CONFIRMATORY FACTOR ANALYSIS

Transcription

1 1 CONFIRMATORY FACTOR ANALYSIS The purpose of confirmatory factor analysis (CFA) is to explain the pattern of associations among a set of observe variables in terms of a smaller number of unerlying latent variables (or factors). Figure 1: Figure 2: j x 1 x 2 l 11 l 21 l 31 l 41 l 52 l 62 l 72 l 82 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x q 11 q 22 q 33 q 44 q 55 q 66 q 77 q 88 In general, the goal of CFA is similar to that of exploratory factor analysis (EFA). However, in EFA the number of factors is not known a priori, an it is also unknown which observe variables loa on which factor(s). In contrast, in CFA the number of factors is etermine a priori, an the researcher also specifies which observe variables loa on which factor(s). I. Specification: x x x where: x q x 1 vector of observe variables in eviation form, x q x n matrix of factor loaings, x n x 1 vector of common factors in eviation form, q x 1 vector of unique (specific plus ranom error) factors. Assuming that E ( ) 0 an that Cov ( x, ') 0, this specification of the moel implies the following structure for the variance-covariance matrix of x: x x x Cov( x, x'),, ) ' where an are the variance-covariance matrices of x an, respectively (see Appenix A for the specification of the moel in Figure 1).

2 2 (1) the basic (congeneric) CFA moel: in this moel we specify that each row of has only one non-zero entry an that Cov(, ') is a iagonal matrix; (2) variations on the basic moel: aitional restrictions may be impose on (e.g., that certain factor loaings are equal across items); if all items loa equally on a given factor, we speak of (essentially) t-equivalent measurement; certain restrictions on may be relaxe (e.g., an item may be allowe to loa on multiple factors); in the basic moel the factors are allowe to be correlate (i.e., they are specifie to be oblique); one coul test whether the factors are uncorrelate (orthogonal) or perfectly correlate; aitional restrictions may be impose on (e.g., that the error variances are equal across items); if all items loa equally on a given factor an all error variances are ientical, we speak of parallel measurement; certain restrictions on may be relaxe (e.g., correlate errors of measurement may be introuce); [for a fairly complicate moel use in MTMM analysis see Appenix B] Moel ientification: A factor analysis moel is sai to be (globally) ientifie if implies that (,, ) (,, ) ,, Note that in orer for a moel to be ientifie, each free parameter has to be ientifie; a moel is exactly (or just) ientifie if all the restrictions impose on the moel are neee to ientify the moel; if there are reunant restrictions, the moel is sai to be overientifie (i.e., it has a positive number of egrees of freeom).

3 3 In orer to achieve ientification, we have to fix the scale of the latent variates: the coefficient relating to x has alreay been set to one in the specification of the moel; the scale of the factors is fixe by setting their variances to one (i.e., has unit iagonal elements) or by constraining one loaing per factor to unity (which correspons to equating the scale of a factor to that of one of its inicators); In aition, various other constraints have to be impose on,, an in orer to ientify the moel. Ientification proceures: [see Appenix C for an example] (i) a necessary conition for ientification is that the number of freely estimate parameters not be greater than the number of istinct elements in the variance-covariance matrix of x; (ii) to show that a moel is ientifie, one has to show that every free parameter in,, an can be expresse as a unique function of the variances an covariances of the observe variables (i.e., the elements of ); (iii) ientification rules for some special cases: three-inicator rule [sufficient but not necessary]: if there are at least three inicators per factor, each inicator loas on one an only one factor, an is iagonal, the factor moel is ientifie; two-inicator rule [sufficient but not necessary]: if there are at least two factors an two inicators per factor, each inicator loas on one an only one factor, the factors are allowe to freely correlate, an is iagonal, the factor moel is ientifie; (iv) if it is too ifficult to check ientification formally, empirical tests of ientification may be use; empirical tests are base on the concept of local ientifiability; one common test is base on whether the inverse of the estimate information matrix exists; Note: In CFA, the factor correlations (containe in the matrix, if the factor variances in the iagonals have been stanarize to 1) are correcte for attenuation ue to measurement error.

4 4 II. Estimation: The goal of estimation is to fin values for the unknown parameters in,, an (i.e.,,, ), base on S (the observe variance-covariance matrix of x), such that the variance-covariance matrix (,, ) implie by the estimate moel parameters is as close as possible to the observe variance/covariance matrix S. To make the concept of closeness between an S operational, we have to choose a iscrepancy function F( S; ), which is a scalar-value function with the following properties (see Browne 1982): (i) F( S; ) 0 (ii) F( S; ) 0 iff S (iii) F( S; ) is continuous in S an Some common iscrepancy functions are the following: (i) Unweighte Least Squares (ULS): F ULS tr 1 2 S 2 this expression minimizes one-half the sum of square resiuals between S an ; provie the moel is ientifie, ULS prouces consistent estimates regarless of the istribution of x; however, the estimates are not asymptotically efficient an they are not scale free; in aition, FULS is not scale invariant (see the iscussion below); (ii) Maximum Likelihoo (ML): F ML log tr S 1 log S q this iscrepancy function is base on the assumption that x has a multivariate normal istribution, which implies that S has a Wishart istribution;

5 5 uner very general conitions, ML estimators are consistent, asymptotically efficient, an asymptotically normally istribute; in aition, FML is scale invariant an ML estimates are scale free (in most cases); this means that F S, FML D S D, D D,, D,, ML where D is a iagonal, nonsingular matrix with positive iagonal elements [see Appenix D for an example]; D (iii) Generalize Least Squares (GLS): F GLS 1 tr 2 S ˆ S 1 2 the assumptions unerlying GLS estimation are slightly less restrictive than those necessary for ML estimation (namely, that fourth-orer cumulants are zero so that there is no excess kurtosis); GLS estimates are also consistent, asymptotically efficient, an asymptotically normally istribute; in aition, FGLS is scale invariant an GLS estimates are scale free (in most cases); (iv) other estimation methos: other estimation methos are available, incluing asymptoticlly istribution-free (ADF) proceures; Estimation problems: nonconvergence: no solution can be foun in a given number of iterations or within a given time limit; it is important that estimation begin at goo starting values (usually supplie automatically); causes of nonconvergence may be poorly specifie moels an small sample sizes with few inicators per factor;

6 6 improper solutions: values of sample estimates that are not possible in the population (e.g., negative error variances, also referre to as Heywoo cases); the causes of improper solutions are similar to those of nonconvergence; III. Testing: 1. Global fit measures: (a) 2 gooness-of-fit test: H 0 :,, perfect fit H A :,, eparture from perfect fit base on the likelihoo ratio criterion, one compares the likelihoo of the hypothesize moel (L0) to the likelihoo of a moel with perfect fit (L1): 2 L 0 ln ~ L1 2 f where f is equal to the number of overientifying restrictions; since (N 1) times the minimum of the fit function (e.g., FML) equals 2ln( L0 / L 1 ), where N is the sample size, we can use a 2 test base on the minimum of the fit function to investigate the null hypothesis that the estimate variance-covariance matrix eviates from the sample variancecovariance matrix only because of sampling error; note that in orer for the 2 test to be applicable an vali, the moel has to have a positive number of overientifying restrictions, the assumptions unerlying the application of the chosen estimation proceure (e.g., multivariate normality in the case of maximum likelihoo estimation) have to be satisfie, an the sample size has to be large (because it is an asymptotic test); in practice, the 2 test is often of limite usefulness because of the following reasons (see Bentler 1990): the assumptions on which its appropriateness is base may not be met, an there is evience that the 2 test is not robust to violations of these assumptions;

7 7 the test is only asymptotically vali an the sample size may be too small to yiel a vali test of moel aequacy; the sample size may be too large so that the test is powerful enough to etect relatively minor or even trivial iscrepancies between the estimate an observe covariance matrices; note the following points: (i) if S, then F ULS F ML F GLS 0 ; (ii) application of the 2 gooness-of-fit test requires that the moel be overientifie; essentially the test assesses the appropriateness of the overientifying restrictions; (b) alternative fit inices: there are many alternative fit inices which assess the fit of the moel in an absolute sense (stan-alone fit inices) or relative to a baseline moel (incremental fit inices); we will iscuss these inices in a separate hanout; 2. Moel moification: (a) moification inices an expecte parameter changes: moification inices (MI s) show the preicte ecrease in the 2 statistic when a fixe parameter is free or an equality constraint is relaxe; expecte parameter changes (EPC s) show the preicte estimate of the parameter when it is freely estimate; stanarize EPC s are available as well; (b) resiual analysis: the size of the resiuals, ( ), is epenent on the appropriateness of s ij ij the hypothesize moel, the scale in which the observe variables are measure, an sampling fluctuation; correlation resiuals (resiuals base on the completely stanarize solution) remove scale epenencies; stanarize resiuals (resiuals ivie by the square root of the estimate asymptotic variance) correct for ifferences in scale an sample size effects; the pattern of over- an unerfitting might suggest moel moifications;

8 8 LISREL an other programs also provie a summary statistic base on the resiuals calle the root mean square resiual (or RMR) as well as a stanarize RMR; 3. Local fit measures: (a) parameter estimates: check whether the estimates are proper an whether they make substantive sense; also, investigate the significance of the parameter estimates base on asymptotic stanar errors; (b) reliability/convergent valiity: iniviual-item reliability (square correlation between a construct xj an one of its inicators xi): IIR xi = λ 2 ij Var(ξ j ) 2 Var(ξ j ) + Var(δ i ) λ ij note: in LISREL iniviual-item reliabilities are calle square multiple correlations; average variance extracte or AVE (proportion of the total variance in all inicators of a construct accounte for by the construct, or the average iniviual-item reliability across all inicators of a construct; see Fornell an Larcker 1981): ( λ 2 ij )Var(ξ j ) AVE(ξ j ) = ( λ 2 ij )Var(ξ j ) + Var(δ i ) or, more simply, AVE(ξ j ) = IIR x i K where K is the number of inicators for the construct in question.

9 9 composite reliability (square correlation between a construct an an unweighte composite of its inicators x i : ( λ ij ) 2 Var(ξ j ) CR xi = ( λ ij ) 2 Var(ξ j ) + Var(δ i ) (c) iscriminant valiity: factor correlations shoul be significantly ifferent from unity (base on the confience interval aroun the estimate factor correlation or a 2 ifference test); average variance extracte shoul be greater than the square correlation between the factors (see Fornell an Larcker 1981);

10 10 Appenix A: Specification of the moel in Figure x x l l l l l l l l x x x x x x x x x 1 = λ 11 ξ 1 + δ 1 x 2 = λ 21 ξ 1 + δ 2 x 3 = λ 31 ξ 1 + δ 3 x 4 = λ 41 ξ 1 + δ 4 x 5 = λ 52 ξ 2 + δ 5 x 6 = λ 62 ξ 2 + δ 6 x 7 = λ 72 ξ 2 + δ 7 x 8 = λ 82 ξ 2 + δ j 21 q q 88 Diag Var(ξ 1 ) = 1 Var(ξ 2 ) = 1 Cov(ξ 1, ξ 2 ) = φ 21 Var(δ i ) = θ ii Cov (δ i, δ j ) = 0

11 11 Appenix B: A moel for MTMM (multi-trait multi-metho) analysis T 1 T 2 T 3 T 1 M 1 T 1 M 2 T 1 M 3 T 2 M 1 T 2 M 2 T 2 M 3 T 3 M 1 T 3 M 2 T 3 M 3 M 1 M 2 M 3

12 12 Appenix C: Ientification of a simple CFA moel j 21 x 1 x 2 l 1 l 2 l 3 l 4 x 1 x 2 x 3 x Rules of covariance algebra: Let X1, X2, an X3 be ranom variables an a, b, c an constants. Then COV (a + bx1, c + X2) = b COV (X1, X2) COV (X1 + X2, X3) = COV (X1, X3) + COV (X2, X3) l q l 21l11 l 21 q22 2 l l j l l j l q l 42l11j 21 l 42l 21j 21 l 42l 32 l 42 q 44

13 13 Appenix D: Scale invariance an scale freeness covariances correlations x1 x2 q x1 x2 q aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t (19) Fitte covariance matrix: aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Fitte correlation matrix: aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Fitte correlation matrix pre- an post-multiplie by stanar eviations: aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t

14 14 EXPLAINING CONSUMERS USAGE OF COUPONS FOR GROCERY SHOPPING (Bagozzi, Baumgartner, an Yi, JCR 1992) Proceure Female staff members at two American universities complete two questionnaires that were sent to them via campus mail. The first questionnaire containe measures of seven beliefs about the consequences of using coupons an corresponing evaluations, as well as measures of attitue towar using coupons, behavioral intentions, an the personality variable of state-/action-orientation. One week later a secon questionnaire was sent to those people who ha participate in the first wave of ata collection. This questionnaire assesse some of the same variables as in wave one as well as people s self-reporte coupon usage uring the past week. Specifically, participants were presente with a table that ha 21 prouct categories as its rows (e.g., cereal, juice rinks, paper towels, snack foos, canne goos) an six sources of coupons as its columns (i.e., irect mail, newspapers, magazines, in or on packages, from store isplays or flyers, from relatives or friens). An aitional row was inclue for other proucts so that responents coul inicate usage in categories not covere by the 21 liste. Participants were aske to state how many coupons they ha use for each category an source combination. Measures (1) Beliefs: perceive likelihoo of the following consequences of using coupons (rate on 7-point unlikely-likely scales): inconveniences: o searching for, gathering, an organizing coupons takes much time an effort; o planning the use of an actually reeeming coupons in the supermarket takes much time an effort; rewars: o using coupons saves much money on the grocery bill; o using coupons leas to feelings of being a thrifty shopper; encumbrances: o in orer to obtain coupons one has to subscribe to extra newspapers, magazines, etc.; o in orer to take avantage of coupon offers one has to purchase nonpreferre brans; o in orer to take avantage of coupon offers one has to shop at multiple supermarkets; (2) Evaluations: how each of the seven consequences of using coupons makes the responent feel, rate on 7-point goo-ba scales; (3) Aact: attitue towar using coupons for shopping in the supermarket uring the upcoming week (assesse on four semantic ifferential scales, i.e., unpleasant-pleasant, ba-goo, foolish-wise, an unfavorable-favorable); measure twice (week 1, week 2); (4) BI: behavioral intentions to use coupons for shopping in the supermarket uring the upcoming week (measure with a 7-point unlikely-likely scale assessing intentions to use coupons an an 11-point no chance-certain scale asking about plans to use coupons); (5) Actual coupon usage: the total number of coupons use across prouct categories an sources; a square root transformation was use to normalize the variable;

15 15 DATA coupon; INFILE ':\ipss\cfa\factor.at'; INPUT i aa1t1 aa2t1 aa3t1 aa4t1 aa1t2 aa2t2 aa3t2 aa4t2; title 'Confirmatory Factor Moel: Congeneric Moel'; title2 '(using the PATH specification in CALIS)'; PROC CALIS DATA=coupon MODIFICATION RESIDUAL; PATH aa1t1 <--- AAT1 = L11, aa2t1 <--- AAT1 = L21, aa3t1 <--- AAT1 = L31, aa4t1 <--- AAT1 = L41, aa1t2 <--- AAT2 = L12, aa2t2 <--- AAT2 = L22, aa3t2 <--- AAT2 = L32, aa4t2 <--- AAT2 = L42; PVAR AAT1 = 1., AAT2 = 1., aa1t1 = th11, aa2t1 = th22, aa3t1 = th33, aa4t1 = th44, aa1t2 = th55, aa2t2 = th66, aa3t2 = th77, aa4t2 = th88; PCOV AAT1 AAT2 = CovAAT1AAT2(0.); run;

16 16 Confirmatory Factor Moel: Congeneric Moel (using the PATH specification in CALIS) Fit Summary Moeling Info Number of Observations 250 Number of Variables 8 Number of Moments 36 Number of Parameters 17 Number of Active Constraints 0 Baseline Moel Function Value Baseline Moel Chi-Square Baseline Moel Chi-Square DF 28 Pr > Baseline Moel Chi-Square <.0001 Absolute Inex Fit Function Chi-Square Chi-Square DF 19 Pr > Chi-Square <.0001 Z-Test of Wilson & Hilferty Hoelter Critical N 103 Root Mean Square Resiual (RMR) Stanarize RMR (SRMR) Gooness of Fit Inex (GFI) Parsimony Inex Ajuste GFI (AGFI) Parsimonious GFI RMSEA Estimate RMSEA Lower 90% Confience Limit RMSEA Upper 90% Confience Limit Probability of Close Fit ECVI Estimate ECVI Lower 90% Confience Limit ECVI Upper 90% Confience Limit Akaike Information Criterion Bozogan CAIC Schwarz Bayesian Criterion McDonal Centrality Incremental Inex Bentler Comparative Fit Inex Bentler-Bonett NFI Bentler-Bonett Non-norme Inex Bollen Norme Inex Rho Bollen Non-norme Inex Delta James et al. Parsimonious NFI

17 17 The CALIS Proceure Covariance Structure Analysis: Maximum Likelihoo Estimation Asymptotically Stanarize Resiual Matrix aa1t1 aa2t1 aa3t1 aa4t1 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Asymptotically Stanarize Resiual Matrix aa1t2 aa2t2 aa3t2 aa4t2 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Average Stanarize Resiual Average Off-iagonal Stanarize Resiual Rank Orer of the 10 Largest Asymptotically Stanarize Resiuals Var1 Var2 Resiual aa3t2 aa3t aa1t2 aa1t aa4t2 aa2t aa1t2 aa2t aa4t1 aa2t aa3t2 aa2t aa3t2 aa1t aa4t1 aa3t aa4t2 aa2t aa3t2 aa1t

18 18 The CALIS Proceure Covariance Structure Analysis: Maximum Likelihoo Estimation PATH List Stanar Path Parameter Estimate Error t Value aa1t1 <=== AAT1 L aa2t1 <=== AAT1 L aa3t1 <=== AAT1 L aa4t1 <=== AAT1 L aa1t2 <=== AAT2 L aa2t2 <=== AAT2 L aa3t2 <=== AAT2 L aa4t2 <=== AAT2 L Variance Parameters Variance Stanar Type Variable Parameter Estimate Error t Value Exogenous AAT AAT Error aa1t1 th aa2t1 th aa3t1 th aa4t1 th aa1t2 th aa2t2 th aa3t2 th aa4t2 th Covariances Among Exogenous Variables Stanar Var1 Var2 Parameter Estimate Error t Value AAT1 AAT2 CovAAT1AAT Square Multiple Correlations Error Total Variable Variance Variance R-Square aa1t aa1t aa2t aa2t aa3t aa3t aa4t aa4t

19 19 Stanarize Results for PATH List Stanar Path Parameter Estimate Error t Value aa1t1 <=== AAT1 L aa2t1 <=== AAT1 L aa3t1 <=== AAT1 L aa4t1 <=== AAT1 L aa1t2 <=== AAT2 L aa2t2 <=== AAT2 L aa3t2 <=== AAT2 L aa4t2 <=== AAT2 L Stanarize Results for Variance Parameters Variance Stanar Type Variable Parameter Estimate Error t Value Exogenous AAT AAT Error aa1t1 th aa2t1 th aa3t1 th aa4t1 th aa1t2 th aa2t2 th aa3t2 th aa4t2 th Stanarize Results for Covariances Among Exogenous Variables Stanar Var1 Var2 Parameter Estimate Error t Value AAT1 AAT2 CovAAT1AAT

20 20 Rank Orer of the 10 Largest LM Stat for Path Relations Parm To From LM Stat Pr > ChiSq Change aa3t1 aa3t < aa3t2 aa3t < aa1t2 aa1t < aa1t1 aa1t aa1t2 aa2t aa4t2 aa2t aa2t1 aa3t aa2t1 aa4t aa1t2 AAT AAT1 aa1t NOTE: No LM statistic in the efault test set for the covariances of exogenous variables is nonsingular. Ranking is not isplaye. Rank Orer of the 10 Largest LM Stat for Error Variances an Covariances Error Error Parm of of LM Stat Pr > ChiSq Change aa3t2 aa3t < aa1t2 aa1t < aa4t2 aa2t aa3t2 aa2t aa2t1 aa1t aa4t1 aa1t aa4t1 aa2t aa3t2 aa1t aa4t2 aa4t aa3t2 aa1t

21 title 'Confirmatory Factor Moel: Moel with correlate errors'; title2 '(using the PATH specification in CALIS)'; PROC CALIS DATA=coupon MODIFICATION RESIDUAL; PATH aa1t1 <--- AAT1 = L11, aa2t1 <--- AAT1 = L21, aa3t1 <--- AAT1 = L31, aa4t1 <--- AAT1 = L41, aa1t2 <--- AAT2 = L12, aa2t2 <--- AAT2 = L22, aa3t2 <--- AAT2 = L32, aa4t2 <--- AAT2 = L42; PVAR AAT1 = 1., AAT2 = 1., aa1t1 = th11, aa2t1 = th22, aa3t1 = th33, aa4t1 = th44, aa1t2 = th55, aa2t2 = th66, aa3t2 = th77, aa4t2 = th88; PCOV aa1t1 aa1t2 = th51(0.), aa2t1 aa2t2 = th62(0.), aa3t1 aa3t2 = th73(0.), aa4t1 aa4t2 = th84(0.), AAT1 AAT2 = CovAAT1AAT2(0.); run; 21

22 22 Confirmatory Factor Moel: Moel with correlate errors (using the PATH specification in CALIS) Fit Summary Moeling Info Number of Observations 250 Number of Variables 8 Number of Moments 36 Number of Parameters 21 Number of Active Constraints 0 Baseline Moel Function Value Baseline Moel Chi-Square Baseline Moel Chi-Square DF 28 Pr > Baseline Moel Chi-Square <.0001 Absolute Inex Fit Function Chi-Square Chi-Square DF 15 Pr > Chi-Square Z-Test of Wilson & Hilferty Hoelter Critical N 233 Root Mean Square Resiual (RMR) Stanarize RMR (SRMR) Gooness of Fit Inex (GFI) Parsimony Inex Ajuste GFI (AGFI) Parsimonious GFI RMSEA Estimate RMSEA Lower 90% Confience Limit RMSEA Upper 90% Confience Limit Probability of Close Fit ECVI Estimate ECVI Lower 90% Confience Limit ECVI Upper 90% Confience Limit Akaike Information Criterion Bozogan CAIC Schwarz Bayesian Criterion McDonal Centrality Incremental Inex Bentler Comparative Fit Inex Bentler-Bonett NFI Bentler-Bonett Non-norme Inex Bollen Norme Inex Rho Bollen Non-norme Inex Delta James et al. Parsimonious NFI

23 23 Asymptotically Stanarize Resiual Matrix aa1t1 aa2t1 aa3t1 aa4t1 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Asymptotically Stanarize Resiual Matrix aa1t2 aa2t2 aa3t2 aa4t2 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Average Stanarize Resiual Average Off-iagonal Stanarize Resiual Rank Orer of the 10 Largest Asymptotically Stanarize Resiuals Var1 Var2 Resiual aa1t2 aa2t aa4t2 aa2t aa1t1 aa1t aa4t1 aa2t aa3t2 aa4t aa4t1 aa3t aa1t2 aa1t aa4t1 aa4t aa3t2 aa2t aa3t1 aa1t

24 24 The CALIS Proceure Covariance Structure Analysis: Maximum Likelihoo Estimation PATH List Stanar Path Parameter Estimate Error t Value aa1t1 <=== AAT1 L aa2t1 <=== AAT1 L aa3t1 <=== AAT1 L aa4t1 <=== AAT1 L aa1t2 <=== AAT2 L aa2t2 <=== AAT2 L aa3t2 <=== AAT2 L aa4t2 <=== AAT2 L Variance Parameters Variance Stanar Type Variable Parameter Estimate Error t Value Exogenous AAT AAT Error aa1t1 th aa2t1 th aa3t1 th aa4t1 th aa1t2 th aa2t2 th aa3t2 th aa4t2 th Covariances Among Exogenous Variables Stanar Var1 Var2 Parameter Estimate Error t Value AAT1 AAT2 CovAAT1AAT Covariances Among Errors Error Error Stanar of of Parameter Estimate Error t Value aa1t1 aa1t2 th aa2t1 aa2t2 th aa3t1 aa3t2 th aa4t1 aa4t2 th

25 25 Square Multiple Correlations Error Total Variable Variance Variance R-Square aa1t aa1t aa2t aa2t aa3t aa3t aa4t aa4t Stanarize Results for PATH List Stanar Path Parameter Estimate Error t Value aa1t1 <=== AAT1 L aa2t1 <=== AAT1 L aa3t1 <=== AAT1 L aa4t1 <=== AAT1 L aa1t2 <=== AAT2 L aa2t2 <=== AAT2 L aa3t2 <=== AAT2 L aa4t2 <=== AAT2 L Stanarize Results for Variance Parameters Variance Stanar Type Variable Parameter Estimate Error t Value Exogenous AAT AAT Error aa1t1 th aa2t1 th aa3t1 th aa4t1 th aa1t2 th aa2t2 th aa3t2 th aa4t2 th Stanarize Results for Covariances Among Exogenous Variables Stanar Var1 Var2 Parameter Estimate Error t Value AAT1 AAT2 CovAAT1AAT

26 26 Stanarize Results for Covariances Among Errors Error Error Stanar of of Parameter Estimate Error t Value aa1t1 aa1t2 th aa2t1 aa2t2 th aa3t1 aa3t2 th aa4t1 aa4t2 th Rank Orer of the 10 Largest LM Stat for Path Relations Parm To From LM Stat Pr > ChiSq Change aa1t2 aa2t aa4t2 aa2t aa2t1 aa1t aa2t1 aa4t aa3t1 aa4t aa3t2 aa4t aa1t2 AAT aa1t2 aa1t aa4t1 aa2t AAT1 aa1t NOTE: No LM statistic in the efault test set for the covariances of exogenous variables is nonsingular. Ranking is not isplaye. Rank Orer of the 10 Largest LM Stat for Error Variances an Covariances Error Error Parm of of LM Stat Pr > ChiSq Change aa2t1 aa1t aa4t2 aa2t aa4t1 aa3t aa4t1 aa3t aa4t1 aa2t aa3t2 aa2t aa2t1 aa1t aa4t1 aa1t aa3t1 aa1t aa4t2 aa2t

27 27 Construct Parameter MEASUREMENT ANALYSIS FOR CFA MODEL Parameter estimate z-value of parameter estimate Iniviualitem reliability Composite reliability (average variance extracte) AAT1.88 (.66) l x l x l x l x q q q q AAT2.91 (.72) l x l x Assessment of iscriminant valiity: l x l x q q q q (1) test of whether 21 = 1: chi-square ifference test: 2 (1) = confience interval: = [.85;.93] Lagrange multiplier test: 2 (1) = (2) Fornell an Larcker criterion: not satisfie here