4.1 ONE FACTOR RANDOMISED BLOCK MODEL Y = S'|A with orthogonal contrasts on A
Three-level factor A and four-level block S.
Model-1 analysis by GLM of terms: S + B + C(B) + S*B + S*C(B);
Model-2 analysis by GLM of terms: S + B + C(B)
Data:
S A B C Y
1 1 2 0 -4.4297
2 1 2 0 4.7513
3 1 2 0 3.2971
4 1 2 0 -1.4606
1 2 -1 1 4.8458
2 2 -1 1 5.1163
3 2 -1 1 6.0739
4 2 -1 1 -1.9225
1 3 -1 -1 6.1542
2 3 -1 -1 10.4794
3 3 -1 -1 12.4438
4 3 -1 -1 9.2150
COMMENT: If A[1] is a control, and A[2], A[3] are treatment levels, contrasts B and
C test for a control-versus-treatment effect, and a between-treatments effect.
SS[B] + SS[C(B)] = SS[A]; likewise DF[B] + DF[C(B)] = 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]:
SS[S*B] + SS[S*C(B)] = SS[S*A]; likewise DF[S*B] + DF[S*C(B)] = DF[S*A].
Model 4.1(i) A is a fixed factor, S is a random blocking factor:
Source DF SS MS F P
1 S 3 74.25 24.75 - -
2 A 2 169.44 84.72 12.36 0.007
3 S*A 6 41.11 6.85 - -
4 P(S*A) 0 - -
Total 11 284.80
Model-1 analysis of orthogonal contrasts, with fixed factors B and C:
Source DF Seq SS Adj SS Seq MS F P
1 S 3 74.25 79.98 24.75 - -
2 B 1 96.36 96.36 96.36 17.37 0.025 = A[1] versus average{A[2],A[3]}
3 C(B) 1 73.08 73.08 73.08 8.96 0.058 = A[2] versus A[3]
4 S*B 3 16.64 16.64 5.55 - -
5 S*C(B) 3 24.47 24.47 8.16 - -
6 P(S*A) 0 - - -
Total 11 284.80
Model-2 analysis of orthogonal contrasts, with fixed factors B and C:
Source DF Seq SS Adj SS Seq MS F P
1 S 3 74.25 74.25 24.75 - -
2 B 1 96.36 96.36 96.36 14.06 0.010 = A[1] versus average{A[2],A[3]}
3 C(B) 1 73.08 73.08 73.08 10.66 0.017 = A[2] versus A[3]
4 S*A 6 41.11 41.11 6.85 - -
5 P(S*A) 0 - - -
Total 11 284.80
__________________________________________________________________
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/