Term 4, 2006BIO656--Multilevel Models Multi-Level Statistical Models If you did not receive the welcome from me, me at:

Term 4, 2006BIO656--Multilevel Models2 ROOM CHANGE, AGAIN! Starting Thursday, March 30 th and henceforth, lectures will be in W2030 Labs will still be in W2009

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Term 4, 2006BIO656--Multilevel Models4 Prerequisites, resources and Grading

Term 4, 2006BIO656--Multilevel Models5 Learning Objectives

Term 4, 2006BIO656--Multilevel Models6 Content & Approach

Term 4, 2006BIO656--Multilevel Models7 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, 2006BIO656--Multilevel Models8 Structure

Term 4, 2006BIO656--Multilevel Models9 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, 2006BIO656--Multilevel Models10 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, 2006BIO656--Multilevel Models11 COMPUTING & DATA We will support WinBUGS, 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 WinBUGS and another “traditional” program (STATA, SAS, R,...) We provide datasets, including some in the WinBUGS examples

Term 4, 2006BIO656--Multilevel Models12 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 WinBugs isn’t very data input friendly And, it’s difficult to produce P-values  

Term 4, 2006BIO656--Multilevel Models13 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, 2006BIO656--Multilevel Models14 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 Models15 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, 2006BIO656--Multilevel Models16 ALCOHOL ABUSE: 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, 2006BIO656--Multilevel Models17 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, 2006BIO656--Multilevel Models18 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, 2006BIO656--Multilevel Models19 We do need MLMs If your question is about variance components, you need to build the multi-level model Y ijkl =  0 +  1 X 1 +  2 X  ijkl Var(Y ijkl ) = Var(  ijkl ) = = V Hospital + V Clinic + V Physician + V Patient + V unexplained These variances depend on what Xs are in the model

Term 4, 2006BIO656--Multilevel Models20 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, 2006BIO656--Multilevel Models21 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, 2006BIO656--Multilevel Models22 MLMs are not and should not be A religion A truth The only way to model multi-level data!

Term 4, 2006BIO656--Multilevel Models23 Improving individual-level estimates Improving individual-level estimates Similar to the BUGS rat data Dependent variable (Y ij ) is weight for rat “i” at age X ij i = 1,..., I (=10); j = 1,..., J (=5) X ij = X j = (-14, -7, 0, 7, 14) = (8-22, 15-22, 22-22, ) Y ij = b i0 + b i1 X j +  ij –As usual, the intercept depends on the centering Analyses  –Each rat has its own line  –All rats follow the same line: b i0 =  0, b i1 =  1 –A compromise between these two

Term 4, 2006BIO656--Multilevel Models24 Each rat has its own (LSE, MLE) line Each rat has its own (LSE, MLE) line (with the population line) Pop line

Term 4, 2006BIO656--Multilevel Models25 A multi-level model: A multi-level model: Each rat has its own line, but the lines come from the same distribution The b i0 are independent Normal(  0,  0 2 ) The b i1 are independent N(  1,  1 2 ) Overdispersion Sample variance of the OLS estimated intercepts: 345 = SE int 2 +  0 2 =  0 2   0 2 = 25,  0 = 5 Sample variance of the OLS estimated slopes 4.25 = SE slope 2 +  1 2 =  1 2   1 2 = 1.00,  1 = 1.00

Term 4, 2006BIO656--Multilevel Models26 A compromise: each rat has its own line, but the lines come from the same distribution Pop line

Term 4, 2006BIO656--Multilevel Models27 ONE-WAY RANDOM EFFECTS ANOVA

Term 4, 2006BIO656--Multilevel Models28 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) Y ij = b j + e ij, e ij (iid) N(0, 4) Conditional Independence  Over-dispersion: Variance of each point is 13 (= 4 + 9) Correlation: Measurements within each cluster are correlated

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Term 4, 2006BIO656--Multilevel Models30 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 True ICC: 0.69 = 9/(9 + 4) = 9/13

Term 4, 2006BIO656--Multilevel Models31 V(b)

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Term 4, 2006BIO656--Multilevel Models37 regression line Pop line 45 o line

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Term 4, 2006BIO656--Multilevel Models44 WEIGHTED MEANS

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Term 4, 2006BIO656--Multilevel Models52 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, 2006BIO656--Multilevel Models53 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, 2006BIO656--Multilevel Models54 PLEASE DO THIS If you did not receive the welcome from me, me at:

Term 4, 2006BIO656--Multilevel Models55 ROOM CHANGE, AGAIN! Starting Thursday, March 30 th and henceforth, lectures will be in W2030 Labs will still be in W2009

Term 4, 2006BIO656--Multilevel Models56 END OF PART I