises that the design should generally be restricted to situations where losses of subjects are expected to be small, and the Cochrane Handbook links the higher dropout risk to the longer duration of crossover trials.

Analysing a crossover trial

When neither carryover nor period effects are thought to be a problem, the Cochrane Handbook gives the paired t-test as an appropriate analysis of continuous data from an AB/BA trial, using each participant's difference between the two treatments. Models that include period terms, often mixed effects models, extend this to unbalanced sequences and more periods, and the CONSORT extension asks that estimation account for correlation between repeated measurements in the same person.

The Handbook warns against the two-stage analysis, in which a carryover test decides whether to keep both periods or only the first. It describes this approach as discredited, citing Freeman, and treats crossover trials reporting only first-period data as at risk of bias.

Crossover trials in meta-analysis and economic evaluation

Crossover results often reach a decision through evidence synthesis. The Cochrane Handbook warns that analysing the treatment periods as if they came from a parallel-group trial is a unit-of-analysis error, giving confidence intervals that are too wide and too little weight to the trial. When only the separate standard deviations are reported, a paired analysis can be approximated by imputing a correlation.

$$SD_{diff} = \sqrt{SD_E^2 + SD_C^2 - 2 \times Corr \times SD_E \times SD_C}$$

where $SD_{diff}$ is the standard deviation of within-participant differences, $SD_E$ and $SD_C$ are the standard deviations under the experimental and control interventions, and $Corr$ is the imputed within-participant correlation. The standard error of the mean difference is then $SD_{diff}/\sqrt{N}$ for $N$ participants. In the Handbook's example, standard deviations of 2.38 and 2.21 with a correlation of 0.68 give an $SD_{diff}$ of about 1.84 and a standard error of about 0.583 for 10 participants, and the Handbook asks for sensitivity analysis across other plausible correlations.

The Handbook also notes that crossover trials may have shorter intervention periods or less severely ill participants than parallel trials, and advises analysing the two designs in separate subgroups. For a cost-effectiveness analysis, short periods are the main limit. The NICE manual (PMG36, section 4.2.22) requires a time horizon long enough to reflect all important differences in costs or outcomes, and section 4.2.24 notes that a lifetime horizon often requires extrapolation beyond the trials. A crossover trial measures response over short treatment periods, so a model has to state separately how long benefit and costs last once treatment continues or stops.

Treatment switching, the other meaning of crossover

In oncology appraisals "crossover" often refers to control-group patients moving onto the experimental treatment during follow-up. NICE DSU Technical Support Document 16 explains that this is common in cancer trials, generally permitted once interim analyses show the new treatment to be effective, and that an intention-to-treat analysis then underestimates the overall survival gain of the new treatment. The document deliberately uses the term "switching", because "crossover" may evoke crossover trials, which it calls a different entity.

TSD 16 reviews adjustment methods, and TSD 24 later extended it without replacing it. Randomisation-based methods such as the rank preserving structural failure time model rely on a common treatment effect for switchers, while inverse probability of censoring weights rely on no unmeasured confounders. PMG36 (section 4.6.26) says simple methods such as censoring or excluding patients who cross over should be avoided because of selection bias, and asks for the choice of method to be justified against the trial's switching mechanism and data. The CONSORT extension draws the same line, separating prespecified crossover designs from trials in which control participants are given the chance to cross over to the experimental treatment. It reports a finding by Zeng and colleagues that almost a quarter of trial records labelled "crossover assignment" let participants change intervention during the trial rather than randomising them to a sequence.

Common misunderstandings about crossover evidence

A smaller sample does not always mean a cheaper or quicker study, since each participant is measured more than once and stays in the trial longer, which is why the saving in the worked example shrinks when $\rho$ is low. A non-significant carryover test does not show that carryover is absent, because in the 2x2 design it rests on a between-person contrast and so lacks power. Pooling crossover arms as if they were parallel groups throws away the pairing and under-weights the trial in a meta-analysis. Finally, a survival trial in which control patients cross over to the new drug is not a crossover design, and its adjustment problem belongs to treatment switching methods rather than to sequence and period analysis.

Sources

• Bell Gorrod H, Latimer NR, Abrams KR. NICE DSU Technical Support Document 24: Adjusting survival time estimates in the presence of treatment switching: an update to TSD 16. Sheffield: Decision Support Unit, ScHARR; April 2024.
• Dwan K, Li T, Altman DG, Elbourne D. CONSORT 2010 statement: extension to randomised crossover trials. BMJ. 2019;366:l4378. doi:10.1136/bmj.l4378
• Higgins JPT, Eldridge S, Li T. Chapter 23: Including variants on randomized trials. In: Higgins JPT, Thomas J, et al., editors. Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 23.2.
• ICH. ICH M13A Guideline on bioequivalence for immediate-release solid oral dosage forms (EMA/CHMP/ICH/953493/2022). Amsterdam: European Medicines Agency; 25 July 2024.
• ICH. ICH Topic E9: Statistical Principles for Clinical Trials (CPMP/ICH/363/96). London: European Medicines Agency; September 1998. Sections 2.3.2 and 3.1.2.
• Latimer NR, Abrams KR. NICE DSU Technical Support Document 16: Adjusting survival time estimates in the presence of treatment switching. Sheffield: Decision Support Unit, ScHARR; July 2014.
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 13 October 2026. Sections 4.2.22, 4.2.24 and 4.6.26.
• Wellek S, Blettner M. On the proper use of the crossover design in clinical trials: part 18 of a series on evaluation of scientific publications. Deutsches Ärzteblatt International. 2012;109(15):276-281. doi:10.3238/arztebl.2012.0276