Multilevel Structural Equation Modeling

Transcription

1 Multilevel Structural Equation Modeling Joop Hox Utrecht University 14_15_mlevsem

2 Multilevel Regression Three level data structure Groups at different levels may have different sizes Response (outcome) variable at lowest level Explanatory variables at all levels The statistical model assumes sampling at all levels 2

3 The Multilevel Regression Model Single Equation Version At the lowest (individual) level we have Y ij = β 0j + β 1j X ij + e ij and at the second (group) level β 0j = γ 00 + γ 01 Z j + u 0j β 1j = γ 10 + γ 11 Z j + u 1j Combining (substitution and rearranging terms) gives Y ij = γ 00 + γ 10 X ij + γ 01 Z j + γ 11 Z j X ij + u 1j X ij + u 0j + e ij 3

4 Why Multilevel SEM? Limitations of the Multilevel Regression Model Does not accommodate measurement error Model is very simple: no latent variables with multiple indicators (=no measurement model), no mediation, no reciprocal effects No overall goodness-of-fit indices Overcome by using the more general Structural Equation Modeling approach Of which multiple regression is a special case 4

5 Multilevel SEM and Multilevel Regression are starting to merge Approaches are Incorporating latent variables and measurement errors into multilevel regression (HLM, MLwiN) Incorporating multilevel structures including random intercepts and slopes into SEM (Mplus, gllamm) (Lisrel, Eqs) 5

6 Statistical Elements of Structural Equation Modeling (SEM) Model specifications: + Data (covariance matrix S) Estimates of model parameters (regression coefficients, variances) 6

7 Statistical Elements of Structural Equation Modeling (SEM) Multivariate normality Observed data summed up perfectly by covariance matrix S (+ means M) Observed covariance matrix S is an estimator of the population covariance matrix Σ Model for Σ contains parameters θ: M(θ) Factor loadings, path coefficients, (co)variances Estimated by Maximum Likelihood Standard errors, overall significance test 7

8 Two Level SEM Assume a population that consists of groups Sampling at two levels: groups and individuals within groups Decompose Total scores T into disaggregated group means B (Between Groups) and individual deviations (Within Groups): W=T-B T=W+B Σ T =Σ W +Σ B S T =S W +S B 8

9 Two-Level CFA: Single Equation Version At the lowest (individual) level we have and at the second (group) level y = µ + Λ η + ε ij j W ij W µ = µ + Λ η + ε j B j B Combining (substitution and rearranging terms) gives y = µ + Λ η + Λ η + ε + ε ij W ij B j B W Note: random intercept model, but in Mplus the factor loadings Λ W can also vary across groups 9

10 Estimating Two-level SEM With Standard Software Muthén showed that analyze S PW with Within model and analyze S * B with same Within model plus (scale factor n) * Between model If groups are not of equal size MUML (Muthén s ML) = ignore problem (Limited Information ML) = Σ MUML approximation is still available but de facto superseded by better estimation methods: ML & WLSM(V) (Mplus only) S PW W S = Σ + nσ * B W B 10

11 Example 2-level 2 CFA: Family IQ Data 60 families, 400 children Scores on 6 intelligence tests Multilevel structure: children nested in families Simulated data (Hox, 2010), patterned after Van Peet, A.A.J. (1992). De potentieeltheorie van intelligentie. [The potentiality theory of intelligence] Amsterdam: University of Amsterdam, Unpublished Ph.D. Thesis 11

12 Path Diagram for Family IQ Data Showing the between & within part separately Note that the 2 nd level variables are actually latent variables They represent the 2 nd level variation of the intercepts of the 1 st level observed variables 12

13 Path Diagram for Family IQ Data Showing Random Intercepts & Slopes Between model Within model 13

14 Example Sibling Data: Raw Data File Variables are: famnr wordlist cards figures matrices animals occup Input to Mplus

15 Example Family IQ Data: Exploratory Analysis N Within =N-G and N Between =G Usually N-G >> G start with analysis of S PW Sibling data: exploratory factor analysis of S PW suggests 2 correlated factors, hence the within model is specified as a CFA with 2 factors 15

16 Example Family IQ Data: Mplus Analysis Muthén s Mplus makes two-level SEM simple Hides all complications from the user Since version 3 full Maximum Likelihood estimation Since version 5 Weighted Least Squares estimation Standard WLSM or WLSMV should be used Faster, but no random slopes User needs only to specify within and between model 16

17 Example Family IQ Data: Mplus Setup TITLE: Two level CFA Family data with robust standard errors DATA: FILE IS FamIQData.dat"; VARIABLE: NAMES ARE family wordlist cards figures matrices animals occup; CLUSTER IS family; ANALYSIS: TYPE IS TWOLEVEL;! Note robust estimation by default Model: %between% general by wordlist* cards figures matrices animals occup; %within% numeric by wordlist* cards figures; percept by matrices* animals occup; OUTPUT: sampstat standardized cinterval modindices(10); 17

18 Family IQ Data Results, ML estimation Table 14.2 Individual and family level estimates, MLR estimation Individual level Family level Numer. Percept. resid. var. General resid. var. Wordlst 3.18 (.30) 6.19 (.78) 3.06 (.37) 1.25 (.53) Cards 3.14 (.23) 5.40 (.65) 3.05 (.43) 1.32 (.62) Matrix 3.05 (.22) 6.42 (.79) 2.63 (.33) 1.94 (.60) Figures 3.10 (.21) 6.85 (.77) 2.81 (.36) 2.16 (.61) Animals 3.19 (.16) 4.88 (.62) 3.20 (.37) 0.66 (.56) Occupat 2.78 (.16) 5.33 (.71) 3.44 (.39) 1.58 (.58) Standard errors in parentheses. Correlation between individual factors:

19 Family IQ Data Results, WLS estimation Table 14.3 Individual and family level estimates, WLSM estimation Individual level Family level Numer. Percept. resid. var. General resid. var. Wordlst 3.25 (.15) 5.67 (.84) 3.01 (.48) 1.51 (.62) Cards 3.14 (.18) 5.44 (.68) 3.03 (.38) 1.25 (.71) Matrix 2.96 (.22) 6.91 (.92) 2.62 (.45) 2.02 (.69) Figures 2.96 (.22) 7.67 (.92) 2.80 (.46) 2.03 (.72) Animals 3.35 (.21) 3.79 (.99) 3.15 (.41) 0.96 (.61) Occupat 2.75 (.24) 5.49 (.94) 3.43 (.44) 1.67 (.63) Standard errors in parentheses. Correlation between individual factors:

20 Mplus Multilevel Features Joop Hox Utrecht University mplusmlev

21 The MplusM system Multilevel Part 21

22 Multilevel Commands VARIABLE: CLUSTER IS <varname>; BETWEEN IS <varname>; WITHIN IS <varname>; ANALYSIS: TYPE IS twolevel; ESTIMATOR IS MLR; Model: %WITHIN% <within model> %BETWEEN% <between model> group identification only on between level only on within level default: robust ML options: ML, WLSM(V) 22

23 Steps in Multilevel SEM 1. Estimate S PW and S B and ICCs 2. Examine ICC for within groups variables 3. (Analysis of pooled within groups covariance matrix) 4. (Analysis of between groups covariance matrix) 5. Simultaneous analysis of between and within level 23

24 Estimate S PW, S B and ICCs <usual commands defining data> CLUSTER IS <varname>; ANALYSIS TYPE IS TWOLEVEL BASIC; SAVEDATA: sample=within.dat; sigb=between.dat; (only if needed) Estimates S W and S B (Maximum Likelihood) Output contains ICCs S W and S B may be written to file 24

25 DATA: (Analyze S W separately) FILE IS within.dat; TYPE IS covariance; NOBS=<N-G> VARIABLE NAMES ARE <list of names> Model: <model commands> Reads S W NOBS = Ncases - Ngroups 25

26 DATA: (Analyze S B separately) FILE IS between.dat; TYPE IS covariance; NOBS=<G> VARIABLE NAMES ARE <list of names> Model: <model commands> Reads S NOBS = Ngroups 26

27 Two-level SEM of raw data <usual commands defining data> CLUSTER IS <varname>; ANALYSIS TYPE IS TWOLEVEL; MODEL: %within% %between% Use raw data instead of covariance matrices Robust χ 2 and robust standard errors Correct χ 2 and standard errors with incomplete data Correct χ 2 and standard errors with complex data 27

28 Two-level SEM of raw data MAR Simultaneous analysis of W & B data Works also with incomplete data or categorical data Choice of estimation methods MLR (ML Robust): generally OK ML: use if multivariate normality is plausible & there is no unobserved heterogeneity (= no omitted level) WLSM(V): generally OK, much faster with large models and incomplete or categorical data WLS: only if (between) sample size is HUGE MCAR 28

29 How about Random Slopes? In multilevel SEM, random slopes can refer to varying loadings and varying path coefficients ANALYSIS: TYPE IS TWOLEVEL RANDOM; ALGORITHM IS INTEGRATION; PROCESSORS IS 2; (or larger) MODEL: %WITHIN% Numeric by wordlist cards matrices; RanLoad cards on numeric; %BETWEEN% RanLoad on general; (varying loading) 29

30 Example of a Multilevel Path Model Data: 1377 pupils in 58 schools DV: GALO school test, Advice for secondaty school type IV: father occupation, father education, mother education IV: denomination (school level only) 30

31 Example of a Multilevel Path Model Pupil Level Model Note Mediation Effect on Advice 31

32 Example of a Multilevel Path Model School Level Model Note two Mediation Effects on Advice 32

33 Example of a Multilevel Path Model Model Fit & Mediation Effect Model Fit (MLR): χ 2 =10.99, df=11, p =.44 CFI/TLI=1.00 RMSEA=0.00 Standardized Direct Effects (between) Advice SESb 0.24 (0.05) Indirect Effects Advice GALO SESb 0.44 (.08) Advice GALO denom 0.15 (.06) Standardized Direct Effects (within) Advice SESw 0.09 (0.02) Indirect Effects Advice GALO SESw 0.28 (.02) 33

34 Example of a Multilevel Path Model Mplus Data Specification VARIABLE: NAMES ARE school gender galo advice feduc meduc foccup denom; USEVARIABLES ARE school galo advice feduc meduc foccup denom; MISSING ARE advice feduc meduc foccup (999); CLUSTER IS school; BETWEEN ARE denom; DEFINE: galo=galo/10;!(rescale) ANALYSIS: TYPE IS TWOLEVEL; ESTIMATOR IS MLR; 34

35 Example of a Multilevel Path Model Mplus Model Specification MODEL: %within% sesw by feduc* meduc foccup; sesw@1; galo on sesw; advice on galo sesw; foccup with feduc; %between% sesb by feduc* meduc foccup; sesb@1; galo on sesb; advice on galo sesb; galo on denom; MODEL INDIRECT: advice IND galo sesw; advice IND galo sesb; advice IND galo denom; 35

36 Useful Resources Guide to web based multilevel SEM resources Mplus homepage: (Discussion) Joop Hox homepage Mehta, P. D., & Neale, M. C. (2005). People are variables too: Multilevel structural equations modeling. Psychological Methods, 10, Curran, P. J. (2003). Have multilevel models been structural equation models all along? Multivariate Behavioral Research, 38,