The MCMC Procedure

Example 59.3 Logistic Regression Model with a Diffuse Prior

This example illustrates how to fit a logistic regression model with a diffuse prior in PROC MCMC. You can also use the BAYES statement in PROC GENMOD. See Chapter 42: The GENMOD Procedure.

The following statements create a SAS data set with measurements of the number of deaths, y, among n beetles that have been exposed to an environmental contaminant x:

title 'Logistic Regression Model with a Diffuse Prior';
data beetles;
   input n y x @@;
6  0  25.7   8  2  35.9   5  2  32.9   7  7  50.4   6  0  28.3
7  2  32.3   5  1  33.2   8  3  40.9   6  0  36.5   6  1  36.5
6  6  49.6   6  3  39.8   6  4  43.6   6  1  34.1   7  1  37.4
8  2  35.2   6  6  51.3   5  3  42.5   7  0  31.3   3  2  40.6

You can model the data points $y_ i$ with a binomial distribution,

\[  y_ i|p_ i \sim \mbox{binomial}(n_ i, p_ i)  \]

where $p_ i$ is the success probability and links to the regression covariate $x_ i$ through a logit transformation:

\[  \mi {logit} (p_ i) = \log \left( \frac{p_ i}{1-p_ i} \right) = \alpha + \beta x_ i  \]

The priors on $\alpha $ and $\beta $ are both diffuse normal:

\begin{eqnarray*}  \alpha & \sim &  \mbox{normal}(0, \mbox{var} = 10000) \\ \beta & \sim &  \mbox{normal}(0, \mbox{var} = 10000) \end{eqnarray*}

These statements fit a logistic regression with PROC MCMC:

ods graphics on;
proc mcmc data=beetles ntu=1000 nmc=20000 propcov=quanew
          diag=(mcse ess) outpost=beetleout seed=246810;
   ods select PostSumInt mcse ess TADpanel;
   parms (alpha  beta) 0;
   prior alpha beta ~ normal(0, var = 10000);
   p = logistic(alpha + beta*x);
   model y ~ binomial(n,p);

The key statement in the program is the assignment to p that calculates the probability of death. The SAS function LOGISTIC does the proper transformation. The MODEL statement specifies that the response variable, y, is binomially distributed with parameters n (from the input data set) and p. The summary statistics table, interval statistics table, the Monte Carlo standard error table, and the effective sample sizes table are shown in Output 59.3.1.

Output 59.3.1: MCMC Results

Logistic Regression Model with a Diffuse Prior

The MCMC Procedure

Posterior Summaries and Intervals
Parameter N Mean Standard
95% HPD Interval
alpha 20000 -11.7689 2.0942 -15.9412 -7.7491
beta 20000 0.2919 0.0541 0.1901 0.4029

Logistic Regression Model with a Diffuse Prior

The MCMC Procedure

Monte Carlo Standard Errors
Parameter MCSE Standard
alpha 0.0418 2.0942 0.0200
beta 0.00109 0.0541 0.0201

Effective Sample Sizes
Parameter ESS Autocorrelation
alpha 2507.0 7.9778 0.1253
beta 2478.5 8.0694 0.1239

The summary statistics table shows that the sample mean of the output chain for the parameter alpha is –11.77. This is an estimate of the mean of the marginal posterior distribution for the intercept parameter alpha. The estimated posterior standard deviation for alpha is 2.09. The two 95% credible intervals for alpha are both negative, which indicates with very high probability that the intercept term is negative. On the other hand, you observe a positive effect on the regression coefficient beta. Exposure to the environment contaminant increases the probability of death.

The Monte Carlo standard errors of each parameter are significantly small relative to the posterior standard deviations. A small MCSE/SD ratio indicates that the Markov chain has stabilized and the mean estimates do not vary much over time. Note that the precision in the parameter estimates increases with the square of the MCMC sample size, so if you want to double the precision, you must quadruple the MCMC sample size.

MCMC chains do not produce independent samples. Each sample point depends on the point before it. In this case, the correlation time estimate, read from the effective sample sizes table, is roughly 8. This means that it takes four observations from the MCMC output to make inferences about alpha with the same precision that you would get from using an independent sample. The effective sample size of around 2500 reflects this loss of efficiency. The coefficient beta has similar efficiency. You can often observe that some parameters have significantly better mixing (better efficiency) than others, even in a single Markov chain run.

Output 59.3.2: Plots for Parameters in the Logistic Regression Example

Trace plots and autocorrelation plots of the posterior samples are shown in Output 59.3.2. Convergence looks good in both parameters; there is good mixing in the trace plot and quick drop-off in the ACF plot.

One advantage of Bayesian methods is the ability to directly answer scientific questions. In this example, you might want to find out the posterior probability that the environmental contaminant increases the probability of death—that is, $\mbox{Pr}(\beta > 0 | \mb {y})$. This can be estimated using the following steps:

proc format;
   value betafmt low-0 = 'beta <= 0' 0<-high = 'beta > 0';

proc freq data=beetleout;
   tables beta /nocum;
   format beta betafmt.;

Output 59.3.3: Frequency Counts

Logistic Regression Model with a Diffuse Prior

The FREQ Procedure

beta Frequency Percent
beta > 0 20000 100.00

All of the simulated values for $\beta $ are greater than zero, so the sample estimate of the posterior probability that $\beta >0$ is 100%. The evidence overwhelmingly supports the hypothesis that increased levels of the environmental contaminant increase the probability of death.

If you are interested in making inference based on any quantities that are transformations of the random variables, you can either do it directly in PROC MCMC or by using the DATA step after you run the simulation. Transformations sometimes can make parameter inference quite formidable using direct analytical methods, but with simulated chains, it is easy to compute chains for any set of parameters. Suppose you are interested in the lethal dose and want to estimate the level of the covariate x that corresponds to a probability of death, p. Abbreviate this quantity as ldp. In other words, you want to solve the logit transformation with a fixed value p. The lethal dose is as follows:

\[  \mbox{\Variable{ldp}} = \frac{\log \left( \frac{p}{1-p}\right) - \alpha }{\beta }  \]

You can obtain an estimate of any ldp by using the posterior mean estimates for $\alpha $ and $\beta $. For example, lp95, which corresponds to $p = 0.95$, is calculated as follows:

\[  \mbox{\Variable{lp95}} = \frac{\log \left( \frac{0.95}{1-0.95}\right) + 11.77}{0.29} = 50.79  \]

where –11.77 and 0.29 are the posterior mean estimates of $\alpha $ and $\beta $, respectively, and 50.79 is the estimated lethal dose that leads to a 95% death rate.

While it is easy to obtain the point estimates, it is harder to estimate other posterior quantities, such as the standard deviation directly. However, with PROC MCMC, you can trivially get estimates of any posterior quantities of lp95. Consider the following program in PROC MCMC:

proc mcmc data=beetles ntu=1000 nmc=20000 propcov=quanew
          outpost=beetleout seed=246810 plot=density
          monitor=(pi30 ld05 ld50 ld95);
   ods select PostSumInt densitypanel;
   parms (alpha  beta) 0;
      c1 = log(0.05 / 0.95);
      c2 = -c1;

   prior alpha beta ~ normal(0, var = 10000);
   pi30 = logistic(alpha + beta*30);
   ld05 = (c1 - alpha) / beta;
   ld50 = - alpha / beta;
   ld95 = (c2 - alpha) / beta;
   pi = logistic(alpha + beta*x);
   model y ~ binomial(n,pi);
ods graphics off;

The program estimates four additional posterior quantities. The three lpd quantities, ld05, ld50, and ld95, are the three levels of the covariate that kills 5%, 50%, and 95% of the population, respectively. The predicted probability when the covariate x takes the value of 30 is pi30. The MONITOR= option selects the quantities of interest. The PLOTS= option selects kernel density plots as the only ODS graphical output, excluding the trace plot and autocorrelation plot.

Programming statements between the BEGINCNST and ENDCNST statements define two constants. These statements are executed once at the beginning of the simulation. The programming statements between the BEGINNODATA and ENDNODATA statements evaluate the quantities of interest. The symbols, pi30, ld05, ld50, and ld95, are functions of the parameters alpha and beta only. Hence, they should not be processed at the observation level and should be included in the BEGINNODATA and ENDNODATA statements. Output 59.3.4 lists the posterior summary and Output 59.3.5 shows the density plots of these posterior quantities.

Output 59.3.4: PROC MCMC Results

Logistic Regression Model with a Diffuse Prior

The MCMC Procedure

Posterior Summaries and Intervals
Parameter N Mean Standard
95% HPD Interval
pi30 20000 0.0524 0.0252 0.0126 0.1022
ld05 20000 29.9310 1.8731 26.2170 33.2648
ld50 20000 40.3781 0.9371 38.5334 42.1808
ld95 20000 50.8251 2.5327 46.2349 55.7958

The posterior mean estimate of lp95 is 50.82, which is close to the estimate of 50.79 by using the posterior mean estimates of the parameters. With PROC MCMC, in addition to the mean estimate, you can get the standard deviation, quantiles, and interval estimates at any level of significance.

From the density plots, you can see, for example, that the sample distribution for $\pi _{30}$ is skewed to the right, and almost all of your posterior belief concerning $\pi _{30}$ is concentrated in the region between zero and 0.15.

Output 59.3.5: Density Plots of Quantities of Interest in the Logistic Regression Example

It is easy to use the DATA step to calculate these quantities of interest. The following DATA step uses the simulated values of $\alpha $ and $\beta $ to create simulated values from the posterior distributions of ld05, ld50, ld95, and $\pi _{30}$:

data transout;
   set beetleout;
   pi30 = logistic(alpha + beta*30);
   ld05 = (log(0.05 / 0.95) - alpha) / beta;
   ld50 = (log(0.50 / 0.50) - alpha) / beta;
   ld95 = (log(0.95 / 0.05) - alpha) / beta;

Subsequently, you can use SAS/INSIGHT, or the UNIVARIATE, CAPABILITY, or KDE procedures to analyze the posterior sample. If you want to regenerate the default ODS graphs from PROC MCMC, see Regenerating Diagnostics Plots.