Module 2: Bayesian Hierarchical Models

1
Module 2 Bayesian Hierarchical Models
Instructor Elizabeth Johnson Course Developed
Francesca Dominici and Michael Griswold The Johns
Hopkins University Bloomberg School of Public
Health
2
Key Points from yesterday

• Multi-level Models
• Have covariates from many levels and their
  interactions
• Acknowledge correlation among observations from
  within a level (cluster)
• Random effect MLMs condition on unobserved
  latent variables to describe correlations
• Random Effects models fit naturally into a
  Bayesian paradigm
• Bayesian methods combine prior beliefs with the
  likelihood of the observed data to obtain
  posterior inferences

3
Bayesian Hierarchical Models

• Module 2
• Example 1 School Test Scores
• The simplest two-stage model
• WinBUGS
• Example 2 Aww Rats
• A normal hierarchical model for repeated measures
• WinBUGS

4
Example 1 School Test Scores
5
Testing in Schools

• Goldstein et al. (1993)
• Goal differentiate between good' and bad
  schools
• Outcome Standardized Test Scores
• Sample 1978 students from 38 schools
• MLM students (obs) within schools (cluster)
• Possible Analyses
• Calculate each schools observed average score
• Calculate an overall average for all schools
• Borrow strength across schools to improve
  individual school estimates

6
Testing in Schools

• Why borrow information across schools?
• Median of students per school 48, Range 1-198
• Suppose small school (N3) has 90, 90,10
  (avg63)
• Suppose large school (N100) has avg65
• Suppose school with N1 has 69 (avg69)
• Which school is better?
• Difficult to say, small N ? highly variable
  estimates
• For larger schools we have good estimates, for
  smaller schools we may be able to borrow
  information from other schools to obtain more
  accurate estimates
• How? Bayes

7
Testing in Schools Direct Estimates
Mean Scores C.I.s for Individual Schools

• Model E(Yij) ?j ? bj

bj
?
8
Fixed and Random Effects

• Standard Normal regression models ?ij N(0,?2)
  
• 1. Yij ? ?ij
• 2. Yij ?j ?ij
• ? bj ?ij

Fixed Effects
X bj X (Xj X)
9
Fixed and Random Effects

• Standard Normal regression models ?ij N(0,?2)
  
• 1. Yij ? ?ij
• 2. Yij ?j ?ij
• ? bj ?ij
• A random effects model
• 3. Yij bj ? bj ?ij, with bj
  N(0,?2) Random Effects

Fixed Effects
X bj X (Xj X)
Represents Prior beliefs about similarities
between schools!
10
Fixed and Random Effects

• Standard Normal regression models ?ij N(0,?2)
  
• 1. Yij ? ?ij
• 2. Yij ?j ?ij
• ? bj ?ij
• A random effects model
• 3. Yij bj ? bj ?ij, with bj
  N(0,?2) Random Effects
• Estimate is part-way between the model and the
  data
• Amount depends on variability (?) and underlying
  truth (?)

Fixed Effects
X bj X (Xj X)
11
Testing in Schools Shrinkage Plot
bj
?
bj
12
Testing in Schools Winbugs

• Data i1..1978 (students), s138 (schools)
• Model
• Yis Normal(?s , ?2y)
• ?s Normal(? , ?2?) (priors on school avgs)
• Note WinBUGS uses precision instead of
• variance to specify a normal distribution!
• WinBUGS
• Yis Normal(?s , ?y) with ?2y 1 / ?y
• ?s Normal(? , ??) with ?2? 1 / ??

13
Testing in Schools Winbugs

• WinBUGS Model
• Yis Normal(?s , ?y) with ?2y 1 / ?y
• ?s Normal(? , ??) with ?2? 1 / ??
• ?y ?(0.001,0.001) (prior on precision)
• Hyperpriors
• Prior on mean of school means
• ? Normal(0 , 1/1000000)
• Prior on precision (inv. variance) of school
  means
• ?? ?(0.001,0.001)
• Using Vague / Noninformative Priors

14
Testing in Schools Winbugs

• Full WinBUGS Model
• Yis Normal(?s , ?y) with ?2y 1 / ?y
• ?s Normal(? , ??) with ?2? 1 / ??
• ?y ?(0.001,0.001)
• ? Normal(0 , 1/1000000)
• ?? ?(0.001,0.001)

15
Testing in Schools Winbugs

• WinBUGS Code
• model
• for( i in 1 N )
• Yi dnorm(mui,y.tau)
• mui lt- alphaschooli
• for( s in 1 M )
• alphas dnorm(alpha.c, alpha.tau)
• y.tau dgamma(0.001,0.001)
• sigma lt- 1 / sqrt(y.tau)
• alpha.c dnorm(0.0,1.0E-6)
• alpha.tau dgamma(0.001,0.001)

16
Testing in Schools Winbugs

• Lets fit this one together!
• All the model, data and inits files are now
  posted on the course webpage for you to use for
  practice!

17
Example 2 Aww, RatsA normal hierarchical model
for repeated measures
18
Improving individual-level estimates

• Gelfand et al (1990)
• 30 young rats, weights measured weekly for five
  weeks
• Dependent variable (Yij) is weight for rat i at
  week j
• Data
• Multilevel weights (observations) within rats
  (clusters)

19
Individual population growth

• Rat i has its own expected growth line
• E(Yij) b0i b1iXj
• There is also an overall, average population
  growth line
• E(Yij) ?0 ?1Xj

Weight
Pop line (average growth)
Individual Growth Lines
Study Day (centered)
20
Improving individual-level estimates

• Possible Analyses
• Each rat (cluster) has its own line
• intercept bi0, slope bi1
• All rats follow the same line
• bi0 ?0 , bi1 ?1
• A compromise between these two
• Each rat has its own line, BUT
• the lines come from an assumed distribution
• E(Yij bi0, bi1) bi0 bi1Xj
• bi0 N(?0 , ?02)
• bi1 N(?1 , ?12)

Random Effects
21
A compromise Each rat has its own line, but
information is borrowed across rats to tell us
about individual rat growth
Weight
Pop line (average growth)
Bayes-Shrunk Individual Growth Lines
Study Day (centered)
22
Rats Winbugs (see help Examples Vol I)

• WinBUGS Model

23
Rats Winbugs (see help Examples Vol I)

• WinBUGS Code

24
Rats Winbugs (see help Examples Vol I)

• WinBUGS Results 10000 updates

25
Interpretation of the results

• Primary parameter of interest is beta.c
• Our estimate is 6.185
• (95 Interval 5.975 6.394)
• We estimate that a typical rats weight will
  increase by 6.2 gm/day
• Among rats with similar growth influences, the
  average weight will increase by 6.2 gm/day
• 95 Interval for the expected growth for a rat is
  5.975 6.394 gm/day

26

• WinBUGS Diagnostics
• MC error tells you to what extent simulation
  error contributes to the uncertainty in the
  estimation of the mean.
• This can be reduced by generating additional
  samples.
• Always examine the trace of the samples.
• To do this select the history button on the
  Sample Monitor Tool.
• Look for
• Trends
• Correlations

27
Rats Winbugs (see help Examples Vol I)

• WinBUGS Diagnostics history

28

• WinBUGS Diagnostics
• Examine sample autocorrelation directly by
  selecting the auto cor button.
• If autocorrelation exists, generate additional
  samples and thin more.

29
Rats Winbugs (see help Examples Vol I)

• WinBUGS Diagnostics autocorrelation

30
WinBUGS provides machinery for Bayesian paradigm
shrinkage estimates in MLMs
Bayes
Weight
Weight
Pop line (average growth)
Pop line (average growth)
Bayes-Shrunk Growth Lines
Individual Growth Lines
Study Day (centered)
Study Day (centered)
31
School Test Scores Revisited
32
Testing in Schools revisited

• Suppose we wanted to include covariate
  information in the school test scores example
• Student-level covariates
• Gender
• London Reading Test (LRT) score
• Verbal reasoning (VR) test category (1, 2 or 3,
  where 1 represents the highest level of
  understanding)
• School -level covariates
• Gender intake (all girls, all boys or mixed)
• Religious denomination (Church of England, Roman
  Catholic, State school or other)

33
Testing in Schools revisited

• Model
• Wow! Can YOU fit this model?
• Yes you can!
• See WinBUGSgthelpgtExamples Vol II for data, code,
  results, etc.
• More Importantly Do you understand this model?

34
Additional Comments

• Y is actually standardized score (difference from
  expected norm in standard deviations)
• What are the fixed effects in the model?
• The ß are the fixed effects (measured both at the
  school and student level)
• Assume these are independent normal

35
Additional Comments

• What are the random effects in the model?
• The a are the random effects (at the school
  level)
• Assume these are multivariate normal
• These may represent a) inherent school
  differences (random intercept) b) inherent school
  difference in terms of LRT and c) inherent school
  differences in terms of VR test
• Fixed effects interpretations are conditional on
  schools where these random effects are similar.
• In this example we also put a model on the
  overall variance we assume that the inverse of
  the between-pupil variance will increase linearly
  with LRT score

36
Some results
37
Some results

• Gamma1 to Gamma3 represent the means of the
  random effects distributions
• Gamma1 is the mean of the random intercept
  distribution hard to interpret in this case
• Gamma2 is the mean of the random effect of LRT
• Among children from schools with similar latent
  effects, a one unit increase in LRT yeilds a 0.03
  standard deviation increase in the childs test
  score.

38
Some results

• Gamma3 is the mean of the random effect for the
  VR test.
• Among children from schools with similar latent
  effects, children with the highest VR scores have
  test scores that are on average 0.95 standard
  deviations greater than children with the lowest
  VR scores (95 CI 0.78 1.12)
• Among children from schools with similar latent
  effects, children with the moderate VR scores
  have test scores that are on average 0.42
  standard deviations greater than children with
  the lowest VR scores (95 CI 0.29 0.54).

39
Some results

• Among children from similar schools, girls have
  average test scores that are 0.17 standard
  deviation greater than boys (95 CI 0.08 0.27)
• Among similar schools, all girls schools have
  average test scores that are 0.12 standard
  deviations greater than mixed schools (95 CI
  -0.15 0.37)

40
Bayesian Concepts

• Frequentist Parameters are the truth
• Bayesian Parameters have a distribution
• Borrow Strength from other observations
• Shrink Estimates towards overall averages
• Compromise between model data
• Incorporate prior/other information in estimates
• Account for other sources of uncertainty
• Posterior ? Likelihood Prior