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PROC MCMC
Masud Rana,1 Rhonda Bryce,1 J. A. Dosman,2 and Punam Pahwa1,3
1Clinical Research Support Unit, College of Medicine2Department of Medicine
3Department of Community Health & EpidemiologyUniversity of Saskatchewan
Saskatoon, Saskatchewan, S7N 5E5, Canada
Saskatoon SAS User Group (SUCCESS)
May 14, 2013
Masud Rana (CRSU) PROC MCMC May 14, 2013 1 / 26
Data
Data
F Forced Expiratory Volume in one second (FEV1) is the volume of airthat can forcibly be blown out in one second, after full inspiration.
F FEV1 is a frequently used index for assessing lung function.
F FEV1 is assumed to be correlated with sex, age, height, weight andsmoking habits.
F In 1978 Labour Canada started the Grain Dust Medical SurveillanceProgram to assess the prevalence of respiratory system among grainworkers and ended in 1993 over five different cycles across Canada.
F Data on 5702 personnel were collected in Cycle 1 of the survey.
Masud Rana (CRSU) PROC MCMC May 14, 2013 6 / 26
Example
Model 1
CURRFEV 1i = β0 + β1 ∗ CURREXPi + β2 ∗ Smokeri + β3 ∗ BASEHTi+
β4 ∗ CURRWTi + β5 ∗ CURRAGEi + εi (1)
where εi ∼ N(0, σ2), i = 1, 2, ......, n.
Prior Distribution
βj ∼ N(0,VAR = 10000), j = 0, 1, ..., 5
σ2 ∼ IGAMMA(SHAPE = 0.01,SCALE = 0.01) (2)
Likelihood Function
CURRFEV 1i ∼ N(β0 + β1 ∗ CURREXPi + β2 ∗ Smokeri + β3 ∗ BASEHTi+
β4 ∗ CURRWTi + β5 ∗ CURRAGEi , σ2) (3)
Masud Rana (CRSU) PROC MCMC May 14, 2013 7 / 26
Example
Random Effects Model
♣ Correlation coefficient between Age and Experience is 0.77.
♣ Regression coefficients are assumed to vary across different regions.
♣ Regions are:
♦ Atlantic: East of Quebec♦ St. Lawrence: Quebec only♦ Great Lakes: Ontario (East of Thunder Bay)♦ Central: Ontario (Thunder Bay and westward), Manitoba and
Saskatchewan♦ Mountain: Alberta, British Columbia, Yukon and North West
Territories
Model 2
CURRFEV 1ij = α0i + α1i ∗ CURRAGEij + α2i ∗ CURREXPij + εij (4)
where εij ∼ N(0, σ2), i = 1, 2, ......, 5, j = 1, 2, .., ni .
Masud Rana (CRSU) PROC MCMC May 14, 2013 16 / 26
Example
Prior Distribution
θi =
α0i
α1i
α2i
∼ MVN
θc =
α0c
α1c
α2c
,Σc
θc ∼ MVN
µ0 =
000
,Σ0 =
1000 0 00 1000 00 0 1000
Σc ∼ IWISHART
3,
1 0 00 1 00 0 1
σ2 ∼ GAMMA (SHAPE = 3,SCALE = 2)
(5)
Likelihood Function
CURRFEV 1ij ∼ N(α0i + α1i ∗ CURRAGEij + α2i ∗ CURREXPij , σ2) (6)
Masud Rana (CRSU) PROC MCMC May 14, 2013 17 / 26
Example
References
Thomas NicholsBayesian Inference.http : //www .fil .ion.ucl .ac .uk/spm/course/slides10−vancouver/08 Bayes.pdf .
Fang ChenThe RANDOM Statement and More: Moving On with PROC MCMCin Proceedings of the SAS Global Forum 2011 Conference.Cary, NC: SAS Institute Inc.
SAS Institute Inc. 2011SAS/STAT 9.3 Users Guide.Cary, NC: SAS Institute Inc.
Masud Rana (CRSU) PROC MCMC May 14, 2013 25 / 26