Estimating Multilevel Linear Models as Structural Equation Models

Transcription

1 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Estimating Multilevel Linear Models as Structural Equation Models Daniel J. Bauer North Carolina State University Patrick J. Curran University of North Carolina Chapel Hill Presentation at the 2002 Meeting of the Psychometric Society June 2-June 23, 2002

2 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Outline Describe Motivation Introduce Multilevel Linear Model (MLM) Show that MLM can be estimated as SEM Show that we can extend MLM within SEM

3 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Motivation Stengths/Limitations of MLMs Optimal For o Obtaining correct SEs for nested data o Estimating & predicting random effects Difficult For o Estimating measurement models o Obtaining explicit tests of mediation Strengths/Limitations of SEM Opposite of above Goals are to combine the strengths of the two models and bridge modeling traditions

4 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 A 2-Level MLM w/ L Covariates Level Model P = j + p= y π 0 π x + pj p r where r MVN 0, rj Level 2 Model 0j 00 u 0j j 0 u j Pj P0 u Pj where 0j, j,, 0P MVN,T

5 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 A 2-Level MLM w/ L Covariates Reduced-Form Equation y P P p 0x p + u u pj x 00 + p + p= p= = β + 00 β r Fixed Coefficients Random Coefficients This is a special case of the linear mixed-effects model of Laird & Ware (982) where y j X j Z j u j r j X j is the design matrix for the fixed effects β Z j is the design matrix for the random effects u j implying that y j MVN X j,z j TZ j rj

6 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 From MLM to SEM In our case, X j = Z j, so y j MVN X j,x j TX j rj Further, if the design is balanced then X j = X and y j MVN X,XTX r This is the same structure as a CFA model where Χ = Λ β = κ Τ = Φ Σ r = Θ δ

7 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 A Classic Case: The Growth Model Multilevel Linear Growth Model Level : y x + ti = π 0 + π i i ti Σ r = DIAG( 0,, 2, 3, 4 ) r ti Level 2: π π 0 = β u i i = β u i 0 + i T = τ τ 00 0 τ Linear Latent Curve Model in SEM r 0 r r 2 r 3 r 4 y 0 y y 2 y 3 y π 0 (β 00 ) π (β 0 ) τ 00 τ τ 0 X i = X = Λ because assuming balanced design. Random coefficients are represented as latent variables.

8 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Balanced Data Example 3 male & 3 female students per school to evaluate effect of sex on language ability Multilevel Linear Model Level : Level 2: lang sex + π π = π 0 + π j j 0 = β u j j = β u i 0 + j r Σr = I T τ = τ 00 0 τ Equivalent SEM r m r m2 r m3 r f r f2 r f3 Lang m Lang m2 Lang m3 Lang f Lang f2 Lang f3 π 0 (β 00 ) π (β 0 ) τ 00 τ τ 0 Order of the 3 males, 3 females w/in units j is arbitrary

9 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Strategies for Imbalanced Data Treat as missing Construct complete-data Σ ˆ ( θ ), ˆ( µ θ ) Compare each y j to submatrices Σ ˆ ( θ ) ˆ( j, µ θ ) j Example: M = max # male students & F = max # female students in any given school. r m r m2 r M r f r f2 r F Lang m Lang m2 Lang M Lang f Lang f2 Lang F π 0 (β 00 ) π (β 0 ) τ 00 τ τ 0

10 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Strategies for Imbalanced Data Compute Σ ˆ ( θ ) j, ˆ( µ θ ) j directly from Λ j Truer to multilevel approach: Λ j = X j X j referred to as definition variables for Λ j Due to Neale Example: S = max # students in any given school r r 2 r 3 r S Lang Lang 2 Lang 3 Lang S sex sex 3j 2j sex Sj sex j π 0 (β 00 ) π (β 0 ) τ 00 τ τ 0

11 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 What to do if you re imbalanced? Both approaches provide computationally equivalent results but Strategy is better for few discrete covariates & complex residual structures. Strategy 2 is better for continuous covariates (highly imbalanced data) & homogeneity of error variance.

12 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Adding Higher-Level Predictors Adding Level 2 Covariates Problem is X j Z j but one Λ j Rovine & Molenaar Solution: Fixed effects factors have means, no variance Random effects factors have variance, no means Define Λ j = BLOCK(X j, Z j ) True to mixed-effects model, non-intuitive. Alternative Solution: Extends approach used w/ latent curve models L2 predictors are fixed X covariates o Effects contained in Γ Computationally equivalent to R & M Solution Both solutions can be extended to 3+ Level models

13 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Expanding the Model: A New Approach to Multilevel CFA Adding a measurement model for item level outcomes Example: Data from High-School & Beyond: Teacher Survey 456 schools; 0,365 teachers o Imbalanced: # teachers/school ranges from to 30 o Let max # teachers = T = 30 9 item measure of teacher perceptions of control o 4 items on control of school policy o 5 items on control of classroom teaching/planning o 6 point Likert scales; Centered at mean Estimating 2-Factor Model

14 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 High-School & Beyond 2-Factor Model τ B B τ w2 τ w τ w w. w T λ 2 λ 3 λ 4 λ 2 λ 3 λ 4 C C 2 C 3 C 4. C T C 2T C 3T r r 2 r 3 r 4 r T r 2T r 3T C 4T r 4T 4 τ B2 τ B22 B 2 τ w2 τ w22 τ w22 w 2. w 2T λ 6 λ 7 λ 8 λ 9 λ 6 λ 7 λ 8 λ 9 C 5 C 6 C 7 C 8 C 9. C 5T C 6T C 7T C 8T r 5 r 6 r 7 r 8 r 9 r 5T r 6T r 7T r 8T C 9T r 9T 9

15 Bauer & Curran, Psychometric Society Paper Presentation, 6/2/02 Empirical Validation Comparing SEM and MLM estimates Parameter Multilevel CFA PROC MIXED λ.0.0 λ (.02342).9932 λ (.02508).492 λ (.0265).2867 λ λ (.0746).9803 λ (.0088).5444 λ (.0532).6080 λ (.0047).4269 Loadings fixed to values from multilevel CFA (Cannot be estimated directly) τ w.54 (.0848).54 (.032) τ w (.0970).6384 (.0292) τ w (.00).2637 (.00885) τ B.2029 (.0726).2027 (.0647) τ B22.6 (.0426).6 (.0379) τ B2.53 (.0240).53 (.0232).2579 (.0260).2579 (.02084) (.0247).4890 (.0243) (.0248).3828 (.02367) (.0224).0047 (.0205) (.0778).964 (.0686) (.0295).5799 (.076) (.00636).3675 (.0065) 8.09 (.0566).09 (.0548) (.0078).460 (.007)