3.3 CROSS-FACTORED WITH NESTING MODEL Y = C|B(A) + e Analysis of terms: C|A + C|B(A) Data: A B C Y A1 B1 1 1 1 -3.8558 1 1 1 1 1 4.4076 1 1 1 1 2 -4.1752 1 1 1 1 2 1.4913 1 1 1 2 1 5.9699 2 2 1 2 1 5.2141 2 2 1 2 2 9.1467 2 2 1 2 2 5.8209 2 2 2 1 1 9.4082 3 3 2 1 1 6.0296 3 3 2 1 2 15.3014 3 3 2 1 2 12.1900 3 3 2 2 1 6.9754 4 4 2 2 1 14.3012 4 4 2 2 2 10.4266 4 4 2 2 2 2.3707 4 4 3 1 1 19.1834 5 5 3 1 1 18.3855 5 5 3 1 2 23.3385 5 5 3 1 2 21.9134 5 5 3 2 1 16.4482 6 6 3 2 1 11.6765 6 6 3 2 2 17.9727 6 6 3 2 2 15.1760 6 6 COMMENT: This example is balanced and therefore orthogonal, so uses Type I (sequential) SS. In the event of non-orthogonality, use Type III (adjusted) SS. Model 3.3(i) A and C are fixed factors, B is a random factor: Restricted Unrestricted Source DF SS MS F P F P 1 A 2 905.27 452.63 8.09 0.062 8.09 0.062 2 B(A) 3 167.83 55.94 4.71 0.021 2.81 0.209 3 C 1 11.80 11.80 0.59 0.497 0.59 0.497 4 C*A 2 10.02 5.01 0.25 0.793 0.25 0.793 5 C*B(A) 3 59.74 19.91 1.68 0.225 1.68 0.225 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 Model 3.3(ii) A is a fixed factor, B and C are random factors: Restricted Unrestricted Source DF SS MS F P F P 1 A 2 905.27 452.63 11.03 0.137* 11.03 0.137* 2 B(A) 3 167.83 55.94 2.81 0.209 2.81 0.209 3 C 1 11.80 11.80 0.59 0.497 2.36 0.265 4 C*A 2 10.02 5.01 0.25 0.793 0.25 0.793 5 C*B(A) 3 59.74 19.91 1.68 0.225 1.68 0.225 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 * Quasi F-ratio. Model 3.3(iii) A and B are random factors, C is a fixed factor: Restricted Unrestricted Source DF SS MS F P F P 1 A 2 905.27 452.63 8.09 0.062 11.03 0.137* 2 B(A) 3 167.83 55.94 4.71 0.021 2.81 0.209 3 C 1 11.80 11.80 2.36 0.265 2.36 0.265 4 C*A 2 10.02 5.01 0.25 0.793 0.25 0.793 5 C*B(A) 3 59.74 19.91 1.68 0.225 1.68 0.225 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 * Quasi F-ratio. Model 3.3(iv) A, B and C are all random factors: Source DF SS MS F P 1 A 2 905.27 452.63 11.03 0.137* 2 B(A) 3 167.83 55.94 2.81 0.209 3 C 1 11.80 11.80 2.36 0.265 4 C*A 2 10.02 5.01 0.25 0.793 5 C*B(A) 3 59.74 19.91 1.68 0.225 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 * Quasi F-ratio. Model 3.3(v) A is a fixed factor, B is a random factor, C is a covariate of the response: Source DF SS MS F P 1 A 2 905.27 452.63 8.09 0.062 2 B(A) 3 167.83 55.94 4.71 0.021 3 C 1 11.80 11.80 0.59 0.497 4 C*A 2 10.02 5.01 0.25 0.792 5 C*B(A) 3 59.74 19.91 1.68 0.225 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 Model 3.3(vi) A and B are random factors, C is a covariate of the response: Source DF SS MS F P 1 A 2 905.27 452.63 8.09 0.062 2 B(A) 3 167.83 55.94 4.71 0.021 3 C 1 11.80 11.80 2.36 0.265 4 C*A 2 10.02 5.01 0.25 0.792 5 C*B(A) 3 59.74 19.91 1.68 0.225 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 Model 3.3(vii) C is a fixed factor, B is a random factor, A is a covariate of the response: Note: A = A1 and B = B1 in dataset above Restricted Unrestricted Source DF SS MS F P F P 1 A 1 816.63 816.63 12.74 0.023 12.74 0.023 2 B(A) 4 256.47 64.12 5.40 0.010 3.89 0.109 3 C 1 11.80 11.80 0.72 0.445 0.72 0.445 4 C*A 1 3.76 3.76 0.23 0.658 0.23 0.658 5 C*B(A) 4 66.00 16.50 1.39 0.295 1.39 0.295 6 S(C*B(A)) 12 142.47 11.87 Total 23 1297.13 Model 3.3(viii) B and C are random factors, A is a covariate of the response: Note: A = A1 and B = B1 in dataset above Source DF SS MS F P 1 A 1 816.627 816.63 37.83 0.025* 2 B(A) 4 256.474 64.12 3.89 0.109 3 C 1 11.801 11.80 0.72 0.445 4 C*A 1 3.764 3.76 0.23 0.658 5 C*B(A) 4 65.996 16.50 1.39 0.295 6 S(C*B(A)) 12 142.472 11.87 Total 23 1297.133 * Quasi F-ratio. Model 3.3(ix) A and C are covariates of the response, B is a random factor: Note: A = A1 and B = B1 in dataset above Source DF SS MS F P 1 A 1 816.627 816.63 12.74 0.023 2 B(A) 4 256.474 64.12 5.40 0.010 3 C 1 11.801 11.80 0.72 0.445 4 C*A 1 3.764 3.76 0.23 0.658 5 C*B(A) 4 65.996 16.50 1.39 0.295 6 S(C*B(A)) 12 142.472 11.87 Total 23 1297.133 __________________________________________________________________ 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/