## The threetreatment twoperiod design Kochs design

Koch's design is the 'obvious' design to use with three treatments and two treatment periods. There are six possible treatment sequences - AB, AC, BA, BC, CA and CB - and in this design all six possibilities are used with equal frequency.

 Within-patient analysis (fixed effects) Between-patient analysis Combined analysis (random effects) Treatment effects Omitting carry-over 2(2 + 3 y) Including carry-over 4 3 3(1 + 2K) 2(1 + 2y)(l + y) 7 + 14 y + 3 y2 Carry-over effects 4 + 2y) 2(1 + 2y)(2 + 3y) 7+ 14y + 3 y2

The variances of treatment differences from within-patient comparisons (fixed effects model), between-patient comparisons and a combination of these (random effects model) have been investigated by Brown and Kempton (1994). They are summarised in Table 7.5 for models both with and without carry-over effects.

In the absence of carry-over, the within-patient (fixed effects) analysis is reasonably efficient in comparison with the fully efficient combined (random effects) analysis. The variance of the treatment estimates is never more than 4/3 of the variance for the combined analysis. This upper limit occurs when y, the ratio of the patient and residual variance components, is zero. For a moderate value of y, such as 2, the ratio of the variances is 1.07:1, and for a high value such as 10, the ratio is 1.02:1. Of course, even such small gains in efficiency should be accepted, because the analysis is no longer difficult or time consuming to perform. The major benefit from recovery of between-patient information, however, occurs if carry-over terms are fitted. This leads to a fourfold increase in treatment variance from the within-patient analysis, compared with the no carryover situation, whereas the corresponding increase from the combined analysis is by a factor of only (8 + 4y)/7 for small y. The estimates of the carry-over effects themselves are even more dramatically affected by recovery of between-patient information. The ratio of the variance estimates from the within-patient analysis to the combined analysis has a maximum value of 7:1 when y = 0, a value of 2.35:1 when y = 2, and even when y = 10 the ratio is 1.33:1.