2006 hopkins epi-biostat summer institute1 module 2: bayesian hierarchical models instructor:...

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


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


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


Example 1: School Test Scores


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


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


0 10 20 30 40

02

04

06

08

01

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school

sco

reTesting in Schools: “Direct Estimates”

Model: E(Yij) = j = + b*j

Mean Scores & C.I.s for Individual Schools

b*j


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



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


j = X (overall avg)

j = Xj (shool avg)


Represents Prior beliefs about similarities between schools!



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 ()


j = X (overall avg)

j = Xj (shool avg)

j = X + bjblup = X + b*j = X + (Xj – X)



Testing in Schools: Shrinkage Plot

0 10 20 30 40

02

04

06

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school

sco

re

Direct Sample EstsBayes Shrunk Ests

b*j

bj


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 /


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



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)


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)

}


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!



Example 2: Aww, Rats…A normal hierarchical model for

repeated measures


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)


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)


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”



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



Rats: Winbugs (see help: Examples Vol I)

WinBUGS Model:


Rats: Winbugs (see help: Examples Vol I) WinBUGS Code:


Rats: Winbugs (see help: Examples Vol I) WinBUGS Results: 10000 updates


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


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


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


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


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


Bayes-Shrunk Growth Lines

WinBUGS provides machinery for Bayesian paradigm “shrinkage estimates” in MLMs


Wei

ght




Wei

ght

Individual Growth Lines

Bayes


School Test Scores Revisited


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)


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?


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


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:


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


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:


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:


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:


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

2006 hopkins epi-biostat summer institute1 module 2: bayesian hierarchical models instructor:...

Documents

bj ij

yij bj

larger schools

smaller schools

school avgsnote

small school n

large school n

random effectsestimate