Power comparison of Prescott’s test and the Mainland-Gart test for cross-over trials with binary outcomes


A classic analysis of binary data from AB/BA cross-over trials is the so-called Mainland-Gart test(Gart, 1969; Mainland, 1963). This is an analysis of discordant pairs of data, that is to say it uses data from those patients that had a response in on period and not another.


An example from my book Cross-over Trials in Clinical Research(Senn, 1993, 2002) illustrates the general idea. Twenty-four patients in a trial in asthma were treated with two beta-agonists, formoterol and salbutamol on two separate occasions. They were allocated in equal numbers to two sequences, formoterol followed by salbutamol (for/sal) or salbutamol followed by formoterol (sal/for) and classified as to whether the ‘response’ was good or bad. If the response was good in period one but not period two the patient was said to ‘prefer’ period one. If the response was good in period two but not period one the patient was said to ‘prefer’ period two. If the response was bad in both periods or good in both periods then no preference was recorded. The outcome is summarised in the table below.



Prefer period 1

No preference

Prefer period 2

Total all preferences











Total both sequences






The Mainland-Gart(Gart, 1969; Mainland, 1963) test ignores the eight patients showing no preference and analyses the resulting two by two data using Fisher’s exact test. If there is an association between sequence and period of preference, as appears to be the case here, then this suggests that there is a difference between treatments.


An alternative test due to Prescott(Prescott, 1981) uses all three columns and has been claimed to have superior power. The origin of this power is debatable and may simply be due to its less discrete data. For reasons outlined in (Senn, 2007) this advantage may not be what it seems.


This project will investigate this point both analytically and by simulation.



Gart, J. J. (1969), An exact test for comparing matched proportions in cross-over designs Biometrika, 56, 75-80.

Mainland, D. (1963). Elementary Medical Statistics. Philadelphia: W.b. Saunders

Prescott, R. J. (1981), The comparison of success rates in cross-over trials in the presence of an order effect Applied Statistics, 30, 9-15.

Senn, S. J. (1993). Cross-over Trials in Clinical Research, First ed. Chichester: John Wiley

Senn, S. J. (2002). Cross-over Trials in Clinical Research, Second ed. Chichester: Wiley

Senn, S. J. (2007), Drawbacks to Noninteger Scoring for Ordered Categorical Data Biometrics, 63, 296-299.