BIO656--Multilevel Models 1Term 4, 2006
Part 2Part 2
• Schematic of the alcohol model• Marginal and conditional models• Variance components• Random Effects and Bayes• General, linear MLMs
BIO656--Multilevel Models 2Term 4, 2006
PLEASE DO THISPLEASE DO THIS
If you did not receive the welcome email from me, email me at: ([email protected])
BIO656--Multilevel Models 3Term 4, 2006
MULTI-LEVEL MODELSMULTI-LEVEL MODELS
• Biological, physical, psycho/social processes that influence health occur at many levels:– Cell Organ Person Family Nhbd
City Society ... Solar system– Crew VesselFleet ...
– Block Block Group Tract ...
– Visit Patient Phy Clinic HMO ...
• Covariates can be at each level• Many “units of analysis”
• More modern and flexible parlance and approach: “many variance components”
BIO656--Multilevel Models 4Term 4, 2006
Factors in Alcohol AbuseFactors in Alcohol Abuse
• Cell: neurochemistry
• Organ: ability to metabolize ethanol
• Person: genetic susceptibility to addiction
• Family: alcohol abuse in the home
• Neighborhood: availability of bars
• Society: regulations; organizations; social norms
BIO656--Multilevel Models 5Term 4, 2006
ALCOHOL ABUSEALCOHOL ABUSEA multi-level, interaction model
• Interaction between prevalence/density of bars & state drunk driving laws
• Relation between alcohol abuse in a family & ability to metabolize ethanol
• Genetic predisposition to addiction
• Household environment
• State regulations about intoxication & job requirements
BIO656--Multilevel Models 6Term 4, 2006
ONE POSSIBLE DIAGRAMONE POSSIBLE DIAGRAM
Personal Income
Family income
Percent poverty in neighborhood
State support ofthe poor
Predictor Variables
Alcoholabuse
Response
BIO656--Multilevel Models 7Term 4, 2006
NOTATIONNOTATION(the reverse order of what I usually use!)
BIO656--Multilevel Models 8Term 4, 2006
X & Y DIAGRAMX & Y DIAGRAM
PersonX.p(sijk)
FamilyX.f(sij)
NeighborhoodX.n(si)
StateX.s(s)
Predictor Variables
ResponseY(sijk)
Response
BIO656--Multilevel Models 9Term 4, 2006
Standard Regression Standard Regression Analysis AssumptionsAnalysis Assumptions
Data follow normal distribution
All the key covariates are included
Xs are measured without error
Responses are independent
BIO656--Multilevel Models 10Term 4, 2006
Non-independence (dependence)Non-independence (dependence)within-cluster correlation
• Two responses from the same family (cluster) tend to be more similar than do two observations from different families
• Two observations from the same neighborhood tend to be more similar than do two observations from different neighborhoods
• Why?
BIO656--Multilevel Models 11Term 4, 2006
EXPANDED DIAGRAMEXPANDED DIAGRAM
Personalincome
Family income
Percent povertyin
neighborhood
State supportfor poor
Predictor Variables
AlcoholAbuse
Genes
Availabilityof bars
Effortson drunkdriving
ResponseUnobserved random intercepts; omitted covariates
BIO656--Multilevel Models 12Term 4, 2006
X & Y EXPANDED DIAGRAMX & Y EXPANDED DIAGRAM
PersonX.p(sijk)
FamilyX.f(sij)
NeighborhoodX.n(si)
StateX.s(s)
Predictor Variables
ResponseY(sijk)
a.f(sij)
a.n(si)
a.s(s)
ResponseUnobserved random intercepts; omitted covariates
BIO656--Multilevel Models 13Term 4, 2006
Variance Inflation and Correlation induced by Variance Inflation and Correlation induced by unmeasured or omitted latent effectsunmeasured or omitted latent effects
• Alcohol usage for family members is correlated because they share an unobserved “family effect” via common– genes, diet, family culture, ...
• Repeated observations within a neighborhood are correlated because neighbors share common– traditions, access to services, stress levels,…
• Including relevant covariates can uncover latent effects, reduce variance and correlation
BIO656--Multilevel Models 14Term 4, 2006
Key Components of aKey Components of aMulti-level ModelMulti-level Model
• Specification of predictor variables (fixed effects) at multiple levels: the “traditional” model– Main effects and interactions at and between levels– With these, it’s already multi-level!
• Specification of correlation among responses within a cluster– via Random effects and other correlation-inducers
• Both the fixed effects and random effects specifications must be informed by scientific understanding, the research question and empirical evidence
BIO656--Multilevel Models 15Term 4, 2006
INFERENTIAL TARGETSINFERENTIAL TARGETS
Marginal mean or other summary “on the margin”• For specified covariate values, the average response
across the population
Conditional mean or other summary conditional on:• Other responses (conditioning on observeds) • Unobserved random effects
BIO656--Multilevel Models 16Term 4, 2006
Marginal Model InferencesMarginal Model InferencesPublic Health Relevant
• Features of the distribution of response averaged over the reference population– Mean response– Variance of the response distribution– Comparisons for different covariates
Examples• Mean alcohol consumption for men compared to
women• Rate of alcohol abuse for states with active addiction
treatment programs versus states without– Association is not causation!
BIO656--Multilevel Models 17Term 4, 2006
Conditional Inferences Conditional Inferences
Conditional on observeds or latent effects• Probability that a person abuses alcohol conditional
on the number of family members who do
• A person’s average alcohol consumption, conditional on the neighborhood average
WarningWarning• For conditional models, don’t put a LHS variable on
the RHS “by hand”• Use the MLM to structure the conditioning
BIO656--Multilevel Models 18Term 4, 2006
The Warning The Warning
Model: Yit = 0 + 1smokingit + eij
Don’t do thisYi(t+1) | Yit = 0 + 1smokingit + Yit + e*i(t+1)
Do this (better still, let probability theory do it)
Yi(t+1) | Yit = 0 + 1smokingi(t+1) + (Yit – 0 - 1smokingit) + e**i(t+1)
BecauseUnless you center the regressor, the smoking effect will not have a marginal model interpretation, will be attenuated, will depend on , won’t be “exportable,” ...
See Louis (1988), Stanek et al. (1989)
BIO656--Multilevel Models 19Term 4, 2006
Homework due datesHomework due dates
• The homework due dates in the syllabus are semi-firm, designed to focus your work in the appropriate time frame.
• We will allow late homework, however so that we can post answers, we need to set an absolute deadline.
• Here are the due dates and absolute deadlines: Due date Absolute deadlineHW1 April 6 Apr 11 before or during classHW2 Apr 18 Apr 21 at the end of the dayHW3 Apr 25 Apr 28 at the end of the dayHW4 May 2 May 5 at the end of the day • Homework can be turned in in class or in Yijie Zhou's mailbox
opposite E3527 Wolfe
BIO656--Multilevel Models 20Term 4, 2006
Random Effects ModelsRandom Effects Models
• Latent effects are unobserved – inferred from the correlation among residuals
• Random effects models prescribe the marginal mean and the source of correlation
• Assumptions about the latent variables determine the nature of the correlation matrix
BIO656--Multilevel Models 21Term 4, 2006
Conditional and Marginal ModelsConditional and Marginal ModelsConditioning on random effects
• For linear models, regression coefficients and their interpretation in conditional & marginal models are identical:
average of linear model = linear model of average
• For non-linear models, coefficients have different meanings and values
- Marginal models: - population-average parameters
- Conditional models:
- Cluster-specific parameters
BIO656--Multilevel Models 22Term 4, 2006
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Death Rates for Coronary Artery Death Rates for Coronary Artery Bypass Graft (CABG)Bypass Graft (CABG)
BIO656--Multilevel Models 27Term 4, 2006
CABAG DEATH RATECABAG DEATH RATE
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BASEBALL DATABASEBALL DATA
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TOXOPLASMOSIS RATESTOXOPLASMOSIS RATES(centered)(centered)
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BIO656--Multilevel Models 34Term 4, 2006
Observed & Predicted Deviations of Annual Charges (in dollars) Observed & Predicted Deviations of Annual Charges (in dollars) for for Specialist Services vs. Primary Care ServicesSpecialist Services vs. Primary Care Services
John Robinson’s researchJohn Robinson’s research
De
via
tio
n, S
pec
ialis
ts’ C
ha
rges
Square (blue) = Posterior Mean of Predicted Deviation
Dot (red) = Posterior Mean of Observed Deviation
-30
-20
-10
0
10
20
30
40
Deviation, Primary Care Charges-60 -40 -20 0 20 40 60
-30
-20
-10
0
10
20
30
40
BIO656--Multilevel Models 35Term 4, 2006
Observed and Predicted Deviations for Observed and Predicted Deviations for Specialist ServicesSpecialist Services::Log(Charges>$0) and Probability of Any Use of ServiceLog(Charges>$0) and Probability of Any Use of Service
John Robinson’s researchJohn Robinson’s research
Me
an
Dev
iati
on
of
Lo
g(C
har
ge
s >
$0
)
Dot (red) = Posterior Mean of Observed Deviation
Square (blue) = Posterior Mean of Predicted Deviation
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Mean Deviation of P(Any Use)-0.16 -0.06 0.04 0.14 0.24
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
BIO656--Multilevel Models 36Term 4, 2006
Informal Information BorrowingInformal Information Borrowing
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DIRECT ESTIMATESDIRECT ESTIMATES
BIO656--Multilevel Models 41Term 4, 2006
A Linear Mixed ModelA Linear Mixed Model
BIO656--Multilevel Models 42Term 4, 2006
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Effect of Regressors at Various LevelsEffect of Regressors at Various Levels
• Including regressors at a level will reduce the size of the variance component at that level
• And, reduce the sum of the variance components
• Including may change “percent accounted for” but sometimes in unpredictable ways
• Except in the perfectly balanced case, including regressors will also affect other variance components
BIO656--Multilevel Models 46Term 4, 2006
““Vanilla” Multi-level ModelVanilla” Multi-level Model(for Patients Physicians Clinics)
• i indexes patient, j physician, k clinic• Yijk = measured value for ith patient, jth physician in the kth clinicPure vanilla Yijk = + ai + bj + ck
• With no replications at the patient level, there is no residual error term
Total Variance
222
2
222)
cba
c
cba
100 :clinic for Percent
V(Y ijk
BIO656--Multilevel Models 47Term 4, 2006
Cascading HierarchiesCascading Hierarchies
BIO656--Multilevel Models 48Term 4, 2006
With a physician-level covariateWith a physician-level covariate
• Xjk is a physician level covariate• This is equivalent to using the full subscript Xijk but
noting that Xijk = Xijk for all i and i
Model with a covariate Yijk = + ai + bj + ck + Xjk
• Compute the total variance and percent accounted for as before, but now there is less overall variability, less at the physician level and, usually, a reallocation of the remaining variance
BIO656--Multilevel Models 49Term 4, 2006
Hypothetical ResultsHypothetical ResultsVariance ComponentVariance ComponentPercent of total VariancePercent of total Variance
BIO656--Multilevel Models 50Term 4, 2006
Hypothetical ResultsHypothetical ResultsVariance ComponentVariance ComponentPercent of total VariancePercent of total Variance
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Random Effects Random Effects should replace “unit of analysis”should replace “unit of analysis”
• Models contain Fixed-effects, Random effects (Variance Components) and other correlation-inducers
• There are many “units” and so in effect no single set of units
• Random Effects induce unexplained (co)variance• Some of the unexplained may be explicable by
including additional covariates• MLMs are one way to induce a structure and
estimate the REs
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