The MCMC Procedure
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Overview
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Getting Started
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Syntax
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DetailsHow PROC MCMC WorksBlocking of ParametersSampling MethodsTuning the Proposal DistributionDirect SamplingConjugate SamplingInitial Values of the Markov ChainsAssignments of ParametersStandard DistributionsUsage of Multivariate DistributionsSpecifying a New DistributionUsing Density Functions in the Programming StatementsTruncation and CensoringSome Useful SAS FunctionsMatrix Functions in PROC MCMCCreate Design MatrixModeling Joint LikelihoodRegenerating Diagnostics PlotsCaterpillar PlotAutocall Macros for PostprocessingGamma and Inverse-Gamma DistributionsPosterior Predictive DistributionHandling of Missing DataFunctions of Random-Effects ParametersFloating Point Errors and OverflowsHandling Error MessagesComputational ResourcesDisplayed OutputODS Table NamesODS Graphics
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ExamplesSimulating Samples From a Known DensityBox-Cox TransformationLogistic Regression Model with a Diffuse PriorLogistic Regression Model with Jeffreys’ PriorPoisson RegressionNonlinear Poisson Regression ModelsLogistic Regression Random-Effects ModelNonlinear Poisson Regression Multilevel Random-Effects ModelMultivariate Normal Random-Effects ModelMissing at Random AnalysisNonignorably Missing Data (MNAR) AnalysisChange Point ModelsExponential and Weibull Survival AnalysisTime Independent Cox ModelTime Dependent Cox ModelPiecewise Exponential Frailty ModelNormal Regression with Interval CensoringConstrained AnalysisImplement a New Sampling AlgorithmUsing a Transformation to Improve MixingGelman-Rubin Diagnostics
- References
This section describes the output that PROC MCMC displays. For a quick reference of all ODS table names, see the section ODS Table Names. ODS tables are arranged under four groups, which are listed in the following sections: Model and Data Related ODS Tables, Sampling Related ODS Tables, Posterior Statistics Related ODS Tables, Convergence Diagnostics Related ODS Tables, and Optimization Related ODS Tables.
The “Missing Data Information” table (ODS table name MISSDATAINFO) displays the name of the response variable that contains missing values, the number of missing observations, the corresponding observation indices in the input data set, and the sampling method used in the simulation for the missing values.
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.
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.
The “Random Effect Observation Information” table (ODS table name REObsInfo) lists the name of the random effect, each subject value, the number of observations in each subject, and their corresponding observation indices in the input data set. You can request this table by specifying the REOBSINFO option.
The “REParameters” table (ODS table name REParameters) lists the name of the random effect, sampling algorithm, the subject variable, the number of subjects, unique values of the subject variable, and the prior distribution. This table is displayed by default if a RANDOM statement is used in the program.
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.
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.
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;
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.
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.
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.
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.
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 in Chapter 7: Introduction to Bayesian Analysis Procedures.
The “Posterior Summaries and Intervals” table (ODS table name PostSumInt) contains a summary of basic point and interval statistics for each parameter. The table lists the number of posterior samples, the posterior mean and standard deviation estimates, and the 95% HPD interval estimates. This table is displayed by default.
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. The table is not displayed by default and can be requested by specifying the option STATISTICS=SUMMARY.
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 STATISTICS=CORR.
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.
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) in Chapter 7: Introduction to Bayesian Analysis Procedures.
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 0.05, and you can change it to other levels by using the STATISTICS= option. The table is not displayed by default and can be requested by specifying the option STATISTICS=INTERVAL.
PROC MCMC has convergence diagnostic tests that check for Markov chain convergence. PROC MCMC produces a number of ODS tables that you can request and save individually. For details in calculation, see the section Statistical Diagnostic Tests in Chapter 7: Introduction to Bayesian Analysis Procedures.
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. The table is not displayed by default and can be requested by specifying the option DIAGNOSTICS=AUTOCORR.
The “Effective Sample Sizes” table (ODS table name ESS) calculates the effective sample size of each parameter. See the section Effective Sample Size in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The table is displayed by default.
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 in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The table is not displayed by default and can be requested by specifying the option DIAGNOSTICS=MCSE.
The “Geweke Diagnostics” table (ODS table name Geweke) lists the result of the Geweke diagnostic test. See the section Geweke Diagnostics in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The table is not displayed by default and can be requested by specifying the option DIAGNOSTICS=GEWEKE.
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 in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The table is not displayed by default and can be requested by specifying DIAGNOSTICS = HEIDEL.
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 in Chapter 7: Introduction to Bayesian Analysis Procedures, for more details. The table is not displayed by default and can be requested by specifying DIAGNOSTICS = RAFTERY.
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.
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 0.05, 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.
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:
The “Input Options” table (ODS table name InputOptions) lists optimization options used in the procedure.
The “Optimization Start” table (ODS table name ProblemDescription) shows the initial state of the optimization.
The “Iteration History” table (ODS table name IterHist) shows iteration history of the optimization.
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.
The “Convergence Status” table (ODS table name ConvergenceStatus) shows whether the convergence criterion is satisfied.
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.