Term 4, 2006BIO656--Multilevel Models 1 Part 2 Schematic of the alcohol model Marginal and...

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


PLEASE DO THISPLEASE DO THIS

If you did not receive the welcome email from me, email me at: ([email protected])


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”


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


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


ONE POSSIBLE DIAGRAMONE POSSIBLE DIAGRAM

Personal Income

Family income

Percent poverty in neighborhood

State support ofthe poor

Predictor Variables

Alcoholabuse

Response


NOTATIONNOTATION(the reverse order of what I usually use!)


X & Y DIAGRAMX & Y DIAGRAM

PersonX.p(sijk)

FamilyX.f(sij)

NeighborhoodX.n(si)

StateX.s(s)

Predictor Variables

ResponseY(sijk)

Response


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


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?


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


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


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


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


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


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!


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


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)


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


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


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


Death Rates for Coronary Artery Death Rates for Coronary Artery Bypass Graft (CABG)Bypass Graft (CABG)


CABAG DEATH RATECABAG DEATH RATE


BASEBALL DATABASEBALL DATA


TOXOPLASMOSIS RATESTOXOPLASMOSIS RATES(centered)(centered)


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


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


Informal Information BorrowingInformal Information Borrowing


DIRECT ESTIMATESDIRECT ESTIMATES


A Linear Mixed ModelA Linear Mixed Model


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


““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


Cascading HierarchiesCascading Hierarchies


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


Hypothetical ResultsHypothetical ResultsVariance ComponentVariance ComponentPercent of total VariancePercent of total Variance


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

Term 4, 2006BIO656--Multilevel Models 1 Part 2 Schematic of the alcohol model Marginal and...

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