MULTILEVEL ANALYSIS

1 MULTILEVEL ANALYSISTom A. B. ~ of StatisticsUniversity of Oxford2012 ForewordThis is a set of slides following Snijders & Bosker (2012).The page headings give the chapter numbers and the page numbers in the :Tom Snijders & Roel Bosker, MULTILEVEL ANALYSIS : An Introduction to Basic and Applied MULTILEVEL ANALYSIS ,2ndedition. Sage, 1-2, 4-6, 8, 10, 13, 14, is an associated ~ data sets and scripts for various software slides arenotself-contained, for understanding them it is necessaryalso to study the corresponding parts of the book!22. MULTILEVEL data and MULTILEVEL analysis72. MULTILEVEL data and MULTILEVEL analysisMultilevel ANALYSIS using the hierarchical linear model :random coefficient regression ANALYSIS for data with several nested level is (potentially) asource of unexplained MULTILEVEL data and MULTILEVEL analysis9 Some examples of units at the macro and micro level:macro-levelmicro-levelschoolsteach ersclassespupilsneighborhoodsfamiliesdis trictsvotersfirmsdepartmentsdepartmentse mployeesfamilieschildrenlittersanimalsdo ctorspatientsinterviewersrespondentsjudg essuspectssubjectsmeasurementsrespondent s = egosalters42.

2 MULTILEVEL data and MULTILEVEL analysis11 12 MULTILEVEL ANALYSIS is a suitable approach to take into account thesocial contextsas well as theindividual hierarchical linear model is a type of regression ANALYSIS for MULTILEVEL datawhere the dependent variable is at the lowest variables can be defined at any level(including aggregates of level-one variables).@@@ ..@@@ ..-xAAAAUZy..-xFigure The structure of macro micro longitudinal data can be regarded as a nested structure;for such data the hierarchical linear model is likewise MULTILEVEL data and MULTILEVEL analysis7 8 Two kinds of argument to choose for a MULTILEVEL ANALYSIS instead of an OLSregression of disaggregated as a nuisanceStandard errors and tests base on OLS regression are suspectbecause the assumption of independent residuals is as an interesting phenomenonIt is interesting in itself to disentangle variability at the various levels.

3 Moreover, this can give insight in the directionswhere further explanation may fruitfully be The random intercept model424. The random intercept modelHierarchical Linear Model:iindicates level-one unit ( , individual);jindicates level-two unit ( , group).Variables for individualiin groupj:Yijdependent variable;xijexplanatory variable at level one;for groupj:zjexplanatory variable at level two;njgroup regression model ofYonXignoring groups :Yij= 0+ 1xij+ regressions:Yij= 0j+ 1jxij+ The random intercept model42 Distinguish two kinds offixed effectsmodels:1. models where group structure is ignored;2.

4 Models with fixed effects for groups: 0jare fixed therandom interceptmodel, the intercepts 0jare random variablesrepresenting random differences between groups:Yij= 0j+ 1xij+ 0j= average intercept 00plus group-dependent deviationU0j: 0j= 00+ this model, the regression coefficient 1is common to all the The random intercept model45In the random intercept model, the constant regression coefficient 1issometimes denoted 10:Substitution yieldsYij= 00+ 10xij+U0j+ the hierarchical linear model, model assumption is that they are independent,normally distributed with expected value 0, and variance 2=var(U0j).

5 The statistical parameter in the model is not their individual values,but their variance The random intercept model45XY 01 03 02 regression line group 1regression line group 3regression line group 2py12R12{Figure Different parallel regression pointy12is indicated with its The random intercept model46 47 Arguments for choosing between fixed (F) and random (R) coefficient models forthe group dummies:1. If groups are unique entities and inference should focus often is the case with a small number of If groups are regarded as sample from a (perhaps hypothetical) population andinference should focus on this population, often is the case with a large number of If level-two effects are to be tested, If group sizes are small and there are many groups, and it is reasonable toassume exchangeability of group-level residuals, thenRmakes better use of If the researcher is interested only inwithin-groupeffects, and is suspiciousabout the model forbetween-groupdifferences, thenFis more If group effectsU0j(etc.)}

6 Are not nearly normally distributed,Ris risky(or use more complicated MULTILEVEL models).114. The random intercept model49; also see 17 18 The empty model (random effects ANOVA) is a modelwithout explanatory variables:Yij= 00+U0j+ decomposition:var(Yij) =var(U0j) +var(Rij) = 20+ between two individuals (i6=i ) in the same groupj:cov(Yij,Yi j) =var(U0j) = 20,and their correlation: (Yij,Yi j) = I(Y) = 20( 20+ 2).This is theintraclass correlation between .05 and .25 in social science research,where the groups represent some kind of social The random intercept model50 Example: 3758 pupils in 211 schools ,Y= language / schools are level-2 Estimates for empty modelFixed EffectCoefficient 00= PartVariance Component variance: 20=var(U0j) variance.

7 2=var(Rij) The random intercept model50 51 Intraclass correlation I= + population of individual valuesYijhas estimated standarddeviation + = of class means 0jhas estimated standard deviation = model becomes more interesting,when alsofixed effectsof explanatory variables are included:Yij= 00+ 10xij+U0j+Rij.(Note the difference between fixed effects of explanatory variablesand fixed effects of group dummies!)144. The random intercept model52 53 Table Estimates for random intercept model with effect for IQFixed 00= 10= Coefficient of PartVariance variance: 20=var(U0j) variance: 2=var(Rij) are two kinds of parameters:1.

8 Fixed effects: regression coefficients (just like in OLS regression);2. random effects: variance components 2and The random intercept model54 55 Table Estimates for ordinary least squares regressionFixed 00= 10= Coefficient of PartVariance variance: 2=var(Rij) model has more structure ( dependence interesting );OLS has misleading standard error for intercept ( dependence nuisance ).164. The random intercept model54 55 4 3 2 10123425305055X= Fifteen randomly chosen regression lines according to the random intercept model ofTable The random intercept model54 59 More explanatory variables:Yij= 00+ 10x1ij+.

9 + p0xpij+ 01z1j+..+ 0qzqj+U0j+ important:difference between within-group and between-group within-group regression coefficient is the regression coefficient within eachgroup, assumed to be the same across the between-group regression coefficient is defined as the regression coefficient forthe regression of the group means ofYon the group means distinction is essential to avoidecological fallacies(p. 15 17 in the book).184. The random intercept model54 59XY"""""""""""""""""""""between-group regression line regression line within group 1regression line within group 3regression linewithin group 2 Figure Different between-group and within-group regression is obtained by havingseparate fixed effectsfor the level-1 variableXand its group mean X.

10 (Alternative:use the within-group deviation variable Xij= (X X)instead ofX.)194. The random intercept model54 59 Table Estimates for random intercept modelwith different within- and between-group regressionsFixed 00= 10= Coefficient of 01= Coefficient ofIQ (group mean) PartVariance variance: 20=var(U0j) variance: 2=var(Rij) The random intercept model53 54In the model with separate effects for the original variablexijand the group meanYij= 00+ 10xij+ +U0j+Rij,the within-group regression coefficient is 10,between-group regression coefficient is 10+ is convenient because the difference between within-group and between-groupcoefficients can be tested by considering the model with separate effects for group-centered variable xijand the group meanYij= 00+ 10 xij+ +U0j+Rij,the within-group regression coefficient is 10.