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Repeated Measurements and Cross-Over Designs by Damaraju Raghavarao, Lakshmi Padgett

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Chapter 5Cross-Over Designs without Residual Effects

5.1 INTRODUCTION

When several treatments are tested in an experiment, it is economical and efficient to use sequences of treatments on each experimental unit. In such cases, the treatment differences can be more accurately estimated by eliminating the differences in experimental units.

Consider an experiment in which v treatments are tested on b experimental units. Let the duration of the experiment be suitably divided into k periods for application of treatments. Each of the v treatments can be used on each experimental unit, or a subset of the v treatments can be used on each experimental unit depending on the relative sizes of v and k. With v = 4 = k = b, using treatments A, B, C, and D, the experimental layout may appear as in Table 5.1.1.

Table 5.1.1 Design for v = b = k = 4

Period Experimental unit
1 2 3 4
1 A B C D
2 B A D C
3 C D A B
4 D C B A

In the layout of Table 5.1.1, unit 1 receives treatments A, B, C, and D in that order in the four periods; unit 2 receives treatments B, A, D, and C in that order in the four periods; etc. With v = 7, k = 3, b = 7, an experimental layout is given in Table 5.1.2.

Table 5.1.2 Design for v = b = 7, k = 3

Period Experimental unit
1 2 3 4 5 6 7
1 A B C D E F G
2 B C D E F G A
3 D E F G A B C

In the layout of Table 5.1.2, unit 1 receives treatments A, B, and D in that order of the seven treatments A–G; unit 2 receives treatments B, C, and ...

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