Chapter TwentyTwo Structural Equation Modeling and Path Analysis

Chapter Twenty-Two Structural Equation Modeling and Path Analysis Copyright © 2010 Pearson Education, Inc. 22 -1

Chapter Outline 1) Objectives 2) Overview 3) Basic Concepts in SEM i. Theory, Model and Path Diagram ii. Exogenous versus Endogenous Constructs iii. Dependence and Correlational Relationships iv. Model Fit Model Identification 4) Statistics Associated with SEM Copyright © 2010 Pearson Education, Inc. 22 -2

Chapter Outline 5) 6) 7) i. 8) Conducting SEM Define the Individual Constructs Specify The Measurement Model Sample Size Requirements Assess Measurement Model Reliability and Validity i. Assess Measurement Model Fit a. b. c. d. e. chi-square (χ2) Absolute Fit Indices: Goodness-of-Fit Absolute Fit Indices: Badness-of-Fit Incremental Fit Indices Parsimony Fit Indices Copyright © 2010 Pearson Education, Inc. 22 -3

Chapter Outline ii. Measurement Model Reliability and Validity a. Reliability b. 9) 10) Discriminant Validity iii. Lack of Validity: Diagnosing Problems Specify the Structural Model Assess Structural Model Validity i. Assessing Fit ii. Comparison with Competing Models iii. Testing Hypothesized Relationships iv. Structural Model Diagnostics Copyright © 2010 Pearson Education, Inc. 22 -4

Chapter Outline 11) 12) 13) 14) 15) 16) 17) Draw Conclusions and Make Recommendations Higher-Order Confirmatory Factor Analysis Relationship of SEM to Other Multivariate Technique Application of SEM: First-Order Factor Model Application of SEM: Second-Order Factor Model Path Analysis Statistical Software Copyright © 2010 Pearson Education, Inc. 22 -5

Structural Equation Modeling (SEM) Structural equation modeling (SEM), a procedure for estimating a series of dependence relationships among a set of concepts or constructs represented by multiple measured variables and incorporated into an integrated model. Copyright © 2010 Pearson Education, Inc. 22 -6

Structural Equation Modeling: Distinctive Aspects 1. Representation of constructs as unobservable or latent factors in dependence relationships. 2. Estimation of multiple and interrelated dependence relationships incorporated in an integrated model. 3. Incorporation of measurement error in an explicit manner. SEM can explicitly account for less than perfect reliability of the observed variables, providing analyses of attenuation and estimation bias due to measurement error. 4. Explanation of the covariance among the observed variables. Copyright © 2010 Pearson Education, Inc. 22 -7

Statistics Associated with SEM Absolute fit indices These indices measure the overall goodness-of-fit or badness-of-fit for both the measurement and structural models. Average variance extracted A measure used to assess convergent and discriminant validity, which is defined as the variance in the indicators or observed variables that is explained by the latent construct. Chi-square difference statistic ( Δχ2) A statistic used to compare two competing, nested SEM models. It is calculated as the difference between the models’ chisquare value. Its degrees of freedom equal the difference in the models’ degrees of freedom. Communality is the variance of a measured variable that is explained by its construct. Copyright © 2010 Pearson Education, Inc. 22 -8

Statistics Associated with SEM Composite reliability (CR) It is defined as the total amount of true score variance in relation to the total score variance. Confirmatory factor analysis (CFA) A technique used to estimate the measurement model. It seeks to confirm if the number of factors (or constructs) and the loadings of observed (indicator) variables on them conform to what is expected on the basis of theory. Construct In SEM, a construct is a latent or unobservable concept that can be defined conceptually but that cannot be measured directly or without error. Also called a factor, a construct is measured by multiple indicators or observed variables. Copyright © 2010 Pearson Education, Inc. 22 -9

Statistics Associated with SEM Endogenous constructs An endogenous construct is the latent, multi-item equivalent of a dependent variable. It is determined by constructs or variables within the model and, thus, it is dependent on other constructs. Estimated covariance matrix Denoted by Σk , it consists of the predicted covariances between all observed variables based on equations estimated in SEM. Exogenous construct An exogenous construct is the latent, multi-item equivalent of an independent variable in traditional multivariate analysis. An exogenous construct is determined by factors outside of the model and it cannot be explained by any other construct or variable in the model. Copyright © 2010 Pearson Education, Inc. 22 -10

Statistics Associated with SEM First-order factor model Covariances between observed variables are explained with a single latent factor or construct layer. Incremental fit indices These measures assess how well a model specified by the researcher fits relative to some alternative baseline model. Typically, the baseline model is a null model in which all observed variables are unrelated to each other. Measurement error It is the degree to which the observed variables do not describe the latent constructs of interest in SEM. Copyright © 2010 Pearson Education, Inc. 22 -11

Statistics Associated with SEM Measurement model The first of two models estimated in SEM. It represents theory that specifies the observed variables for each construct and permits the assessment of construct validity. Modification index An index calculated for each possible relationship that is not freely estimated but is fixed. The index shows the improvement in the overall model χ2 if that path was freely estimated. Nested model A model is nested within another model if it has the same number of constructs and variables and can be derived from the other model by altering relationships, as by adding or deleting relationships. Copyright © 2010 Pearson Education, Inc. 22 -12

Statistics Associated with SEM Nonrecursive model A structural model that contains feedback loops or dual dependencies. Parsimony fit indices The parsimony fit indices are designed to assess fit in relation to model complexity and are useful in evaluating competing models. These are goodness-of-fit measures and can be improved by a better fit or by a simpler, less complex model that estimates fewer parameters. Parsimony ratio Is calculated as the ratio of degrees of freedom used by the model to the total degrees of freedom available. Copyright © 2010 Pearson Education, Inc. 22 -13

Statistics Associated with SEM Path analysis A special case of SEM with only single indicators for each of the variables in the causal model. In other words, path analysis is SEM with a structural model, but no measurement model. Path diagram A graphical representation of a model showing the complete set of relationships amongst the constructs. Dependence relationships are portrayed by straight arrows and correlational relationships by curved arrows. Residuals In SEM, the residuals are the differences between the observed and estimated covariance matrices. Recursive model A structural model that does not contain any feedback loops or dual dependencies. Copyright © 2010 Pearson Education, Inc. 22 -14

Statistics Associated with SEM Sample covariance matrix Denoted by S, it consists of the variances and covariances for the observed variables. Second-order factor model There are two levels or layers. A second-order latent construct causes multiple first-order latent constructs, which in turn cause the observed variables. Thus, the first-order constructs now act as indicators or observed variables for the second order factor. Squared multiple correlations Similar to communality, these values denote the extent to which an observed variable’s variance is explained by a latent construct or factor. Copyright © 2010 Pearson Education, Inc. 22 -15

Statistics Associated with SEM Standardized residuals Used as a diagnostic measure of model fit, these are residuals, each divided by its standard error. Structural error is the same as an error term in regression analysis. In the case of completely standardized estimates, squared multiple correlation is equal to 1 – the structural error. Structural model The second of two models estimated in SEM. It represents theory that specifies how the constructs are related to each other, often with multiple dependence relationships. Copyright © 2010 Pearson Education, Inc. 22 -16

Statistics Associated with SEM Structural relationship Dependence relationship between an endogenous construct and another exogenous or endogenous construct. Unidimensionality A notion that a set of observed variables represent only one underlying construct. All cross-loadings are zero. Copyright © 2010 Pearson Education, Inc. 22 -17

Exogenous and Endogenous Constructs Exogenous constructs are the latent, multi-item equivalent of independent variables. They use a variate (linear combination) of measures to represent the construct, which acts as an independent variable in the model. Multiple measured variables (X) represent the exogenous constructs (ξ). Endogenous constructs are the latent, multi-item equivalent to dependent variables. These constructs are theoretically determined by factors within the model. Multiple measured variables (Y) represent the endogenous constructs (η). Copyright © 2010 Pearson Education, Inc. 22 -18

SEM Models can be represented visually with a path diagram. Dependence relationships are represented with singleheaded straight arrows. Correlational (covariance) relationships are represented with two-headed curved arrows. Copyright © 2010 Pearson Education, Inc. 22 -19

Dependence and Correlational Relationships in SEM Fig. 22. 1 (a) Dependence Relationship Endogenous Construct: C 2 Exogenous Construct: C 1 X 2 X 3 Copyright © 2010 Pearson Education, Inc. Y 1 Y 2 Y 3 22 -20

Dependence and Correlational Relationships in SEM Fig. 22. 1 Cont. Exogenous Construct: C 1 X 2 Copyright © 2010 Pearson Education, Inc. Exogenous Construct: C 2 X 3 X 4 X 5 X 6 22 -21

Conducting SEM Steps in SEM Step Step 1: 2: 3: 4: 5: 6: Define the Individual Constructs Specify the Measurement Model Assess Measurement Model Reliability and Validity Specify the Structural Model Assess Structural Model Validity Draw Conclusions and Make Recommendations Copyright © 2010 Pearson Education, Inc. 22 -22

The Process for Structural Equation Modeling Define the Individual Constructs Fig. 22. 2 Develop and Specify the Measurement Model Assess Measurement Model Reliability and Validity Refine Measures and Design a New Study NO Measurement Model Valid? YES Specify the Structural Model Assess Structural Model Validity Refine Model and Test with New Data No Structural Model Valid? YES Draw Conclusions and Make Recommendations Copyright © 2010 Pearson Education, Inc. 22 -23

Path Diagram of a Simple Measurement Model Fig. 22. 3 Φ 21 C 2 ξ 2 C 1 ξ 1 λ X 4, 2 λх3, 1 λ x 1, 1 λ X 2, 1 λ x 6, 2 λ X 5, 2 X 1 X 2 X 3 X 4 X 5 X 6 δ 1 δ 2 δ 3 δ 4 δ 5 δ 6 Copyright © 2010 Pearson Education, Inc. 22 -24

A Simple Measurement Model In Figure 22. 3, ξ 1 represents the latent construct C 1 , ξ 2 represents the latent construct C 2, x 1 - x 6 represent the measured variables, λx 1, 1 - λχ6, 2 represent the relationships between the latent constructs and the respective measured items (i. e; factor loadings), and δ 1 - δ 6 represent the errors. Copyright © 2010 Pearson Education, Inc. 22 -25

Types of Fit Measures 1. Absolute fit measures overall goodness- or badness-of-fit for both the structural and measurement models. This type of measure does not make any comparison to a specified null model (incremental fit measure) or adjusts for the number of parameters in the estimated model (parsimonious fit measure). 2. Incremental fit measures goodness-of-fit that compares the current model to a specified “null” (independence) model to determine the degree of improvement over the null model. 3. Parsimonious fit measures goodness-of-fit representing the degree of model fit per estimated coefficient. This measure attempts to correct for any “overfitting” of the model and evaluates the parsimony of the model compared to the goodness-of-fit. Copyright © 2010 Pearson Education, Inc. 22 -26

A Classification of Fit Measures Fig. 22. 4 Absolute Fit Indices Goodness-of-Fit Badness-of-Fit Incremental Fit Indices Goodness-of-Fit Parsimony Fit Indices Goodness-of-Fit • GFI • x 2 • NFI § PGFI • AGFI • RMSR • NNFI § PNFI • SRMR • CFI • RMSEA • TLI • RNI Copyright © 2010 Pearson Education, Inc. 22 -27

Assessing Model Fit: Chi-square Copyright © 2010 Pearson Education, Inc. 22 -28

Assessing Model Fit: Fit Indexes Multiple fit indices should be used to assess a model’s goodness of fit. They should include: • The χ2 value and the associated df • Two absolute fit indices (GFI, AGFI, RMSEA, or SRMR) • • • One goodness-of-fit index (GFI, AGFI) • One badness-of-fit index (RMSR, SRMR, RMSEA) One incremental fit index (CFI, TLI, NFI, NNFI, RNI) One parsimony fit index for models of different complexities (PGFI, PNFI) Copyright © 2010 Pearson Education, Inc. 22 -29

Composite Reliability (CR) CR = p (∑ λi )2 i=1 p p (∑λi)2 + (∑δi) i=1 Where CR = Composite reliability λ = completely standardized factor loading δ = Error variance p = number of indicators or observed variables Copyright © 2010 Pearson Education, Inc. 22 -30

Average Variance Extracted (AVE) p 2 Σ λί i=1 AVE = p 2 Σ λί + i=1 p Σ δi i=1 Where AVE = Average variance extracted λ = Completely Standardized factor loading δ = Error variance p = number of indicators or observed variables Copyright © 2010 Pearson Education, Inc. 22 -31

Path Diagram of a Simple Structural Model Fig. 22. 5 γ 1, 1 Construct 1: C 1 ξ 1 λx 1, 1 λx 3, 1 λx 2, 1 X 1 δ 1 X 2 δ 2 Copyright © 2010 Pearson Education, Inc. X 3 δ 3 Construct 2: C 2 η 1 λy 3, 1 λy 1, 1 λy 2, 1 Y 2 ε 1 ε 2 Y 3 ε 3 22 -32

Chi-Square Difference Test Δχ2Δdf = χ2 df(M 1) - χ2 df(M 2) and Δdf Copyright © 2010 Pearson Education, Inc. = df(M 1) –df(M 2) 22 -33

First-Order Model of IUIPC Fig. 22. 6 Ф 3, 1 Φ 2, 1 COL X 1 X 2 X 3 X 4 Ф 3, 2 CON X 5 Copyright © 2010 Pearson Education, Inc. X 6 AWA X 7 X 8 X 9 X 10 22 -34

Second-Order Model of IUIPC Fig. 22. 7 X 1, 1 Layer 1 X 2 X 3, 1 X 2, 1 COL X 1 Layer 2 IUIPC CON X 3 X 4 X 5 X 6 AWA X 7 X 8 X 9 X 10 Legend: First-Order Factors: COL = Collection CON = Control AWA = Awareness Second-Order Factor : IUIPC = Internet Users Information Privacy Concerns Copyright © 2010 Pearson Education, Inc. 22 -35

Measurement Model for TAM Fig. 22. 8 Ф 3, 1 PU PU 1 PU 2 PU 3 Φ 2, 1 Ф 3, 2 PE PE 1 Copyright © 2010 Pearson Education, Inc. PE 2 INT PE 3 INT 1 INT 2 INT 3 22 -36

Structural Model for TAM Fig. 22. 9 Perceived Usefulness PU 0. 60 0. 46 Intention to Use INT 0. 40 Perceived Ease Of Use PE Copyright © 2010 Pearson Education, Inc. 0. 28 22 -37

TAM: Means, Standard Deviations, and Correlations Table 22. 1 Copyright © 2010 Pearson Education, Inc. 22 -38

TAM: Results of Measurement Model Table 22. 2 Copyright © 2010 Pearson Education, Inc. 22 -39

Banking Application: Measurement Model Fig. 22. 10 TANG ξ 1 x 2 δ 1 δ 2 x 3 REL ξ 2 x 4 δ 3 δ 4 X 5 δ 5 X 6 δ 6 X 7 δ 7 RESP ξ 3 X 8 δ 8 X 9 δ 9 x 10 δ 10 X 11 δ 11 Copyright © 2010 Pearson Education, Inc. ASS 4 ξ 4 x 12 δ 12 x 13 x 14 X 15 δ 13 δ 14 δ 15 EMP ξ 5 X 16 X 17 x 18 x 19 δ 16 δ 17 δ 18 δ 19 ATT ξ 6 x 20 x 21 x 22 SAT ξ 7 x 23 x 24 x 25 PAT ξ 8 x 26 x 27 x 28 X 2 9 δ 20 δ 21 δ 22 δ 23 δ 24 δ 25 δ 26 δ 27 δ 28 δ 29 δ 30 22 -40 x 30

Banking Application: Structure Model Fig. 22. 11 TANG Service Attitude REL RESP Service Quality Patronage Intention ASSU Service Satisfaction EMP TANG = tangibility; REL= reliability; RESP = responsiveness; ASSU = assurance; EMP = empathy Copyright © 2010 Pearson Education, Inc. 22 -41

Psychometric Properties of Measurement Model Table 22. 3 When it comes to…. My Perception of My Bank’s Service. Loadings TANG 1: Modern equipments TANG 2: Visual appeal of physical facilities TANG 3: Neat, professional appearance of employees TANG 4: Visual appeal of materials associated with the service REL 1: Keeping a promise by a certain time REL 2: Performing service right the first time REL 3: Providing service at the promised time REL 4: Telling customers the exact time the service will be performed Copyright © 2010 Pearson Education, Inc. 0. 71 0. 80 0. 76 0. 72 0. 79 0. 83 0. 91 0. 81 Measurement Error 0. 49 0. 36 0. 42 0. 48 0. 37 0. 31 0. 18 0. 34 22 -42

Psychometric Properties of Measurement Model Table 22. 3 Cont. When it comes to…. My Perception of My Bank’s Service. Loadings RESP 1: Keeping a promise by a certain time RESP 2: Willingness to always help customers RESP 3: Responding to customer requests despite being busy ASSU 1: Employees instilling confidence in customers ASSU 2: Customers’ safety feelings in transactions (e. g. physical, financial, emotional, etc. ) ASSU 3: Consistent courtesy to customers ASSU 4: Employees’ knowledge to answer customer questions Copyright © 2010 Pearson Education, Inc. Measurement Error 0. 73 0. 89 0. 81 0. 71 0. 47 0. 21 0. 35 0. 49 0. 80 0. 86 0. 36 0. 26 22 -43

Psychometric Properties of Measurement Model Table 22. 3 Cont. When it comes to…. My Perception of My Bank’s Service. Loadings EMP 1: Giving customers individual attention EMP 2: Dealing with customers with care EMP 3: Having customers’ best interests at heart EMP 4: Understanding specific needs of customers Overall attitude toward your bank (items reverse coded): ATT 1: Favorable 1 ---2 ---3 ---4 ---5 ---6 ---7 Unfavorable ATT 2: Good 1 ---2 ---3 ---4 ---5 ---6 ---7 Bad ATT 3: Positive 1 ---2 ---3 ---4 ---5 ---6 ---7 Negative ATT 4: Pleasant 1 ---2 ---3 ---4 ---5 ---6 ---7 Unpleasant SAT 1: I believe I am satisfied with my bank’s services SAT 2: Overall, I am pleased with my bank’s services Copyright © 2010 Pearson Education, Inc. Measurement Error 0. 80 0. 84 0. 87 0. 37 0. 29 0. 24 0. 95 0. 93 0. 10 0. 14 22 -44

Psychometric Properties of Measurement Model Table 22. 3 Cont. When it comes to…. My Perception of My Bank’s Service. Loadings SAT 3: Using services from my bank is usually a satisfying experience SAT 4: My feelings toward my bank’s services can best be characterized as PAT 1: The next time my friend needs the services of a bank I will recommend my bank PAT 2: I have no regrets of having patronized my bank in the past PAT 3: I will continue to patronize the services of my bank in the future Copyright © 2010 Pearson Education, Inc. Measurement Error 0. 88 0. 23 0. 92 0. 15 0. 88 0. 22 0. 89 0. 88 0. 20 0. 22 22 -45

Goodness of Fit Statistics Measurement Model Table 22. 4 -----------------------Degrees of Freedom = 377 Minimum Fit Function Chi-Square = 767. 77 (P = 0. 0) Chi-Square for Independence Model with 435 Degrees of Freedom = 7780. 15 Root Mean Square Error of Approximation (RMSEA) = 0. 064 Standardized RMR = 0. 041 Normed Fit Index (NFI) = 0. 90 Non-Normed Fit Index (NNFI) = 0. 94 Comparative Fit Index (CFI) = 0. 95 -----------------------Copyright © 2010 Pearson Education, Inc. 22 -46

Measurement Model: Construct Reliability, Average Variance Extracted & Correlation Matrix Table 22. 5 Construct Reliability 1. TANG Average Variance Extracted Correlation Matrix 1 2 3 4 0. 56 0. 75 5 6 7 8 0. 84 2. REL 0. 90 0. 77 0. 84 3. RESP 0. 85 0. 66 0. 65 0. 76 0. 81 4. ASSU 0. 87 0. 63 0. 73 0. 80 0. 92 0. 80 5. EMP 0. 91 0. 71 0. 69 0. 75 0. 85 0. 90 0. 85 6. ATT 0. 97 0. 90 0. 42 0. 46 0. 52 0. 54 0. 58 0. 95 7. SAT 0. 85 0. 83 0. 56 0. 67 0. 69 0. 72 0. 91 8. PAT 0. 92 0. 78 0. 50 0. 55 0. 57 0. 62 0. 66 0. 89 TANG = tangibles; REL = reliability; RESP = responsiveness; ASSU = assurance; EMP = empathy; ATT = attitude; SAT = satisfaction; PAT = patronage. Value on the diagonal of the correlation matrix is the square root of AVE. Copyright © 2010 Pearson Education, Inc. 22 -47

Goodness of Fit Statistics (Structural Model) Table 22. 6 ----------------------------------------Degrees of Freedom = 396 Minimum Fit Function Chi-Square = 817. 16 (P = 0. 0) Chi-Square for Independence Model with 435 Degrees of Freedom = 7780. 15 Root Mean Square Error of Approximation (RMSEA) = 0. 065 Standardized RMR = 0. 096 Normed Fit Index (NFI) = 0. 89 Non-Normed Fit Index (NNFI) = 0. 94 Comparative Fit Index (CFI) = 0. 94 -------------------------------------------------------Copyright © 2010 Pearson Education, Inc. 22 -48

Banking Application: Structural Coefficients Table 22. 7 Second Order Loading Estimates Dimensions of Service Quality T-values TANG g 11 0. 82 l fixed to 1 REL g 21 0. 85 13. 15 RESP g 31 0. 93 13. 37 ASSU g 41 0. 98 16. 45 EMP g 51 0. 93 15. 18 Copyright © 2010 Pearson Education, Inc. 22 -49

Banking Application: Structural Coefficients Table 22. 7 Cont. Structural Coefficient Estimates Consequences of Service Quality T-Values SQ ATT g 61 0. 60 10. 25 SQ SAT g 71 0. 45 8. 25 ATT SAT b 76 0. 47 8. 91 ATT PAT b 86 0. 03 0. 48 SAT PAT b 87 0. 88 13. 75 Copyright © 2010 Pearson Education, Inc. 22 -50

Path Analysis Path analysis A special case of SEM with only single indicators for each of the variables in the causal model. In other words, path analysis is SEM with a structural model, but no measurement model. Copyright © 2010 Pearson Education, Inc. 22 -51

Diagram for Path Analysis Fig. 22. 12 X 1 Attitude B Y 1 Purchase Intention A X 2 Emotion Copyright © 2010 Pearson Education, Inc. C 22 -52

Path Analysis: Calculating Structural Coefficients Figure 22. 12 portrays a simple model with two exogenous constructs X 1 and X 2 causally related to the endogenous construct Y 1. The correlational path A is X 1 correlated with X 2. Path B is the effect of X 1 predicting Y 1, and path C shows the effect of X 2 predicting Y 1. The value for Y 1 can be modeled as: Y 1 = b 1 X 1 + b 2 X 2 Note that this is similar to a regression equation. The direct and indirect paths in our model can now be identified. Direct Paths A=X 1 to X 2 B=X 1 to Y 1 C=X 2 to Y 1 Copyright © 2010 Pearson Education, Inc. Indirect Paths AC = X 1 to Y 1 (Via X 2) AB = X 2 to Y 1 (Via X 1) 22 -53

Path Analysis: Calculating Structural Coefficients The unique correlations among the three constructs can be shown to be composed of direct and indirect paths as follows: Corrx 1, x 2 = A Corrx 1, y 1 = B + AC Corrx 2, y 1 = C + AB The correlation of X 1 and X 2 is simply equal to A. The Correlation of X 1 and Y 1 ( Corrx 1, y 1 ) can be represented by two paths: B and AC. B represents the direct path from X 1 to Y 1. AC is a compound path that follows the curved arrow from X 1 to X 2 and then to Y 1. Similarly, the correlation of X 2 and Y 1 can be shown to consist of two causal paths: C and AB. Copyright © 2010 Pearson Education, Inc. 22 -54

Path Analysis Bivariate Correlations X 1 X 2 Y 1 1. 0 X 2 . 40 Y 1 . 50 Copyright © 2010 Pearson Education, Inc. 1. 0. 60 1. 0 22 -55

Path Analysis Correlations as Compound Paths Corr X 1, X 2 =A Corr X 1 , Y 1 = B+AC Corr X 2 , Y 1 = C+AB Copyright © 2010 Pearson Education, Inc. 22 -56

Path Analysis: Calculating Structural Coefficients. 40 = A. 50 = B+AC. 60 = C+AB Substituting A =. 40. 50 = B+. 40 C. 60 = C+. 40 B Solving for B and C B =. 310 C =. 476 Copyright © 2010 Pearson Education, Inc. 22 -57

Statistical Software • LISREL, LInear Structural RELations • AMOS, Analysis of Moment Structures, is an added module to SPSS • CALIS is offered by SAS • EQS, an abbreviation for equations • Mplus is another software Selection of a specific computer program should be based on availability and user’s preference. Copyright © 2010 Pearson Education, Inc. 22 -58

Copyright © 2010 Pearson Education, Inc. 22 -59

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright © 2010 Pearson Education, Inc. 22 -60