Post on 28-Dec-2015
2006 Hopkins Epi-Biostat Summer Institute 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
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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
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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
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Example 1: School Test Scores
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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:1. Calculate each school’s observed average score
2. Calculate an overall average for all schools
3. Borrow strength across schools to improve individual school estimates
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Testing in Schools Why borrow information across schools?
Median # of students per school: 48, Range: 1-198 Suppose small school (N=3) has: 90, 90,10 (avg=63) Suppose large school (N=100) has avg=65 Suppose school with N=1 has: 69 (avg=69) 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
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0 10 20 30 40
02
04
06
08
01
00
school
sco
reTesting in Schools: “Direct Estimates”
Model: E(Yij) = j = + b*j
Mean Scores & C.I.s for Individual Schools
b*j
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Standard Normal regression models: ij ~ N(0,2)
1. Yij = + ij
2. Yij = j + ij
= + b*j + ij
Fixed and Random Effects
j = X (overall avg)
j = Xj (school avg)
= X + b*j = X + (Xj – X) Fixed Effects
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Standard Normal regression models: ij ~ N(0,2)
1. Yij = + ij
2. Yij = j + ij
= + b*j + ij
A random effects model:
3. Yij | bj = + bj + ij, with: bj ~ N(0,2) Random
Effects
Fixed and Random Effects
j = X (overall avg)
j = Xj (shool avg)
= X + b*j = X + (Xj – X) Fixed Effects
Represents Prior beliefs about similarities between schools!
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Standard Normal regression models: ij ~ N(0,2)
1. Yij = + ij
2. Yij = j + ij
= + b*j + 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 and Random Effects
j = X (overall avg)
j = Xj (shool avg)
j = X + bjblup = X + b*j = X + (Xj – X)
= X + b*j = X + (Xj – X) Fixed Effects
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Testing in Schools: Shrinkage Plot
0 10 20 30 40
02
04
06
08
01
00
school
sco
re
Direct Sample EstsBayes Shrunk Ests
b*j
bj
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Testing in Schools: Winbugs
Data: i=1..1978 (students), s=1…38 (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 /
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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)
HyperpriorsPrior 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
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Testing in Schools: Winbugs
Full WinBUGS Model: Yis ~ Normal(s , y) with: 2
y = 1 / y
s ~ Normal( , ) with: 2 = 1 /
y ~ (0.001,0.001)
~ Normal(0 , 1/1000000)
~ (0.001,0.001)
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Testing in Schools: Winbugs WinBUGS Code:model
{for( i in 1 : N ) {
Y[i] ~ dnorm(mu[i],y.tau)mu[i] <- alpha[school[i]] }
for( s in 1 : M ) {alpha[s] ~ dnorm(alpha.c, alpha.tau)}
y.tau ~ dgamma(0.001,0.001)sigma <- 1 / sqrt(y.tau)alpha.c ~ dnorm(0.0,1.0E-6)alpha.tau ~ dgamma(0.001,0.001)
}
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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!
Testing in Schools: Winbugs
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Example 2: Aww, Rats…A normal hierarchical model for
repeated measures
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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)
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Individual & population growth
Pop line(average growth)
Individual Growth Lines
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
Wei
ght
Study Day (centered)
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Improving individual-level estimates Possible Analyses
1. Each rat (cluster) has its own line:
intercept= bi0, slope= bi1
2. All rats follow the same line:
bi0 = 0 , bi1 = 1
3. 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”
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Pop line(average growth)
Bayes-Shrunk Individual Growth Lines
A compromise: Each rat has its own line, but information is borrowed across rats to tell us about individual rat growth
Wei
ght
Study Day (centered)
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Rats: Winbugs (see help: Examples Vol I)
WinBUGS Model:
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Rats: Winbugs (see help: Examples Vol I) WinBUGS Code:
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Rats: Winbugs (see help: Examples Vol I) WinBUGS Results: 10000 updates
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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” rat’s 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
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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
mean
iteration
1 250 500 750 1000
110.0
120.0
130.0
140.0
150.0
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Rats: Winbugs (see help: Examples Vol I) WinBUGS Diagnostics: history
alpha0
iteration
1001 2500 5000 7500 10000
90.0
100.0
110.0
120.0
130.0
beta.c
iteration
1001 2500 5000 7500 10000
5.5
5.75
6.0
6.25
6.5
6.75
sigma
iteration
1001 2500 5000 7500 10000
4.0
5.0
6.0
7.0
8.0
9.0
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WinBUGS Diagnostics:
Examine sample autocorrelation directly by selecting the ‘auto cor’ button.
If autocorrelation exists, generate additional samples and thin more.
mean
lag
0 20 40
-1.0 -0.5 0.0 0.5 1.0
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Rats: Winbugs (see help: Examples Vol I) WinBUGS Diagnostics: autocorrelation
alpha0
lag
0 20 40
-1.0 -0.5 0.0 0.5 1.0
beta.c
lag
0 20 40
-1.0 -0.5 0.0 0.5 1.0
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Bayes-Shrunk Growth Lines
WinBUGS provides machinery for Bayesian paradigm “shrinkage estimates” in MLMs
Pop line(average growth)
Wei
ght
Study Day (centered)
Pop line(average growth)
Study Day (centered)
Wei
ght
Individual Growth Lines
Bayes
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School Test Scores Revisited
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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)
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Testing in Schools revisited Model
Wow! Can YOU fit this model? Yes you can! See WinBUGS>help>Examples Vol II for data,
code, results, etc. More Importantly: Do you understand this model?
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Additional Comments: Y is actually standardized score
(difference from expected norm in standard deviations)
What are the fixed effects in the model?The β are the fixed effects (measured both at
the school and student level)Assume these are independent normal
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What are the random effects in the model? The α 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
Additional Comments:
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Some results: node mean sd MC error 2.50% median 97.50%beta[1] 2.62E-04 9.87E-05 2.73E-06 6.95E-05 2.63E-04 4.58E-04beta[2] 0.4163 0.06504 0.00332 0.2875 0.4182 0.537beta[3] 0.1715 0.04775 0.001163 0.07816 0.1714 0.2663beta[4] 0.1192 0.134 0.006156 -0.1459 0.1206 0.3731beta[5] 0.06045 0.1044 0.004469 -0.15 0.06354 0.2612beta[6] -0.2839 0.1818 0.005977 -0.6371 -0.2868 0.07477beta[7] 0.1497 0.1062 0.00392 -0.05925 0.1487 0.3657beta[8] -0.1574 0.1763 0.006249 -0.4984 -0.1595 0.1949gamma[1] -0.6726 0.1003 0.006384 -0.8611 -0.674 -0.4734gamma[2] 0.03135 0.01022 1.31E-04 0.01128 0.03127 0.05167gamma[3] 0.9511 0.09027 0.004472 0.7763 0.9532 1.119max.var 0.6228 0.06987 7.49E-04 0.4967 0.6186 0.7709min.var 0.5138 0.05349 6.45E-04 0.4181 0.5113 0.6276phi -0.00266 0.002843 3.28E-05 -0.00831 -0.00265 0.002981theta 0.5792 0.03313 3.67E-04 0.5154 0.5795 0.6435
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Gamma[1] to Gamma[3] represent the means of the random effects distributions
Gamma[1] is the mean of the random intercept distribution; hard to interpret in this case
Gamma[2] 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 child’s test score.
Some results:
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Gamma[3] 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).
Some results:
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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)
Some results:
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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