Such constraints have no impact on what linear

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Unformatted text preview: and only if C (X ) ∩ C (M ) = {0} and rank(M) = p − rank(X). Such constraints have no impact on what linear functions of β are estimable or on inferences about estimable functions of β . Thus, the same analysis results are obtained with or without constraints. Copyright c 2010 Dept. of Statistics (Iowa State University) Statistics 511 9 / 14 In 511, we often start with a non-full-rank design matrix for ease and symmetry when specifying a model. For example, it is nice to be able to write yijk = µ + αi + βj + ijk for i = 1, 2, 3; j = 1, 2, 3, 4; k = 1, . . . , 10 if we want to specify a two-factor additive model. The design matrix that matches this model specification does not have full-column rank. R (and other statistical software) will automatically pick a full-column-rank design matrix, which is equivalent to placing certain constraints on the solutions to the normal equations. Copyright c 2010 Dept. of Statistics (Iowa State University) Statistics 511 10 / 14 It does not matter which full-column-rank design matrix (or corresponding constraint set) is chosen as long as the column space of the selected design matrix is the same as the c...
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This document was uploaded on 03/27/2014.

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