Rat tumor example

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Rat tumor example
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SAT coaching example

• Goal The Educational Testing Service (ETS) wants
  to analyze the effects of special coaching
  programs on SAT-V scores
• Separate randomized controlled experiments were
  performed at eight high schools
• For each school, the estimated coaching effect
  and its standard error were obtained.

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The Model
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Two Non-hierarchical Approaches

• Each school is analyzed separately
• All schools are pooled
• The hierarchical model provides a compromise

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p(µ,t)? 1
µ overall trtmt effect t heterogeneity among
schools
? j N( µ, t )
? j effect at school j
yj N( ?j , sj2), sj2 known
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Computation
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Posterior inference strategy

• ? of interest
• µ, t niusance
• joint p(?,µ,t y)?
• conditional of ? p(?µ,t, y)?
• marginal of niusance p(µ,ty)?
• marginal of ? p(? y)(integrate the product of
  the above two, or simulate from the above two)

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R code can refer to the code for rat tumor
example. Winbugs can do this as well.
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Estimated School Effects
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Applications of BHM Meta-analysis

• Meta-analysis aims to summarize and integrate
  findings from multiple studies.
• It involves combining information from several
  parallel data sources, so is closely connected to
  hierarchical models.
• Counterpart of a frequent test of heterogeneity
• ty0 ?

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Clinical trial Heart attack patients receiving
beta-blockers
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Conditional posterior of treatment effect

• treatment effect (log odds ratio) of study j
• ?jf,yN(yj,sj2)

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Microarray shrinking variance estimators

• Differential gene expression
• large p ( of genes), small n (sample size for
  each gene)
• traditional t statistic for gene i
• Variance estimate si for gene i can borrow from
  other genes

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Empirical Bayes

• Instead of doing a full Bayes model like we did,
  plug in maximum posterior estimate of
  hyper-parameter f.

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Full Bayes
p(µ,t)? 1
µ overall trtmt effect t heterogeneity among
schools
? j N( µ, t )
? j effect at school j
yj N( ?j , sj2), sj2 known
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Empirical Bayes
µµ,tt
µ overall trtmt effect t heterogeneity among
schools
? j N( µ, t )
? j effect at school j
yj N( ?j , sj2), sj2 known
µargmaxµp(µy)
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Empirical Bayes Methods

• Point estimates of ?s work well
• Shape spread not good
• underestimate spread
• cannot examine joint posterior
• say ?2-?1