The GLIMMIX Procedure

Satterthwaite Degrees of Freedom Approximation

The DDFM=SATTERTHWAITE option in the MODEL statement requests that denominator degrees of freedom in t tests and F tests be computed according to a general Satterthwaite approximation.

The general Satterthwaite approximation computed in PROC GLIMMIX for the test

\[  H \colon \bL \left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right] = \mb {0}  \]

is based on the F statistic

\[  F = \frac{\left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right] \bL (\mb {LCL})^{-1} \bL \left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right] }{r}  \]

where $r = \mr {rank}(\mb {LCL}’)$ and $\bC $ is the approximate variance matrix of $[\widehat{\bbeta }’,\widehat{\bgamma }’-\bgamma ’]’$. 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 $ \mb {LCL}’ = \bU ’ \bD \bU $, where $\bU $ is an orthogonal matrix of eigenvectors and $\bD $ is a diagonal matrix of eigenvalues, both of dimension $r \times r$. Define $\mb {b}_ j$ to be the jth row of $\mb {UL}$, and let

\[  \nu _ j = \frac{2 (D_ j)^2}{ \mb {g}_ j \mb {A} \mb {g}_ j}  \]

where $D_ j$ is the jth diagonal element of $\bD $ and $\mb {g}_ j$ is the gradient of $\mb {b}_ j \mb {C} \mb {b}_ j’$ with respect to $\btheta $, evaluated at $\widehat{\btheta }$. The matrix $\bA $ is the asymptotic variance-covariance matrix of $\widehat{\btheta }$, which is 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

\[  E = \sum _{j=1}^{r} \frac{\nu _ j}{\nu _ j - 2}I(\nu _ j > 2)  \]

where the indicator function eliminates terms for which $\nu _ j \le 2$. The degrees of freedom for F are then computed as

\[  \nu = \frac{2 E}{E - \mr {rank}(\mb {L})}  \]

provided E > r; otherwise $\nu $ is set to 0.

In the one-dimensional case, when PROC GLIMMIX computes a t test, the Satterthwaite degrees of freedom for the t statistic

\[  t = \frac{\mb {l} \left[ \begin{array}{c} \widehat{\bbeta } \\ \widehat{\bgamma } \end{array} \right]}{\mb {l}\mb {Cl}}  \]

are computed as

\[  \nu = \frac{2 (\mb {l} \mb {Cl})^2}{ \mb {g} \mb {Ag}}  \]

where $\mb {g}$ is the gradient of $\mb {l}’\mb {Cl}$ with respect to $\btheta $, evaluated at $\widehat{\btheta }$.

The calculation of Satterthwaite degrees of freedom requires extra memory to hold q matrices that are the size of the mixed model equations, where q is the number of covariance parameters. Extra computing time is also required to process these matrices. The implemented Satterthwaite method is intended to produce an accurate F approximation; however, the results can differ from those produced by PROC GLM. Also, the small-sample properties of this approximation have not been extensively investigated for the various models available with PROC GLIMMIX.