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The GLIMMIX Procedure |
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
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is based on the statistic
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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
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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
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where the indicator function eliminates terms for which . The degrees of freedom for
are then computed as
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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
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are computed as
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where is the gradient of
with respect to
, evaluated at
.
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