4.2 TWO FACTOR RANDOMISED BLOCK MODEL Y = S'|A|B with orthogonal contrasts on A Three-level factor A crossed with two-level factor B and four-level block S. Model-1 analysis by GLM of terms: S|B|C + S|B|D(C) - S*B*D(C); Model-2 analysis by GLM of terms: S + B + C + D(C) + B*C + B*D(C). Data: S A C D B Y 1 1 2 0 1 4.5924 2 1 2 0 1 -0.5488 3 1 2 0 1 6.1605 4 1 2 0 1 2.3374 1 1 2 0 2 5.1873 2 1 2 0 2 3.3579 3 1 2 0 2 6.3092 4 1 2 0 2 3.2831 1 2 -1 1 1 7.3809 2 2 -1 1 1 9.2085 3 2 -1 1 1 13.1147 4 2 -1 1 1 15.2654 1 2 -1 1 2 12.4188 2 2 -1 1 2 14.3951 3 2 -1 1 2 8.5986 4 2 -1 1 2 3.4945 1 3 -1 -1 1 21.3220 2 3 -1 -1 1 25.0426 3 3 -1 -1 1 22.6600 4 3 -1 -1 1 24.1283 1 3 -1 -1 2 16.5927 2 3 -1 -1 2 10.2129 3 3 -1 -1 2 9.8934 4 3 -1 -1 2 10.0203 COMMENT: If A[1] is a control, and A[2], A[3] are treatment levels, contrasts C and D test for a control-versus-treatment effect, and a between-treatments effect. SS[C] + SS[D(C)] = SS[A]; likewise DF[C] + DF[D(C)] = DF[A]. Model-1 analysis declines to assume equal block-by-contrast interactions, so tests each treatment contrast against its interaction with block. Model-2 analysis assumes equal or negligible block-by-contrast interactions, so tests each treatment contrast against the pooled error MS[S*A + S*B + S*B*A]: SS[S*C] + SS[S*D(C)] = SS[S*A]; DF[S*C] + DF[S*D(C)] = DF[S*A]. SS[S*B*C] + SS[S*B*D(C)] = SS[S*B*A]; DF[S*B*C] + DF[S*B*D(C)] = DF[S*B*A]. Model 4.2(i) A and B are fixed factors, S is a random blocking factor: Restricted Model_1 Model_2 Source DF SS MS F P F P 1 S 3 9.07 3.02 - - 0.27 0.850 2 A 2 745.36 372.68 67.58 <0.001 32.67 <0.001 3 B 1 91.65 91.65 4.13 0.135 8.03 0.013 4 B*A 2 186.37 93.18 7.82 0.021 8.17 0.004 5 S*A 6 33.09 5.51 - - - - 6 S*B 3 66.51 22.17 - - - - 7 S*B*A 6 71.51 11.92 - - - - 8 P(S*B*A) 0 - - Total 23 1203.56 Model-1 analysis of orthogonal contrasts, with fixed factors C and D: Source DF Seq SS Adj SS Seq MS F P 1 S 3 9.07 16.32 3.02 - - 2 C 1 549.39 549.39 549.39 67.28 0.004 = A[1] versus average{A[2],A[3]} 3 D(C) 1 195.97 195.97 195.97 68.44 0.004 = A[2] versus A[3] 4 B 1 91.65 35.54 91.65 4.13 0.135 5 B*C 1 84.50 84.50 84.50 8.18 0.065 = B*(A[1] versus average{A[2],A[3]}) 6 B*D(C) 1 101.86 101.86 101.86 7.54 0.071 = B*(A[2] versus A[3]) 7 S*C 3 24.50 24.50 8.17 - - 8 S*D(C) 3 8.59 8.59 2.86 - - 9 S*B 3 66.51 36.92 22.17 - - 10 S*B*C 3 30.98 30.98 10.33 - - 11 S*B*D(C) 3 40.53 40.53 13.51 12 P(S*B*A) 0 - - Total 23 1203.56 Model-2 analysis of orthogonal contrasts, with fixed factors C and D: Source DF Seq SS Adj SS Seq MS F P 1 S 3 9.07 16.32 3.02 - - 2 C 1 549.39 549.39 549.39 48.16 <0.001 = A[1] versus average{A[2],A[3]} 3 D(C) 1 195.97 195.97 195.97 17.18 0.001 = A[2] versus A[3] 4 B 1 91.65 35.54 91.65 8.03 0.013 5 B*C 1 84.50 84.50 84.50 7.41 0.016 = B*(A[1] versus average{A[2],A[3]}) 6 B*D(C) 1 101.86 101.86 101.86 8.93 0.009 = B*(A[2] versus A[3]) 7 S*A 6 33.09 5.51 - - - 8 S*B 3 66.51 22.17 - - - 9 S*B*A 6 71.51 11.92 - - - 10 P(S*B*A) 0 - - Total 23 1203.56 __________________________________________________________________ Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press. http://www.southampton.ac.uk/~cpd/anovas/datasets/