1. SEM: Confirmatory Factor Analysis

2. LEARNING OBJECTIVES Upon completing this chapter, you should be able to do the following:  Distinguish between exploratory factor analysis and confirmatory factor analysis.  Assess the construct validity of a measurement model.  Know how to represent a measurement model using a path diagram. SEM – Confirmatory Factor Analysis

3. LEARNING OBJECTIVES continued... Upon completing this chapter, you should be able to do the following:  Understand the basic principles of statistical identification and know some of the primary causes of SEM identification problems.  Understand the concept of model fit as it applies to measurement models and be able to assess the fit of a confirmatory factor analysis model. SEM – Confirmatory Factor Analysis

4. What is it? What is it? Why use it? Why use it? Confirmatory Factor Analysis Overview

5. Confirmatory Factor Analysis... is similar to EFA in some respects, but philosophically it is quite different. With CFA, the researcher must specify both the number of factors that exist within a set of variables and which factor each variable will load highly on before results can be computed. So the technique does not assign variables to factors. Instead the researcher must be able to make this assignment before any results can be obtained. SEM is then applied to test the extent to which a researcher’s a- priori pattern of factor loadings represents the actual data. Confirmatory Factor Analysis Defined

6. Review of and Contrast with Exploratory Factor Analysis EFA (exploratory factor analysis) explores the data and provides the researcher with information about how many factors are needed to best represent the data. With EFA, all measured variables are related to every factor by a factor loading estimate. Simple structure results when each measured variable loads highly on only one factor and has smaller loadings on other factors (i.e., loadings <.40). The distinctive feature of EFA is that the factors are derived from statistical results, not from theory, and so they can only be named after the factor analysis is performed. EFA can be conducted without knowing how many factors really exist or which variables belong with which constructs. In this respect, CFA and EFA are not the same.

8. CFA and Construct Validity One of the biggest advantages of CFA/SEM is its ability to assess the construct validity of a proposed measurement theory. Construct validity... is the extent to which a set of measured items actually reflect the theoretical latent construct they are designed to measure. Construct validity is made up of four important components: 1.Convergent validity – three approaches: oFactor loadings. oVariance extracted. oReliability. 2.Discriminant validity. 3.Nomological validity. 4.Face validity.

9. Rules of Thumb 13–1 Construct Validity: Convergent and Discriminant Validity Standardized loading estimates should be.5 or higher, and ideally.7 or higher. AVE should be.5 or greater to suggest adequate convergent validity. AVE estimates for two factors also should be greater than the square of the correlation between the two factors to provide evidence of discriminant validity. Construct reliability should be.7 or higher to indicate adequate convergence or internal consistency.

10. Confirmatory Factor Analysis Stages Stage 1: Defining Individual Constructs Stage 2: Developing the Overall Measurement Model Stage 3: Designing a Study to Produce Empirical Results Stage 4: Assessing the Measurement Model Validity Stage 5: Specifying the Structural Model Stage 6: Assessing Structural Model Validity Note: CFA involves stages 1 – 4 above. SEM is stages 5 and 6.

11. Stage 1: Defining Individual Constructs List constructs that will comprise the measurement model. List constructs that will comprise the measurement model. Determine if existing scales/constructs are available or can be modified to test your measurement model. Determine if existing scales/constructs are available or can be modified to test your measurement model. If existing scales/constructs are not available, then develop new scales. If existing scales/constructs are not available, then develop new scales.

12. Rules of Thumb 13–2 Defining Individual Constructs All constructs must display adequate construct validity, whether they are new scales or scales taken from previous research. Even previously established scales should be carefully checked for content validity. Content validity should be of primary importance and judged both qualitatively (e.g., expert’s opinions) and empirically (e.g., unidimensionality and convergent validity). A pre-test should be used to purify measures prior to confirmatory testing.

13. Stage 2: Developing the Overall Measurement Model Key Issues... Unidimensionality – no cross loadings Unidimensionality – no cross loadings Congeneric measurement model – no covariance between or within construct error variances Congeneric measurement model – no covariance between or within construct error variances Items per construct – identification Items per construct – identification Reflective vs. formative measurement models Reflective vs. formative measurement models

15. Stage 2: A Congeneric Measurement Model Compensation X1X1 X2X2 X3X3 X4X4 e1e1 e2e2 e3e3 e4e4 L x1 L x 2 Lx 3Lx 3 L x 4 Teamwork X5X5 X6X6 X7X7 X8X8 e5e5 e6e6 e7e7 e8e8 L x 5 L 6 Lx 7Lx 7 L x 8 Each measured variable is related to exactly one construct.

16. X5X5 X6X6 X7X7 X8X8 δ5δ5 δ6δ6 δ7δ7 δ8δ8 λ x5,2 λ x6,2 λ x7,2 λ x8,2 X1X1 X2X2 X3X3 X4X4 λ x1,1 λ x2,1 λ x3,1 λ x4,1 λ x3,2 λ x5,1 δ1δ1 δ2δ2 δ3δ3 δ4δ4 Ф 21 θ δ 2,1 θ δ 7,4 Figure 11.2 A Measurement Model with Hypothesized Cross- Loadings and Correlated Error Variance Each measured variable is not related to exactly one construct – errors are not independent. Stage 2: A Measurement Model that is Not Congeneric Compensation Teamwork

18. Stage 2: A 4-Item Construct is Over-Identified Measured Items X 1 =Cheerful X 2 =Stimulated X 3 =Lively X 4 =Bright Loading Estimates λ x 1,1 =0.78 λ x 2,1 =0.89 λ x 3,1 =0.83 λ x 4,1 =0.87 Error Variance Estimates θ δ 1,1 =0.39 θ δ 2,2 =0.21 θ δ 3,3 =0.31 θ δ 4,4 =0.24 Eight paths to estimate 10 unique variance-covariance terms ξ1ξ1 X1X1 X2X2 X3X3 X4X4 δ1δ1 δ2δ2 δ3δ3 δ4δ4 λ x 1,1 λ x 2,1 λ x 3,1 λ x 4,1 θ δ 1,1 θ δ 2,2 θ δ 3,3 θ δ 4,4 Symmetric Covariance Matrix: X1 X2 X3 X4 ------------------------- X1 2.01 X2 1.43 2.01 X3 1.31 1.56 2.24 X4 1.36 1.54 1.57 2.00 Model Fit: χ 2 = 14.9 df = 2 p =.001 CFI =.99

19. Rules of Thumb 13–3 Developing the Overall Measurement Model In standard CFA applications testing a measurement theory, within and between error covariance terms should be fixed at zero and not estimated. In standard CFA applications testing a measurement theory, all measured variables should be free to load only on one construct. Latent constructs should be indicated by at least three measured variables, preferably four or more. In other words, latent factors should be statistically identified. Formative factors are not latent and are not validated as are conventional reflective factors. As such, they present greater difficulties with statistical identification and should be used cautiously.

20. Formative Constructs Formative factors are not latent and are not validated as are conventional reflective factors. Internal consistency and reliability are not important. The variables that make up a formative factor should explain the largest portion of variation in the formative construct itself and should relate highly to other constructs that are conceptually related (minimum correlation of.5): oFormative factors present greater difficulties with statistical identification. oAdditional variables or constructs must be included along with a formative construct in order to achieve an over-identified model. oA formative factor should be represented by the entire population of items that form it. Therefore, items should not be dropped because of a low loading. oWith reflective models, any item that is not expected to correlate highly with the other indicators of a factor should be deleted.

22. Key Issues... Measurement scales in CFA SEM/CFA and sampling Specifying the model: oWhich indicators belong to each construct? oSetting the scale to “1” for one indicator on each construct Issues in identification Problems in estimation... oHeywood cases. oIllogical standardized parameters. Stage 3: Designing a Study to Produce Empirical Results

23. Rules of Thumb 13–4 Designing a Study to Provide Empirical Results The ‘scale’ of a latent construct can be set by either: oFixing one loading and setting its value to 1, or oFixing the construct variance and setting its value to 1. Congeneric, reflective measurement models in which all constructs have at least three item indicators are statistically identified in models with two or more constructs. The researcher should check for errors in the specification of the measurement model when identification problems are indicated. Models with large samples (more than 300) that adhere to the three indicator rule generally do not produce Heywood cases.

24. Order Condition – the net degrees of freedom for a model are greater than zero. Order Condition – the net degrees of freedom for a model are greater than zero. Rank Condition – each parameter estimated is uniquely, algebraically defined. Rank Condition – each parameter estimated is uniquely, algebraically defined. Identification Recognizing Identification Problems: 1.Very large standard errors 2.Inability to invert the information matrix (no solution can be found) 3.Wildly unreasonable estimates including negative error variances 4.Unstable parameter estimates

25. Stage 4: Assessing Measurement Model Validity Key Issues... Assessing fit – GOF and path estimates (significance and size) Assessing fit – GOF and path estimates (significance and size) Construct validity Construct validity Diagnosing problems Diagnosing problems oStandardized residuals oModification indices oSpecification searches

26. Rules of Thumb 13–5 Assessing Measurement Model Validity Loading estimates can be statistically significant but still be too low to qualify as a good item (standardized loadings below |.5|). In CFA, items with low loadings become candidates for deletion. Completely standardized loadings above +1.0 or below -1.0 are out of the feasible range and can be an important indicator of some problem with the data. Typically, standardized residuals less than |2.5| do not suggest a problem. o Standardized residuals greater than |4.0| suggest a potentially unacceptable degree of error that may call for the deletion of an offending item. o Standardized residuals between |2.5| and |4.0| deserve some attention, but may not suggest any changes to the model if no other problems are associated with those items.

27. Rules of Thumb 13–5 continued... Assessing Measurement Model Validity The researcher should use the modification indices only as a guideline for model improvements of those relationships that can theoretically be justified. Specification searches based on purely empirical grounds are discouraged because they are inconsistent with the theoretical basis of CFA and SEM. CFA results suggesting more than minor modification should be re-evaluated with a new data set (e.g., if more than 20% of the measured variables are deleted, then the modifications can not be considered minor).

28. HBAT CFA/SEM Case Study HBAT employs thousands of workers in different operations around the world. Like many firms, one of their biggest management problems is attracting and keeping productive employees. The cost to replace and retrain employees is high. Yet the average new person hired works for HBAT less than three years. In most jobs, the first year is not productive, meaning the employee is not contributing as much as the costs associated with employing him/her. After the first year, most employees become productive. HBAT management would like to understand the factors that contribute to employee retention. A better understanding can be obtained if the key constructs are measured accurately. Thus, HBAT is interested in developing and testing a measurement model made up of constructs that impact employees’ attitudes and opinions about remaining with HBAT. HBAT initiated a research project to study the employee retention/turnover problem. Preliminary research discovered that a large number of employees are exploring job options with the intention of leaving HBAT should an acceptable offer be obtained from another firm. Based on published literature and some preliminary interviews with employees, an employee retention/turnover study was designed focusing on five key constructs. The five constructs are defined as:  Job Satisfaction (JS) – reactions / beliefs about one’s job situation.  Organizational Commitment (OC) – the extent to which an employee identifies and feels part of HBAT.  Staying Intentions (SI) – the extent to which an employee intends to continue working for HBAT and is not participating in activities that make quitting more likely.  Environmental Perceptions (EP) – beliefs an employee has about their day-to-day, physical working conditions.  Employee Attitudes toward Coworkers (AC) – attitudes an employee has toward the coworkers he/she interacts with on a regular basis.

29. Attitudes toward Coworkers JS4 JS3 JS5 JS2 JS1 OC1 OC2OC3 OC4 AC3AC2AC4AC1 SI2 SI3 SI1 SI4 EP2EP1EP3 Note: Measured variables are shown as a box with labels corresponding to those shown in the HBAT questionnaire. Latent constructs are an oval. Each measured variable has an error term, but the error terms are not shown. Two headed connections indicate covariance between constructs. One headed connectors indicate a causal path from a construct to an indicator (measured) variable. In CFA all connectors between constructs are two-headed covariances / correlations. EP4 Organizational Commitment Staying Intentions Job Satisfaction Environmental Perceptions Measurement Theory Model for HBAT 5 Construct CFA

30. Theoretically-Based HBAT Employee Retention SEM Model JS OC SI EP AC Hypotheses: H1: EP +  JS H2: EP +  OC H3: AC +  JS H4: AC +  OC H5: JS +  OC H6: JS +  SI H7: OC +  SI Note: observable indicator variables are not shown to simplify the model.

32. Copyright © 2010 Pearson Education, Inc., publishing as Prentice-Hall. 13-32 CFA Learning Checkpoint 1.What is the difference between EFA and CFA? 2.Describe the four stages of CFA. 3.What is the difference between reflective and formative measurement models? 4.What is “statistical identification” and how can it be avoided? 5.How do you decide if CFA is successful?