This section illustrates some basic features of PROC MCMC by using a linear regression model. The model is as follows:

for the observations .

The following statements create a SAS data set with measurements of Height and Weight for a group of children:

   title 'Simple Linear Regression';
    
   data Class;
      input Name $ Height Weight @@;
      datalines;
   Alfred  69.0 112.5   Alice  56.5  84.0   Barbara 65.3  98.0
   Carol   62.8 102.5   Henry  63.5 102.5   James   57.3  83.0
   Jane    59.8  84.5   Janet  62.5 112.5   Jeffrey 62.5  84.0
   John    59.0  99.5   Joyce  51.3  50.5   Judy    64.3  90.0
   Louise  56.3  77.0   Mary   66.5 112.0   Philip  72.0 150.0
   Robert  64.8 128.0   Ronald 67.0 133.0   Thomas  57.5  85.0
   William 66.5 112.0
   ;

The equation of interest is as follows:

The observation errors, , are assumed to be independent and identically distributed with a normal distribution with mean zero and variance .

The likelihood function for each of the Weight, which is specified in the MODEL statement, is as follows:

where denotes a conditional probability density and is the normal density. There are three parameters in the likelihood: , , and . You use the PARMS statement to indicate that these are the parameters in the model.

Suppose that you want to use the following three prior distributions on each of the parameters:

where indicates a prior distribution and is the density function for the inverse-gamma distribution. The normal priors on and have large variances, expressing your lack of knowledge about the regression coefficients. The priors correspond to an equal-tail credible intervals of approximately for and . Priors of this type are often called vague or diffuse priors. See the section Prior Distributions for more information. Typically diffuse prior distributions have little influence on the posterior distribution and are appropriate when stronger prior information about the parameters is not available.

A frequently used diffuse prior for the variance parameter is the inverse-gamma distribution. With a shape parameter of and a scale parameter of , this prior corresponds to an equal-tail credible interval of , with the mode at for . Alternatively, you can use any other positive prior, meaning that the density support is positive on this variance component. For example, you can use the gamma prior.

According to Bayes’ theorem, the likelihood function and prior distributions determine the posterior (joint) distribution of , , and as follows:

You do not need to know the form of the posterior distribution when you use PROC MCMC. PROC MCMC automatically obtains samples from the desired posterior distribution, which is determined by the prior and likelihood you supply.

The following statements fit this linear regression model with diffuse prior information:

   ods graphics on;
   proc mcmc data=class outpost=classout nmc=50000 thin=5 seed=246810;
      parms beta0 0 beta1 0;
      parms sigma2 1;
      prior beta0 beta1 ~ normal(mean = 0, var = 1e6);
      prior sigma2 ~ igamma(shape = 3/10, scale = 10/3);
      mu = beta0 + beta1*height;
      model weight ~ n(mu, var = sigma2);
   run;
   ods graphics off;
    

The ODS GRAPHICS ON statement invokes the ODS Graphics environment and displays the diagnostic plots, such as the trace and autocorrelation function plots of the posterior samples. For more information about ODS, see Chapter 21, Statistical Graphics Using ODS.

The PROC MCMC statement invokes the procedure and specifies the input data set class. The output data set classout contains the posterior samples for all of the model parameters. The NMC= option specifies the number of posterior simulation iterations. The THIN= option controls the thinning of the Markov chain and specifies that one of every 5 samples is kept. Thinning is often used to reduce the correlations among posterior sample draws. In this example, 10,000 simulated values are saved in the classout data set. The SEED= option specifies a seed for the random number generator, which guarantees the reproducibility of the random stream. For more information about Markov chain sample size, burn-in, and thinning, see the section Burn-in, Thinning, and Markov Chain Samples.

The PARMS statements identify the three parameters in the model: beta0, beta1, and sigma2. Each statement also forms a block of parameters, where the parameters are updated simultaneously in each iteration. In this example, beta0 and beta1 are sampled jointly, conditional on sigma2; and sigma2 is sampled conditional on fixed values of beta0 and beta1. In simple regression models such as this, you expect the parameters beta0 and beta1 to have high posterior correlations, and placing them both in the same block improves the mixing of the chain—that is, the efficiency that the posterior parameter space is explored by the Markov chain. For more information, see the section Blocking of Parameters. The PARMS statements also assign initial values to the parameters (see the section Initial Values of the Markov Chains). The regression parameters are given 0 as their initial values, and the scale parameter sigma2 starts at value 1. If you do not provide initial values, the procedure chooses starting values for every parameter.

The PRIOR statements specify prior distributions for the parameters. The parameters beta0 and beta1 both share the same prior—a normal prior with mean and variance . The parameter sigma2 has an inverse-gamma distribution with a shape parameter of 3/10 and a scale parameter of 10/3. For a list of standard distributions that PROC MCMC supports, see the section Standard Distributions.

The mu assignment statement calculates the expected value of Weight as a linear function of Height. The MODEL statement uses the shorthand notation, n, for the normal distribution to indicate that the response variable, Weight, is normally distributed with parameters mu and sigma2. The functional argument MEAN= in the normal distribution is optional, but you have to indicate whether sigma2 is a variance (VAR=), a standard deviation (SD=), or a precision (PRECISION=) parameter. See Table 52.2 in the section MODEL Statement for distribution specifications.

The distribution parameters can contain expressions. For example, you can write the MODEL statement as follows:

   model weight ~ n(beta0 + beta1*height, var = sigma2);

Before you do any posterior inference, it is essential that you examine the convergence of the Markov chain (see the section Assessing Markov Chain Convergence). You cannot make valid inferences if the Markov chain has not converged. A very effective convergence diagnostic tool is the trace plot. Although PROC MCMC produces graphs at the end of the procedure output (see Figure 52.6), you should visually examine the convergence graph first.

The first table that PROC MCMC produces is the "Number of Observations" table, as shown in Figure 52.1. This table lists the number of observations read from the DATA= data set and the number of non-missing observations used in the analysis.