SAS/STAT 9.2 User's Guide: The MIXED Procedure (Book Excerpt)

3940 ✦ Chapter 56: The MIXED Procedure
In selecting the base distribution, PROC MIXED makes use of the fact that the fixed-effects parameters
can be analytically integrated out of the joint posterior, leaving the marginal posterior density
of the variance components. In order to better approximate the marginal posterior density of the
variance components, PROC MIXED transforms them by using the MIVQUE(0) equations. You
can display the selected transformation with the PTRANS option or specify your own with the
TDATA= option. The density of the transformed parameters is then approximated by a product of
inverted gamma densities (see Gelfand et al. 1990).
To determine the parameters for the inverted gamma densities, PROC MIXED evaluates the logarithm
of the posterior density over a grid of points in each of the transformed parameters, and you
can display the results of this search with the PSEARCH option. PROC MIXED then performs a
linear regression of these values on the logarithm of the inverted gamma density. The resulting base
densities are displayed in the “Base Densities” table; for ODS purposes, the name of this table is
“BaseDen.” You can input different base densities with the BDATA= option.
At the end of the sampling, the “Acceptance Rates” table displays the acceptance rate computed
as the number of accepted samples divided by the total number of samples generated. For ODS
purposes, the name of the “Acceptance Rates” table is “AcceptanceRates.”
The OUT= option specifies the output data set containing the posterior sample. PROC MIXED automatically
includes all variance component parameters in this data set (labeled COVP1–COVPn),
the Type 3 F statistics constructed as in Ghosh (1992) discussing Schervish (1992) (labeled T3Fn),
the log values of the posterior (labeled LOGF), the log of the base sampling density (labeled
LOGG), and the log of their ratio (labeled LOGRATIO). If you specify the SOLUTION option
in the MODEL statement, the data set also contains a random sample from the posterior density
of the fixed-effects parameters (labeled BETAn); and if you specify the SOLUTION option in
the RANDOM statement, the table contains a random sample from the posterior density of the
random-effects parameters (labeled GAMn). PROC MIXED also generates additional variables
corresponding to any CONTRAST, ESTIMATE, or LSMEANS statement that you specify.
Subsequently, you can use SAS/INSIGHT or the UNIVARIATE, CAPABILITY, or KDE procedure
to analyze the posterior sample.
The prior density of the variance components is, by default, a noninformative version of Jeffreys’
prior (Box and Tiao 1973). You can also specify informative priors with the DATA= option or a
flat (equal to 1) prior for the variance components. The prior density of the fixed-effects parameters
is assumed to be flat (equal to 1), and the resulting posterior is conditionally multivariate normal
(conditioning on the variance component parameters) with mean .X 0 V 1 X/ X 0 V 1 y and variance
.X 0 V 1 X/ .
The distribution argument in the PRIOR statement determines the prior density for the variance
component parameters of your mixed model. Valid values are as follows.
DATA=
enables you to input the prior densities of the variance components used by the sampling
algorithm. This data set must contain the Type and Parm1–Parmn variables, where n is the
largest number of parameters among each of the base densities. The format of the DATA=
data set matches that created by PROC MIXED in the “Base Densities” table, so you can
output the densities from one run and use them as input for a subsequent run.