A165438 Number a(n) of alternative sets of orthogonal contrasts available to partition variation between n levels of a categorical factor in analysis of variance, with each set described by a unique general linear model.

1, 3, 4, 8, 15, 34, 69, 152, 332, 751, 1698, 3905, 9020, 21051, 49356, 116505, 276217, 658091, 1573835, 3778152, 9098915, 21980209, 53241777, 129294912, 314714273, 767700735, 1876437054, 4595005570, 11271747564, 27695048780

Offset

3,2

Comments

Each set has n-1 orthogonal contrasts.

References

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.

Links

Table of n, a(n) for n=3..32.

C. P. Doncaster, Contrast sets

C. P. Doncaster, Orthogonal contrasts

C. P. Doncaster & A. J. H. Davey, Analysis of Variance and Covariance

Formula

For n=5,6,7: a(n) = -mod(n,2)*a([n-mod{n,2}]/2) + sum_{k=3..n-1} a[k]

For n>7: a(n) = -mod(n,2)*a([n-mod{n,2}]/2) + 2*a(n-1) + b(n) - b(n-1)

where b(m) = 0^mod(log(m,2),1) + mod(m-1,2)*0.5*a([m-mod{m,2}]/2)*(a[{m-mod(m,2)}/2]-1)

+ sum_{k=3..(m-1-mod[m-1,2])/2} a(m-k)*(a[k]-1)

Example

A factor 'A' with n = 5 levels, has a(5) = 4 alternative sets of orthogonal
contrasts, each with n - 1 = 4 contrasts. The corresponding alternative
general linear models describing contrasts 'B', 'C', 'D', 'E' are:
B + C(B) + D(B) + E(D B)
B + C(B) + D(C B) + E(D C B)
B + C(B) + D(C B) + E(C B)
B + C(B) + D(B) + E(B)

Crossrefs

Sequence in context: A337438 A007486 A027977 * A293781 A202025 A227615

Adjacent sequences: A165435 A165436 A165437 * A165439 A165440 A165441

Keyword

nonn

Author

C. Patrick Doncaster (cpd(AT)soton.ac.uk), Sep 18 2009

Extensions

Corrected and edited by C. P. Doncaster (cpd(AT)soton.ac.uk), Mar 02 2010

Status

approved