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### Part 2

### Effect of Regressors at Various Levels

Schematic of the alcohol model

Marginal and conditional models

Variance components

Random Effects and Bayes

General, linear MLMs

BIO656--Multilevel Models

PLEASE DO THIS

If you did not receive the welcome email from me,

email me at: (tlouis@jhsph.edu)

BIO656--Multilevel Models

MULTI-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

Factors 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

ALCOHOL 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

ONE POSSIBLE DIAGRAM

Predictor Variables

Response

Personal

Income

Family

income

Alcohol

abuse

Percent poverty

in neighborhood

State support of

the poor

BIO656--Multilevel Models

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

BIO656--Multilevel Models

X & Y DIAGRAM

Predictor Variables

Response

Person

X.p(sijk)

Family

X.f(sij)

Response

Y(sijk)

Neighborhood

X.n(si)

State

X.s(s)

BIO656--Multilevel Models

Standard Regression Analysis Assumptions

Data follow normal distribution

All the key covariates are included

Xs are measured without error

Responses are independent

BIO656--Multilevel Models

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

EXPANDED DIAGRAM

Unobserved random intercepts; omitted covariates

Predictor Variables

Response

Personal

income

Genes

Family

income

Alcohol

Abuse

Availability

of bars

Percent poverty

in

neighborhood

Efforts

on drunk

driving

State support

for poor

BIO656--Multilevel Models

X & Y EXPANDED DIAGRAM

Unobserved random intercepts; omitted covariates

Predictor Variables

Response

Person

X.p(sijk)

a.f(sij)

Family

X.f(sij)

Response

Y(sijk)

Neighborhood

X.n(si)

a.n(si)

State

X.s(s)

a.s(s)

BIO656--Multilevel Models

Variance Inflation and Correlation induced by unmeasured 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

Key Components of aMulti-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

INFERENTIAL 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

Marginal 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

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

Warning

- For conditional models, don’t put a LHS variable on the RHS “by hand”
- Use the MLM to structure the conditioning

BIO656--Multilevel Models

The Warning

Model: Yit = 0 + 1smokingit + eij

Don’t do this

Yi(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)

Because

Unless 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

Homework 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 deadline

HW1 April 6 Apr 11 before or during class

HW2 Apr 18 Apr 21 at the end of the day

HW3 Apr 25 Apr 28 at the end of the day

HW4 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

Random 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

Conditional 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

Death Rates for Coronary Artery Bypass Graft (CABG)

BIO656--Multilevel Models

BIO656--Multilevel Models

BIO656--Multilevel Models

Observed & Predicted Deviations of Annual Charges (in dollars) for Specialist Services vs. Primary Care ServicesJohn Robinson’s research

Dot (red) = Posterior Mean of Observed Deviation

Square (blue) = Posterior Mean of Predicted Deviation

Deviation, Specialists’ Charges

BIO656--Multilevel Models

Observed and Predicted Deviations for Specialist Services:Log(Charges>$0) and Probability of Any Use of ServiceJohn Robinson’s research

Mean Deviation of Log(Charges >$0)

Dot (red) = Posterior Mean of Observed Deviation

Square (blue) = Posterior Mean of Predicted Deviation

BIO656--Multilevel Models

Informal Information Borrowing

BIO656--Multilevel Models

BIO656--Multilevel Models

BIO656--Multilevel Models

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

“Vanilla” 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 clinic

Pure vanilla

Yijk = + ai + bj + ck

- With no replications at the patient level, there is no residual error term

Total Variance

BIO656--Multilevel Models

With 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

Random Effects 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

BIO656--Multilevel Models

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