Bayesian Analysis of a Linear Regression Model 
Neter et al. (1996) describe a study of 54 patients undergoing a certain kind of liver operation in a surgical unit. The data set Surg contains survival time and certain covariates for each patient. Observations for the first 20 patients in the data set Surg are shown in Figure 39.7.
Obs  x1  x2  x3  x4  y  logy  Logx1 

1  6.7  62  81  2.59  200  2.3010  1.90211 
2  5.1  59  66  1.70  101  2.0043  1.62924 
3  7.4  57  83  2.16  204  2.3096  2.00148 
4  6.5  73  41  2.01  101  2.0043  1.87180 
5  7.8  65  115  4.30  509  2.7067  2.05412 
6  5.8  38  72  1.42  80  1.9031  1.75786 
7  5.7  46  63  1.91  80  1.9031  1.74047 
8  3.7  68  81  2.57  127  2.1038  1.30833 
9  6.0  67  93  2.50  202  2.3054  1.79176 
10  3.7  76  94  2.40  203  2.3075  1.30833 
11  6.3  84  83  4.13  329  2.5172  1.84055 
12  6.7  51  43  1.86  65  1.8129  1.90211 
13  5.8  96  114  3.95  830  2.9191  1.75786 
14  5.8  83  88  3.95  330  2.5185  1.75786 
15  7.7  62  67  3.40  168  2.2253  2.04122 
16  7.4  74  68  2.40  217  2.3365  2.00148 
17  6.0  85  28  2.98  87  1.9395  1.79176 
18  3.7  51  41  1.55  34  1.5315  1.30833 
19  7.3  68  74  3.56  215  2.3324  1.98787 
20  5.6  57  87  3.02  172  2.2355  1.72277 
Consider the model
where Y is the survival time, LogX1 is log(bloodclotting score), X2 is a prognostic index, X3 is an enzyme function test score, X4 is a liver function test score, and is an error term.
A question of scientific interest is whether blood clotting score has a positive effect on survival time. Using PROC GENMOD, you can obtain a maximum likelihood estimate of the coefficient and construct a null point hypothesis to test whether is equal to 0. However, if you are interested in finding the probability that the coefficient is positive, Bayesian analysis offers a convenient alternative. You can use Bayesian analysis to directly estimate the conditional probability, , using the posterior distribution samples, which are produced as part of the output by PROC GENMOD.
The example that follows shows how to use PROC GENMOD to carry out a Bayesian analysis of the linear model with a normal error term. The SEED= option is specified to maintain reproducibility; no other options are specified in the BAYES statement. By default, a uniform prior distribution is assumed on the regression coefficients. The uniform prior is a flat prior on the real line with a distribution that reflects ignorance of the location of the parameter, placing equal likelihood on all possible values the regression coefficient can take. Using the uniform prior in the following example, you would expect the Bayesian estimates to resemble the classical results of maximizing the likelihood. If you can elicit an informative prior distribution for the regression coefficients, you should use the COEFFPRIOR= option to specify it. A default noninformative gamma prior is used for the scale parameter .
You should make sure that the posterior distribution samples have achieved convergence before using them for Bayesian inference. PROC GENMOD produces three convergence diagnostics by default. If ODS Graphics is enabled as specified in the following SAS statements, diagnostic plots are also displayed. See the section Assessing Markov Chain Convergence for more information about convergence diagnostics and their interpretation.
Summary statistics of the posterior distribution samples are produced by default. However, these statistics might not be sufficient for carrying out your Bayesian inference, and further processing of the posterior samples might be necessary. The following SAS statements request the Bayesian analysis, and the OUTPOST= option saves the samples in the SAS data set PostSurg for further processing:
ods graphics on; proc genmod data=Surg; model y = Logx1 X2 X3 X4 / dist=normal; bayes seed=1 OutPost=PostSurg; run; ods graphics off;
The results of this analysis are shown in the following figures.
The "Model Information" table in Figure 39.8 summarizes information about the model you fit and the size of the simulation.
Model Information  

Data Set  WORK.SURG  
BurnIn Size  2000  
MC Sample Size  10000  
Thinning  1  
Sampling Algorithm  Conjugate  
Distribution  Normal  
Link Function  Identity  
Dependent Variable  y  Survival Time 
The "Analysis of Maximum Likelihood Parameter Estimates" table in Figure 39.9 summarizes maximum likelihood estimates of the model parameters.
Analysis Of Maximum Likelihood Parameter Estimates  

Parameter  DF  Estimate  Standard Error  Wald 95% Confidence Limits  
Intercept  1  730.559  85.4333  898.005  563.112 
Logx1  1  171.8758  38.2250  96.9561  246.7954 
x2  1  4.3019  0.5566  3.2109  5.3929 
x3  1  4.0309  0.4996  3.0517  5.0100 
x4  1  18.1377  12.0721  5.5232  41.7986 
Scale  1  59.8591  5.7599  49.5705  72.2832 
Note:  The scale parameter was estimated by maximum likelihood. 
Since no prior distributions for the regression coefficients were specified, the default noninformative uniform distributions shown in the "Uniform Prior for Regression Coefficients" table in Figure 39.10 are used. Noninformative priors are appropriate if you have no prior knowledge of the likely range of values of the parameters, and if you want to make probability statements about the parameters or functions of the parameters. See, for example, Ibrahim, Chen, and Sinha (2001) for more information about choosing prior distributions.
Uniform Prior for Regression Coefficients 


Parameter  Prior 
Intercept  Constant 
Logx1  Constant 
x2  Constant 
x3  Constant 
x4  Constant 
The default noninformative gamma prior distribution for the normal scale parameter is shown in the "Independent Prior Distributions for Model Parameters" table in Figure 39.11.
Independent Prior Distributions for Model Parameters  

Parameter  Prior Distribution  Hyperparameters  
Shape  Scale  
Dispersion  Inverse Gamma  2.001  0.0001 
By default, the maximum likelihood estimates of the regression parameters are used as the starting values for the simulation when noninformative prior distributions are used. These are listed in the "Initial Values and Seeds" table in Figure 39.12.
Initial Values of the Chain  

Chain  Seed  Intercept  Logx1  x2  x3  x4  Dispersion 
1  1  730.559  171.8758  4.301896  4.030878  18.1377  3223.694 
Summary statistics for the posterior sample are displayed in the "Fit Statistics," "Descriptive Statistics for the Posterior Sample," "Interval Statistics for the Posterior Sample," and "Posterior Correlation Matrix" tables in Figure 39.13, Figure 39.14, Figure 39.15, and Figure 39.16, respectively.
Fit Statistics  

DIC (smaller is better)  608.411 
pD (effective number of parameters)  6.571 
Posterior Summaries  

Parameter  N  Mean  Standard Deviation 
Percentiles  
25%  50%  75%  
Intercept  10000  730.1  91.0133  789.6  729.6  670.5 
Logx1  10000  171.7  40.3792  144.3  171.8  198.6 
x2  10000  4.3000  0.5989  3.8990  4.2932  4.6951 
x3  10000  4.0310  0.5354  3.6645  4.0265  4.3910 
x4  10000  18.0888  12.8949  9.4919  18.0430  26.7881 
Dispersion  10000  3795.9  770.4  3247.6  3694.7  4238.2 
Posterior Intervals  

Parameter  Alpha  EqualTail Interval  HPD Interval  
Intercept  0.050  908.7  551.0  906.2  549.2 
Logx1  0.050  92.4773  251.6  94.2813  253.0 
x2  0.050  3.1062  5.4839  3.1747  5.5328 
x3  0.050  2.9812  5.1041  2.9532  5.0612 
x4  0.050  7.2646  43.6506  5.9839  44.6427 
Dispersion  0.050  2569.0  5548.5  2389.4  5308.8 
Posterior Correlation Matrix  

Parameter  Intercept  Logx1  x2  x3  x4  Dispersion 
Intercept  1.000  0.856  0.580  0.712  0.579  0.002 
Logx1  0.856  1.000  0.285  0.490  0.636  0.009 
x2  0.580  0.285  1.000  0.302  0.492  0.007 
x3  0.712  0.490  0.302  1.000  0.616  0.004 
x4  0.579  0.636  0.492  0.616  1.000  0.002 
Dispersion  0.002  0.009  0.007  0.004  0.002  1.000 
Since noninformative prior distributions were used, the posterior sample means, standard deviations, and interval statistics shown in Figure 39.13 and Figure 39.14 are consistent with the maximum likelihood estimates shown in Figure 39.9.
By default, PROC GENMOD computes three convergence diagnostics: the lag1, lag5, lag10, and lag50 autocorrelations (Figure 39.17); Geweke diagnostic statistics (Figure 39.18); and effective sample sizes (Figure 39.19). There is no indication that the Markov chain has not converged. See the section Assessing Markov Chain Convergence for more information about convergence diagnostics and their interpretation.
Posterior Autocorrelations  

Parameter  Lag 1  Lag 5  Lag 10  Lag 50 
Intercept  0.0050  0.0023  0.0138  0.0032 
Logx1  0.0030  0.0063  0.0070  0.0034 
x2  0.0113  0.0046  0.0235  0.0139 
x3  0.0019  0.0064  0.0073  0.0047 
x4  0.0001  0.0084  0.0050  0.0084 
Dispersion  0.0019  0.0088  0.0297  0.0025 
Geweke Diagnostics  

Parameter  z  Pr > z 
Intercept  0.8783  0.3798 
Logx1  1.4800  0.1389 
x2  0.0438  0.9651 
x3  0.1000  0.9204 
x4  0.8893  0.3739 
Dispersion  0.1011  0.9195 
Effective Sample Sizes  

Parameter  ESS  Autocorrelation Time 
Efficiency 
Intercept  10000.0  1.0000  1.0000 
Logx1  10000.0  1.0000  1.0000 
x2  10232.2  0.9773  1.0232 
x3  10000.0  1.0000  1.0000 
x4  10000.0  1.0000  1.0000 
Dispersion  10000.0  1.0000  1.0000 
Trace, autocorrelation, and density plots for the seven model parameters, shown in Figure 39.20 through Figure 39.25, are useful in diagnosing whether the Markov chain of posterior samples has converged. These plots show no evidence that the chain has not converged. See the section Visual Analysis via Trace Plots for help with interpreting these diagnostic plots.
Suppose, for illustration, a question of scientific interest is whether blood clotting score has a positive effect on survival time. Since the model parameters are regarded as random quantities in a Bayesian analysis, you can answer this question by estimating the conditional probability of being positive, given the data, , from the posterior distribution samples. The following SAS statements compute the estimate of the probability of being positive:
data Prob; set PostSurg; Indicator = (logX1 > 0); label Indicator= 'log(Blood Clotting Score) > 0'; run; proc Means data = Prob(keep=Indicator) n mean; run;
As shown in Figure 39.26, there is a 1.00 probability of a positive relationship between the logarithm of a blood clotting score and survival time, adjusted for the other covariates.
Analysis Variable : Indicator log(Blood Clotting Score) > 0 


N  Mean 
10000  0.9999000 