The number of alternative sets of orthogonal contrasts refers to the number of different statistical models available to describe the orthogonal and balanced partitioning of variation between i levels of a categorical factor. For eight and more levels, the number of sets increases supra-exponentially with the number of factor levels (sequence A165438 in OEIS). An ANOVA factor with 3 levels has 1 set of orthogonal contrasts; a factor with 4 levels has 3 contrast sets. A factor with i = 5, 6, or 7 levels has n_{i} contrast sets, given by:
_{}
A factor with i > 7 levels has n_{i} contrast sets, given by:
_{}
where
Levels |
Sets |
3 |
1 |
4 |
3 |
5 |
4 |
6 |
8 |
7 |
15 |
8 |
34 |
9 |
69 |
10 |
152 |
11 |
332 |
12 |
751 |
13 |
1698 |
14 |
3905 |
15 |
9020 |
16 |
21051 |
17 |
49356 |
18 |
116505 |
19 |
276217 |
20 |
658091 |
21 |
1573835 |
22 |
3778152 |
23 |
9098915 |
24 |
21980209 |
25 |
53241777 |
26 |
129294912 |
27 |
314714273 |
28 |
767700735 |
29 |
1876437054 |
30 |
4595005570 |
For example, a factor A with i = 6 levels has n_{6} = 8 alternative sets of orthogonal contrasts, each with i - 1 = 5 contrasts. The corresponding alternative general linear models describing contrasts B, C, D, E, F are:
1. Y = B + C(B) + D(C B) + E(B) + F(E B) + ε
2. Y = B + C(B) + D(B) + E(D B) + F(E D B) + ε
3. Y = B +
C(B) + D(B) + E(D B) + F(D B) + ε
4. Y = B +
C(B) + D(B) + E(B) + F(B) + ε
5. Y = B +
C(B) + D(C B) + E(C B) + F(E C B) + ε
6. Y = B +
C(B) + D(C B) + E(D C B) + F(E D C B) + ε
7. Y = B +
C(B) + D(C B) + E(D C B) + F(D C B) + ε
8. Y = B +
C(B) + D(C B) + E(C B) + F(C B) + ε
where Y is the response and ε is the residual error from the main effect model Y = A + ε.
Program Contrasts.exe identifies the coefficients for every set of balanced orthogonal contrasts on a factor with any number of levels up to a maximum of 12. For a chosen set or range of sets, it stores contrast coefficients in a text file (Contrasts.txt) for any specified number of replicates, and will identify the unique GLM model for analysing the set with sequential SS, after each data line has been tagged with the response value for the replicate.
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.