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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.

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 .. i + ε .. ) 2 i =1 = a since ∑τ i 1 E (anµ + ε .. ) 2 an = 0. Upon squaring the term in parentheses and taking expectation, we obtain i =1 E FG 1 y IJ = 1 [(anµ) H an K an 2 .. 2 + anσ 2 ] = anµ 2 + σ 2 since the expected value of the cross-product is zero. Therefore, 1 a 2 1 E ( SSTreatments ) = E ( ∑ yi . 2 ) n i =1 an a = anµ 2 + n∑ τ i2 + aσ 2 − (anµ 2 + σ 2 ) i =1 a = σ 2 (a − 1) + n∑ τ i2 i =1 Consequently the expected value of the mean square for treatments is E ( MSTreatments ) = E FG SS IJ H a −1 K Treatments a = σ 2 (a − 1) + n∑ τ i2 i =1 a −1 a σ2 + n ∑ τ i2 i =1 a −1 This is the result given in the textbook.

87 (3)(5) The complete analysis of variance is shown in Table 6. Illumination levels are significantly different at 1 percent. 95 19 a Significant at 1 percent. 87a S4-4. Lattice Designs Consider a balanced incomplete block design with k2 treatments arranged in b = k(k + 1) blocks with k runs per block and r = k + 1 replicates. Such a design is called a balanced lattice. An example is shown in Table 7 for k2=9 treatments in 12 blocks of 3 runs each. Notice that the blocks can be grouped into sets such that each set contains a complete replicate.

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