Satterthwaite Degrees of Freedom Approximation |
The DDFM=SATTERTHWAITE option in the MODEL statement requests denominator degrees of freedom in tests and tests computed according to a general Satterthwaite approximation. The DDFM=KENWARDROGER option also entails the computation of Satterthwaite-type degrees of freedom.
The general Satterthwaite approximation computed in PROC GLIMMIX for the test
is based on the statistic
where , and is the approximate variance matrix of ; see the section Estimated Precision of Estimates and the section Aspects Common to Adaptive Quadrature and Laplace Approximation.
The approximation proceeds by first performing the spectral decomposition , where is an orthogonal matrix of eigenvectors and is a diagonal matrix of eigenvalues, both of dimension . Define to be the th row of , and let
where is the th diagonal element of and is the gradient of with respect to , evaluated at . The matrix is the asymptotic variance-covariance matrix of , obtained from the second derivative matrix of the likelihood equations. You can display this matrix with the ASYCOV option in the PROC GLIMMIX statement.
Finally, let
where the indicator function eliminates terms for which . The degrees of freedom for are then computed as
provided ; otherwise is set to zero.
In the one-dimensional case, when PROC GLIMMIX computes a test, the Satterthwaite degrees of freedom for the statistic
are computed as
where is the gradient of with respect to , evaluated at .