It is now some years since Peter Freeman showed convincingly in the pages of this journal that the two-stage analysis of the AB/BA cross-over was extremely misleading. More recently, however, Jones and Lewis have claimed that the view in that paper was too pessimistic and have written about it. 'We show that the properties of this procedure are not as bad as some authors have suggested' (p. 1026). It is the purpose of this letter to point out that their investigation was flawed for the purpose of examining the performance of the two-stage analysis. I also make the necessary correction. This supports Freeman's original conclusion.
Jones and Lewis simulate data from an AB/BA cross-over trial with a variance of 144, a treatment effect of 5, 22 patients per sequence and a correlation between repeat measures of 2/3. They show that for values of the carry-over of more than 2•0, the two-stage analysis is more powerful than the simple within-patient analysis (CROS). This comparison is illegitimate, however, since the two-stage analysis will have a type I error rate in excess of 5 per cent (I calculate it to be about 8•7 per cent for this example).
In Table I, I have calculated (by double integration using MATHCAD PLUS 6.0 ) the asymptotic power of such a cross-over trial for various analyses. These results agree closely with those of Jones and Lewis (my values, being asymptotic, are, as one might expect, slightly higher). In addition to giving results for a standard within-patient analysis (CROS), between-patient analysis (PAR) and uncorrected two-stage analysis (2SU), using a pre-test for carry-over at the usual 10 per cent level, I also give results for a corrected two-stage procedure (2SC). This carries out the PAR test given that carry-over has been 'detected', not at the 5 per cent level but at the 0•5 per cent level. This is an appropriate value for a correlation between repeat measures of 1 if one wishes the resulting test for treatment to have a conditionally correct size, and should be conservative in general. In fact in this case it produces a type I error rate of 5 per cent to 4 significant figures for the overall two-stage procedure.