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 in ). 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 ni contrast sets, given by:
A factor with i > 7 levels has ni 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 n6 = 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 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 model for analysing the set with , 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.