Jump to first page Bayesian Approach FOR MIXED MODEL Bioep740 Final Paper Presentation By Qiang Ling

Jump to first page Introduction of Bayesian n Latent variables n Prior distribution n posterior distribution

Jump to first page What is meant by Latent variables Conceive population parameters as varying over time or other variables. Such as human body temperature. And over given days in the year, there may be a difference in the number of deer ticks in the grass[1].

Jump to first page What is PRIOR Distribution Prior distribution is representing the belief of model parameters distribution prior to carrying out experiments. Such as we know the human temperature is positive. That is a prior distribution. Or we may even to specify human temperature uniformly distributed at [35-40] interval.

Jump to first page 2 kinds of prior distribution n Noninformative ( contains no prior information on the parameters). 1. Jeffreys Prior 2. Flat Prior n Informative( Based on prior knowledge of the parameters, such as human body temperature) 1. Example: Normal Distribution

Jump to first page What is posterior Distribution n Updated parameter distribution n Used to estimate parameters of interest n generated by combining prior distribution and the likelihood function

Jump to first page What is the Difference between Classical Statistics Method vs Bayesian Distribution of data, not parameters are used to estimated parameters. Based on ML & REML, only the parameters values that maximize the Likelihood and SE are obtained. Posterior density is fully evaluated based on ML and prior distribution

Jump to first page MODEL BUILDING PSU SSU *Both N and M are infinite

Jump to first page LET

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Use first measure of DBP Data To estimate overall mean of DBP, and Use those estimations above to generate a data set of 3 trials, each trial contains 30 clinical centers and 10 repeat measures.

Jump to first page Mathematical Aspects of Bayesian

Jump to first page Prior distribution Joint density function of observed data conditional on model paremeters are all possible values of

Jump to first page ==> Let So distribution integrates to one

Jump to first page Flat Prior Since: So:

Jump to first page Jeffreys Prior Define AND THEN Jeffreys prior:

Jump to first page NormalPrior Normal Prior Prior Joint Density Function:

Jump to first page Posterior Density Define TEHN

Jump to first page RESULTS

Jump to first page Estimation of mean and variance of BDP data PROC MIXED DATA=Dbp2; CLASS center; MODEL dbp=/s; RANDOM center/s; RUN;

Jump to first page Estimation of DBP data Covariance Parameter Estimates Cov Parm Estimate centre Residual Effect Estimate Error DF t Value Pr > |t| Intercept <.0001 Solution for fixed Effect:

Jump to first page DATA d1; FILE PRINT; * Route output to Log Window; %LET tmean=90; *overall Mean of DBP; %LET ncenter=30; *Number of centers; %LET nmeasure=10; *Number of time measures; %LET err_v=81.08; *Residual error variance at a time point; %LET within_v=3.49; %LET nrep=3; *Number of replications of the simulation; *Loop begins HERE; DO trial=1 to &nrep; DO center=1 to &ncenter; m=&tmean; ec=sqrt(&within_v)*rannor(1234); DO rep=1 to &nmeasure; err=sqrt(&err_v)*rannor(3456); SBP=m+ec+err; OUTPUT; END; DROP m ec err;

Jump to first page SAS program for Bayesican PROC MIXED DATA=d1; BY trial; CLASS center; MODEL SBP = /s; RANDOM center/s; PRIOR FLAT / NSAMPLE=5500 SEED=52201 OUT=bayes4; RUN; FLAT PRIOR Can input DATA= for the prior density or just leave it blank for default Jeffrey prior

Jump to first page Posterior Sampling Information Prior Flat Algorithm Independence Chain Sample Size 5500 Seed Base Densities Type Parm1 Parm2 ig ig

Jump to first page INPUT data for Norm prior DATA nor_dist; INPUT TYPE $ parm1 parm2; CARDS; n n ;

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Discussion(1) n Bayesian works fine with mixed data n Bayesian estimation of variance is biased but less variance than using

Jump to first page Discussion(2) n If Normal prior is used properly, we should see a posterior variance smaller than both prior variance and variance of likelihood n Bayesian approach for mixed model with 2 random effects will require much longer computer time and depend on the speed of computer