lecture 4: generalized linear mixed modelss3ri, 21 - 22 november 2013 1/23 lecture 4: generalized...
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Lecture 4: Generalized Linear Mixed Models
Lecture 4: Generalized Linear Mixed Models
Dankmar Bohning
Southampton Statistical Sciences Research InstituteUniversity of Southampton, UK
S3RI, 21 - 22 November 2013
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
An example with two nested random effects
an example with a random effect and several fixed effects
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
An example: health awareness study
I three states in the US participated in a health awareness study
I each state independently devised a health awareness program
I three cities within each state were selected for participationand five households within each city were randomly selectedto evaluate the effectiveness of the program
I a composite index (a count number) was formed (the largethe index, the greater the awareness)
the data have the following hierarchical structure:
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
data:
household
state city 1 2 3 4 5
1 1 42 56 35 40 281 2 26 38 42 35 531 3 34 51 60 29 442 1 47 58 39 62 652 2 56 43 65 70 592 3 68 51 49 71 573 1 19 36 24 12 333 2 18 40 27 31 233 3 16 28 45 30 21
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
Poisson model with random effect for statefor the health awareness index Yijk for household k, in city j , andstate i :
log E [Yijk ]|αi= log µijk = µ + αi
with
I a state random effect αi ∼ N(0, σ2S)
I and a Poisson error Yijk ∼ Po(µijk)
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
Poisson model with random effect for state
I let P(Yijk = y) = Po(y |µijk) = Po(y |µ + αi )
I likelihoodL =
∏i ,j ,k
Po(yijk |µ + αi )
(in the fixed effect case)
I but αi ∼ N(0, σ2S), e.g. normal random, so
L =∏i
∫αi
∏j ,k
Po(yijk |µ + αi )φ(αi )dαi
I where φ(αi ) is a normal density with mean 0 and variance σ2S
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
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Lecture 4: Generalized Linear Mixed Models
An example with one random effect
Note: log-likelihood calculations are based on the Laplacian approximation.
LR test vs. Poisson regression: chibar2(01) = 154.39 Prob>=chibar2 = 0.0000
sd(_cons) .307096 .1278477 .1358028 .6944475
state: Identity
Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]
_cons 39.81456 7.124649 20.59 0.000 28.03645 56.54067
index IRR Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -183.93181 Prob > chi2 = .
Integration points = 1 Wald chi2(0) = .
max = 15
avg = 15.0
Obs per group: min = 15
Group variable: state Number of groups = 3
Mixed-effects Poisson regression Number of obs = 45
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Lecture 4: Generalized Linear Mixed Models
An example with two nested random effects
Poisson model with random effect for state and randomeffect for city nested within state
I let P(Yijk = y) = Po(y |µijk) = Po(y |µ + αi + βj(i))
I where βj(i) ∼ N(0, σ2T ), e.g. normal random
I likelihood
L =∏i
∫αi
∫βj
∏j
∏k
Po(yijk |µ + αi + βj(i))φ(βj)dβj
φ(αi )dαi
I where φ(αi ) is a normal density with mean 0 and variance σ2S
I where φ(βj) is a normal density with mean 0 and variance σ2T
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Lecture 4: Generalized Linear Mixed Models
An example with two nested random effects
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Lecture 4: Generalized Linear Mixed Models
An example with two nested random effects
Note: log-likelihood calculations are based on the Laplacian approximation.
Note: LR test is conservative and provided only for reference.
LR test vs. Poisson regression: chi2(2) = 154.39 Prob > chi2 = 0.0000
sd(_cons) 7.65e-12 .0492277 0 .
city: Identity
sd(_cons) .3070964 .127848 .1358029 .694449
state: Identity
Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]
_cons 39.81457 7.124658 20.59 0.000 28.03644 56.54069
index IRR Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -183.93181 Prob > chi2 = .
Wald chi2(0) = .
city 9 5 5.0 5 1
state 3 15 15.0 15 1
Group Variable Groups Minimum Average Maximum Points
No. of Observations per Group Integration
Mixed-effects Poisson regression Number of obs = 45
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
Meta-Analysis on BCG vaccine against tuberculosis
Colditz et al. 1974, JAMA provide a meta-analysis to examine theefficacy of BCG vaccine against tuberculosis
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
Data on the meta-analysis of BCG and TB
the data contain the following details
I 13 studiesI each study contains:
I TB cases for BCG interventionI number at risk for BCG interventionI TB cases for controlI number at risk for control
I also two covariates are given: year of study and latitudeexpressed in degrees from equator
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
intervention controlstudy year latitude TB cases total TB cases total
1 1933 55 6 306 29 3032 1935 52 4 123 11 1393 1935 52 180 1541 372 14514 1937 42 17 1716 65 16655 1941 42 3 231 11 2206 1947 33 5 2498 3 23417 1949 18 186 50634 141 273388 1950 53 62 13598 248 128679 1950 13 33 5069 47 580810 1950 33 27 16913 29 1785411 1965 18 8 2545 10 62912 1965 27 29 7499 45 727713 1968 13 505 88391 499 88391
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
Data analysis on the meta-analysis of BCG and TB
these kind of data can be analyzed by taking
I TB case as disease occurrence response
I intervention as exposure (fixed effect)
I study as random effect
I latitude and year as further fixed effects
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
Mixed Logistic Regression Model
logpxij
1− pxij
= µ + αi + βINTER × INTERij + βLAT × LATij
where αi ∼ N(0, σ2S)
I each trial arm within each study contributes a binomiallikelihood
I (nij
yij
)p
yijxij (1− pxij )
nij−yij
I where
pxij =exp(µ + αi + βINTER × INTERij + βLAT × LATij)
1 + exp(µ + αi + βINTER × INTERij + βLAT × LATij)
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
Mixed Logistic Likelihood
I
L =∏i
∫αi
∏j
(nij
yij
)p
yijxij (1− pxij )
nij−yij φ(αi )dαi
I where φ(αi ) is a normal density with mean 0 and variance σ2S
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
.
Note: log-likelihood calculations are based on the Laplacian approximation.
LR test vs. logistic regression: chibar2(01) = 3259.71 Prob>=chibar2 = 0.0000
sd(_cons) 1.410135 .2813828 .953694 2.085029
study: Identity
Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]
_cons .0149296 .005892 -10.65 0.000 .0068885 .0323576
intervention .6203579 .0255761 -11.58 0.000 .5722016 .672567
cases Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -196.3842 Prob > chi2 = 0.0000
Integration points = 1 Wald chi2(1) = 134.12
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
LR test vs. logistic regression: chibar2(01) = 2180.88 Prob>=chibar2 = 0.0000
sd(_cons) 1.027901 .2080747 .6912662 1.528471
study: Identity
Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]
_cons .001693 .0012119 -8.91 0.000 .0004162 .0068863
latitude 1.064961 .0201333 3.33 0.001 1.026223 1.105162
intervention .6206475 .0255857 -11.57 0.000 .5724728 .6728762
cases Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -192.38326 Prob > chi2 = 0.0000
Integration points = 1 Wald chi2(2) = 145.29
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
LR test vs. logistic regression: chibar2(01) = 2019.42 Prob>=chibar2 = 0.0000
sd(_cons) .9696404 .197926 .6499213 1.44664
study: Identity
Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]
_cons .0363575 .0948862 -1.27 0.204 .0002183 6.054397
year .9560516 .035339 -1.22 0.224 .8892379 1.027886
latitude 1.037569 .0289224 1.32 0.186 .9824025 1.095833
intervention .6207321 .0255885 -11.57 0.000 .5725521 .6729664
cases Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -191.68717 Prob > chi2 = 0.0000
Integration points = 1 Wald chi2(3) = 148.12
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
LR test vs. logistic regression: chibar2(01) = 2346.37 Prob>=chibar2 = 0.0000
sd(_cons) 1.035618 .2104227 .6954207 1.542238
study: Identity
Random-effects Parameters Estimate Std. Err. [95% Conf. Interval]
_cons .7934227 .9916217 -0.19 0.853 .0684969 9.190487
year .9208262 .0232455 -3.27 0.001 .8763747 .9675324
intervention .6206583 .025586 -11.57 0.000 .5724831 .6728877
cases Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -192.50774 Prob > chi2 = 0.0000
Integration points = 1 Wald chi2(2) = 144.88
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Lecture 4: Generalized Linear Mixed Models
an example with a random effect and several fixed effects
model evaluation
model log L AIC BIC
intervention -196.3842 398.7684 402.5427+ latitude -192.3833 392.7665 397.7989
+ year -191.6872 393.3743 399.6648- latitude -192.5077 393.0155 398.0479
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