A165438 - OEIS

%I #1 Jun 01 2010 03:00:00

%S 1,3,4,8,15,34,69,152,332,751,1698,3905,9020,21051,49356,116505,

%T 276217,658091,1573835,3778152,9098915,21980209,53241777,129294912,

%U 314714273,767700735,1876437054,4595005570,11271747564,27695048780

%N 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.

%C Each set has n-1 orthogonal contrasts.

%D 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.

%H C. P. Doncaster, <a href="http://www.soton.ac.uk/~cpd/anovas/datasets/Contrast%20sets.htm">Contrast sets</a>

%H C. P. Doncaster, <a href="http://www.soton.ac.uk/~cpd/anovas/datasets/Orthogonal%20contrasts.htm">Orthogonal contrasts</a>

%H C. P. Doncaster & A. J. H. Davey, <a href="http://www.soton.ac.uk/~cpd/anovas/datasets/">Analysis of Variance and Covariance</a>

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

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

%F 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)

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

%e A factor 'A' with n = 5 levels, has a(5) = 4 alternative sets of orthogonal

%e contrasts, each with n - 1 = 4 contrasts. The corresponding alternative

%e general linear models describing contrasts 'B', 'C', 'D', 'E' are:

%e B + C(B) + D(B) + E(D B)

%e B + C(B) + D(C B) + E(D C B)

%e B + C(B) + D(C B) + E(C B)

%e B + C(B) + D(B) + E(B)

%K nonn

%O 3,2

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

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