By J. A. John, E. R. Williams (auth.)

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**Extra resources for Cyclic and Computer Generated Designs**

**Example text**

The efficiency factors of a block design are then obtained from a comparison of the variances of estimated treatment contrasts in the design with those in an orthogonal design using the same number of experimental units, under the assumption that the error variance a 2 is the same for both designs. This is equivalent to making comparisons against the design that would have been obtained if no blocking had been used. 6. The concept of an efficiency factor should not be confused with the measure of the gain (or loss) resulting from the use of a block design or of a design of a given size, the so-called efficiency of a design.

As a consequence, all the information on treatment effects is contained in comparisons within blocks. Similarly, block parameters can be estimated and tested without the need to eliminate treatment effects. 46) since knowledge of the incidence matrix N gives the design directly; remember that the (ij)th element of N gives the number of times the ith treatment occurs in the jth block. 46), = (3, 3)', N=(~ ~) Thus, both treatments 0 and 1 occur once in the first block and twice in the second block.

Let treatments 0 and i occur together in Ai blocks (i = 1,2, ... , v-I) and let Ao = r. Then the concurrence matrix NN' will take 51 EFFICIENCY FACTORS OF CYCLIC DESIGNS the form NN'= AO Al A2 AV -l AO Al Av-2 Av-l AO Al A2 AV -l Av-2 Av-3 A3 AO where Ai = Av-i (i > 0) since NN' is symmetric. This circulant matrix can be specified by the elements in the first row, since the other rows are obtained from the first row by a cyclical rotation. Let r h be the v x v circulant matrix whose first row has 1 in the (h + 1)th column and zero elsewhere.