The MCMC Procedure
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Overview
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Getting Started
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Syntax
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Details
How PROC MCMC Works Blocking of Parameters Sampling Methods Tuning the Proposal Distribution Conjugate Sampling Initial Values of the Markov Chains Assignments of Parameters Standard Distributions Usage of Multivariate Distributions Specifying a New Distribution Using Density Functions in the Programming Statements Truncation and Censoring Some Useful SAS Functions Matrix Functions in PROC MCMC Create Design Matrix Modeling Joint Likelihood Regenerating Diagnostics Plots Caterpillar Plot Posterior Predictive Distribution Handling of Missing Data Floating Point Errors and Overflows Handling Error Messages Computational Resources Displayed Output ODS Table Names ODS Graphics
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Examples
Simulating Samples From a Known Density Box-Cox Transformation Logistic Regression Model with a Diffuse Prior Logistic Regression Model with Jeffreys’ Prior Poisson Regression Nonlinear Poisson Regression Models Logistic Regression Random-Effects Model Nonlinear Poisson Regression Random-Effects Model Multivariate Normal Random-Effects Model Change Point Models Exponential and Weibull Survival Analysis Time Independent Cox Model Time Dependent Cox Model Piecewise Exponential Frailty Model Normal Regression with Interval Censoring Constrained Analysis Implement a New Sampling Algorithm Using a Transformation to Improve Mixing Gelman-Rubin Diagnostics
- References
| Displayed Output |
This section describes the displayed output from PROC MCMC. For a quick reference of all ODS table names, see the section ODS Table Names. ODS tables are arranged under four groups, listed in the following sections: Sampling Related ODS Tables, Posterior Statistics Related ODS Tables, Convergence Diagnostics Related ODS Tables, and Optimization Related ODS Tables.
Sampling Related ODS Tables
Burn-In History
The "Burn-In History" table (ODS table name BurnInHistory) shows the scales and acceptance rates for each parameter block in the burn-in phase. The table is not displayed by default and can be requested by specifying the option MCHISTORY=BRIEF | DETAILED.
Number of Observation Table
The "NObs" table (ODS table name NOBS) shows the number of observations that is in the data set and the number of observations that is used in the analysis. By default, observations with missing values are not used (see the section Handling of Missing Data for more details). This table is displayed by default.
Parameters
The "Parameters" table (ODS table name Parameters) shows the name of each parameter, the block number of each parameter, the sampling method used for the block, the initial values, and the prior or hyperprior distributions. This table is displayed by default.
REParameters
The "REParameters" table (ODS table name REParameters) lists the name of the random effect, the subject variable, number of clusters (levels), and the prior distribution. This table is displayed by default if a RANDOM statement is used in the program.
Parameters Initial Value Table
The "Parameters Initial" table (ODS table name ParametersInit) shows the value of each parameter after the tuning phase. This table is not displayed by default and can be requested by specifying the option INIT=PINIT.
Posterior Samples
The "Posterior Samples" table (ODS table name PosteriorSample) stores posterior draws of all parameters. It is not printed by PROC MCMC. You can create an ODS output data set of the chain by specifying the following:
ODS OUTPUT PosteriorSample = SAS-data-set;
Sampling History
The "Sampling History" table (ODS table name SamplingHistory) shows the scales and acceptance rates for each parameter block in the main sampling phase. The table is not displayed by default and can be requested by specifying the option MCHISTORY=BRIEF | DETAILED.
Tuning Covariance
The "Tuning Covariance" table (ODS table name TuneCov) shows the proposal covariance matrices for each parameter block after the tuning phase. The table is not displayed by default and can be requested by specifying the option INIT=PINIT. For more details about proposal tuning, see the section Tuning the Proposal Distribution.
Tuning History
The "Tuning History" table (ODS table name TuningHistory) shows the number of tuning phases used in establishing the proposal distribution. The table also displays the scales and acceptance rates for each parameter block at each of the tuning phases. For more information about the self-adapting proposal tuning algorithm used by PROC MCMC, see the section Tuning the Proposal Distribution. The table is not displayed by default and can be requested by specifying the option MCHISTORY=BRIEF | DETAILED.
Tuning Probability Vector
The "Tuning Probability" table (ODS table name TuneP) shows the proposal probability vector for each discrete parameter block (when the option DISCRETE=GEO is specified and the geometric proposal distribution is used for discrete parameters) after the tuning phase. The table is not displayed by default and can be requested by specifying the option INIT=PINIT. For more information about proposal tuning, see the section Tuning the Proposal Distribution.
Posterior Statistics Related ODS Tables
PROC MCMC calculates some essential posterior statistics and outputs them to a number of ODS tables that you can request and save individually. For details of the calculations, see the section Summary Statistics.
Summary Statistics
The "Posterior Summaries" table (ODS table name PostSummaries) contains basic statistics for each parameter. The table lists the number of posterior samples, the posterior mean and standard deviation estimates, and the percentile estimates. This table is displayed by default.
Correlation Matrix
The "Posterior Correlation Matrix" table (ODS table name Corr) contains the posterior correlation of model parameters. The table is not displayed by default and can be requested by specifying the option STATS=CORR.
Covariance Matrix
The "Posterior Covariance Matrix" table (ODS table name Cov) contains the posterior covariance of model parameters. The table is not displayed by default and can be requested by specifying the option STATISTICS=COV.
Deviance Information Criterion
The "Deviance Information Criterion" table (ODS table name DIC) contains the DIC of the model. The table is not displayed by default and can be requested by specifying the option DIC. For details of the calculations, see the section Deviance Information Criterion (DIC).
Interval Statistics
The "Posterior Intervals" table (ODS table name PostIntervals) contains the equal-tail and highest posterior density (HPD) interval estimates for each parameter. The default 
value is 
, and you can change it to other levels by using the STATISTICS= option. This table is displayed by default.
Convergence Diagnostics Related ODS Tables
PROC MCMC has convergence diagnostic tests that check for Markov chain convergence. The procedure produces a number of ODS tables that you can request and save individually. For details in calculation, see the section Statistical Diagnostic Tests.
Autocorrelation
The "Autocorrelations" table (ODS table name AUTOCORR) contains the first order autocorrelations of the posterior samples for each parameter. The "Parameter" column states the name of the parameter. By default, PROC MCMC displays lag 1, 5, 10, and 50 estimates of the autocorrelations. You can request different autocorrelations by using the DIAGNOSTICS = AUTOCORR(LAGS=) option. This table is displayed by default.
Effective Sample Size
The "Effective Sample Sizes" table (ODS table name ESS) calculates the effective sample size of each parameter. See the section Effective Sample Size for more details. The table is displayed by default.
Monte Carlo Standard Errors
The "Monte Carlo Standard Errors" table (ODS table name MCSE) calculates the standard errors of the posterior mean estimate. See the section Standard Error of the Mean Estimate for more details. The table is displayed by default.
Geweke Diagnostics
The "Geweke Diagnostics" table (ODS table name Geweke) lists the result of the Geweke diagnostic test. See the section Geweke Diagnostics for more details. The table is displayed by default.
Heidelberger-Welch Diagnostics
The "Heidelberger-Welch Diagnostics" table (ODS table name Heidelberger) lists the result of the Heidelberger-Welch diagnostic test. The test is consisted of two parts: a stationary test and a half-width test. See the section Heidelberger and Welch Diagnostics for more details. The table is not displayed by default and can be requested by specifying DIAGNOSTICS = HEIDEL.
Raftery-Lewis Diagnostics
The "Raftery-Lewis Diagnostics" table (ODS table name Raftery) lists the result of the Raftery-Lewis diagnostic test. See the section Raftery and Lewis Diagnostics for more details. The table is not displayed by default and can be requested by specifying DIAGNOSTICS = RAFTERY.
Summary Statistics for Prediction
The "Posterior Summaries for Prediction" table (ODS table name PredSummaries) contains basic statistics for each prediction. The table lists the number of posterior samples, the posterior mean and standard deviation estimates, and the percentile estimates. This table is displayed by default if any PREDDIST statement is used in the program.
Interval Statistics for Prediction
The "Posterior Intervals for Prediction" table (ODS table name PredIntervals) contains the equal-tail and highest posterior density (HPD) interval estimates for each prediction. The default value is , and you can change it to other levels by using the STATISTICS option in a PREDDIST statement, or the STATISTICS= option in the PROC MCMC statement if the option is not specified in a statement. This table is displayed by default if any PREDDIST statement is used in the program.
Optimization Related ODS Tables
PROC MCMC can perform optimization on the joint posterior distribution. This is requested by the PROPCOV= option. The most commonly used optimization method is the quasi-Newton method: PROPCOV=QUANEW(ITPRINT). The ITPRINT option displays the ODS tables, listed as follows:
Input Options
The "Input Options" table (ODS table name InputOptions) lists optimization options used in the procedure.
Optimization Start
The "Optimization Start" table (ODS table name ProblemDescription) shows the initial state of the optimization.
Iteration History
The "Iteration History" table (ODS table name IterHist) shows iteration history of the optimization.
Optimization Results
The "Optimization Results" table (ODS table name IterStop) shows the results of the optimization, includes information about the number of function calls, and the optimized objective function, which is the joint log posterior density.
Convergence Status
The "Convergence Status" table (ODS table name ConvergenceStatus) shows whether the convergence criterion is satisfied.
Parameters Value After Optimization Table
The "Parameter Values After Optimization" table (ODS table name OptiEstimates) lists the parameter values that maximize the joint log posterior. These are the maximum a posteriori point estimates, and they are used to start the Markov chain.
Covariance Matrix After Optimization Table
The "Proposal Covariance" table (ODS table name OptiCov) lists covariance matrices for each block parameter by using quadrature approximation at the posterior mode. These covariance matrices are used in the proposal distribution.