Example 37.10 Bayesian Analysis of a Poisson Regression Model

This example illustrates a Bayesian analysis of a log-linear Poisson regression model. Consider the following data on patients from clinical trials. The data set is a subset of the data described in Ibrahim, Chen, and Lipsitz (1999).

data Liver;
input X1-X6 Y;
datalines;
19.1358    50.0110     51.000      0        0       1        3
23.5970    18.4959      3.429      0        0       1        9
20.0474    56.7699      3.429      1        1       0        6
28.0277    59.7836      4.000      0        0       1        6
28.6851    74.1589      5.714      1        0       1        1
18.8092    31.0630      2.286      0        1       1       61
28.7201    52.9178     37.286      1        0       1        6
21.3669    61.6603     54.143      0        1       1        6
23.7332    42.2904      0.571      1        0       1       21

   ... more lines ...   

19.1327    65.3425      2.571      1        0       0       1
17.3010    51.4493      4.429      1        0       0       6
;
run;

The primary interest is in prediction of the number of cancerous liver nodes when a patient enters the trials, by using six other baseline characteristics. The number of nodes is modeled by a Poisson regression model with the six baseline characteristics as covariates. The response and regression variables are as follows:

Two analyses are performed using PROC GENMOD. The first analysis uses noninformative normal prior distributions, and the second analysis uses an informative normal prior for one of the regression parameters.

In the following BAYES statement, COEFFPRIOR=NORMAL specifies a noninformative independent normal prior distribution with zero mean and variance for each parameter.

The initial analysis is performed using PROC GENMOD to obtain Bayesian estimates of the regression coefficients by using the following SAS statements:

ods graphics ON;
proc genmod data=Liver;
   model Y = X1-X6 / dist=Poisson link=log;
   bayes seed=1 coeffprior=normal;
run;

Maximum likelihood estimates of the model parameters are computed by default. These are shown in the "Analysis of Maximum Likelihood Parameter Estimates" table in Output 37.10.1.

Noninformative independent normal prior distributions with zero means and variances of were used in the initial analysis. These are shown in Output 37.10.2.

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 Output 37.10.4, Output 37.10.5, Output 37.10.6, and Output 37.10.7, respectively. Since noninformative prior distributions for the regression coefficients were used, the mean and standard deviations of the posterior distributions for the model parameters are close to the maximum likelihood estimates and standard errors.

Posterior sample autocorrelations for each model parameter are shown in Output 37.10.8. The autocorrelation after 10 lags is negligible for all parameters, indicating good mixing in the Markov chain.

The p-values for the Geweke test statistics shown in Output 37.10.9 all indicate convergence of the MCMC. See the section Assessing Markov Chain Convergence for more information about convergence diagnostics and their interpretation.

The effective sample sizes for each parameter are shown in Output 37.10.10.

Trace, autocorrelation, and density plots for the seven model parameters are shown in Output 37.10.11 through Output 37.10.17. All indicate satisfactory convergence of the Markov chain.

Output 37.10.11 Diagnostic Plots for Intercept

Output 37.10.12 Diagnostic Plots for X1

Output 37.10.13 Diagnostic Plots for X2

Output 37.10.14 Diagnostic Plots for X3

Output 37.10.15 Diagnostic Plots for X4

Output 37.10.16 Diagnostic Plots for X5

Output 37.10.17 Diagnostic Plots for X6

In order to illustrate the use of an informative prior distribution, suppose that researchers expect that a unit increase in body mass index (X1) will be associated with an increase in the mean number of nodes of between 10% and 20%, and they want to incorporate this prior knowledge in the Bayesian analysis. For log-linear models, the mean and linear predictor are related by . If and are two values of body mass index, and are the two mean values, and all other covariates remain equal for the two values of X1, then

so that for a unit change in X1,

If , then , or . This gives you guidance in specifying a prior distribution for the for body mass index. Taking the mean of the prior normal distribution to be the midrange of the values of , and taking to be the extremes of the range, an is the resulting prior distribution. The second analysis uses this informative normal prior distribution for the coefficient of and uses independent noninformative normal priors with zero means and variances equal to for the remaining model regression parameters.

In the following BAYES statement, COEFFPRIOR=NORMAL(INPUT=NormalPrior) specifies the normal prior distribution for the regression coefficients with means and variances contained in the data set NormalPrior.

An analysis is performed using PROC GENMOD to obtain Bayesian estimates of the regression coefficients by using the following SAS statements:

data NormalPrior;
  input _type_ $ Intercept X1-X6;
  datalines;
Var  1e6   0.0005      1e6      1e6      1e6      1e6      1e6
Mean 0.0   0.1385      0.0      0.0      0.0      0.0      0.0
;
run;

proc genmod data=Liver;
   model Y = X1-X6 / dist=Poisson link=log;
   bayes seed=1 plots=none coeffprior=normal(input=NormalPrior) ;
run;
ods graphics off;

The prior distributions for the regression parameters are shown in Output 37.10.18.

Goodness-of-fit, summary, and interval statistics are shown in Output 37.10.20. Except for X1, the statistics shown in Output 37.10.20 are very similar to the previous statistics for noninformative priors shown in Output 37.10.4 through Output 37.10.7. The point estimate for X1 is now positive. This is expected because the prior distribution on is quite informative. The distribution reflects the belief that the coefficient is positive. The distribution places the majority of its probability density on positive values. As a result, the posterior density of places more likelihood on positive values than in the noninformative case.