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 observed variables in terms of a smaller number of underlying latent variables (or factors). Figure 1: Figure 2: j x 1 x 2 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 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 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d d d d d d d d 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, and it is also unknown which observed variables load on which factor(s). In contrast, in CFA the number of factors is determined a priori, and the researcher also specifies which observed variables load on which factor(s). I. Specification: x x x d where: x q x 1 vector of observed variables in deviation form, x q x n matrix of factor loadings, x n x 1 vector of common factors in deviation form, d q x 1 vector of unique (specific plus random error) factors. Assuming that E ( d) 0 and that Cov ( x, d') 0, this specification of the model implies the following structure for the variance-covariance matrix of x: x d x x Cov( x, x'),, ) ' d where and d are the variance-covariance matrices of x and d, respectively (see Appendix A for the specification of the model in Figure 1).

2 2 (1) the basic (congeneric) CFA model: in this model we specify that each row of has only one non-zero entry and that d Cov( d, d ') is a diagonal matrix; (2) variations on the basic model: additional restrictions may be imposed on (e.g., that certain factor loadings are equal across items); if all items load equally on a given factor, we speak of (essentially) t-equivalent measurement; certain restrictions on may be relaxed (e.g., an item may be allowed to load on multiple factors); in the basic model the factors are allowed to be correlated (i.e., they are specified to be oblique); one could test whether the factors are uncorrelated (orthogonal) or perfectly correlated; additional restrictions may be imposed on d (e.g., that the error variances are equal across items); if all items load equally on a given factor and all error variances are identical, we speak of parallel measurement; certain restrictions on d may be relaxed (e.g., correlated errors of measurement may be introduced); [for a fairly complicated model used in MTMM analysis see Appendix B] Model identification: A factor analysis model is said to be (globally) identified if implies that d d (,, ) (,, ) d,, d Note that in order for a model to be identified, each free parameter has to be identified; a model is exactly (or just) identified if all the restrictions imposed on the model are needed to identify the model; if there are redundant restrictions, the model is said to be overidentified (i.e., it has a positive number of degrees of freedom).

3 3 In order to achieve identification, we have to fix the scale of the latent variates: the coefficient relating d to x has already been set to one in the specification of the model; the scale of the elements of x is fixed by setting their variances to one (i.e., has unit diagonal elements) or by constraining one loading per factor to unity (which corresponds to equating the scale of a factor to that of one of its indicators); In addition, various other constraints have to be imposed on,, and d in order to identify the model. Identification procedures: [see Appendix C for an example] (i) a necessary condition for identification is that the number of freely estimated parameters not be greater than the number of distinct elements in the variance-covariance matrix of x; (ii) to show that a model is identified, one has to show that every free parameter in,, and d can be expressed as a unique function of the variances and covariances of the observed variables (i.e., the elements of ); (iii) identification rules for some special cases: three-indicator rule [sufficient but not necessary]: if there are at least three indicators per factor, each indicator loads on one and only one factor, and d is diagonal, the factor model is identified; two-indicator rule [sufficient but not necessary]: if there are at least two factors and two indicators per factor, each indicator loads on one and only one factor, the factors are allowed to freely correlate, and d is diagonal, the factor model is identified; (iv) if it is too difficult to check identification formally, empirical tests of identification may be used; empirical tests are based on the concept of local identifiability; one common test is based on whether the inverse of the estimated information matrix exists; Note: In CFA, the factor correlations (contained in the matrix, if the factor variances in the diagonals have been standardized to 1) are corrected for attenuation due to measurement error.

4 4 II. Estimation: The goal of estimation is to find values for the unknown parameters in,, and d (i.e.,,, d ), based on S (the observed variance-covariance matrix of x), such that the variance-covariance matrix (,, d ) implied by the estimated model parameters is as close as possible to the observed variance/covariance matrix S. To make the concept of closeness between and S operational, we have to choose a discrepancy function F( S; ), which is a scalar-valued 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 and Some common discrepancy functions are the following: (i) Unweighted Least Squares (ULS): F ULS tr 1 2 S 2 this expression minimizes one-half the sum of squared residuals between S and ; provided the model is identified, ULS produces consistent estimates regardless of the distribution of x; however, the estimates are not asymptotically efficient and they are not scale free; in addition, FULS is not scale invariant (see the discussion below); (ii) Maximum Likelihood (ML): F ML log tr S 1 log S q this discrepancy function is based on the assumption that x has a multivariate normal distribution, which implies that S has a Wishart distribution;

5 5 under very general conditions, ML estimators are consistent, asymptotically efficient, and asymptotically normally distributed; in addition, FML is scale invariant and 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 diagonal, nonsingular matrix with positive diagonal elements [see Appendix D for an example]; D (iii) Generalized Least Squares (GLS): F GLS 1 tr 2 S ˆ S 1 2 the assumptions underlying GLS estimation are slightly less restrictive than those necessary for ML estimation (namely, that fourth-order cumulants are zero so that there is no excess kurtosis); GLS estimates are also consistent, asymptotically efficient, and asymptotically normally distributed; in addition, FGLS is scale invariant and GLS estimates are scale free (in most cases); (iv) other estimation methods: several other estimation methods are available, including asymptotic distribution-free (ADF) procedures; Estimation problems: nonconvergence: no solution can be found in a given number of iterations or within a given time limit; it is important that estimation begin at good starting values (usually supplied automatically); causes of nonconvergence may be poorly specified models and small sample sizes with few indicators per factor;

6 6 improper solutions: values of sample estimates that are not possible in the population (e.g., negative error variances, also referred to as Heywood cases); the causes of improper solutions are similar to those of nonconvergence; III. Testing: 1. Global fit measures: (a) 2 goodness-of-fit test: H 0 :,, perfect fit H A :,, departure from perfect fit based on the likelihood ratio criterion, one compares the likelihood of the hypothesized model (L0) to the likelihood of a model with perfect fit (L1): 2 L 0 ln ~ L1 2 df where df is equal to the number of overidentifying 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 based on the minimum of the fit function to investigate the null hypothesis that the estimated variance-covariance matrix deviates from the sample variancecovariance matrix only because of sampling error; note that in order for the 2 test to be applicable and valid, the model has to have a positive number of overidentifying restrictions, the assumptions underlying the application of the chosen estimation procedure (e.g., multivariate normality in the case of maximum likelihood estimation) have to be satisfied, and the sample size has to be large (because it is an asymptotic test); in practice, the 2 test is often of limited usefulness because of the following reasons (see Bentler 1990): the assumptions on which its appropriateness is based may not be met, and there is evidence that the 2 test is not robust to violations of these assumptions;

7 7 the test is only asymptotically valid and the sample size may be too small to yield a valid test of model adequacy; the sample size may be too large so that the test is powerful enough to detect relatively minor or even trivial discrepancies between the estimated and observed covariance matrices; note the following points: (i) if S, then F ULS F ML F GLS 0 ; (ii) application of the 2 goodness-of-fit test requires that the model be overidentified; essentially the test assesses the appropriateness of the overidentifying restrictions; (b) alternative fit indices: there are many alternative fit indices which assess the fit of the model in an absolute sense (stand-alone fit indices) or relative to a baseline model (incremental fit indices); we will discuss these indices in a separate handout; (2) Local fit measures: (a) parameter estimates: check whether the estimates are proper and whether they make substantive sense; also, investigate the significance of the parameter estimates based on asymptotic standard errors; (b) reliability: individual-item reliability (squared correlation between a construct xj and one of its indicators xi): ii = l ij 2 var(xj)/[l ij 2 var(xj) + qii] note: in LISREL individual-item reliabilities are called squared multiple correlations; composite reliability (squared correlation between a construct and an unweighted composite of its indicators x = x1 + x xk): c = (l ij ) 2 var(xj)/[(l ij ) 2 var(xj) + qii]

8 8 average variance extracted (proportion of the total variance in all indicators of a construct accounted for by the construct; see Fornell and Larcker 1981): ave = (l ij 2 ) var(xj)/[(l ij 2 ) var(xj) + qii] (c) discriminant validity: factor correlations should be significantly different from unity (based on the confidence interval around the estimated factor correlation or a 2 difference test); average variance extracted should be greater than the squared correlation between the factors (see Fornell and Larcker 1981); Model modification: (a) modification indices and expected parameter changes: modification indices (MI s) show the predicted decrease in the 2 statistic when a fixed parameter is freed or an equality constraint is relaxed; expected parameter changes (EPC s) show the predicted estimate of the parameter when it is freely estimated; standardized EPC s are available as well; (b) residual analysis: the size of the residuals, ( ), is dependent on the appropriateness of s ij ij the hypothesized model, the scale in which the observed variables are measured, and sampling fluctuation; correlation residuals (residuals based on the completely standardized solution) remove scale dependencies; standardized residuals (residuals divided by the square root of the estimated asymptotic variance) correct for differences in scale and sample size effects; the pattern of over- and underfitting might suggest model modifications; LISREL and other programs also provide a summary statistic based on the residuals called the root mean squared residual (or RMR) as well as a standardized RMR;

9 9 Appendix A: Specification of the model in Figure d d d d d d d d x x l l l l l l l l x x x x x x x x 1 1 j 21 q q 88 d Diag

10 10 Appendix B: A model for MTMM (multi-trait multi-method) 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

11 11 Appendix C: Identification of a simple CFA model j 21 x 1 x 2 l 1 l 2 l 3 l 4 x 1 x 2 x 3 x 4 d 1 d 2 d 3 d 4 Rules of covariance algebra: Let X1, X2, and X3 be random variables and a, b, c and d constants. Then COV (a + bx1, c + dx2) = b d 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

12 12 Appendix D: Scale invariance and scale freeness covariances correlations x 1 x 2 q x 1 x 2 q aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t (19) Fitted covariance matrix: aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Fitted correlation matrix: aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Fitted correlation matrix pre- and post-multiplied by standard deviations: aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t

13 13 EXPLAINING CONSUMERS USAGE OF COUPONS FOR GROCERY SHOPPING (Bagozzi, Baumgartner, and Yi, JCR 1992) 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, participants were presented with a table that had 21 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 21 listed. Participants were asked to state how many coupons they had used for each category and source combination. Measures (1) 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 1, 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 11-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;

14 14 SIMPLIS SPECIFICATION: INITIAL MODEL Title Confirmatory factor model (attitude toward using coupons measured at two points in time) Observed Variables id Raw Data from File=d:\m554\eden2\cfa.dat Latent Variables Sample Size 250 Relationships aa1t1 aa2t1 aa3t1 aa4t1 = AAT1 aa1t2 aa2t2 aa3t2 aa4t2 = AAT2 Set the Variance of AAT1 to 1 Set the Variance of AAT2 to 1 Options sc rs mi End of Problem Sample Size = 250 Confirmatory factor model (attitude toward using coupons measured at two points Covariance Matrix aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Total Variance = Generalized Variance = Largest Eigenvalue = Smallest Eigenvalue = Condition Number = 5.602

15 Confirmatory factor model (attitude toward using coupons measured at two points 15 Number of Iterations = 5 LISREL Estimates (Maximum Likelihood) Measurement Equations aa1t1 = 1.099*AAT1, Errorvar.= 0.649, R² = Standerr (0.0730) (0.0702) Z-values P-values aa2t1 = 1.102*AAT1, Errorvar.= 0.496, R² = Standerr (0.0685) (0.0575) Z-values P-values aa3t1 = 0.935*AAT1, Errorvar.= 0.736, R² = Standerr (0.0708) (0.0739) Z-values P-values aa4t1 = 1.214*AAT1, Errorvar.= 0.558, R² = Standerr (0.0742) (0.0663) Z-values P-values aa1t2 = 1.196*AAT2, Errorvar.= 0.546, R² = Standerr (0.0727) (0.0611) Z-values P-values aa2t2 = 1.164*AAT2, Errorvar.= 0.410, R² = Standerr (0.0674) (0.0491) Z-values P-values aa3t2 = 0.985*AAT2, Errorvar.= 0.545, R² = Standerr (0.0658) (0.0564) Z-values P-values aa4t2 = 1.233*AAT2, Errorvar.= 0.487, R² = Standerr (0.0722) (0.0573) Z-values P-values Correlation Matrix of Independent Variables AAT AAT (0.020)

16 Log-likelihood Values 16 Estimated Model Saturated Model Number of free parameters(t) ln(L) AIC (Akaike, 1974)* BIC (Schwarz, 1978)* *LISREL uses AIC= 2t - 2ln(L) and BIC = tln(n)- 2ln(L) Goodness of Fit Statistics Degrees of Freedom for (C1)-(C2) 19 Maximum Likelihood Ratio Chi-Square (C1) (P = ) Browne's (1984) ADF Chi-Square (C2_NT) (P = ) Estimated Non-centrality Parameter (NCP) Percent Confidence Interval for NCP ( ; ) Minimum Fit Function Value Population Discrepancy Function Value (F0) Percent Confidence Interval for F0 (0.127 ; 0.338) Root Mean Square Error of Approximation (RMSEA) Percent Confidence Interval for RMSEA ( ; 0.133) P-Value for Test of Close Fit (RMSEA < 0.05) Expected Cross-Validation Index (ECVI) Percent Confidence Interval for ECVI (0.339 ; 0.550) ECVI for Saturated Model ECVI for Independence Model Chi-Square for Independence Model (28 df) Normed Fit Index (NFI) Non-Normed Fit Index (NNFI) Parsimony Normed Fit Index (PNFI) Comparative Fit Index (CFI) Incremental Fit Index (IFI) Relative Fit Index (RFI) Critical N (CN) Root Mean Square Residual (RMR) Standardized RMR Goodness of Fit Index (GFI) Adjusted Goodness of Fit Index (AGFI) Parsimony Goodness of Fit Index (PGFI) 0.494

17 Confirmatory factor model (attitude toward using coupons measured at two points 17 Fitted Covariance Matrix aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Fitted Residuals aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Summary Statistics for Fitted Residuals Smallest Fitted Residual = Median Fitted Residual = Largest Fitted Residual = Stemleaf Plot Standardized Residuals aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Summary Statistics for Standardized Residuals Smallest Standardized Residual = Median Standardized Residual = Largest Standardized Residual = 4.853

18 18 Stemleaf Plot Largest Negative Standardized Residuals Residual for aa4t2 and aa2t Largest Positive Standardized Residuals Residual for aa1t2 and aa1t Residual for aa3t2 and aa3t Confirmatory factor model (attitude toward using coupons measured at two points Modification Indices and Expected Change Modification Indices for LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Expected Change for LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Standardized Expected Change for LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Completely Standardized Expected Change for LAMBDA-X

19 19 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t No Non-Zero Modification Indices for PHI The Modification Indices Suggest to Add an Error Covariance Between and Decrease in Chi-Square New Estimate aa1t2 aa1t aa3t2 aa3t Modification Indices for THETA-DELTA aa1t1 - - aa2t aa3t aa4t aa1t aa2t aa3t aa4t Expected Change for THETA-DELTA aa1t1 - - aa2t aa3t aa4t aa1t aa2t aa3t aa4t Completely Standardized Expected Change for THETA-DELTA aa1t1 - - aa2t aa3t aa4t aa1t aa2t aa3t aa4t Maximum Modification Index is for Element ( 7, 3) of THETA-DELTA

20 Confirmatory factor model (attitude toward using coupons measured at two points 20 Standardized Solution LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t PHI AAT AAT Confirmatory factor model (attitude toward using coupons measured at two points Completely Standardized Solution LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t PHI AAT AAT THETA-DELTA Time used seconds

21 21 SIMPLIS SPECIFICATION: MODIFIED MODEL Title Confirmatory factor model (attitude toward using coupons measured at two points in time) Observed Variables id Raw Data from File=d:\m554\eden2\cfa.dat Latent Variables Sample Size 250 Relationships aa1t1 aa2t1 aa3t1 aa4t1 = AAT1 aa1t2 aa2t2 aa3t2 aa4t2 = AAT2 Set the Variance of AAT1 to 1 Set the Variance of AAT2 to 1 Set the Error Covariance of aa1t1 and aa1t2 free Set the Error Covariance of aa2t1 and aa2t2 free Set the Error Covariance of aa3t1 and aa3t2 free Set the Error Covariance of aa4t1 and aa4t2 free Options sc rs mi wp Path Diagram End of Problem Sample Size = 250 Confirmatory factor model (attitude toward using coupons measured at two points Covariance Matrix aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Total Variance = Generalized Variance = Largest Eigenvalue = Smallest Eigenvalue = Condition Number = 5.602

22 LISREL Estimates (Maximum Likelihood) 22 Measurement Equations aa1t1 = 1.084*AAT1, Errorvar.= 0.665, R² = Standerr (0.0732) (0.0728) Z-values P-values aa2t1 = 1.108*AAT1, Errorvar.= 0.483, R² = Standerr (0.0688) (0.0598) Z-values P-values aa3t1 = 0.923*AAT1, Errorvar.= 0.755, R² = Standerr (0.0710) (0.0763) Z-values P-values aa4t1 = 1.221*AAT1, Errorvar.= 0.546, R² = Standerr (0.0746) (0.0693) Z-values P-values aa1t2 = 1.188*AAT2, Errorvar.= 0.563, R² = Standerr (0.0730) (0.0638) Z-values P-values aa2t2 = 1.170*AAT2, Errorvar.= 0.395, R² = Standerr (0.0675) (0.0508) Z-values P-values aa3t2 = 0.976*AAT2, Errorvar.= 0.555, R² = Standerr (0.0657) (0.0578) Z-values P-values aa4t2 = 1.236*AAT2, Errorvar.= 0.480, R² = Standerr (0.0725) (0.0596) Z-values P-values Error Covariance for aa1t2 and aa1t1 = (0.0498) Error Covariance for aa2t2 and aa2t1 = (0.0390) Error Covariance for aa3t2 and aa3t1 = (0.0496) Error Covariance for aa4t2 and aa4t1 = (0.0456) 1.105

23 Correlation Matrix of Independent Variables 23 AAT AAT (0.020) Log-likelihood Values Estimated Model Saturated Model Number of free parameters(t) ln(L) AIC (Akaike, 1974)* BIC (Schwarz, 1978)* *LISREL uses AIC= 2t - 2ln(L) and BIC = tln(n)- 2ln(L) Goodness of Fit Statistics Degrees of Freedom for (C1)-(C2) 15 Maximum Likelihood Ratio Chi-Square (C1) (P = ) Browne's (1984) ADF Chi-Square (C2_NT) (P = ) Estimated Non-centrality Parameter (NCP) Percent Confidence Interval for NCP (1.143 ; ) Minimum Fit Function Value Population Discrepancy Function Value (F0) Percent Confidence Interval for F0 ( ; 0.122) Root Mean Square Error of Approximation (RMSEA) Percent Confidence Interval for RMSEA ( ; ) P-Value for Test of Close Fit (RMSEA < 0.05) Expected Cross-Validation Index (ECVI) Percent Confidence Interval for ECVI (0.233 ; 0.350) ECVI for Saturated Model ECVI for Independence Model Chi-Square for Independence Model (28 df) Normed Fit Index (NFI) Non-Normed Fit Index (NNFI) Parsimony Normed Fit Index (PNFI) Comparative Fit Index (CFI) Incremental Fit Index (IFI) Relative Fit Index (RFI) Critical N (CN) Root Mean Square Residual (RMR) Standardized RMR Goodness of Fit Index (GFI) Adjusted Goodness of Fit Index (AGFI) Parsimony Goodness of Fit Index (PGFI) 0.406

24 Confirmatory factor model (attitude toward using coupons measured at two points 24 Fitted Covariance Matrix aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Fitted Residuals aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Summary Statistics for Fitted Residuals Smallest Fitted Residual = Median Fitted Residual = Largest Fitted Residual = Stemleaf Plot Standardized Residuals aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t

25 Summary Statistics for Standardized Residuals 25 Smallest Standardized Residual = Median Standardized Residual = Largest Standardized Residual = Stemleaf Plot Largest Positive Standardized Residuals Residual for aa1t2 and aa2t Modification Indices and Expected Change Modification Indices for LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Expected Change for LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t

26 Standardized Expected Change for LAMBDA-X 26 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Completely Standardized Expected Change for LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t No Non-Zero Modification Indices for PHI The Modification Indices Suggest to Add an Error Covariance Between and Decrease in Chi-Square New Estimate aa1t2 aa2t Modification Indices for THETA-DELTA aa1t1 - - aa2t aa3t aa4t aa1t aa2t aa3t aa4t Expected Change for THETA-DELTA aa1t1 - - aa2t aa3t aa4t aa1t aa2t aa3t aa4t

27 Completely Standardized Expected Change for THETA-DELTA 27 aa1t1 - - aa2t aa3t aa4t aa1t aa2t aa3t aa4t Maximum Modification Index is for Element ( 5, 2) of THETA-DELTA Confirmatory factor model (attitude toward using coupons measured at two points Standardized Solution LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t PHI AAT AAT Confirmatory factor model (attitude toward using coupons measured at two points Completely Standardized Solution LAMBDA-X aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t

28 PHI 28 AAT AAT THETA-DELTA aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Time used seconds

29 29 LISREL SPECIFICATION: MODIFIED MODEL confirmatory factor model (attitude toward using coupons measured at two points in time) DA NI=9 NO=0 LA id ra fi=d:\m554\eden2\cfa.dat se / MO nx=8 nk=2 ph=st td=sy,fr fr lx 1 1 lx 2 1 lx 3 1 lx 4 1 lx 5 2 lx 6 2 lx 7 2 lx 8 2 pa td lk PD ou sc rs mi wp confirmatory factor model Number of Input Variables 9 Number of Y - Variables 0 Number of X - Variables 8 Number of ETA - Variables 0 Number of KSI - Variables 2 Number of Observations 250 confirmatory factor model Covariance Matrix aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Total Variance = Generalized Variance = Largest Eigenvalue = Smallest Eigenvalue = Condition Number = 5.602

30 confirmatory factor model 30 Parameter Specifications LAMBDA-X aat1 aat2 aa1t1 1 0 aa2t1 2 0 aa3t1 3 0 aa4t1 4 0 aa1t2 0 5 aa2t2 0 6 aa3t2 0 7 aa4t2 0 8 PHI aat1 aat2 aat1 0 aat2 9 0 THETA-DELTA aa1t1 10 aa2t aa3t aa4t aa1t aa2t aa3t aa4t

31 confirmatory factor model 31 Number of Iterations = 18 LISREL Estimates (Maximum Likelihood) LAMBDA-X aat1 aat2 aa1t (0.073) aa2t (0.069) aa3t (0.071) aa4t (0.074) aa1t (0.073) aa2t (0.067) aa3t (0.066) aa4t (0.072) PHI aat1 aat2 aat aat (0.020)

32 THETA-DELTA 32 aa1t (0.073) aa2t (0.060) aa3t (0.076) aa4t (0.069) aa1t (0.050) (0.064) aa2t (0.039) (0.051) aa3t (0.049) (0.058) aa4t (0.046) (0.059) Squared Multiple Correlations for X - Variables

33 Log-likelihood Values 33 Estimated Model Saturated Model Number of free parameters(t) ln(L) AIC (Akaike, 1974)* BIC (Schwarz, 1978)* *LISREL uses AIC= 2t - 2ln(L) and BIC = tln(n)- 2ln(L) Goodness of Fit Statistics Degrees of Freedom for (C1)-(C2) 15 Maximum Likelihood Ratio Chi-Square (C1) (P = ) Browne's (1984) ADF Chi-Square (C2_NT) (P = ) Estimated Non-centrality Parameter (NCP) Percent Confidence Interval for NCP (1.143 ; ) Minimum Fit Function Value Population Discrepancy Function Value (F0) Percent Confidence Interval for F0 ( ; 0.122) Root Mean Square Error of Approximation (RMSEA) Percent Confidence Interval for RMSEA ( ; ) P-Value for Test of Close Fit (RMSEA < 0.05) Expected Cross-Validation Index (ECVI) Percent Confidence Interval for ECVI (0.233 ; 0.350) ECVI for Saturated Model ECVI for Independence Model Chi-Square for Independence Model (28 df) Normed Fit Index (NFI) Non-Normed Fit Index (NNFI) Parsimony Normed Fit Index (PNFI) Comparative Fit Index (CFI) Incremental Fit Index (IFI) Relative Fit Index (RFI) Critical N (CN) Root Mean Square Residual (RMR) Standardized RMR Goodness of Fit Index (GFI) Adjusted Goodness of Fit Index (AGFI) Parsimony Goodness of Fit Index (PGFI) 0.406

34 34 Completely Standardized Solution LAMBDA-X aat1 aat2 aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t PHI aat1 aat2 aat aat THETA-DELTA aa1t aa2t aa3t aa4t aa1t aa2t aa3t aa4t Time used seconds

35 MEASUREMENT ANALYSIS FOR CFA MODEL 35 Construct Parameter Parameter estimate z-value of parameter estimate Individualitem reliability Composite reliability (average variance extracted) AAT1.88 (.66) l x l x l x l x q d q d q d q d AAT2.91 (.72) l x l x l x l x q d q d q d q d Assessment of discriminant validity: (1) test of whether 21 = 1: chi-square difference test: 2 (1) = confidence interval: = [.85;.93] Lagrange multiplier test: 2 (1) = (2) Fornell and Larcker criterion: not satisfied here