Hierarchical Linear Models

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Hierarchical Linear Models
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Revisit Hierarchical Model
hyper-prior

• Rat tumor experiments, j1,2,,71

hyper-parameters
super-population prior
parameters
likelihood
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Meta-analysis of beta-blocker
hyper-prior

• Beta-blocker trials, j1,2,,22

?0, t
hyper-parameters
super-population prior
?22
parameters
likelihood
y22
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In the form of a HLM
hyper-prior
?0, t
ß0, t
hyper-parameters
super-population prior
?22
ß1 ß2 ß3
ß (ß1,ß2, , ß22)
likelihood
y22
known
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1 set of random effects
super-population prior
ß0
ß0
hyper-prior
Noninformative
Informative Empirical
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WinBUGS code for 1 set of RE

• model
• for( i in 1 N )
• Yi dnorm(betai,tau.yi)
• betai dnorm(beta0,tau.beta0
• beta0 dflat()
• tau.beta0 dchisq(2)
• data
• list(N22,Yc(), tau.yc())

likelihood
super-population prior
hyper-prior
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The School Example

• Yij bj ? bj ?ij,
• with ?ij N(0,sy2) bj N(0,sb2)
• for student i in school j. There is a random
  school effect
• YN(Xß, sy2I)

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• model
• for( i in 1 N ) N number of students
• Yidnorm(bi,tau.y)1/sigma.y2
• bi lt- betaschooli
• for( j in 1 M ) M no. of schools
• betaj dnorm(mu, tau.b)
• sigma.y1/sqrt(tau.y)
• log(sigma.y) dflat() or invGamma on s2
• mu dflat() or mu dnorm(0.0,1.0E-6)
• tau.b dflat() or dchisq(2)
• data
• list(N, M,Y)

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The School Example, 2 levels

• Yijk ? bj ck ?ijk,
• ?ijk N(0,sy2), bj N(0,sb2), ck N(0,sc2)
• Student i is in school j class k. There is a
  random school effect random class effect.
  Assuming classes are exchangeable
• YN(Xß, sy2I)

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• model
• for( i in 1 N ) N number of students
• Yidnorm(mi,tau.y)1/sigma.y2
• mi lt- mubschoolicclassi
• for( j in 1 M ) M no. of schools
• bj dnorm(0, tau.b)
• for( k in 1 K ) K no. of classes
• cj dnorm(0, tau.c)
• sigma.y1/sqrt(tau.y)
• log(sigma.y) dflat() or Gamma on tau.y
• mu dflat() or mu dnorm(0.0,1.0E-6)
• tau.b dflat() or dchisq(2)
• tau.c dflat()

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fixed
random
Same as setting these sß2 to 8
ß0
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Several clusters
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Weight Growth of Rats
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E(Yij ai, ßi) ai ßiXj , i-th rat, j-th
week ai N(a0 , ?a ) ßi N(?0 , ?ß)
Random Effects
a (a1, ,a30), ß(ß1, ,ß30) can be correlated
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Model
can be noninformative
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Gibbs
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• model
• likelihood
• for(i in 1N)
• for(j in 1T)
• Yi,j dnorm(mui,tau.y)
• mui,jlt- thetai,1 thetai,2 xj
  theta(alpha,beta)
• super-population
• thetai,12dmnorm(ehta12,
  tau.theta12,12)
• prior
• tau.ydgamma(0.001,0.001) or log(1/tau.y)dflat(
  )
• hyper-priors
• for (k in 12) ehtakdflat() or another
  dmnorm()
• R21,12 lt- c(10000, 5000)
• R22,12 lt- c(5000, 10000)
• tau.theta12,12 dwish(R212, 12,2)

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HLM
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General Case
likelihood
super-popn
hyperprior
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One Big Linear Regression