Kuehl+-+Chapter+16+Notes+-+Overheads+(06) - Chapter 16...

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1 Chapter 16 - Crossover Designs Chapter 16 - Crossover Designs Sequence Period 1 2 3 4 5 6 IA BCAB C washout period I IB CACA B washout period I I IC ABBC A ijklm i (i)j k l m (ijk)l y = : + S + A + P + T + C + e where : is the general mean i S is the fixed effect of the i treatment sequence th (i)j A A is the random effect of the j animal in i trt sequence NID(0, F ) th th 2 k P is the fixed effect of the of the k period th l T is the fixed effect of the l treatment th m C is the fixed carryover effect from the previous period (=0 if period =1) (ijk)l e is the random error with NID(0, F ) 2

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2 Chapter 16 - Crossover Designs The main advantage is the increased precision of treatment comparisons. When treatments are compared on the same experimental unit the between-unit variation is removed from the error. In other words the experimental unit acts like a block. Sequence 1 (A-B-C) 11 = 1 1 A E(y ) : +S +P +T 1 2 = 12B1 1 E(y ) : +S +P +T +C (here C is the carryover of treatment A) 1 3 = 13C2 2 E(y ) : +S +P +T +C (here C is the carryover of treatment B) Sequence 2 (B-C-A) 21 2 1 B E(y ) : +S +P +T 2 2 = 22C2 1 E(y ) : +S +P +T +C (here C is the carryover of treatment B) 2 3 = 23A3 2 E(y ) : +S +P +T +C (here C is the carryover of treatment C) Note the book uses 8 (p 526) where I am using C to denote the carryover effect.
3 Chapter 16 - Crossover Designs This crossover is a balanced row-column or LR design if no carryover exits. Sequence Period 1 2 3 4 5 6 IABCABC I IBCACAB I I I CABBCA 1 2 3 4 5 6 AB CA B C ITT TT T T BACBACCAABBC II T + C CBACBABCCAAB III If the carryover effects are real then the direct trt effects and the carryover effects are not orthogonal because they do not occur in all ii possible combinations (i.e. T and C never in the same observation).

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4 Chapter 16 - Crossover Designs Analysis of Crossover Designs 16.2 pp 524-530 The crossover designs present some of the same questions we saw with repeated measures. Whether or not a univariate analysis of variance should be used depends upon the relationships among the variances and covariances of the experimental errors for the repeated measures. The univariate analysis can be used if the assumptions of independence, compound symmetry or the Huynh-Feldt condition are met. The book assumes the HF condition has been met.
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Kuehl+-+Chapter+16+Notes+-+Overheads+(06) - Chapter 16...

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