• Q 1.  I feel that I would have a better understanding of cross-over trials if I could consider a concrete example in some detail. Is there a good reference you can direct me to assist me in this way?

A. You may wish to consider the following reference:

Jones, B (2008) The cross-over trial: a subtle knife. Significance, 5(3): 135-137.

This reference provides an example of a cross-over trial involving the comparison of two drugs for the treatment of symptoms of chronic pulmonary obstructive disease (COPD).  The study outcome is measured in terms of the mean morning peak flow meter reading (PEFR), with higher readings indicating a better response to treatment. The design is a called a 2 x 2 design because each of 2 groups of patients receives each of 2 possible treatments.

Notice some key features of the study described in this article.

1. The order in which patients receive treatments in Group II is the reverse of that in Group I.
2. Patients are randomly allocated to either Group I or Group II with the intention that they remain within that group for the duration of the experiment;
3. The cross-over therefore refers to the change of drug after a pre-defined period rather than a switching of patients between groups;
4. The key hypothesis to be tested is that Drug A is superior to Drug B;
5. Once the data on mean PEFRs for all patients have been collected, four grand means can be obtained:
   m₁₁ (the mean response for treatment A within Group 1),
   m₁₂ (the mean response for treatment B within Group 1),
   m₂₁ (the mean response for treatment A within Group 2)
   and
   m₂₂ (the mean response for treatment B within Group 2).
6. To estimate the treatment difference, we don’t simply compare the readings under each treatment using a t-test;
7. Instead the effect of time must be corrected for appropriately;
8. The appropriate estimate of treatment difference, mean_diff, is therefore

0.5 x [(m₁₁ – m₁₂) – (m₂₁– m₂₂)].

9. You should obtain a 95% confidence interval (CI) for this treatment difference.

The formulae for the lower and upper limits of the CI are as shown

(mean_diff – 1.96 SE mean_diff, mean_diff + 1.96 SE mean_diff), where ‘SE mean_diff‘

denotes the standard error of as an estimate of the true treatment mean difference.

10. The formula for SE mean_diff is provided in the above reference as √ (2((1-ρ) ⁄ N)σ), where N is the total number of patients involved in the trial, σ is the sample standard deviation for the PEFR readings and ρ is the Pearson Correlation Coefficient for the correlation between paired readings across all patients.

10. For there to be evidence of a true difference between the treatments at the 5% level of significance, the CI you calculate must not include zero; otherwise you would be 95% certain that the range of possible values for this difference includes the value zero (the value indicating no difference whatsoever). Of course, had you wished to assess the bioequivalence of two drugs (in the sense that the effects are so similar that one drug may be substituted for the other), finding that the CI included zero would not be a concern. However, before conducting the study, you would also require to have pre-determined how wide you would have been willing to allow this CI to be to allow you to conclude from a clinical point of view that the drugs are interchangeable.

It is possible to construct line graphs like those shown in Figure 1 and 2 of the above reference using the statistical package SPSS.

• Q 2. What are the key advantages of a cross-over design?

A. The cross-over design takes into consideration the potential influence of any time factor in the behaviour of any treatment. In our 2 x 2 cross-over example above, Drug A is allocated to Group 1 during the first 4 weeks and Group 2 during the 2^nd 4 weeks, whilst Drug B is allocated to Group 2 during the first 4 weeks and Group 1 during the 2^nd 4 weeks of treatment. This means that each drug has been allocated during identical time periods, thus controlling for potential confounding factors such as humidity or temperature or staff-changeovers, from one treatment interval to another.

Also, consider a parallel design (in which all members of Group I would receive the same treatment for the duration of the trial and all members of Group II would receive the remaining treatment for the duration of the trail and all patients are to be allocated randomly to each of Groups I and II). Here, the treatment difference would be estimated by comparing the mean of the mean PEFR readings for Groups I and II directly, typically by means of a t-test. However, in order to obtain as small a standard error for this estimate as that obtained via a cross-over design, considerably more patients would be required – and this is not always guaranteed. In the above reference, it is illustrated that the higher the correlation between the paired data for the cross-over design the greater the factor by which the sample size would require to be increased in a parallel groups design. In fact, even when the correlation coefficient is 0, this factor is 2!

• Q 3. Are there any disadvantages of a cross-over design?

A. Care needs to be taken in ensuring that a wash-out period is allowed for between treatments, where the length of that period is obtained by taking into consideration the half-life of the treatments administered to the patients. The lag-time (time taken for treatment to manifest itself caused by movement of treatment through biological pathways) may also need to be taken into consideration.

Through neglect of the above considerations, the estimate of treatment difference may be flawed by failure to allow sufficient time to eliminate a carry-over effect from one treatment allocation to another. The author of the above paper recommends a period of 5 half-lives as an estimate for a suitable wash-out period.

A key assumption of the cross-over design is that within a given patient group the medical state of the patient does not change from the start of one treatment allocation to another. Therefore, such a design would not be appropriate where treatments were capable of curing a disease. As recommended by the author in the above reference, this sort of design is best suited for trials involving stable chronic conditions such as mountain sickness, COPD, asthma, stage-fright, hypertension, migraine and heart problems.

Cross-Over Trials by Margaret MacDougall is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.