Analogous to a Latin square, a Greco-Latin square design can be used when it is suspected that three extraneous factors may affect the response variable and all four factors (the three extraneous ones and the one of interest) have the same number of levels. In a Latin square, each level of the factor of interest (C) appears once in each row (with each level of A) and once in each column (with each level of B). In a Greco-Latin square, each level of factor D appears once in each row, in each column, and also with each level of the third extraneous factor C. Alternatively, the design can be used when the four factors are all of equal interest, the number of levels of each is N, and resources are available only for N2 observations. A 5 × 5 square is pictured in (a), with (k, l) in each cell denoting the kth level of C and lth level of D. In (b) we present data on weight loss in silicon bars used for semiconductor material as a function of volume of etch (A), color of nitric acid in the etch solution (B), size of bars (C), and time in the etch solution (D) (from "Applications of Analytic Techniques to the Semiconductor Industry," Fourteenth Midwest Quality Control Conference, 1959).

Let xij(kd) denote the observed weight loss when factor A is at level i, B is at level j, C is at level k, and D is at level l. Assuming no interaction between factors, the total sum of squares SST (with N2 - 1df) can be partitioned into SSA, SSB, SSC, SSD, and SSE. Give expressions for these sums of squares, including computing formulas, obtain the ANOVA table for the given data, and test each of the four main effect hypotheses using α = .05.

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(C,D)1 2 3 5 1 (2,3) (3, 5 (4,2 5, 4) 2 (2,2 3,4 (4, 1 5,3) (1, 5) A 3 3, 3) 4, 5) (5,2),4) (2, 1) 65 82 108 101 126 84 109 73 97 83 05 129 89 89 52 119 72 76 117 84 97 59 94 78 106