Multilevel Modeling Soc 543 Fall 2004

Presentation overview What is multilevel modeling? Problems with not using multilevel models Benefits of using multilevel models Basic multilevel model Variation one: person and time Variation two: person, time, and space

Multilevel models Units of analysis are nested within higher-level units of analysis Students within schools Observations with person

Problems without MLM If we ignore higher-level units of analysis => cannot account for context (individualistic approach) If we ignore individual-level observation and rely on higher-level units of analysis, we may commit ecological fallacy (aggregated data approach) Without explicit modeling, sampling errors at second level may be large =>unreliable slopes Homoscedasticity and no serial correlation assumptions of OLS are violated (an efficiency problem). No distinction between parameter and sampling variances

Advantages of MLM Cross-level comparisons Controls for level differences

General MLM Example: Raudenbush and Bryk, 1986 Dependent variable: ContinuousObserved

General MLM High school and beyond (HSB) survey 10,231 students from 82 Catholic and 94 public schools Dependent variable: standardized math achievement score Independent variable: SES

General MLM Variability among schools Level one: within schools math ij =  0j +  1j (SES ij - SES j ) + r ij

General MLM Variability among schools Level two: between schools  0j =  00 + u 0j  1j =  10 + u 1j

General MLM Variability among schools Combined model math ij =  00 + u 0j +  10 (SES ij - SES j ) + u 1j (SES ij - SES j ) + r ij =  00 +  10 (SES ij - SES j ) + u 0j + v ij + u 0j + v ij (Easy interpretation given the “centering” parameterization)

General MLM Variability among schools Combined model math ij =  +  (SES ij - SES j ) + u 0j + v ij There is a positive relation between SES and math score

General MLM Variability among schools Results: math score means school means are different 90% of the variance is parameter variance 10% is sampling variance Results: math score-SES relation school relations are different 35% is parameter variance (this requires additional assumption and analysis) 65% is sampling variance

General MLM Covariates at level 2 Level one: within schools math ij =  0j +  1j (SES ij - SES j ) + r ij

General MLM Covariates at level 2 Level two: between schools  0j =  00 +  01 sect j + u 0j  1j =  10 +  11 sect j + u 1j

General MLM Covariates at level 2 Combined model: math ij =  00 +  01 sect j +  10  SES ij - SES j ) +  11 sect j (SES ij - SES j ) + r ij + v j

General MLM Combined model: math ij =  +  sect j +  SES ij - SES j ) -  sect j (SES ij - SES j ) + r ij + v j

General MLM Variability as a function of sector Results: math score means 80.7% is parameter variance differences in school means is not entirely accounted for by sector Results: SES-math score relation 9.7% is parameter variance differences in school SES-math score relation may be accounted for by sector

General MLM Sector effects Cannot say that previous relations are causal – may be selection effects Use example of homework to explain sector differences

General MLM Sector effects Results: school SES is strongly related to mean math score, but SES composition accounts for Catholic difference schools with lower SES had weaker SES- math score relation than higher SES schools

General MLM Sector effects Results: variation in SES-math score relation may be accounted for by school SES variation in mean math score is not entirely accounted for by school SES

MLM with person and time When observations are repeated for the same units, we also have a nested structure. Examining within-person changes over time – growth curve analysis. Growth curves may be similar across persons within a class. Example: Muthén and Muthén Dependent variable: categorical, latent

Muthen and Muthen NLSY N=7326 (part 1); N=924 (part 2); N=922 (part 3); N=1225 (part 4) Dependent variables: antisocial behavior (excluding alcohol use) during past year, in 17 dichotomous items; alcohol use during past year, in 22 dichotomous items

MLM with person and time Part 1: latent class determination by latent class analysis and factor analysis It’s a cross-sectional analysis of baseline data in It found 4 latent classes.

MLM with person and time Part 2: growth curve determination by latent class growth curve analysis and growth mixture modeling It uses longitudinal information. Different growth curves are allowed and estimated for different latent classes. Growth mixture modeling is a generalization of latent class growth analysis, in allowing growth variance within class GMM yields a 4-class solution.

MLM with person and time Part 3: latent class relation to growth curve model by general growth mixture modeling (GGMM) What’s new is to the ability to predict a categorical outcome variable from latent classes. The example also illustrates how covariates that predict membership in classes (Table 4).

MLM with person and time Part 4: latent class relation to growth curve model by GGMM Multiple (2) latent class variables. The first one comes from Part 1; the second one comes from Part 2. It bridges the two component parts, asking how the first class membership affects membership in the second class scheme.

MLM with person, time & space Example: Axinn and Yabiku Dependent variable: dichotomous, observed Hazard model with event history

MLM with person, time & space Chitwan Valley Family Study (CVFS) 171 neighborhoods (5-15 household cluster) Dependent variable: initiated contraception to terminate childbearing

MLM with person, time & space Age 0Age 12Birth of 1 st child Contraceptive use or end of observation Time-invariant childhood community context Time-varying contemporary community context Time-invariant early life nonfamily experiences Time-varying contemporary nonfamily experiences

MLM with person, time & space Level one: Logit(p tij ) =  0j +  1 C j +  X +  D +  Z Logit(p tij ) =  0j +  1 C j +  2 X ij +  3 D jt +  4 Z ijt C: time-invariant community var. D: time-variant community var. X: time-invariant personal var. Z: time-variant personal var. (Note that there is no interaction across levels)

Multi-level Hazard Models There is a general problem with non-linear multi-level models. Unbiasedness breaks down. Special attention needs to be paid to estimation of hazard models in a multi- level setting. See Barber et al (2000).