Part 2

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

2. PLEASE DO THIS If you did not receive the welcome email from me, email me at: (tlouis@jhsph.edu) BIO656--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” BIO656--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 BIO656--Multilevel Models

5. 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

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

7. NOTATION(the reverse order of what I usually use!) BIO656--Multilevel Models

8. 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

9. 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

10. 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

11. 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

12. 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

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 BIO656--Multilevel Models

14. 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

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 BIO656--Multilevel Models

16. 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

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 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

18. 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

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 BIO656--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 BIO656--Multilevel Models

21. 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

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26. Death Rates for Coronary Artery Bypass Graft (CABG) BIO656--Multilevel Models

27. CABAG DEATH RATE BIO656--Multilevel Models

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29. BASEBALL DATA BIO656--Multilevel Models

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31. TOXOPLASMOSIS RATES (centered) BIO656--Multilevel Models

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34. 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

35. 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

36. Informal Information Borrowing BIO656--Multilevel Models

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40. DIRECT ESTIMATES BIO656--Multilevel Models

41. A Linear Mixed Model BIO656--Multilevel Models

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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 BIO656--Multilevel Models

46. “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

47. Cascading Hierarchies          BIO656--Multilevel Models

48. With a physician-level covariate • Xjk is a physician level covariate • This is equivalent to using the full subscript Xijk but noting that Xijk = Xijk 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

49. Hypothetical Results Variance ComponentPercent of total Variance BIO656--Multilevel Models

50. Hypothetical Results Variance ComponentPercent of total Variance BIO656--Multilevel Models