term 4, 2006bio656--multilevel models1 140.656 multi-level statistical models if you did not receive...

Term 4, 2006 BIO656--Multilevel Models 1

140.656 140.656 Multi-Level Statistical ModelsMulti-Level Statistical Models

If you did not receive the welcome email from me, email me at: ([email protected])

mailto:[email protected]


ROOM CHANGE, AGAIN!ROOM CHANGE, AGAIN!

• Starting Thursday, March 30th and henceforth, lectures will be in W2030

• Labs will still be in W2009


Prerequisites, resources and GradingPrerequisites, resources and Grading


Learning ObjectivesLearning Objectives


Content & ApproachContent & Approach


Approach 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!


StructureStructure


RULES FOR HOMEWORK,RULES FOR HOMEWORK,MID-TERM AND PROJECTMID-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


Handouts and the WebHandouts 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


COMPUTING & DATACOMPUTING & 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


WHY BUGS?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


STATISTICAL MODELSSTATISTICAL 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


MULTI-LEVEL MODELSMULTI-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”


Example: Alcohol AbuseExample: 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


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


Many names for similar, Many names for similar, but not identical but not identical

models, analyses and goalsmodels, analyses and goals

• Multi-Level Models• Random effects models

• Mixed models

• Random coefficient models

• Hierarchical models• Bayesian Models


We don’t need MLMsWe 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


We do need MLMsWe do need MLMs

• If your question is about variance components, you need to build the multi-level model

Yijkl = 0 + 1X1 + 2 X2 + ... + ijkl

Var(Yijkl) = Var(ijkl) =

= VHospital + VClinic + VPhysician + VPatient + Vunexplained

• These variances depend on what Xs are in the model


We do need MLMsWe 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


MLMs are effective in producing MLMs are effective in producing “working models” that “working models” that

incorporate stochastic realitiesincorporate 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 neededBut, MLMs can be fragile and care is needed


MLMs are not and should not beMLMs are not and should not be

• A religion

• A truth

• The only way to model multi-level data!


Improving individual-level estimatesImproving individual-level estimatesSimilar 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 = bi0 + bi1 Xj + ij

– As usual, the intercept depends on the centering

• Analyses– Each rat has its own line – All rats follow the same line: bi0 = 0 , bi1 = 1 – A compromise between these two


Each rat has its own (LSE, MLE) lineEach rat has its own (LSE, MLE) line(with the population line)

Pop line


A multi-level model:A multi-level model: Each rat has its own line,

but the lines come from the same distribution

• The bi0 are independent Normal(0, 02)

• The bi1 are independent N(1, 12)

Overdispersion• Sample variance of the OLS estimated intercepts: 345 = SEint

2 + 02 = 320 + 0

2 02 = 25, 0 = 5

• Sample variance of the OLS estimated slopes 4.25 = SEslope

2 + 12 = 3.25 + 1

2 12 = 1.00, 1 = 1.00


A compromise: each rat has its own line,A compromise: each rat has its own line, butbut the lines come from the same distribution the lines come from the same distribution

Pop line


ONE-WAY RANDOM EFFECTS ANOVAONE-WAY RANDOM EFFECTS ANOVA


Simulated “Neighborhood Clustering”Simulated “Neighborhood Clustering”• Random mean for each of 10 neighborhoods (J=10) b1, b2, ..., b10 (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


Intra-class Correlation (ICC)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


V(b)


regressionline

Pop line

45o line


WEIGHTED MEANSWEIGHTED MEANS


INFERENCE SPACEINFERENCE SPACE(Sanders)(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– .........


Random Effects Random Effects should replace “unit of analysis”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


PLEASE DO THISPLEASE DO THIS

If you did not receive the welcome email from me, email me at: ([email protected])

mailto:[email protected]


ROOM CHANGE, AGAIN!ROOM CHANGE, AGAIN!

• Starting Thursday, March 30th and henceforth, lectures will be in W2030

• Labs will still be in W2009


END OF PART IEND OF PART I

term 4, 2006bio656--multilevel models1 140.656 multi-level statistical models if you did not receive...

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