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Search: id:A165438
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| A165438 |
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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. |
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+0
1
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| 1, 3, 4, 8, 15, 33, 68, 150, 328, 741, 1675, 3850, 8889, 20730, 48576, 114589, 271501, 646419, 1544882, 3706104, 8919257, 21531209, 52117708, 126476078, 307635447, 749900303, 1831623560, 4482062843, 10986820829, 26975597625
(list; graph; listen)
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OFFSET
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3,2
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COMMENT
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Each set has n-1 orthogonal contrasts.
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REFERENCES
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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.
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LINKS
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C. P. Doncaster, Contrast sets
C. P. Doncaster, Orthogonal contrasts
C. P. Doncaster & A. J. H. Davey, Analysis of Variance and Covariance
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FORMULA
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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(n) = mod(n-1,2)*0.5*a([n-mod{n,2}]/2)*(a[{n-mod(n,2)}/2]-1)
+ sum_{k=3..(n-1-mod[n-1,2])/2} a(n-k)*(a[k]-1)
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EXAMPLE
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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)
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CROSSREFS
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Sequence in context: A042981 A007486 A027977 this_sequence A049894 A155861 A153057
Adjacent sequences: A165435 A165436 A165437 this_sequence A165439 A165440 A165441
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KEYWORD
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nonn
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AUTHOR
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C. Patrick Doncaster (cpd(AT)soton.ac.uk), Sep 18 2009
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