part 05: basic bayes

BIO656--Multilevel Models 1Term 4, 2006

Part 05: Basic BayesPart 05: Basic Bayes


TOXOPLASMOSIS RATES(centered)


The essential Bayes/Frequentist difference

• As a Bayesian you can say, “Conditional on the data, the probability that the parameter is in the interval is 0.95” (you’ve always wanted to be able to say this!)


SURGICALSURGICALHospital # of ops # of deathsA [1] 47 0B 148 18C 119 8D 810 46E 211 8F 196 13G 148 9H 215 31I 207 14J 97 8K 256 29L 360 24


model {for( i in 1 : N ) {p[i] ~ dbeta(1.0, 1.0) #need to specify the priorr[i] ~ dbin(p[i], n[i])}righttail<-step(p[1]-3/n[1])}# Also run with p[i] ~ dbeta(0.25,0.25)

““Surgical” Beta-Binomial ModelSurgical” Beta-Binomial Model(no combining; stand alone)


Beta mean sd 2.5% median 97.5%(1,1) 0.020 0.019 0.0003 0.0006 0.014(.25,.25) 0.005 0.010 0.0002 0.0010 0.034 MLE 0 0.078

““Surgical” Results for p[1]Surgical” Results for p[1](no combining; stand alone)

p[1] sample: 2000

-0.05 0.0 0.05 0.1

0.0

20.0

40.0

p[1] sample: 1000

-0.05 0.0 0.05

0.0 100.0

200.0 300.0

Beta(1,1) Beta(.25, .25)


Allows learningabout theprior


model {for( i in 1 : N ) { b[i] ~ dnorm(mu,tau) # tau = 1/var

r[i] ~ dbin(p[i],n[i])logit(p[i]) <- b[i]}

popmn <- exp(mu) / (1 + exp(mu))mu ~ dnorm(0.0,1.0E-6)sigma <- 1 / sqrt(tau)tau ~ dgamma(alphatau, betatau) mutau<-1alphatau<-.001betatau<-alphatau/mutau

}

““Surgical” Beta-binomial modelSurgical” Beta-binomial model(combine evidence; “estimate” the prior)(combine evidence; “estimate” the prior)


node mean sd 2.5% median 97.5%popmn 0.073 0.010 0.053 0.073 0.095 p[1] 0.053 0.020 0.018 0.052 0.094

p[1] sample: 6000

-0.05 0.0 0.05 0.1 0.15

0.0 5.0 10.0 15.0 20.0

““Surgical” ResultsSurgical” Results(combine evidence)


Beta mean sd 2.5% median 97.5%Comb 0.053 0.020 0.0180 0.0520 0.094(1,1) 0.020 0.019 0.0003 0.0006 0.014(.25,.25) 0.005 0.010 0.0002 0.0010 0.034 MLE 0 0.078

““Surgical” Results for p[1]Surgical” Results for p[1](stand alone & combine)

p[1] sample: 2000

-0.05 0.0 0.05 0.1

0.0

20.0

40.0

p[1] sample: 1000

-0.05 0.0 0.05

0.0 100.0

200.0 300.0

1,1 .25, .25

p[1] sample: 6000

-0.05 0.0 0.05 0.1 0.15

0.0 5.0 10.0 15.0 20.0

Comb


BACK TO HISTORICAL CONTROLSBACK TO HISTORICAL CONTROLS


BLOCKERBLOCKERStudy deaths/n

TreatedTreated ControlControl1 3/38 3/392 7/114 14/1163 5/69 11/934 102/1533 27/1520.....20 32/209 40/21821 27/391 43/36422 22/680 39/674


delta =


Meta Analysis on the control probability (pc)Meta Analysis on the control probability (pc)(Historical Controls)

model { for( i in 1 : Num ) { rc[i] ~ dbin(pc[i], nc[i])

rt[i] ~ dbin(pt[i], nt[i])logit(pc[i]) <- mu[i]logit(pt[i]) <- mu[i] + delta[i]mu[i] ~ dnorm(d, tau)delta[i] ~ dnorm(0.0,1.0E-5) }

d ~ dnorm(0.0,1.0E-6)mutau<-1alphatau<-.0001betatau<-alphatau/mutau tau ~ dgamma(alphatau, betatau) delta.new ~ dnorm(d, tau) sigma <- 1 / sqrt(tau) }


Meta Analysis on delta = log(OR)Meta Analysis on delta = log(OR)model { for( i in 1 : Num ) {

rc[i] ~ dbin(pc[i], nc[i])rt[i] ~ dbin(pt[i], nt[i])logit(pc[i]) <- mu[i]logit(pt[i]) <- mu[i] + delta[i]mu[i] ~ dnorm(0.0,1.0E-5)delta[i] ~ dnorm(d, tau) }

d ~ dnorm(0.0,1.0E-6)mutau<-1alphatau<-.0001betatau<-alphatau/mutau tau ~ dgamma(alphatau, betatau) delta.new ~ dnorm(d, tau) sigma <- 1 / sqrt(tau) }


node mean sd 2.5% median 97.5%pc[1] 0.08684 0.03338 0.03255 0.08207 0.1623pt[1] 0.06883 0.02772 0.0256 0.06517 0.1321delta[1] -0.2441 0.126 -0.4995 -0.2479 0.03553

Meta-analysis on pcMeta-analysis on pcnode mean sd 2.5% median 97.5%pc[1] 0.09164 0.03144 0.04031 0.08749 0.164pt[1] 0.07797 0.04282 0.01743 0.07022 0.1779delta[1] -0.2504 0.762 -1.846 -0.2192 1.13

Meta-analysis on delta

BlockerBlockerMLE: pc = 0.079, pt = 0.077, delta = -0.28


SummarySummaryCarefully specified and applied, the Bayesian approach is very effective in• Structuring designs and analyses• Structuring complicated models and goals

– we’ll see more of this in ranking• Incorporating all relevant uncertainties• Improving estimates• Communicating in a more “scientific” manner

However,• The approach is no panacea and must be used

carefully• Traditional values still apply

Space-age methods will not rescue stone-age data

part 05: basic bayes

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