140 656 MultiLevel Statistical Models If you did

140. 656 Multi-Level Statistical Models If you did not receive the welcome email from me, email me at: (tlouis@jhsph. edu) Term 4, 2006 BIO 656 --Multilevel Models 1

ROOM CHANGE, AGAIN! • Starting Thursday, March 30 th and henceforth, lectures will be in W 2030 • Labs will still be in W 2009 Term 4, 2006 BIO 656 --Multilevel Models 2

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Prerequisites, resources and Grading Term 4, 2006 BIO 656 --Multilevel Models 4

Learning Objectives Term 4, 2006 BIO 656 --Multilevel Models 5

Content & Approach Term 4, 2006 BIO 656 --Multilevel Models 6

Approach • Lectures include basic illustrations and case studies, structuring an approach and interpreting results – Labs address computing and amplify on the foregoing • My approach is formal, but not “mathematical” • To understand MLMs, you need a very good understanding on single-level models – If you understand these, you are ready to multi -level! Term 4, 2006 BIO 656 --Multilevel Models 7

Structure Term 4, 2006 BIO 656 --Multilevel Models 8

RULES FOR HOMEWORK, MID-TERM AND PROJECT Homework • Must be individually prepared, but you can get help • Homework due dates should be honored. • Turn in hard copy for grading The in-class, midterm • Must be prepared absolutely independently • During the exam, no advice or information can be obtained from others • You can use your notes and reference materials The term project • Must be individually prepared, but you can get help • Must be electronically submitted Term 4, 2006 BIO 656 --Multilevel Models 9

Handouts and the Web • Virtually all course materials will be on the web • Check frequently for updates • I’ve provided hard copy of the general information sheet • However, other lectures will be on the web in powerpoint format and won’t be handed out • Download to your computer so you have an electronic version each part • Print if you need hard copy, but do it 4 or 6 to a page to save paper • More generally, try to “go electronic” printing sparingly Term 4, 2006 BIO 656 --Multilevel Models 10

COMPUTING & DATA • We will support Win. BUGS, Stata • We provide partial support for SAS, which should be used only by current SAS users; we aren’t teaching it from scratch • Some homeworks require use of Win. BUGS and another “traditional” program (STATA, SAS, R, . . . ) • We provide datasets, including some in the Win. BUGS examples Term 4, 2006 BIO 656 --Multilevel Models 11

WHY BUGS? • Freeware! • In MLMs, it’s important to see distributions – e. g. , Skewness of sampling distribution of variance component estimates • It’s important to incorporate all uncertainties in estimating random effects • Note that Win. Bugs isn’t very data input friendly • And, it’s difficult to produce P-values Term 4, 2006 BIO 656 --Multilevel Models 12

STATISTICAL MODELS • A statistical model is an approximation • Almost never is there a “correct” or “best” model, no holy grail • A model is a tool for structuring a statistical approach and addressing a scientific question • An effective model combines the data with prior information to address a question Term 4, 2006 BIO 656 --Multilevel Models 13

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 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, 2006 BIO 656 --Multilevel Models 14

Example: 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, 2006 BIO 656 --Multilevel Models 15

ALCOHOL ABUSE: A multi-level, interaction model • Interaction between existence of bars & state, drunk driving laws • Alcohol abuse in a family & ability to metabolize ethanol • Genetic predisposition to addiction & household environment • State regulations about intoxication & job requirements Term 4, 2006 BIO 656 --Multilevel Models 16

Many names for similar, but not identical models, analyses and goals • Multi-Level Models • Random effects models • Mixed models • Random coefficient models • Hierarchical models • Bayesian Models Term 4, 2006 BIO 656 --Multilevel Models 17

We don’t need MLMs • If your question is about slopes on regressors, you can run a standard regression and (usually) get valid slope estimates Y = 0 + 1(areal monitor) + 2(home monitor) +. . . Y = 0 + 1(zipcode income) + 2(personal income) +. . . logit(P) =. . . • Analysis can be followed by computing a “robust” SE to get valid inferences Term 4, 2006 BIO 656 --Multilevel Models 18

We do need MLMs • If your question is about variance components, you need to build the multi-level model Yijkl = 0 + 1 X 1 + 2 X 2 +. . . + ijkl Var(Yijkl) = Var( ijkl) = = VHospital + VClinic + VPhysician + VPatient + Vunexplained • These variances depend on what Xs are in the model Term 4, 2006 BIO 656 --Multilevel Models 19

We do need MLMs • To create a broad class of correlation structures – Longitudinal correlations – Nested correlations • To structure improving unit-level estimates (latent effects) and to make unit-level predictions Term 4, 2006 BIO 656 --Multilevel Models 20

MLMs are effective in producing “working models” that incorporate stochastic realities • Producing efficient population estimates • Broadening the inference beyond “these units” • Protecting against some types of informative missing data processes • Producing correlation structures • Generating “overdispersed” versions of standard models • Structuring estimation of latent effects But, MLMs can be fragile and care is needed Term 4, 2006 BIO 656 --Multilevel Models 21

MLMs are not and should not be • A religion • A truth • The only way to model multi-level data! Term 4, 2006 BIO 656 --Multilevel Models 22

Improving individual-level estimates Similar to the BUGS rat data • Dependent variable (Yij) is weight for rat “i” at age Xij i = 1, . . . , I (=10); j = 1, . . . , J (=5) Xij = Xj = (-14, -7, 0, 7, 14) = (8 -22, 15 -22, 22 -22, 29 -22 36 -22) Yij = bi 0 + bi 1 Xj + ij – As usual, the intercept depends on the centering • Analyses – Each rat has its own line – All rats follow the same line: bi 0 = 0 , bi 1 = 1 – A compromise between these two Term 4, 2006 BIO 656 --Multilevel Models 23

Each rat has its own (LSE, MLE) line (with the population line) Pop line Term 4, 2006 BIO 656 --Multilevel Models 24

A multi-level model: Each rat has its own line, but the lines come from the same distribution • The bi 0 are independent Normal( 0, 02) • The bi 1 are independent N( 1, 12) Overdispersion • Sample variance of the OLS estimated intercepts: 345 = SEint 2 + 02 = 320 + 02 = 25, 0 = 5 • Sample variance of the OLS estimated slopes 4. 25 = SEslope 2 + 12 = 3. 25 + 12 = 1. 00, 1 = 1. 00 Term 4, 2006 BIO 656 --Multilevel Models 25

A compromise: each rat has its own line, but the lines come from the same distribution Pop line Term 4, 2006 BIO 656 --Multilevel Models 26

ONE-WAY RANDOM EFFECTS ANOVA Term 4, 2006 BIO 656 --Multilevel Models 27

Simulated “Neighborhood Clustering” • Random mean for each of 10 neighborhoods (J=10) b 1, b 2, . . . , b 10 (iid) N(10, 9) • Random deviation from neighborhood mean for each of 10 persons in each neighborhood (n=10) Yij = bj + eij, eij (iid) N(0, 4) Conditional Independence Over-dispersion: Variance of each point is 13 (= 4 + 9) Correlation: Measurements within each cluster are correlated Term 4, 2006 BIO 656 --Multilevel Models 28

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Intra-class Correlation (ICC) • Correlation of two observations in the same cluster: ICC = Var(Between)/ Var(Total) = 1 – Var(Within)/Var(Total) Estimated ICC: 0. 67 = (9. 8 -3. 2)/9. 8 True ICC: 0. 69 = 9/(9 + 4) = 9/13 Term 4, 2006 BIO 656 --Multilevel Models 30

V(b) Term 4, 2006 BIO 656 --Multilevel Models 31

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45 o line regression line Pop line Term 4, 2006 BIO 656 --Multilevel Models 37

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WEIGHTED MEANS Term 4, 2006 BIO 656 --Multilevel Models 44

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INFERENCE SPACE (Sanders) • The choice between fixed and random effects depends in part on the reference population (the inference space) –These studies or people – Studies or people like these –. . Term 4, 2006 BIO 656 --Multilevel Models 52

Random Effects should replace “unit of analysis” • Models contain Fixed-effects, Random effects (via 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, 2006 BIO 656 --Multilevel Models 53

PLEASE DO THIS If you did not receive the welcome email from me, email me at: (tlouis@jhsph. edu) Term 4, 2006 BIO 656 --Multilevel Models 54

ROOM CHANGE, AGAIN! • Starting Thursday, March 30 th and henceforth, lectures will be in W 2030 • Labs will still be in W 2009 Term 4, 2006 BIO 656 --Multilevel Models 55

END OF PART I Term 4, 2006 BIO 656 --Multilevel Models 56