Term 4, 2006BIO656--Multilevel Models 1 Part 2 Schematic of the alcohol model Marginal and conditional models Variance components Random Effects and Bayes General, linear MLMs

Term 4, 2006BIO656--Multilevel Models 2 PLEASE DO THIS If you did not receive the welcome from me, me at:

Term 4, 2006BIO656--Multilevel Models 3 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  Vessel  Fleet ... –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”

Term 4, 2006BIO656--Multilevel Models 4 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

Term 4, 2006BIO656--Multilevel Models 5 ALCOHOL ABUSE ALCOHOL ABUSE A 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

Term 4, 2006BIO656--Multilevel Models 6 ONE POSSIBLE DIAGRAM Personal Income Family income Percent poverty in neighborhood State support of the poor Predictor Variables Alcohol abuse Response

Term 4, 2006BIO656--Multilevel Models 7 NOTATION NOTATION ( the reverse order of what I usually use!)

Term 4, 2006BIO656--Multilevel Models 8 X & Y DIAGRAM Person X.p(sijk) Family X.f(sij) Neighborhood X.n(si) State X.s(s) Predictor Variables Response Y(sijk) Response

Term 4, 2006BIO656--Multilevel Models 9 Standard Regression Analysis Assumptions Data follow normal distribution All the key covariates are included Xs are measured without error Responses are independent

Term 4, 2006BIO656--Multilevel Models 10 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?

Term 4, 2006BIO656--Multilevel Models 11 EXPANDED DIAGRAM Personal income Family income Percent poverty in neighborhood State support for poor Predictor Variables Alcohol Abuse Genes Availability of bars Efforts on drunk driving Response Unobserved random intercepts; omitted covariates

Term 4, 2006BIO656--Multilevel Models 12 X & Y EXPANDED DIAGRAM Person X.p(sijk) Family X.f(sij) Neighborhood X.n(si) State X.s(s) Predictor Variables Response Y(sijk) a.f(sij) a.n(si) a.s(s) Response Unobserved random intercepts; omitted covariates

Term 4, 2006BIO656--Multilevel Models 13 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

Term 4, 2006BIO656--Multilevel Models 14 Key Components of a Multi-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

Term 4, 2006BIO656--Multilevel Models 15 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

Term 4, 2006BIO656--Multilevel Models 16 Marginal Model Inferences Marginal Model Inferences Public 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!

Term 4, 2006BIO656--Multilevel Models 17 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 averageWarning For conditional models, don’t put a LHS variable on the RHS “by hand” Use the MLM to structure the conditioning

Term 4, 2006BIO656--Multilevel Models 18 The Warning Model: Y it =  0 +  1 smoking it + e ij Don’t do this Y i(t+1) | Y it =  0 +  1 smoking it +  Y it + e* i(t+1) Do this (better still, let probability theory do it) Y i(t+1) | Y it =  0 +  1 smoking i(t+1) +  (Y it –  0 -  1 smoking it ) + 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)

Term 4, 2006BIO656--Multilevel Models 19 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

Term 4, 2006BIO656--Multilevel Models 20 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

Term 4, 2006BIO656--Multilevel Models 21 Conditional and Marginal Models Conditional and Marginal Models Conditioning 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

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Term 4, 2006BIO656--Multilevel Models 26 Death Rates for Coronary Artery Bypass Graft (CABG)

Term 4, 2006BIO656--Multilevel Models 27 CABAG DEATH RATE

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Term 4, 2006BIO656--Multilevel Models 29 BASEBALL DATA

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Term 4, 2006BIO656--Multilevel Models 31 TOXOPLASMOSIS RATES (centered)

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Term 4, 2006BIO656--Multilevel Models 34 Observed & Predicted Deviations of Annual Charges (in dollars) for Specialist Services vs. Primary Care Services John Robinson’s research Deviation, Specialists’ Charges Square (blue) = Posterior Mean of Predicted Deviation Dot (red) = Posterior Mean of Observed Deviation

Term 4, 2006BIO656--Multilevel Models 35 Observed and Predicted Deviations for Specialist Services: Log(Charges>$0) and Probability of Any Use of Service John Robinson’s research Mean Deviation of Log(Charges >$0) Dot (red) = Posterior Mean of Observed Deviation Square (blue) = Posterior Mean of Predicted Deviation

Term 4, 2006BIO656--Multilevel Models 36 Informal Information Borrowing

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Term 4, 2006BIO656--Multilevel Models 40 DIRECT ESTIMATES

Term 4, 2006BIO656--Multilevel Models 41 A Linear Mixed Model

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Term 4, 2006BIO656--Multilevel Models 45 Effect 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

Term 4, 2006BIO656--Multilevel Models 46 “Vanilla” Multi-level Model “Vanilla” Multi-level Model (for Patients  Physicians  Clinics) i indexes patient, j physician, k clinic Y ijk = measured value for i th patient, j th physician in the k th clinic Pure vanilla Y ijk =  + a i + b j + c k With no replications at the patient level, there is no residual error term Total Variance

Term 4, 2006BIO656--Multilevel Models 47          Cascading Hierarchies

Term 4, 2006BIO656--Multilevel Models 48 With a physician-level covariate X jk is a physician level covariate This is equivalent to using the full subscript X ijk but noting that X ijk = X ijk for all i and i Model with a covariate Yijk =  + a i + b j + c k + X jk 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

Term 4, 2006BIO656--Multilevel Models 49 Hypothetical Results Variance Component Percent of total Variance

Term 4, 2006BIO656--Multilevel Models 50 Hypothetical Results Variance Component Percent of total Variance

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Term 4, 2006BIO656--Multilevel Models 54 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