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/