The BGLIMM Procedure

PREDDIST Statement

  • PREDDIST <'label'> OUTPRED=SAS-data-set <NSIM=n> <COVARIATES=SAS-data-set><STATISTICS=options>;

The PREDDIST statement creates a new SAS data set that contains random samples from the posterior predictive distribution of the response variable. The posterior predictive distribution is the distribution of unobserved observations (prediction) conditional on the observed data. Let bold y be the observed data, bold upper X be the covariates, theta be the parameter, and bold y Subscript pred be the unobserved data. The posterior predictive distribution is defined as

StartLayout 1st Row 1st Column p left-parenthesis bold y Subscript pred Baseline vertical-bar bold y comma bold upper X right-parenthesis 2nd Column equals 3rd Column integral p left-parenthesis bold y Subscript pred Baseline comma theta vertical-bar bold y comma bold upper X right-parenthesis d theta 2nd Row 1st Column Blank 2nd Column equals 3rd Column integral p left-parenthesis bold y Subscript pred Baseline vertical-bar theta comma bold y comma bold upper X right-parenthesis p left-parenthesis theta vertical-bar bold y comma bold upper X right-parenthesis d theta EndLayout

Given the assumption that the observed and unobserved data are conditional independent given theta, the posterior predictive distribution can be further simplified as

p left-parenthesis bold y Subscript pred Baseline vertical-bar bold y comma bold upper X right-parenthesis equals integral p left-parenthesis bold y Subscript pred Baseline vertical-bar theta right-parenthesis p left-parenthesis theta vertical-bar bold y comma bold upper X right-parenthesis d theta

The posterior predictive distribution is an integral of the likelihood function p left-parenthesis bold y Subscript pred Baseline vertical-bar theta right-parenthesis with respect to the posterior distribution p left-parenthesis theta vertical-bar bold y right-parenthesis. The PREDDIST statement generates samples from a posterior predictive distribution on the basis of draws from the posterior distribution of theta.

You can specify the following options:

COVARIATES=SAS-data-set

names the SAS data set that contains the sets of explanatory variable values for which the predictions are established. This data set must contain data that have the same variable names that are used in the likelihood function. If you omit this option, the DATA= data set that you specify in the PROC BGLIMM statement is used instead.

ILINK

outputs the inverse link function of the linear predictor for each observation.

LINP

outputs the linear predictors.

MILINK

outputs the inverse link function of the marginal linear predictor for each observation.

MLINP

outputs the marginal linear predictor for each observation.

NSIM=n

specifies the number of simulated predicted values. By default, n is the same as the NMC= option value that you specify in the PROC BGLIMM statement.

OUTPRED=SAS-data-set

creates an output data set to contain the samples from the posterior predictive distribution. The output variable names are listed as resp_1–resp_m, where resp is the name of the response variable and m is the number of observations in the COVARIATES= data set in the PREDDIST statement. If the COVARIATES= data set is not specified, m is the number of observations in the DATA= data set that you specify in the PROC BGLIMM statement.

Table 8 displays the keywords for the variables to be included in the OUTPRED= data set.

Table 8: Keywords for Variables in OUTPRED= Data Set

Keyword Description Expression Variable Name
ILINK Mean using inverse link g Superscript negative 1 Baseline left-parenthesis eta Subscript i Baseline right-parenthesis ILink
LINP Linear predictor eta Subscript i Baseline equals bold x Subscript i Baseline bold-italic beta plus bold z Subscript i Baseline bold-italic gamma Subscript i Linp
MILINK Marginal mean using inverse link g Superscript negative 1 Baseline left-parenthesis eta Subscript i Superscript m Baseline right-parenthesis MILnk
MLINP Marginal linear predictor eta Subscript i Superscript m Baseline equals bold x Subscript i Baseline bold-italic beta MLinp


STATISTICS<(global-options)> =  NONE | ALL |stats-request
STATS<(global-options)> =  NONE | ALL |stats-request

specifies options for calculating posterior statistics. This option works in exactly the same way as the STATISTICS= option in the PROC BGLIMM statement. By default, the STATS option takes the same value that you specify in the STATISTICS= option in the PROC BGLIMM statement.

Last updated: December 09, 2022