3.2 THREE FACTOR FULLY CROSS-FACTORED MODEL Y = C|B|A + e Analysis of terms: C|B|A Data: A B C Y 1 1 1 -3.8558 1 1 1 4.4076 1 1 2 -4.1752 1 1 2 1.4913 1 2 1 5.9699 1 2 1 5.2141 1 2 2 9.1467 1 2 2 5.8209 2 1 1 9.4082 2 1 1 6.0296 2 1 2 15.3014 2 1 2 12.1900 2 2 1 6.9754 2 2 1 14.3012 2 2 2 10.4266 2 2 2 2.3707 3 1 1 19.1834 3 1 1 18.3855 3 1 2 23.3385 3 1 2 21.9134 3 2 1 16.4482 3 2 1 11.6765 3 2 2 17.9727 3 2 2 15.1760 COMMENT: This example is balanced and therefore orthogonal, so uses Type I (sequential) SS. In the event of non-orthogonality, for categorical fixed effects get Type II adjusted SS for main effects by running the model three times and each time using only the SS of the last entered main effect and two-way interaction, or for random effects use Type III adjusted SS. Model 3.2(i) A, B and C are all fixed factors: Source DF SS MS F P 1 A 2 905.27 452.63 38.12 <0.001 2 B 1 0.19 0.19 0.02 0.902 3 B*A 2 167.65 83.82 7.06 0.009 4 C 1 11.80 11.80 0.99 0.338 5 C*A 2 10.02 5.01 0.42 0.665 6 C*B 1 10.90 10.90 0.92 0.357 7 C*B*A 2 48.84 24.42 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 Model 3.2(ii) 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 5.40 0.156 5.40 0.156 2 B 1 0.19 0.19 0.02 0.902 0.00 0.966* 3 B*A 2 167.65 83.82 7.06 0.009 3.43 0.226 4 C 1 11.80 11.80 1.08 0.487 1.08 0.487 5 C*A 2 10.02 5.01 0.21 0.830 0.21 0.830 6 C*B 1 10.90 10.90 0.92 0.357 0.45 0.573 7 C*B*A 2 48.84 24.42 2.06 0.171 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 * Quasi F-ratio. Model 3.2(iii) 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 7.03 0.239* 7.03 0.239* 2 B 1 0.19 0.19 0.02 0.917 0.00 0.966* 3 B*A 2 167.65 83.82 3.43 0.226 3.43 0.226 4 C 1 11.80 11.80 1.08 0.487 x x 5 C*A 2 10.02 5.01 0.21 0.830 0.21 0.830 6 C*B 1 10.90 10.90 0.92 0.357 0.45 0.573 7 C*B*A 2 48.84 24.42 2.06 0.171 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 * Quasi F-ratio. x Not an exact F-test; estimated denominator MS is <0. Model 3.2(iv) A, B and C are all random factors: Source DF SS MS F P 1 A 2 905.27 452.63 7.03 0.239* 2 B 1 0.19 0.19 0.00 0.966* 3 B*A 2 167.65 83.82 3.43 0.226 4 C 1 11.80 11.80 x x 5 C*A 2 10.02 5.01 0.21 0.830 6 C*B 1 10.90 10.90 0.45 0.573 7 C*B*A 2 48.84 24.42 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 * Quasi F-ratio. x Not an exact F-test; estimated denominator MS is <0. Model 3.2(v) A and B are fixed factors, C is a covariate of the response: Source DF SS MS F P 1 A 2 905.27 452.63 38.12 <0.001 2 B 1 0.19 0.19 0.02 0.902 3 B*A 2 167.65 83.82 7.06 0.009 4 C 1 11.80 11.80 0.99 0.338 5 C*A 2 10.02 5.01 0.42 0.665 6 C*B 1 10.90 10.90 0.92 0.357 7 C*B*A 2 48.84 24.42 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 Model 3.2(vi) A is a fixed factor, B is a random factor, C is a covariate of the response: Restricted Unrestricted Source DF SS MS F P F P 1 A 2 905.27 452.63 5.40 0.156 5.40 0.156 2 B 1 0.19 0.19 0.02 0.902 0.00 0.966 3 B*A 2 167.65 83.82 7.06 0.009 7.06 0.009 4 C 1 11.80 11.80 1.08 0.487 1.08 0.487 5 C*A 2 10.02 5.01 0.21 0.830 0.21 0.830 6 C*B 1 10.90 10.90 0.92 0.357 0.92 0.357 7 C*B*A 2 48.84 24.42 2.06 0.171 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 Model 3.2(vii) 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 5.40 0.156 2 B 1 0.19 0.19 0.00 0.967 3 B*A 2 167.65 83.82 7.06 0.009 4 C 1 11.80 11.80 x x 5 C*A 2 10.02 10.02 1.23 0.326 6 C*B 1 10.90 10.90 2.68 0.128 7 C*B*A 2 48.84 24.42 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 x Not an exact F-test; estimated denominator MS is <0. Model 3.2(viii) A is a fixed factor, B and C are covariates of the response: Source DF SS MS F P 1 A 2 905.27 452.63 38.12 <0.001 2 B 1 0.19 0.19 0.02 0.902 3 B*A 2 167.65 83.82 7.06 0.009 4 C 1 11.80 11.80 0.99 0.338 5 C*A 2 10.02 5.01 0.42 0.665 6 C*B 1 10.90 10.90 0.92 0.357 7 C*B*A 2 48.84 24.42 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 Model 3.2(ix) A is a random factor, B and C are covariates of the response: Source DF SS MS F P 1 A 2 905.27 452.63 38.12 <0.001 2 B 1 0.19 0.19 0.00 0.966 3 B*A 2 167.65 83.82 7.06 0.009 4 C 1 11.80 11.80 2.36 0.265 5 C*A 2 10.02 5.01 0.42 0.665 6 C*B 1 10.90 10.90 0.45 0.573 7 C*B*A 2 48.84 24.42 2.06 0.171 8 S(C*B*A) 12 142.47 11.87 Total 23 1297.13 COMMENT: Covariates A, B and C (below) are all orthogonal so use Type I SS. In the event of non-orthogonality,get Type II SS for main effects by running the model three times and each time using only the SS of the last entered main effect and two-way interaction. Model 3.2(x) A, B and C are all covariates of the response: Source DF SS MS F P 1 A 1 901.12 901.12 71.08 0.000 2 B 1 0.19 0.19 0.01 0.905 3 B*A 1 155.20 155.20 12.24 0.003 4 C 1 11.80 11.80 0.93 0.349 5 C*A 1 9.24 9.24 0.73 0.406 6 C*B 1 10.90 10.90 0.86 0.368 7 C*B*A 1 5.85 5.85 0.46 0.506 8 S(C*B*A) 16 202.84 12.68 Total 23 1297.13 __________________________________________________________________ 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/