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If blocking was unnecessary and the experiment was run as a completely randomized design with b replicates we would have had a(b - 1) degrees of freedom for error. Thus, incorrectly blocking has cost a(b - 1) – (a - 1)(b - 1) = b - 1 degrees of freedom for error, and the test on treatment means has been made less sensitive needlessly. However, if block effects really are large, then the experimental error may be so inflated that significant differences in treatment means could possibly remain undetected.

Since there are 4 treatments, there are only 3 of the x’s. The coded data that is used as input to Minitab is shown below: x1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x2 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 x3 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 Etch rate 575 542 530 539 570 565 593 590 579 610 600 651 610 637 629 725 700 715 685 The Regression Module in Minitab was run using the above spreadsheet where x1 through x3 were used as the predictors and the variable “Etch rate” was the response. The output is shown below.

And H. O. Hartley (1948). “Construction of Youden Squares”. Journal of the Royal Statistical Society Series B, Vol. 10, pp. 262-264. Chapter 5 Supplemental Text Material S5-1. Expected Mean Squares in the Two-factor Factorial Consider the two-factor fixed effects model yij = µ + τ i + β j + (τβ ) ij + ε ijk R| ij == 11,,22,,"",,ab S| Tk = 1,2,", n given as Equation (5-1) in the textbook. We list the expected mean squares for this model, but do not develop them. It is relatively easy to develop the expected mean squares from direct application of the expectation operator.

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