THE GENERAL STRUCTURAL EQUATION MODEL WITH LATENT VARIATES

Transcription

1 THE GENERAL STRUCTURAL EQUATION MODEL WITH LATENT VARIATES I. Specification: A full structural equation model with latent variables consists of two parts: a latent variable model (which specifies the relations among the latent variables of substantive interest) η = Β η + Γξ + ς a measurement model (which links the latent variables to observed variables); = Λ η + = Λ ξ + where: η m vector of endogenous latent variates in deviation form, ξ n vector of eogenous latent variates in deviation form, Β m m matri of coefficients showing the influence of endogenous latent variates on each other, Γ m n matri of coefficients showing the influence of eogenous latent variates on endogenous latent variates, ζ m vector of errors in equations, p vector of observed measures (indicators) of the endogenous latent variates in deviation form, q vector of observed measures (indicators) of the eogenous latent variates in deviation form, Λ p m matri of factor loadings for ; Λ q n matri of factor loadings for ; p vector of unique factors corresponding to ; q vector of unique factors corresponding to ; assumptions: () ( I Β) is nonsingular (2) E( ς ) = 0, E ( ) = 0, E ( ) = 0 (3) Cov ( ξ, ς ') = 0, Cov ( ξ, ') = 0, Cov ( ξ, ') = 0 (4) Cov ( ς, ') = 0, Cov ( ς, ') = 0, Cov (, ') = 0

2 2 let Cov(, ') = Θ Cov(, ') = Θ Cov( ξ, ξ ') = Φ Cov( ς, ς ') = Ψ then the implied variance-covariance matri of the observed variables is: Σ( Λ, Λ, Θ, Θ, Β, Γ, Φ, Ψ) Σ Σ = Σ Σ where: ( I ) ( ) ( I ) E( ηξ ) ( I ) '( I ) ' ' Σ = Λ Β ΓΦΓ' + Ψ Β ' Λ ' + Θ Σ = Λ ' Λ ' = Λ Β ΓΦΛ ' Σ = Λ ΦΓ Β Λ Σ = Λ Φ Λ ' + Θ Figure : ϕ θ θ λ 2 ξ γ ψ ψ 22 ψ 33 θ 33 θ 44 θ 55 θ 66 θ ϕ λ 63 ϕ 2 λ 42 ϕ 32 λ 73 η ξ 2 η 2 ξ 3 ϕ 33 ϕ 22 γ 3 γ 2 ζ ζ 2 β 2 λ λ 62 4 λ 2 λ θ θ θ 33 θ θ 55 θ 66 β 32 η 3 7 ζ 3 [see Appendi A for the specification of this model]

3 3 distinguish between the following two tpes of models: recursive models: Β is a lower triangular matri and Ψ is diagonal (i.e., there are no reciprocal paths, feedback loops, or correlated disturbances); nonrecursive models: Β is not lower triangular and/or Ψ is not diagonal (i.e., there are reciprocal paths, feedback loops, and/or correlated disturbances); note that when Λ = I, Λ = I Θ Θ m n, = 0, = 0 the specification of the model reduces to the usual structural equation model with observed variables; Wh is it important to consider measurement error? (i) consequences in bivariate correlation and regression: consider the following model: η = γξ + ζ = η + = ξ + correlation: from classical reliabilit theor we know that ρ ( η ξ ), = Var Var ρ(, ) ( η ) ( ) Var Var ( ξ ) ( ) since the denominator is smaller than or equal to one, ρ( η, ξ) ρ(, ) ; regression: notice that Cov ( η, ξ) = γϕ so that Cov( η, ξ) γ = ϕ

4 4 what would happen if we mistakenl assumed that η and ξ were measured without error b and ; in this case we would regress on and get Cov(, ) Cov( η, ξ) ϕ γ * = = = γ Var( ) Var( ξ) + Var( ) ϕ + θ thus, for θ > 0, γ * < γ ; it can also be shown that if we have onl a sample of observations on and, ρ (, ) and γ * are inconsistent estimators of ρ( η, ξ ) and γ ; (ii) consequences in multiple regression in general, Γ * is an inconsistent estimator of Γ; the multiple correlation coefficient is attenuated if eplanator variables are measured with error; Identification: (i) a necessar condition for identification is that the number of parameters to be freel estimated not be greater than the number of distinct elements in the variance-covariance matri of and ; (ii) a sufficient condition for identification is the two-step rule: in the first step, ignore the particular structural specification of interest and consider a confirmator factor analsis model in which the factors are allowed to correlate freel; identif the free parameters in Λ, Λ, Θ, Θ, Φ, the variance-covariance matri of η, and the variance-covariance matri of η and ξ (see the handout on confirmator factor analsis for details); if the model in step one is identified, proceed to the second step; assume that η and ξ are directl observable and show that the structural model is identified in this case: null B rule [sufficient but not necessar]: if B is zero, the elements in Γ, Φ, and Ψ are identified;

5 5 recursive rule [sufficient but not necessar]: if B can be written as a lower-triangular matri and Ψ is diagonal, the latent variable model is identified; rank condition: assumes that all elements of Ψ are freel estimated; solve for the free parameters in Β, Γ, and Ψ in terms of the known elements of the variance-covariance matri of η and ξ; (b) Estimation: the goal is to find values for the unknown parameters in Λ, Λ, Θ, Θ, Β, Γ, Φ and Ψ, based on S, such that the variance-covariance matri Σ implied b the estimated parameters will be as close as possible to the observed variance-covariance matri S; as in the estimation of confirmator factor models, the most commonl used estimation procedures are ULS, ML, and GLS (refer to the handout on factor analsis for details); (c) Testing:. Global fit measures: (a) χ 2 goodness-of-fit test: test of the hpothesis that the specified model is correct against the alternative that Σ is unconstrained; (b) alternative fit indices: we will discuss these in a separate handout;

6 6 2. Local fit measures: (a) parameter estimates and associated standard errors (b) reliabilit and discriminant validit (c) eplained variation for each structural equation: R 2 ηi = ψ ii Var ( η ) i 3. Model modification: (a) modification indices (b) residual analsis

7 7 Appendi A: Specification of the model in Figure η η η β β η η η γ γ γ ξ ξ ξ ς ς ς = = + λ λ λ λ ξ ξ ξ = + λ λ λ λ η η η Φ = ϕ ϕ ϕ ϕ ϕ ϕ [ ] θ... θ 77 Diag = Θ [ ] θ θ Diag = Θ Ψ = ψ ψ ψ

8 EXPLAINING CONSUMERS USAGE OF COUPONS FOR GROCERY SHOPPING (cf. Bagozzi, Baumgartner, and Yi, JCR 992) 8 Procedure Female staff members at two American universities completed two questionnaires that were sent to them via campus mail. The first questionnaire contained measures of seven beliefs about the consequences of using coupons and corresponding evaluations, as well as measures of attitude toward using coupons, behavioral intentions, and the personalit variable of state-/action-orientation. One week later a second questionnaire was sent to those people who had participated in the first wave of data collection. This questionnaire assessed some of the same variables as in wave one as well as people s self-reported coupon usage during the past week. Specificall, subjects were presented with a table that had 2 product categories as its rows (e.g., cereal, juice drinks, paper towels, snack foods, canned goods) and si sources of coupons as its columns (i.e., direct mail, newspapers, magazines, in or on packages, from store displas or flers, from relatives or friends). An additional row was included for other products so that respondents could indicate usage in categories not covered b the 2 listed. Subjects were asked to state how man coupons the had used for each categor and source combination. Measures () beliefs: perceived likelihood of the following consequences of using coupons (rated on 7-point unlikel-likel scales): inconveniences: o searching for, gathering, and organizing coupons takes much time and effort; o planning the use of and actuall redeeming coupons in the supermarket takes much time and effort; rewards: o using coupons saves much mone on the grocer bill; o using coupons leads to feelings of being a thrift shopper; encumbrances: o in order to obtain coupons one has to subscribe to etra newspapers, magazines, etc.; o in order to take advantage of coupon offers one has to purchase nonpreferred brands; o in order to take advantage of coupon offers one has to shop at multiple supermarkets; (2) evaluations: how each of the seven consequences of using coupons makes the respondent feel, rated on 7-point good-bad scales; (3) Aact: attitude toward using coupons for shopping in the supermarket during the upcoming week (assessed on four semantic differential scales, i.e., unpleasant-pleasant, bad-good, foolish-wise, and unfavorable-favorable); measured twice (week, week 2); (4) BI: behavioral intentions to use coupons for shopping in the supermarket during the upcoming week (measured with a 7-point unlikel-likel scale assessing intentions to use coupons and an -point no chance-certain scale asking about plans to use coupons); (5) actual coupon usage: the total number of coupons used across product categories and sources; a square root transformation was used to normalize the variable;

9 9 SIMPLIS SPECIFICATION: Title A general structural equation model (eplaining coupon usage) Observed Variables id be be2 be3 be4 be5 be6 be7 aat aa2t aa3t aa4t bi bi2 bh Raw Data from File=d:\EDEN\sem.dat Latent Variables Sample Size 250 Relationships be = *INCONV be2 = INCONV be3 = *REWARDS be4 = REWARDS be5 = *ENCUMBR be6 = ENCUMBR be7 = ENCUMBR aat = *AACT aa2t = AACT aa3t = AACT aa4t = AACT bi = *BI bi2 = BI bh = *BH AACT = BI = AACT BH = BI Set the Error Variance of bh to zero Options sc rs mi wp Path Diagram End of Problem

10 0 Covariance Matri aat aa2t aa3t aa4t bi bi2 bh be be2 be aat.86 aa2t.20.7 aa3t aa4t bi bi bh be be be be be be be Covariance Matri be4 be5 be6 be be4.87 be be be Total Variance = Generalized Variance = Largest Eigenvalue = 4.39 Smallest Eigenvalue = Condition Number = 5.643

11 LISREL Estimates (Maimum Likelihood) Measurement Equations aat =.000*AACT, Errorvar.= 0.679, R² = Standerr (0.0748) Z-values P-values aa2t =.036*AACT, Errorvar.= 0.445, R² = Standerr (0.069) (0.0577) Z-values P-values aa3t = 0.849*AACT, Errorvar.= 0.76, R² = Standerr (0.0698) (0.0773) Z-values P-values aa4t =.05*AACT, Errorvar.= 0.594, R² = Standerr (0.0756) (0.0723) Z-values P-values bi =.000*BI, Errorvar.= 0.969, R² = Standerr (0.37) Z-values 7.05 P-values bi2 =.090*BI, Errorvar.= 0.248, R² = Standerr (0.0575) (0.27) Z-values P-values bh =.000*BH, R² =.000 be =.000*INCONV, Errorvar.= 0.559, R² = Standerr (0.68) Z-values P-values 0.00 be2 = 0.983*INCONV, Errorvar.= 0.6, R² = Standerr (0.0866) (0.64) Z-values P-values be3 =.000*REWARDS, Errorvar.= 0.45, R² = Standerr (0.77) Z-values P-values 0.0 be4 = 0.824*REWARDS, Errorvar.= 0.964, R² = Standerr (0.9) (0.45) Z-values P-values

12 2 be5 =.000*ENCUMBR, Errorvar.= 2.785, R² = 0.24 Standerr (0.279) Z-values P-values be6 =.726*ENCUMBR, Errorvar.=.85, R² = Standerr (0.273) (0.336) Z-values P-values be7 =.483*ENCUMBR, Errorvar.=.92, R² = Standerr (0.235) (0.279) Z-values P-values Structural Equations AACT = *INCONV *REWARDS *ENCUMBR, Errorvar.= 0.686, R² = 0.48 Standerr (0.0579) (0.0808) (0.0969) (0.05) Z-values P-values BI =.03*AACT, Errorvar.=.528, R² = Standerr (0.0) (0.97) Z-values P-values BH = 0.492*BI, Errorvar.=.42, R² = Standerr (0.0486) (0.3) Z-values P-values NOTE: R² for Structural Equations are Haduk's (2006) Blocked-Error R² Reduced Form Equations AACT = *INCONV *REWARDS *ENCUMBR, Errorvar.= 0.686, R² = 0.48 Standerr (0.0579) (0.0808) (0.0969) Z-values P-values BI = *INCONV *REWARDS *ENCUMBR, Errorvar.= 2.362, R² = Standerr (0.067) (0.0949) (0.07) Z-values P-values BH = *INCONV *REWARDS *ENCUMBR, Errorvar.=.984, R² = Standerr (0.0352) (0.0506) (0.0527) Z-values P-values

13 3 Covariance Matri of Independent Variables INCONV 2.05 (0.282) REWARDS (0.27) (0.230) ENCUMBR (0.47) (0.00) (0.247) Covariance Matri of Latent Variables AACT.79 BI BH INCONV REWARDS ENCUMBR Log-likelihood Values Estimated Model Saturated Model Number of free parameters(t) ln(L) AIC (Akaike, 974)* BIC (Schwarz, 978)* *LISREL uses AIC= 2t - 2ln(L) and BIC = tln(n)- 2ln(L)

14 4 Goodness of Fit Statistics Degrees of Freedom for (C)-(C2) 70 Maimum Likelihood Ratio Chi-Square (C) (P = ) Browne's (984) ADF Chi-Square (C2_NT) (P = ) Estimated Non-centralit Parameter (NCP) Percent Confidence Interval for NCP (2.698 ; ) Minimum Fit Function Value Population Discrepanc Function Value (F0) Percent Confidence Interval for F0 (0.008 ; 0.24) Root Mean Square Error of Approimation (RMSEA) Percent Confidence Interval for RMSEA (0.024 ; ) P-Value for Test of Close Fit (RMSEA < 0.05) 0.87 Epected Cross-Validation Inde (ECVI) Percent Confidence Interval for ECVI (0.57 ; 0.774) ECVI for Saturated Model ECVI for Independence Model 2.58 Chi-Square for Independence Model (9 df) Normed Fit Inde (NFI) Non-Normed Fit Inde (NNFI) Parsimon Normed Fit Inde (PNFI) Comparative Fit Inde (CFI) Incremental Fit Inde (IFI) Relative Fit Inde (RFI) Critical N (CN) Root Mean Square Residual (RMR) 0.33 Standardized RMR Goodness of Fit Inde (GFI) Adjusted Goodness of Fit Inde (AGFI) Parsimon Goodness of Fit Inde (PGFI) 0.633

15 5 Summar Statistics for Standardized Residuals Smallest Standardized Residual = Median Standardized Residual = Largest Standardized Residual = Stemleaf Plot Largest Negative Standardized Residuals Residual for be3 and aat Residual for be6 and be Largest Positive Standardized Residuals Residual for be3 and bh Residual for be4 and bh Modification Indices and Epected Change Modification Indices for LAMBDA-Y aat aa2t aa3t aa4t bi bi bh Epected Change for LAMBDA-Y aat aa2t aa3t aa4t bi bi bh

16 6 Standardized Epected Change for LAMBDA-Y aat aa2t aa3t aa4t bi bi bh Completel Standardized Epected Change for LAMBDA-Y aat aa2t aa3t aa4t bi bi bh Modification Indices for LAMBDA-X be be be be be be be Epected Change for LAMBDA-X be be be be be be be

17 7 Standardized Epected Change for LAMBDA-X be be be be be be be Completel Standardized Epected Change for LAMBDA-X be be be be be be be The Modification Indices Suggest to Add the Path to from Decrease in Chi-Square New Estimate AACT BI Modification Indices for BETA AACT BI BH Epected Change for BETA AACT BI BH Standardized Epected Change for BETA AACT BI BH The Modification Indices Suggest to Add the Path to from Decrease in Chi-Square New Estimate BH REWARDS

18 8 Modification Indices for GAMMA AACT BI BH Epected Change for GAMMA AACT BI BH Standardized Epected Change for GAMMA AACT BI BH No Non-Zero Modification Indices for PHI The Modification Indices Suggest to Add an Error Covariance Between and Decrease in Chi-Square New Estimate BI AACT Modification Indices for PSI AACT - - BI BH Epected Change for PSI AACT - - BI BH Standardized Epected Change for PSI AACT - - BI BH The Modification Indices Suggest to Add an Error Covariance Between and Decrease in Chi-Square New Estimate be6 be

19 9 Modification Indices for THETA-EPS aat aa2t aa3t aa4t bi bi2 bh aat - - aa2t aa3t aa4t bi bi bh Epected Change for THETA-EPS aat aa2t aa3t aa4t bi bi2 bh aat - - aa2t aa3t aa4t bi bi bh Completel Standardized Epected Change for THETA-EPS aat aa2t aa3t aa4t bi bi2 bh aat - - aa2t aa3t aa4t bi bi bh Modification Indices for THETA-DELTA-EPS aat aa2t aa3t aa4t bi bi2 bh be be be be be be be

20 20 Epected Change for THETA-DELTA-EPS aat aa2t aa3t aa4t bi bi2 bh be be be be be be be Completel Standardized Epected Change for THETA-DELTA-EPS aat aa2t aa3t aa4t bi bi2 bh be be be be be be be Modification Indices for THETA-DELTA be be2 be3 be4 be5 be6 be be - - be be be be be be Epected Change for THETA-DELTA be be2 be3 be4 be5 be6 be be - - be be be be be be

21 2 Completel Standardized Epected Change for THETA-DELTA be be2 be3 be4 be5 be6 be be - - be be be be be be Maimum Modification Inde is 2.65 for Element ( 3, 2) of GAMMA A general structural equation model (eplaining coupon usage) Standardized Solution LAMBDA-Y aat aa2t aa3t aa4t bi bi bh LAMBDA-X be be be be be be be BETA AACT BI BH GAMMA AACT BI BH

22 22 Correlation Matri of ETA and KSI AACT.000 BI BH INCONV REWARDS ENCUMBR PSI Note: This matri is diagonal Regression Matri ETA on KSI (Standardized) AACT BI BH A general structural equation model (eplaining coupon usage) Completel Standardized Solution LAMBDA-Y aat aa2t aa3t aa4t bi bi bh LAMBDA-X be be be be be be be

23 23 BETA AACT BI BH GAMMA AACT BI BH Correlation Matri of ETA and KSI AACT.000 BI BH INCONV REWARDS ENCUMBR PSI Note: This matri is diagonal THETA-EPS aat aa2t aa3t aa4t bi bi2 bh W_A_R_N_I_N_G: THETA-EPS is not positive definite THETA-DELTA be be2 be3 be4 be5 be6 be Regression Matri ETA on KSI (Standardized) AACT BI BH Time used 0.03 seconds

24 24 LISREL SPECIFICATION: A general structural equation model (eplaining coupon usage) DA NI=5 NO=0 LA id be be2 be3 be4 be5 be6 be7 aat aa2t aa3t aa4t bi bi2 bh ra fi=d:\eden\sem.dat se / MO n=7 nk=3 n=7 ne=3 td=di,fr te=di,fr ga=fu,fi be=fu,fi ph=fr ps=di,fr va l l 3 2 l 5 3 fr l 2 l 4 2 l 6 3 l 7 3 va l l 5 2 l 7 3 fr l 2 l 3 l 4 l 6 2 fi te 7 7 pa ga pa be lk inconv rewards encumbr le aact bi bh ou sc

25 25 LOCAL FIT INDICES FOR THE MEASUREMENT MODEL construct parameter parameter estimate standardized parameter estimate z-value individualitem reliabilit composite reliabilit (average variance etracted) inconveniences.88 (.78) λ λ θ θ rewards.76 (.6) λ λ θ θ encumbrances.70 (.45) λ λ λ θ θ θ attitudes.88 (.66) λ λ λ λ θ θ θ θ intentions.9 (.84) λ λ θ behavior θ λ θ