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J + (τβ ) .. It turns out that the only hypotheses that can be tested in an effects model must involve estimable functions. Therefore, when we test the hypothesis of no interaction, we are really testing the null hypothesis H0 :(τβ ) ij − (τβ ) i . − (τβ ) . 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.

Analysis of Variance for the Partially Balanced Incomplete Block Design with Two Associate Classes Source of Variation Sum of Squares Degrees of Freedom Treatments (adjusted) a ∑ τ Q i Blocks i i =1 a-1 y 2 .. 1 b 2 ∑ y . j − bk k j =1 b-1 Error Subtraction Total ∑ ∑ y 2 ij − i j bk-b-a+1 y 2 .. bk bk-1 We may show that the variance of any contrast of the form τ u − τ v is 2( k − ci )σ 2 V (τ u − τ v ) = r ( k − 1) where treatments u and v are ith associates (i = 1, 2). This indicates that comparisons between treatments are not all estimated with the same precision.

23 (E) Considering this design as a balanced incomplete block, we find a = b = 5, r= k = 4, and k = 3. Also, SS T = ∑ ∑ ∑ y 2 ijh − i j 1 4 Q1 = 12 1 4 Q2 = 2 - h (31) 2 y 2 ... 37 (3)(5) b SS Days = ∑ i =1 y 2 i .. y 2 ... h y 2 ... (6) 2 + (9) 2 + (7) 2 + (9) 2 (31) 2 − = − = 135 . 53 Block or day effects may be assessed by computing the adjusted sum of squares for blocks. 87 (3)(5) The complete analysis of variance is shown in Table 6. Illumination levels are significantly different at 1 percent.

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