PROC MCMC Getting Started Example 2: Behrens-Fisher Problem

This script performs a Bayesian analysis to compare the means of two independent groups assumed to follow normal distributions with different variances. It first creates an internal dataset. Then, it uses the MCMC procedure to estimate the model parameters (means and variances by group) and the difference of means ('mudif'). Finally, it uses PROC FREQ to analyze the probability that the difference of means is positive or negative from the posterior sample.

Data Analysis

Type : CREATION_INTERNE

Data is generated directly within the script via a DATA step with the 'datalines' command.

1 Code Block

DATA STEP Data

Explanation :
Sets the title and creates the 'behrens' dataset containing the response variable 'y' and the group indicator 'ind'. Data is read in-line.

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1	title 'The Behrens-Fisher Problem';
2
3	DATA behrens;
4	INPUT y ind @;
5	DATALINES;
6	121 1 94 1 119 1 122 1 142 1 168 1 116 1
7	172 1 155 1 107 1 180 1 119 1 157 1 101 1
8	145 1 148 1 120 1 147 1 125 1 126 2 125 2
9	130 2 130 2 122 2 118 2 118 2 111 2 123 2
10	126 2 127 2 111 2 112 2 121 2
11	;

2 Code Block

PROC MCMC Data

Explanation :
Executes the Markov Chain Monte Carlo (MCMC) simulation. Defines the parameters (mu1, mu2, sig21, sig22), priors (non-informative here), and the conditional model structure (different means and variances depending on the 'ind' group). The difference of means 'mudif' is calculated at each iteration. Results are stored in the 'postout' dataset.

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1	PROC MCMC DATA=behrens outpost=postout seed=123
2	nmc=40000 monitor=(_parms_ mudif)
3	statistics(alpha=0.01);
4	ods select PostSumInt;
5	parm mu1 0 mu2 0;
6	parm sig21 1;
7	parm sig22 1;
8	prior mu: ~ general(0);
9	prior sig21 ~ general(-log(sig21), lower=0);
10	prior sig22 ~ general(-log(sig22), lower=0);
11	mudif = mu1 - mu2;
12	IF ind = 1 THEN DO;
13	mu = mu1;
14	s2 = sig21;
15	END;
16	ELSE DO;
17	mu = mu2;
18	s2 = sig22;
19	END;
20	model y ~ normal(mu, var=s2);
21	RUN;

3 Code Block

PROC FORMAT

Explanation :
Creates a custom format 'diffmt' to categorize numeric values into two groups: less than or equal to 0, and strictly greater than 0.

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1
2	PROC FORMAT;
3	value diffmt low-0 = 'mu1 - mu2 <= 0' 0<-high = 'mu1 - mu2 > 0';
4	RUN;
5

4 Code Block

PROC FREQ

Explanation :
Uses PROC FREQ on the simulation output data ('postout') to calculate the frequency of the difference of means ('mudif') according to the defined format, allowing estimation of the probability that the difference is positive or negative.

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1	PROC FREQ DATA = postout;
2	tables mudif /nocum;
3	FORMAT mudif diffmt.;
4	RUN;

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