PRIOR Statement
- PRIOR < distribution > < / options > ;
The PRIOR statement enables you to carry out a sampling-based
Bayesian analysis in PROC MIXED. It currently operates only with
variance component models. The analysis produces a SAS data set
containing a pseudo-random sample from the joint posterior density
of the variance components and other parameters in the mixed model.
The posterior analysis is performed after all other PROC MIXED
computations. It begins with the "Posterior Sampling
Information" table, which provides basic information about the
posterior sampling analysis, including the prior densities, sampling
algorithm, sample size, and random number seed. For ODS purposes,
the name of this table is "Posterior."
By default, PROC MIXED uses an independence chain algorithm in order
to generate the posterior sample (Tierney 1994). This algorithm
works by generating a pseudo-random proposal from a convenient base
distribution, chosen to be as close as possible to the posterior.
The proposal is then retained in the sample with probability
proportional to the ratio of weights constructed by taking the ratio
of the true posterior to the base density. If a proposal is not
accepted, then a duplicate of the previous observation is added to
the chain.
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 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 (refer
to 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 label 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 III 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 procedures 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'V-1X)-X'V-1y and variance
(X'V-1X)-.
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.
- JEFFREYS
-
specifies a noninformative reference version of Jeffreys' prior
constructed using the square root of the determinant of the expected
information matrix as in (1.3.92) of Box and Tiao (1973).
This is the default prior.
- FLAT
-
specifies a prior density equal to 1 everywhere, making the
likelihood function the posterior.
You can specify the following options in the PRIOR statement after a
slash (/).
- ALG=IC | INDCHAIN
- ALG=IS | IMPSAMP
- ALG=RS | REJSAMP
- ALG=RWC | RWCHAIN
-
specifies the algorithm used for generating the posterior sample.
The ALG=IC option requests an independence chain algorithm, and it is
the default. The option ALG=IS requests importance sampling, ALG=RS
requests rejection sampling, and ALG=RWC requests a random walk
chain. For more information on these techniques, refer to Ripley
(1987), Smith and Gelfand (1992), and Tierney (1994).
- BDATA=
-
enables you to input the base densities 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 BDATA= 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.
- GRID=(value-list)
-
specifies a grid of values over which to evaluate the posterior
density. The value-list syntax is the same as in the
PARMS statement,
and you must specify an output data set name with the
OUTG= option.
- GRIDT=(value-list)
-
specifies a transformed grid of values over which to evaluate the
posterior density. The value-list syntax is the same
as in the PARMS statement,
and you must specify an output data set name with the
OUTGT= option.
- IFACTOR=number
-
is an alias for the SFACTOR= option.
- LOGNOTE=number
-
instructs PROC MIXED to write a note to the SAS log after it
generates the sample corresponding to each multiple of number.
This is useful for monitoring the progress of CPU-intensive runs.
- LOGRBOUND=number
-
specifies the bounding constant for rejection sampling. The value
of number equals the maximum of log(f/g) over the variance
component parameter space, where f is the posterior density and
g is the product inverted gamma densities used to perform
rejection sampling.
When performing the rejection sampling, you may encounter the
message
WARNING: The log ratio bound of LL was violated at sample XX.
When this occurs, PROC MIXED reruns an optimization algorithm to
determine a new log upper bound and then restarts the rejection
sampling. The resulting OUT= data set contains all observations
that have been generated; therefore, assuming that you have requested N samples,
you should retain only the final N observations in this data set for
analysis purposes.
- NSAMPLE=number
-
specifies the number of posterior samples to generate. The
default is 1000, but more accurate results are obtained
with larger samples such as 10000.
- NSEARCH=number
-
specifies the number of posterior evaluations PROC MIXED
makes for each transformed parameter in determining
the parameters for the inverted gamma densities. The default
is 20.
- OUT=SAS-data-set
-
creates an output data set containing the sample from the
posterior density.
- OUTG=SAS-data-set
-
creates an output data set from the grid evaluations
specified in the GRID= option.
- OUTGT=SAS-data-set
-
creates an output data set from the transformed grid
evaluations specified in the GRIDT= option.
- PSEARCH
-
displays the search used to determine the parameters for the
inverted gamma densities. For ODS purposes,
the name of the table is "Search."
- PTRANS
-
displays the transformation of the variance components. For ODS
purposes, the name of the table is "Trans."
- SEED=number
-
specifies a starting value for the random number generation
in PROC MIXED. The computer clock time is the default.
You should use a seed whenever you want to duplicate the
sample in another run of PROC MIXED.
- SFACTOR=number
-
enables you to adjust the range over which PROC MIXED searches the
transformed parameters in order to determine the parameters for the
inverted gamma densities. PROC MIXED determines the range by first
transforming the estimates from the standard PROC MIXED analysis
(REML, ML, or MIVQUE0, depending upon which estimation method you
select). It then multiplies and divides the transformed estimates
by 2*number to obtain upper and lower bounds, respectively.
Transformed values that produce negative variance components in the
original scale are not included in the search. The default value is
1; number must be greater than 0.5.
- TDATA=
-
enables you to input the transformation of the covariance parameters
used by the sampling algorithm. This data set should contain
the CovP1 -CovPn variables. The format of the TDATA= data set
matches that created by PROC MIXED in the "Trans" table, so you can
output the transformation from one run and use is as input for a
subsequent run.
- TRANS=EXPECTED
- TRANS=MIVQUE0
- TRANS=OBSERVED
-
specifies the particular algorithm used to determine the transformation
of the covariance parameters. The default is MIVQUE0, indicating a
transformation based on the MIVQUE(0) equations. The other two
options indicate the type of Hessian matrix used in constructing
the transformation via a Cholesky root.
- UPDATE=number
-
is an alias for the LOGNOTE= option.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
