About the Supplemental Text Material - all supplements

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E (ε .. ) = 0, E (ε ij2 ) = σ 2 , E (ε i2. 2 ) = anσ 2 Now 1 a 2 1 E ( SSTreatments ) = E ( ∑ yi . 2. ) n i =1 an Consider the first term on the right hand side of the above expression: 1 a 2 1 a E ( ∑ yi . ) = ∑ E (nµ + nτ i + ε i . ) 2 n i =1 n i =1 Squaring the expression in parentheses and taking expectation results in a 1 a 2 1 2 2 E ( ∑ yi . ) = [a (nµ ) + n ∑ τ i2 + anσ 2 ] n i =1 n i =1 a = anµ 2 + n∑ τ i2 + aσ 2 i =1 because the three cross-product terms are all zero. Now consider the second term on the right hand side of E ( SSTreatments ) : F 1 I 1 E (anµ + n∑ τ EG y J = H an K an a 2 ..

000 Notice that the ANOVA table in this regression output is identical (apart from rounding) to the ANOVA display in Table 3-4. Therefore, testing the hypothesis that the regression coefficients β 1 = β 2 = β 3 = β 4 = 0 in this regression model is equivalent to testing the null hypothesis of equal treatment means in the original ANOVA model formulation. Also note that the estimate of the intercept or the “constant” term in the above table is the mean of the 4th treatment. Furthermore, each regression coefficient is just the difference between one of the treatment means and the 4th treatment mean.

In general, a Youden square is a symmetric balanced incomplete block design in which rows correspond to blocks and each treatment occurs exactly once in each column or “position” of the block. Thus, it is possible to construct Youden squares from all symmetric balanced incomplete block designs, as shown by Smith and Hartley (l948). A table of Youden squares is given in Davies (1956), and other types of incomplete Latin squares are discussed by Cochran and Cox (1957, Chapter 13). Row 1 2 3 4 5 Table 4.

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