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Multilevel Modeling: Introduction Chongming Yang, Ph.D Social Science Research Institute Social Capital Group Meeting, Spring 2008

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In the past twenty years we have witnessed a paradigm shift in the analysis of correlational data. Confirmatory factor analysis and structural equation modeling have replaced exploratory factor analysis and multiple regression as the standard methods. We are currently in the early stages of a paradigm shift in the analysis of experimental data. Multilevel modeling is replacing ANOVA. Certainly ANOVA will remain a basic tool in the social psychological research, but it can no longer be considered the only technique Kenny, D.A. Kashy, D.A., & Bolger, N. (1998). Data analysis in psychology. In D.T. Gilbert, S.T. Fiske, & G. Lindzey (Eds.) The Handbook of Social Psychology, Vol. 1 (pp233-265). New York: McGraw-Hill. New Paradigm in Data Analysis

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Alternative Labels Hierarchical Linear Model (HLM) Random Coefficient Model Variance Component Model Multilevel Model Contextual Analysis Mixed Linear Model

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Hierarchical Data Structure Response (outcome) variable at lowest level Grouping at higher levels Explanatory (predictive) variables at all levels Assuming sampling at all levels

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Two Types Persons nested within a group Repeated measures nested within a person

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Example of Multilevel Data ClassStudent idMath(yr1)Verb(yr1)sesMath(yr2) 1178727080 1265605667 1380786381 1485807585 2192908090 2291928192 2393918393 2490928291 2594938595

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Properties of Hierarchical Data Observations are interdependent, more similar within groups than from different groups due to shared history, contextual effects, etc. Errors are not independent (longitudinal data)

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Standard Modeling Assumptions Independent observations Independent errors Equal variances of errors for all observations

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Consequences of Ignoring Hierarchical Data Properties Smaller standard errors for regression coefficients, thus Spurious effects

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Design-based Approach Apply standard analysis with sampling weights to adjust standard errors, common in survey research

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Design Effects of Two-level Data Intraclass Correlation = between-level variance/total variance Design Effect n/[1+(n-1) ] where n = average cluster size (=>2 warrants a multilevel analysis)

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Another Look ClassStudent idMath(y1)Verbal(y1)sesTeachers Competence 117872704 126560564 138078634 148580754 219290803 229192813 239391833 249092823 259493853

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Intercepts & Slopes for Each Class X y 0

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Class Level Summary Classinterceptslope … 19.722.50 213.513.26 37.644.07 416.250.92 513.171.27 611.213.85 79.054.21 817.111.32 915.322.11

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Modeling Intercepts & Slopes 0 = g 0 + u 0 1 = g 10 + u 1 when variances of u 0 and u 1 are zero, there are no group differences in 0 and 1. Thus variances of u 0 and u 1 are very important parameters.

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Model-based Approach Multilevel Modeling (Multiple Equations) Multilevel Model: y i = 0 + 1 x i + r i 0 = g 00 + u 0 1 = g 10 + u 1 (Single Equation) Mixed Model: y i = g 00 +u 0 +g 10 x i +u 1 x i +r i

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Multilevel Modeling (with 2 nd Level Predictors) (Multiple Equations) Multilevel Model: y i = 0 + 1 x i + r i 0 = g 00 + g 01 z j + u 0 (main effects) 1 = g 10 + g 11 z j + u 1 (cross-level interaction) (Single Equation) Mixed Model: y i = g 00 +g 01 z j +u 0 +g 10 x i +g 11 z j x i +u 1 x i +r i

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Rearranged Single Equation y i = [g 00 + g 10 x i + g 01 z j + g 11 z j x i ] (fixed effects) + [u 1 x i + u 0 + r i ] (random effects) Parameters to be estimated: intercept: g 00 slopes: g 10, g 01, g 11 variances: r, u 0, u 1 covariances: among r s (in longitudinal data), u 0 & u 1, g s

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Fixed or Random? FixedRandom Effect All levels are present in the experiment Random selection of all possible levels Variable Known Values: e.g. gender has a expectation (mean) and variance Coefficient GenderA probability function of others variables, has a variance, e.g. 1st level coefficients

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Cross-level Interaction Appears in a single equation as product term, not in multiple equations The effect of a lower level variable depends on upper level variables Example: The effect of students aptitude on math achievement depends teachers competence

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Estimation Restricted Maximum Likelihood: Variance components are included in the likelihood function, regression coefficients are estimated in a second step (less biased against variance) Full Maximum Likelihood: Both variance components and regression coefficients are included in the likelihood function (variances are slightly underestimated.

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Deviance -2 times log-likelihood Function, 2 distribution, can be used for model comparison, The smaller, the better fit

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Explore HLM Program Create MDM Specify and run a model Interpret parameters in the output

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Model Exploration Procedures 1.Start with an intercept-only model (Calculate intraclass correlation) 2.Add 1 st level predictors for a fixed model (Test individual slopes) 3.Model intercept by 2 nd level predictors (Test significance & amount of variance explained) 4.Random coefficient model (Test variance component of 1 st level slopes one by one) 5.Model Random slopes predicted by higher level variables (Test significance and amount variance explained)

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Longitudinal Data Time y 0 1234

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Unconditional Growth Model 1 st level: Occasion y = p 0 + p 1 t + r 2 nd level: Person p 0 = g 00 + u 0 p 1 = g 10 + u 1

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Parameters to Interpret Means of Intercept (g 00 ) & Slope(g 10 ) Variances of Intercept (u 0 ) & Slope (u 1 ) Covariance/Correlation of Intercept & Slope (u 0 & u 1 )

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Extended Model Occasion level: Time-variant covariate x y = p 0 + p 1 t + p 3 x + r Person level: time-invariant covariate z p 0 = g 00 + g 01 z j + u 0 p 1 = g 10 + g 11 z j + u 1 p 2 = g 20 + g 21 z j + u 2

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Nonlinear Growth (by Recoding T Variable) Linear: 0, 1, 2, 3… (0, 1, 2.5, 3.5…) Quadratic: 0, 1, 4, 9… Logarithmic: 0, 0.69, 1.10, 1.39… Exponential: 0, 1.72, 6.39, 19.09

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Explore HLM Program Chapter 4 Example Create MDM Specify and run a model Interpret parameters in the output

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Explore the SAS program Identify levels of the variables in the data Identify which variables could have main and/or interaction effects Identify random coefficients and then their variances in the output

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Minimum Sample Size Cluster level: > 20 Individual level: =>1

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Obtain Standardized Coefficients Standardize continuous variables to obtain standardized coefficients

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Further Topics Categorical dependent variables Multivariate dependent variables Latent variables + mediating effects (multilevel structural equation modeling) Power & Sample Size...

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Further Resources http://gseweb.harvard.edu/~faculty/singer/ www.ats.ucla.edu/stat/sas/default.htm SSRI consultants …

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