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0 These are the “usual constraints” imposed on the normal equations. Furthermore, the tests on main effects become H0 :τ 1* = τ *2 =" = τ *a = 0 and H0 : β 1* = β *2 =" = β b* = 0 This is the way that these hypotheses are stated in the textbook, but of course, without the “stars”. S5-4. Regression Model Formulation of the Two-factor Factorial We noted in Chapter 3 that there was a close relationship between ANOVA and regression, and in the Supplemental Text Material for Chapter 3 we showed how the single-factor ANOVA model could be formulated as a regression model.

The partial balance is easily seen, since, for example, treatment 2 appears in the same block with treatments 1, 3, 5, and 8, but does not appear at all with treatments 4, 6, 7, and 9. A lattice design with k2 treatments in 3k blocks grouped into three replicates is called a triple lattice. An example would be the first three replicates in Table 7. A lattice design for k2 treatments in 4k blocks arranged in four replicates is called a quadruple lattice. Block 1 2 3 Block 1 2 3 1 4 7 1 2 3 Table 7.

J + (τβ ) .. = 0 for all i , j When we test hypotheses on main effects A and B we are really testing the null hypotheses H0 :τ 1 + (τβ )1. = τ 2 + (τβ ) 2. =" = τ a + (τβ ) a . b That is, we are not really testing a hypothesis that involves only the equality of the treatment effects, but instead a hypothesis that compares treatment effects plus the average interaction effects in those rows or columns. Clearly, these hypotheses may not be of much interest, or much practical value, when interaction is large.

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