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); y y = Λ η + ε x x = Λ ξ + δ where: η m x vector of endogenous latent variates in deviation form, ξ n x vector of exogenous latent variates in deviation form, Β m x m matrix of coefficients showing the influence of endogenous latent variates on each other, Γ m x n matrix of coefficients showing the influence of exogenous latent variates on endogenous latent variates, ζ m x vector of errors in equations, y p x vector of observed measures (indicators) of the endogenous latent variates in deviation form, x q x vector of observed measures (indicators) of the exogenous latent variates in deviation form, Λ y p x m matrix of factor loadings for y; Λ x q x n matrix of factor loadings for x; ε p x vector of unique factors corresponding to y; δ q x vector of unique factors corresponding to x; assumptions: () ( I Β) is nonsingular (2) E( ς ) = 0, E ( ε ) = 0, E ( δ ) = 0 (3) Cov ( ξ, ς ') = 0, Cov ( ξ, ε ') = 0, Cov ( ξ, δ ') = 0 (4) Cov ( ς, ε ') = 0, Cov ( ς, δ ') = 0, Cov ( ε, δ ') = 0

2 2 let Cov( δ, δ ') = Θ Cov( ε, ε ') = Θ δ ε Cov( ξ, ξ ') = Φ Cov( ς, ς ') = Ψ then the implied variance-covariance matrix of the observed variables is: x y δ ε Σ( Λ, Λ, Θ, Θ, Β, Γ, Φ, Ψ) Σ Σ yy yx = Σ Σ xy xx where: ( I ) ( ) ( I ) E( ηξ ) ( I ) y '( I ) ' ' y y Σ = Λ Β ΓΦΓ' + Ψ Β ' Λ ' + Θ yy y x y x Σ = Λ ' Λ ' = Λ Β ΓΦΛ ' yx x Σ = Λ ΦΓ Β Λ xy x x Σ = Λ Φ Λ ' + Θ xx δ ε Figure : ϕ δ θ δ x δ θ 22 δ 2 x 2 x λ 2 ξ γ ψ ψ 22 ψ 33 δ θ 33 δ θ 44 δ θ 55 δ θ 66 δ θ 77 δ 3 δ 4 δ 5 δ 6 δ 7 x 3 ϕ 3 x 4 x 5 x 6 x 7 x λ 63 ϕ 2 x λ 42 ϕ 32 x λ 73 η ξ 2 η 2 ξ 3 ϕ 33 ϕ 22 γ 3 γ 2 ζ ζ 2 β 2 y y λ λ 62 4 y y λ 2 λ 3 y 5 y 6 y y 2 ε ε 2 ε θ ε θ 22 y 3 y 4 ε 3 ε 4 ε θ 33 ε θ 44 ε 5 ε 6 ε θ 55 ε θ 66 β 32 η 3 y 7 ζ 3 [see Appendix A for the specification of this model]

3 3 distinguish between the following two types of models: recursive models: Β is a lower triangular matrix 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 y x Λ = I, Λ = I Θ Θ m n, δ ε = 0, = 0 the specification of the model reduces to the usual structural equation model with observed variables; Why is it important to consider measurement error? (i) consequences in bivariate correlation and regression: consider the following model: η = γξ + ζ y = η + ε x = ξ + δ correlation: from classical reliability theory we know that ρ ( η ξ ), = Var Var ρ( y, x) ( η ) ( y ) Var Var ( ξ ) ( x) since the denominator is smaller than or equal to one, ρ( η, ξ) ρ( y, x) ; regression: notice that Cov ( η, ξ) = γϕ so that Cov( η, ξ) γ = ϕ

4 4 what would happen if we mistakenly assumed that η and ξ were measured without error by y and x; in this case we would regress y on x and get Cov( y, x) Cov( η, ξ) ϕ γ * = = = γ Var( x) Var( ξ) + Var( δ ) ϕ + θ thus, for θ > 0, γ * < γ ; it can also be shown that if we have only a sample of observations on y and x, ρ ( y, x) 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 explanatory variables are measured with error; Identification: (i) a necessary condition for identification is that the number of parameters to be freely estimated not be greater than the number of distinct elements in the variance-covariance matrix of y and x; (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 confirmatory factor analysis model in which the factors are allowed to correlate freely; identify the free parameters in x y δ ε Λ, Λ, Θ, Θ, Φ, the variance-covariance matrix of η, and the variance-covariance matrix of η and ξ (see the handout on confirmatory factor analysis for details); if the model in step one is identified, proceed to the second step; assume that η and ξ are directly observable and show that the structural model is identified in this case: null B rule [sufficient but not necessary]: if B is zero, the elements in Γ, Φ, and Ψ are identified;

5 5 recursive rule [sufficient but not necessary]: if B can be written as a lower-triangular matrix and Ψ is diagonal, the latent variable model is identified; rank condition: assumes that all elements of Ψ are freely estimated; solve for the free parameters in Β, Γ, and Ψ in terms of the known elements of the variance-covariance matrix of η and ξ; (b) Estimation: the goal is to find values for the unknown parameters in Λ x, Λ y, Θ δ, Θ ε, Β, Γ, Φ and Ψ, based on S, such that the variance-covariance matrix Σ implied by the estimated parameters will be as close as possible to the observed variance-covariance matrix S; as in the estimation of confirmatory factor models, the most commonly used estimation procedures are ULS, ML, and GLS (refer to the handout on factor analysis for details); (c) Testing:. Global fit measures: (a) χ 2 goodness-of-fit test: test of the hypothesis 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) reliability and discriminant validity (c) explained variation for each structural equation: R 2 ηi = ψ ii Var ( η ) i 3. Model modification: (a) modification indices (b) residual analysis

7 7 Appendix A: Specification of the model in Figure η η η β β η η η γ γ γ ξ ξ ξ ς ς ς = + + x x x x x x x = + λ λ λ λ ξ ξ ξ δ δ δ δ δ δ δ y y y y y y y = + λ λ λ λ η η η ε ε ε ε ε ε Φ = ϕ ϕ ϕ ϕ ϕ ϕ [ ] θ... θ 77 δ Diag = Θ [ ] θ θ ε 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 personality 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. Specifically, 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 six sources of coupons as its columns (i.e., direct mail, newspapers, magazines, in or on packages, from store displays or flyers, from relatives or friends). An additional row was included for other products so that respondents could indicate usage in categories not covered by the 2 listed. Subjects were asked to state how many coupons they had used for each category and source combination. Measures () beliefs: perceived likelihood of the following consequences of using coupons (rated on 7-point unlikely-likely scales): inconveniences: o searching for, gathering, and organizing coupons takes much time and effort; o planning the use of and actually redeeming coupons in the supermarket takes much time and effort; rewards: o using coupons saves much money on the grocery bill; o using coupons leads to feelings of being a thrifty shopper; encumbrances: o in order to obtain coupons one has to subscribe to extra 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 unlikely-likely 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 (explaining coupon usage) Observed Variables id be be2 be3 be4 be5 be6 be7 aat aa2t aa3t aa4t bi bi2 bh Raw Data from File=d:\m554\eden2\sem.dat Latent Variables INCONV REWARDS ENCUMBR AACT BI BH 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 = INCONV REWARDS ENCUMBR 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 Matrix 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 Matrix be4 be5 be6 be be4.87 be be be

11 LISREL Estimates (Maximum Likelihood) Measurement Equations aat =.00*AACT, Errorvar.= 0.68, R 2 = 0.63 (0.075) 9.06 aa2t =.04*AACT, Errorvar.= 0.44, R 2 = 0.74 (0.069) (0.058) aa3t = 0.85*AACT, Errorvar.= 0.76, R 2 = 0.53 (0.070) (0.077) aa4t =.0*AACT, Errorvar.= 0.59, R 2 = 0.7 (0.076) (0.072) bi =.00*BI, Errorvar.= 0.97, R 2 = 0.75 (0.4) 7.04 bi2 =.09*BI, Errorvar.= 0.25, R 2 = 0.93 (0.058) (0.3) bh =.00*BH,, R 2 =.00 be =.00*INCONV, Errorvar.= 0.56, R 2 = 0.79 (0.7) 3.32 be2 = 0.98*INCONV, Errorvar.= 0.6, R 2 = 0.77 (0.087) (0.6) be3 =.00*REWARDS, Errorvar.= 0.45, R 2 = 0.75 (0.8) 2.55 be4 = 0.82*REWARDS, Errorvar.= 0.96, R 2 = 0.48 (0.2) (0.5) be5 =.00*ENCUMBR, Errorvar.= 2.78, R 2 = 0.24 (0.28) 9.97 be6 =.73*ENCUMBR, Errorvar.=.85, R 2 = 0.59 (0.27) (0.34) be7 =.48*ENCUMBR, Errorvar.=.92, R 2 = 0.50 (0.24) (0.28)

12 2 Structural Equations AACT = *INCONV *REWARDS *ENCUMBR, Errorvar.= 0.69, R 2 = 0.42 (0.058) (0.08) (0.097) (0.) BI =.0*AACT, Errorvar.=.53, R 2 = 0.48 (0.) (0.20) BH = 0.49*BI, Errorvar.=.4, R 2 = 0.34 (0.049) (0.3) Reduced Form Equations AACT = *INCONV *REWARDS *ENCUMBR, Errorvar.= 0.69, R 2 = 0.42 (0.058) (0.08) (0.097) BI = - 0.3*INCONV *REWARDS *ENCUMBR, Errorvar.= 2.36, R 2 = 0.20 (0.067) (0.095) (0.) BH = - 0.5*INCONV *REWARDS *ENCUMBR, Errorvar.=.98, R 2 = (0.035) (0.05) (0.053) Covariance Matrix of Independent Variables INCONV REWARDS ENCUMBR INCONV 2. (0.28) 7.42 REWARDS (0.3) (0.23) ENCUMBR (0.5) (0.0) (0.25) Covariance Matrix of Latent Variables AACT BI BH INCONV REWARDS ENCUMBR AACT.8 BI BH INCONV REWARDS ENCUMBR

13 3 Goodness of Fit Statistics Degrees of Freedom = 70 Minimum Fit Function Chi-Square = (P = 0.03) Normal Theory Weighted Least Squares Chi-Square = (P = 0.037) Estimated Non-centrality Parameter (NCP) = Percent Confidence Interval for NCP = (.60 ; 5.68) Minimum Fit Function Value = 0.38 Population Discrepancy Function Value (F0) = Percent Confidence Interval for F0 = ( ; 0.2) Root Mean Square Error of Approximation (RMSEA) = Percent Confidence Interval for RMSEA = ( ; 0.054) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.89 Expected Cross-Validation Index (ECVI) = Percent Confidence Interval for ECVI = (0.57 ; 0.77) ECVI for Saturated Model = 0.84 ECVI for Independence Model = 2.6 Chi-Square for Independence Model with 9 Degrees of Freedom = Independence AIC = Model AIC = Saturated AIC = Independence CAIC = Model CAIC = Saturated CAIC = Normed Fit Index (NFI) = 0.97 Non-Normed Fit Index (NNFI) = 0.99 Parsimony Normed Fit Index (PNFI) = 0.75 Comparative Fit Index (CFI) = 0.99 Incremental Fit Index (IFI) = 0.99 Relative Fit Index (RFI) = 0.96 Critical N (CN) = Root Mean Square Residual (RMR) = 0.3 Standardized RMR = Goodness of Fit Index (GFI) = 0.95 Adjusted Goodness of Fit Index (AGFI) = 0.92 Parsimony Goodness of Fit Index (PGFI) = 0.63

14 4 Summary Statistics for Standardized Residuals Smallest Standardized Residual = -3.2 Median Standardized Residual = 0.00 Largest Standardized Residual = 3.72 Stemleaf Plot Largest Negative Standardized Residuals Residual for be3 and aat Residual for be6 and be3-3.2 Largest Positive Standardized Residuals Residual for be3 and bh 3.72 Residual for be4 and bh 2.68

15 5 A general structural equation model (explaining coupon usage) Modification Indices and Expected Change Modification Indices for LAMBDA-Y AACT BI BH aat aa2t aa3t aa4t bi bi bh Expected Change for LAMBDA-Y AACT BI BH aat aa2t aa3t aa4t bi bi bh Standardized Expected Change for LAMBDA-Y AACT BI BH aat aa2t aa3t aa4t bi bi bh Completely Standardized Expected Change for LAMBDA-Y AACT BI BH aat aa2t aa3t aa4t bi bi bh

16 6 Modification Indices for LAMBDA-X INCONV REWARDS ENCUMBR be be be be be be be Expected Change for LAMBDA-X INCONV REWARDS ENCUMBR be be be be be be be Standardized Expected Change for LAMBDA-X INCONV REWARDS ENCUMBR be be be be be be be Completely Standardized Expected Change for LAMBDA-X INCONV REWARDS ENCUMBR 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

17 7 Modification Indices for BETA AACT BI BH AACT BI BH Expected Change for BETA AACT BI BH AACT BI BH Standardized Expected Change for BETA AACT BI BH AACT BI BH The Modification Indices Suggest to Add the Path to from Decrease in Chi-Square New Estimate BH REWARDS Modification Indices for GAMMA INCONV REWARDS ENCUMBR AACT BI BH Expected Change for GAMMA INCONV REWARDS ENCUMBR AACT BI BH Standardized Expected Change for GAMMA INCONV REWARDS ENCUMBR 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

18 8 Modification Indices for PSI AACT BI BH AACT - - BI BH Expected Change for PSI AACT BI BH AACT - - BI BH Standardized Expected Change for PSI AACT BI BH AACT - - BI BH The Modification Indices Suggest to Add an Error Covariance Between and Decrease in Chi-Square New Estimate be6 be Modification Indices for THETA-EPS aat aa2t aa3t aa4t bi bi2 bh aat - - aa2t aa3t aa4t bi bi bh Expected Change for THETA-EPS aat aa2t aa3t aa4t bi bi2 bh aat - - aa2t aa3t aa4t bi bi bh

19 9 Completely Standardized Expected 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 Expected Change for THETA-DELTA-EPS aat aa2t aa3t aa4t bi bi2 bh be be be be be be be Completely Standardized Expected Change for THETA-DELTA-EPS aat aa2t aa3t aa4t bi bi2 bh be be be be be be be

20 20 Modification Indices for THETA-DELTA be be2 be3 be4 be5 be6 be be - - be be be be be be Expected Change for THETA-DELTA be be2 be3 be4 be5 be6 be be - - be be be be be be Completely Standardized Expected Change for THETA-DELTA be be2 be3 be4 be5 be6 be be - - be be be be be be Maximum Modification Index is 2.67 for Element ( 3, 2) of GAMMA

21 2 A general structural equation model (explaining coupon usage) Standardized Solution LAMBDA-Y AACT BI BH aat aa2t aa3t aa4t bi bi bh LAMBDA-X INCONV REWARDS ENCUMBR be be be be be be be BETA AACT BI BH AACT BI BH GAMMA INCONV REWARDS ENCUMBR AACT BI BH Correlation Matrix of ETA and KSI AACT BI BH INCONV REWARDS ENCUMBR AACT.00 BI BH INCONV REWARDS ENCUMBR

22 22 PSI Note: This matrix is diagonal. AACT BI BH Regression Matrix ETA on KSI (Standardized) INCONV REWARDS ENCUMBR AACT BI BH A general structural equation model (explaining coupon usage) Completely Standardized Solution LAMBDA-Y AACT BI BH aat aa2t aa3t aa4t bi bi bh LAMBDA-X INCONV REWARDS ENCUMBR be be be be be be be BETA AACT BI BH AACT BI BH

23 23 GAMMA INCONV REWARDS ENCUMBR AACT BI BH Correlation Matrix of ETA and KSI AACT BI BH INCONV REWARDS ENCUMBR AACT.00 BI BH INCONV REWARDS ENCUMBR PSI Note: This matrix is diagonal. AACT BI BH THETA-EPS aat aa2t aa3t aa4t bi bi2 bh THETA-DELTA be be2 be3 be4 be5 be6 be Regression Matrix ETA on KSI (Standardized) INCONV REWARDS ENCUMBR AACT BI BH Time used: 0.30 Seconds

24 24 LISREL SPECIFICATION: A general structural equation model (explaining 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:\m554\eden2\sem.dat se / MO nx=7 nk=3 ny=7 ne=3 td=di,fr te=di,fr ga=fu,fi be=fu,fi ph=fr ps=di,fr va lx lx 3 2 lx 5 3 fr lx 2 lx 4 2 lx 6 3 lx 7 3 va ly ly 5 2 ly 7 3 fr ly 2 ly 3 ly 4 ly 6 2 fi te 7 7 pa ga pa be lk inconv rewards encumbr le aact bi bh ou sc

25 25 A general structural equation model (explaining coupon usage) Number of Input Variables 5 Number of Y - Variables 7 Number of X - Variables 7 Number of ETA - Variables 3 Number of KSI - Variables 3 Number of Observations 250 Parameter Specifications LAMBDA-Y aact bi bh aat aa2t 0 0 aa3t aa4t bi bi bh LAMBDA-X inconv rewards encumbr be be be be be be be BETA aact bi bh aact bi bh GAMMA inconv rewards encumbr aact 2 3 bi bh 0 0 0

26 26 PHI inconv rewards encumbr inconv 4 rewards 5 6 encumbr PSI aact bi bh THETA-EPS aat aa2t aa3t aa4t bi bi THETA-EPS bh THETA-DELTA be be2 be3 be4 be5 be THETA-DELTA be

27 27 LISREL Estimates (Maximum Likelihood) LAMBDA-Y aact bi bh aat aa2t (0.07) 4.97 aa3t (0.07) 2.4 aa4t (0.08) 4.58 bi bi (0.06) 8.9 bh LAMBDA-X inconv rewards encumbr be be (0.09).32 be be (0.2) 6.89 be be (0.27) 6.30 be (0.24) 6.30

28 28 BETA aact bi bh aact bi (0.) 0.04 bh (0.05) 0.0 GAMMA inconv rewards encumbr aact (0.06) (0.08) (0.0) bi bh Covariance Matrix of ETA and KSI aact bi bh inconv rewards encumbr aact.8 bi bh inconv rewards encumbr PHI inconv rewards encumbr inconv 2. (0.28) 7.42 rewards (0.3) (0.23) encumbr (0.5) (0.0) (0.25)

29 29 PSI Note: This matrix is diagonal. aact bi bh (0.) (0.20) (0.3) Squared Multiple Correlations for Structural Equations aact bi bh Squared Multiple Correlations for Reduced Form aact bi bh Reduced Form inconv rewards encumbr aact (0.06) (0.08) (0.0) bi (0.07) (0.0) (0.) bh (0.04) (0.05) (0.05) THETA-EPS aat aa2t aa3t aa4t bi bi (0.07) (0.06) (0.08) (0.07) (0.4) (0.3) THETA-EPS bh

30 30 Squared Multiple Correlations for Y - Variables aat aa2t aa3t aa4t bi bi Squared Multiple Correlations for Y - Variables bh THETA-DELTA be be2 be3 be4 be5 be (0.7) (0.6) (0.8) (0.5) (0.28) (0.34) THETA-DELTA be (0.28) 6.87 Squared Multiple Correlations for X - Variables be be2 be3 be4 be5 be Squared Multiple Correlations for X - Variables be

31 3 Goodness of Fit Statistics Degrees of Freedom = 70 Minimum Fit Function Chi-Square = (P = 0.03) Normal Theory Weighted Least Squares Chi-Square = (P = 0.037) Estimated Non-centrality Parameter (NCP) = Percent Confidence Interval for NCP = (.60 ; 5.68) Minimum Fit Function Value = 0.38 Population Discrepancy Function Value (F0) = Percent Confidence Interval for F0 = ( ; 0.2) Root Mean Square Error of Approximation (RMSEA) = Percent Confidence Interval for RMSEA = ( ; 0.054) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.89 Expected Cross-Validation Index (ECVI) = Percent Confidence Interval for ECVI = (0.57 ; 0.77) ECVI for Saturated Model = 0.84 ECVI for Independence Model = 2.6 Chi-Square for Independence Model with 9 Degrees of Freedom = Independence AIC = Model AIC = Saturated AIC = Independence CAIC = Model CAIC = Saturated CAIC = Normed Fit Index (NFI) = 0.97 Non-Normed Fit Index (NNFI) = 0.99 Parsimony Normed Fit Index (PNFI) = 0.75 Comparative Fit Index (CFI) = 0.99 Incremental Fit Index (IFI) = 0.99 Relative Fit Index (RFI) = 0.96 Critical N (CN) = Root Mean Square Residual (RMR) = 0.3 Standardized RMR = Goodness of Fit Index (GFI) = 0.95 Adjusted Goodness of Fit Index (AGFI) = 0.92 Parsimony Goodness of Fit Index (PGFI) = 0.63

32 32 LOCAL FIT INDICES FOR THE MEASUREMENT MODEL construct parameter parameter standardized z-value individual- composite reliability estimate parameter item (average variance estimate reliability extracted) inconveniences.88 (.78) λ x λ x θ δ θ δ rewards.76 (.6) λ x λ x θ δ θ δ encumbrances.70 (.45) λ x λ x λ x θ δ θ δ θ δ attitudes.88 (.66) λ y λ y λ y λ y θ ε θ ε θ ε θ ε intentions.9 (.84) λ y λ y θ ε θ ε behavior -- λ y θ ε

33 33

34 34 AMOS output: d be d2 d3 be2 be3 Incon e aat e2 aa2t z z2 z3 d4 d5 d6 be4 be5 be6 Rew Encum aa3t e3 aa bi B aa4t e4 bi e5 bi2 e6 bh e7 0 d7 be7 Notes for Model (Default model) Computation of degrees of freedom (Default model) Number of distinct sample moments: 05 Number of distinct parameters to be estimated: 35 Result (Default model) Minimum was achieved Chi-square = Degrees of freedom = 70 Probability level =.03 Degrees of freedom (05-35): 70

35 35 Estimates (Group number - Default model) Maximum Likelihood Estimates Regression Weights: (Group number - Default model) Estimate S.E. C.R. P Label aa <--- Rew *** aa <--- Incon *** aa <--- Encum bi <--- aa *** B <--- bi *** be <--- Incon.000 be2 <--- Incon *** be5 <--- Encum.000 be6 <--- Encum *** be7 <--- Encum *** be3 <--- Rew.000 be4 <--- Rew *** aat <--- aa.000 aa3t <--- aa *** aa4t <--- aa *** aa2t <--- aa *** bi <--- bi.000 bi2 <--- bi *** bh <--- B.000 Standardized Regression Weights: (Group number - Default model) Estimate aa <--- Rew.465 aa <--- Incon aa <--- Encum bi <--- aa.696 B <--- bi.580 be <--- Incon.889 be2 <--- Incon.877 be5 <--- Encum.49 be6 <--- Encum.766 be7 <--- Encum.709 be3 <--- Rew.864 be4 <--- Rew.695 aat <--- aa.797 aa3t <--- aa.727 aa4t <--- aa.84 aa2t <--- aa.860 bi <--- bi.868 bi2 <--- bi.966 bh <--- B.000

36 36 Covariances: (Group number - Default model) Estimate S.E. C.R. P Label Incon <--> Rew Encum <--> Rew Incon <--> Encum *** Correlations: (Group number - Default model) Estimate Incon <--> Rew -.05 Encum <--> Rew Incon <--> Encum.494 Variances: (Group number - Default model) Estimate S.E. C.R. P Label Incon *** Encum *** Rew *** z *** z *** z *** e7.000 d *** d *** d d *** d *** d *** d *** e *** e *** e *** e *** e *** e

37 37 Squared Multiple Correlations: (Group number - Default model) Estimate aa.48 bi.484 B.337 bh.000 bi2.934 bi.753 aa2t.740 aa4t.708 aa3t.528 aat.635 be4.483 be3.746 be7.503 be6.587 be5.24 be2.769 be.790 Modification Indices (Group number - Default model) Covariances: (Group number - Default model) M.I. Par Change z2 <--> z z3 <--> Rew z3 <--> Incon e7 <--> Rew e7 <--> Incon e <--> Rew d3 <--> z d3 <--> e d6 <--> Rew d6 <--> z d6 <--> e d6 <--> d Variances: (Group number - Default model) M.I. Par Change

38 38 Regression Weights: (Group number - Default model) M.I. Par Change bi <--- Encum bi <--- Incon B <--- Rew bh <--- Rew bh <--- be bh <--- be bh <--- be aa2t <--- be aat <--- Rew aat <--- be aat <--- be aat <--- be be3 <--- B be3 <--- bh be7 <--- be be6 <--- Rew be6 <--- be Model Fit Summary CMIN Model NPAR CMIN DF P CMIN/DF Default model Saturated model Independence model RMR, GFI Model RMR GFI AGFI PGFI Default model Saturated model Independence model Baseline Comparisons Model NFI Delta RFI rho IFI Delta2 TLI rho2 CFI Default model Saturated model Independence model

39 39 Parsimony-Adjusted Measures Model PRATIO PNFI PCFI Default model Saturated model Independence model NCP Model NCP LO 90 HI 90 Default model Saturated model Independence model FMIN Model FMIN F0 LO 90 HI 90 Default model Saturated model Independence model RMSEA Model RMSEA LO 90 HI 90 PCLOSE Default model Independence model AIC Model AIC BCC BIC CAIC Default model Saturated model Independence model ECVI Model ECVI LO 90 HI 90 MECVI Default model Saturated model Independence model HOELTER Model HOELTER.05 HOELTER.0 Default model Independence model 6 8