3.1 TWO FACTOR FULLY CROSS-FACTORED MODEL Y = B|A + e Analysis of terms: B|A Data: A B Y 1 1 4.5924 1 1 -0.5488 1 1 6.1605 1 1 2.3374 1 2 5.1873 1 2 3.3579 1 2 6.3092 1 2 3.2831 2 1 7.3809 2 1 9.2085 2 1 13.1147 2 1 15.2654 2 2 12.4188 2 2 14.3951 2 2 8.5986 2 2 3.4945 3 1 21.3220 3 1 25.0426 3 1 22.6600 3 1 24.1283 3 2 16.5927 3 2 10.2129 3 2 9.8934 3 2 10.0203 COMMENT: This example is balanced and therefore orthogonal, so uses Type I (sequential) SS. In the event of non-orthogonality, for fixed effects get Type II adjusted SS for main effects by running the model twice and each time using only the SS of the last entered main effect, or for random effects use Type III adjusted SS. Model 3.1(i) A and B are both fixed factors: Source DF SS MS F P 1 A 2 745.36 372.68 37.23 <0.001 2 B 1 91.65 91.65 9.16 0.007 3 B*A 2 186.37 93.18 9.31 0.002 4 S(B*A) 18 180.18 10.01 Total 23 1203.56 Model 3.1(ii) A is a fixed factor, B is a random factor: Restricted Unrestricted Source DF SS MS F P F P 1 A 2 745.36 372.68 4.00 0.200 4.00 0.200 2 B 1 91.65 91.65 9.16 0.007 0.98 0.426 3 B*A 2 186.37 93.18 9.31 0.002 9.31 0.002 4 S(B*A) 18 180.18 10.01 Total 23 1203.56 Model 3.1(iii) A and B are both random factors: Source DF SS MS F P 1 A 2 745.36 372.68 4.00 0.200 2 B 1 91.65 91.65 0.98 0.426 3 B*A 2 186.37 93.18 9.31 0.002 4 S(B*A) 18 180.18 10.01 Total 23 1203.56 Model 3.1(iv) A is a fixed factor, B is a covariate of the response: Source DF SS MS F P 1 A 2 745.36 372.68 37.23 <0.001 2 B 1 91.65 91.65 9.16 0.007 3 B*A 2 186.37 93.18 9.31 0.002 4 S(B*A) 18 180.18 10.01 Total 23 1203.56 Model 3.1(v) A is a random factor, B is a covariate of the response: Source DF SS MS F P 1 A 2 745.36 372.68 37.23 <0.001 2 B 1 91.65 91.65 0.98 0.426 3 B*A 2 186.37 93.18 9.31 0.002 4 S(B*A) 18 180.18 10.01 Total 23 1203.56 Model 3.1(vi) A and B are both covariates of the response: Source DF SS MS F P 1 A 1 745.20 745.20 75.46 <0.001 2 B 1 91.65 91.65 9.28 0.006 3 B*A 1 169.19 169.19 17.13 0.001 4 S(B*A) 20 197.52 9.88 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/