Lecture 1 Introduction to Multi-level Models

1 11 Lecture 1 Introduction to multi -levelModelsCourse web ~fdominic/teaching/bio656 Background on MLMs Main Ideas Accounting for Within-Cluster Associations Marginal & Conditional Models A Simple Example Key MLM components23 The Main Biological, psychological and social processes thatinfluence health occur at many levels : Cell Organ Person Family Neighborhood City Society An analysis of risk factors should consider: Each of these levels Their interactionsMulti-level Models Main IdeaHealthOutcome35 Example: Alcohol Abuse1. Cell: Neurochemistry2. Organ: Ability to metabolize ethanol3. Person: Genetic susceptibility to addiction4. Family: Alcohol abuse in the home5. Neighborhood: Availability of bars6. Society: Regulations; organizations; social normsLevel:6 Example: Alcohol Abuse; Interactions betweenLevels5 Availability of bars and6 State laws about drunk driving4 Alcohol abuse in the family and2 Person s ability to metabolize ethanol3 Genetic predisposition to addiction and4 Household environment6 State regulations about intoxication and3 Job requirementsLevel:47 Notation:PopulationNeighborhood: i=1.

2 ,IsState: s=1,..,SFamily: j=1,..,JsiPerson: k=1,..,KsijOutcome: YsijkPredictors: XsijkPerson: sijk( y1223 , x1223 )8 Notation (cont.)59 Multi-level Models : IdeaPredictor VariablesAlcoholAbuseResponse Person s Income Family Income Percent poverty in neighborhood State support of the Rose is a Rose is Multi-level model Random effects model Mixed model Random coefficient model Hierarchical modelMany names for similar Models , analyses, and on Statistical Models A statistical model is an approximation to reality There is not a correct model ; ( forget the holy grail ) A model is a tool for asking a scientific question; ( screw-driver vs. sludge-hammer ) A useful model combines the data with priorinformation to address the question of interest. Many Models are better than Linear Models (GLMs) g( ) = 0 + 1*X1 +.

3 + p*XpLog RelativeRiskLog Odds RatioChange inavg(Y) per unitchange in XCoef InterpPoissonlog( )Count/Timesto eventsLog-linearBinomial logBinary(disease)LogisticGaussian Continuous(ounces)LinearDistributiong( )ResponseModel ( = E(Y|X) = mean ) (1- )713 Since: E(y|Age+1,Gender) = 0 + 1(Age+1) + 2 GenderAnd: E(y|Age ,Gender) = 0 + 1 Age + 2 Gender E(y) = 1 Generalized Linear Models (GLMs) g( ) = 0 + 1*X1 + .. + p*XpGaussian Linear:E(y) = 0 + 1 Age + 2 GenderExample: Age & Gender 1 = Change in Average Response per 1 unit increase in Age, Comparing people of the SAME Linear Models (GLMs) g( ) = 0 + 1*X1 + .. + p*XpBinary Logistic: log{odds(Y)} = 0 + 1 Age + 2 GenderExample: Age & Gender 1 = log-OR of + Response for a 1 unit increase in Age, Comparing people of the SAME : log{odds(y|Age+1,Gender)} = 0 + 1(Age+1) + 2 GenderAnd: log{odds(y|Age ,Gender)} = 0 + 1 Age + 2 Gender log-Odds = 1log-OR = 1815 Generalized Linear Models (GLMs) g( ) = 0 + 1*X1 +.

4 + p*XpCounts Log-linear: log{E(Y)} = 0 + 1 Age + 2 GenderExample: Age & Gender 1 = log-RR for a 1 unit increase in Age, Comparing people of the SAME : Verify Tonight16D. Responses are independentB. All the key covariates are included in the model Quiz : Most Important Assumptions ofRegression Analysis?A. Data follow normal distributionB. All the key covariates are included in the modelC. Xs are fixed and knownD. Responses are independent917 Non-independent responses(Within-Cluster Correlation) Fact: two responses from the same familytend to be more like one another than twoobservations from different families Fact: two observations from the sameneighborhood tend to be more like oneanother than two observations from differentneighborhoods Why?18 Why? (Family Wealth Example)GODG randparentsParentsYouGreat-GrandparentsG reat-GrandparentsYouParentsGrandparents1 019 Key Components of Multi-level Models Specification of predictor variables from multiplelevels (Fixed Effects) Variables to include Key interactions Specification of correlation among responsesfrom same clusters (Random Effects) Choices must be driven by scientificunderstanding, the research question andempirical (within-cluster associations)1121 Multi-level analyses Multi-level analyses of social/behavioralphenomena: an important idea Multi-level Models involve predictors frommulti- levels and their interactions They must account for associations amongobservations within clusters ( levels ) to makeefficient and valid with Correlated DataMust take account of correlation to.

5 Obtain valid inferences standard errors confidence intervals Make efficient inferences1223 Logistic Regression Example:Cross-over trial Response: 1-normal; 0- alcohol dependence Predictors: period (x1); treatment group (x2) Two observations per person (cluster) Parameter of interest: log odds ratio ofdependence: treatment vs placeboMean model : log{odds(AD)} = 0 + 1 Period + 2 Trt24 Results: estimate, (standard error) ( ) ( ) ( )Ordinary ( ) ( ) ( ) InterceptAccount forcorrelationVariableSimilar Estimates,WRONG Standard Errors (& Inferences) for OLR( 0 )( 2 )( 1 )1325 Simulated Data: Non-ClusteredCluster Number (Neighborhood)Alcohol Consumption (ml/day)26 Simulated Data: ClusteredCluster Number (Neighborhood)Alcohol Consumption (ml/day)1427 Within-Cluster Correlation Correlation of two observations fromsame cluster = Non-Clustered = ( ) / = 0 Clustered = ( ) / = Var - Var Within Tot Var28 Models for Clustered Data Models are tools for inference Choice of model determined by scientific question Scientific Target for inference?

6 Marginal mean: Average response across the population Conditional mean: Given other responses in the cluster(s) Given unobserved random effects We will deal mainly with conditional Models (but we ll mention some important differences)1529 Marginal vs Conditional Models Focus is on the mean model : E(Y|X) Group comparisons are of main interest, with high alcohol use with low alcohol use Within-cluster associations are accounted forto correct standard errors, but are not of model Interpretations log{ odds(AlcDep) } = 0 + 1 Period + 2pl = + ( )Period + ( )plTRT Effect: (placebo vs. trt)OR = exp( ) = , 95% CI ( , )Risk of Alcohol Dependence is almost twice as highon placebo, regardless of, (adjusting for), time periodSince: log{odds(AlcDep|Period, pl)} = 0 + 1 Period + 2 And: log{odds(AlcDep|Period, trt)} = 0 + 1 Period log-Odds = 2 OR = exp( 2 )WHY?

7 32 Random Effects Models Conditional on unobserved latentvariables or random effects Alcohol use within a family is relatedbecause family members share anunobserved family effect : common genes,diets, family culture and other unmeasuredfactors Repeated observations within aneighborhood are correlated becauseneighbors share: common traditions,access to services, stress levels ,..1733 Random Effects model InterpretationsSince: log{odds(AlcDepi|Period, pl, bi) )} = 0 + 1 Period + 2 + biAnd: log{odds(AlcDep|Period, trt, bi) )} = 0 + 1 Period + bi log-Odds = 2 OR = exp( 2 )WHY? In order to make comparisons we must keep thesubject-specific latent effect (bi) the same. In a Cross-Over trial we have outcome data for eachsubject on both placebo & treatment In other study designs we may vs.

8 Random Effects Models For linear Models , regression coefficients inrandom effects Models and marginal Models areidentical:average of linear function = linear function of average For non-linear Models , (logistic, log-linear,..)coefficients have different meanings/values, andaddress different questions- Marginal Models -> population-averageparameters- Random effects Models -> cluster-specificparameters1835 Marginal -vs- Random Intercept Models ;Cross-over ( ) ( ) OR(assoc.) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )InterceptMarginal (GEE)LogisticRegressionVariable36 Comparison of Marginal and RandomEffect Logistic Regressions Regression coefficients in the random effectsmodel are roughly times as large Marginal: population odds (prevalencewith/prevalence without) of AlcDep is exp(.57) = for placebo than on active drug;population-average parameter Random Effects: a person s odds of AlcDep isexp( )= times greater on placebo than onactive drug;cluster-specific, here person-specific, parameterWhich model is better?

9 They ask different : Forests & TreesMulti-Level Models : Explanatory variables from multiple levels person, family, n bhd, state, .. Interactions Take account of correlation amongresponses from same clusters: observations on the same person, family,.. Marginal: GEE, MMM Conditional: RE, GLMMR emainder of thecourse will focus Points Multi-level Models : Have covariates from many levels and their interactions Acknowledge correlation among observations fromwithin a level (cluster) Random effect MLMs condition on unobserved latentvariables to account for the correlation Assumptions about the latent variables determine thenature of the within cluster correlations Information can be borrowed across clusters ( levels ) toimprove individual estimates2039 Examples of two-level data Studies of health services: assessment of quality of care areoften obtained from patients that are clustered within are level 1 data and hospitals are level 2 data.

10 In developmental toxicity studies: pregnant mice (dams) areassigned to increased doses of a chemical and examined forevidence of malformations (a binary response). Data collected indevelopmental toxicity studies are clustered. Observations onthe fetuses (level 1 units) nested within dams/litters (level 2data) The level signifies the position of a unit of observation withinthe hierarchy40 Examples of three-level data Observations might be obtained inpatients nested within clinics, that inturn, are nested within different regionsof the country. Observations are obtained on children(level 1) nested within classrooms (level2), nested within schools (level 3).