This document provides an overview of the important steps in conducting a Confirmatory Factor Analysis (CFA) using Structural Equation Modeling (SEM). It outlines the 5 key steps as model specification, identification, estimation, assessment of model fit, and potential model re-specification. For model specification, the document emphasizes that SEM is a confirmatory technique that requires a theoretically-driven model delineating relationships among variables. It also defines exogenous and endogenous variables. For model identification, it notes that a unique parameter solution is required. The document discusses maximum likelihood estimation and several absolute and relative fit indices for assessing model fit. It concludes by noting guidelines for potential model modification.

1. Structural Equation Modelling
(SEM)
An Introduction (Part 3)

2. CFA Models: Important Steps
• Model Specification
• Model Identification
• Model Estimation
• Assessment of Model Fit
• Model Re-specification

3. Step 1: Model Specification
• SEM is a confirmatory technique and it
Needs a model that delineates the relationships among variables
Requires a model that is based on theory (Bollen & Long, 1993)

4. Step 1: Model Specification
• Exogenous variables
• Variables whose causes are unknown and/or not included in the
model
• Variables that explain other variables in the model (i.e. independent
variables (IVs))
• Endogenous variables
• Variables that serve as DVs in a model
• May also serve as IVs

5. Step 2: Model Identification
• Model must be specified so that there are enough pieces of information to give unique
estimates for all parameters
• SEM involves estimating unknown parameters (e.g., factor loadings, path coefficients)
based on known parameters (i.e., covariances)
• Identification involves whether a unique solution for a model can be obtained
• Requires more known vs. unknown parameters
• Identification is a property of the model, not the data
 Does not depend on sample size
 i.e., if a model is not identified, it remains so regardless of whether the sample size is
100, 1000, 10,000, etc.

6. Step 3: Model Estimation
• Over-identified models have infinite # of solutions.
• Parameters need to be estimated based on a mathematical criterion.
• Goal is to minimize differences between the observed and implied covariance
matrices.
• Process begins with initial estimates- start values.
• Is an iterative process – will stop when a minimum fitting criterion is
reached.
 When the difference between the observed and implied covariance
matrices are minimized

7. Step 4: Assessing Model Fit
• Absolute fit
• Relative (Comparative) fit

8. Common Absolute Fit Indices
• Model X2*
• Non-significant X2 (p>0.05) indicates good fit
• Root Mean Squared Error of Approximation (RMSEA)
• Acceptable fit < 0.10; good fit < 0.05
• Goodness of Fit (GFI)
• > 0.90 is considered good fit

9. Common Relative Fit Indices
• Normed Fit Index (NFI)
• Incremental Fit Index (IFI)
• Comparative Fit Index (CFI)
• All range 0-1
• Generally, >0.90 is considered good

10. SEM Model Fit: Rules of Thumb
• Will often see/hear reference to 0.90 or above indicating acceptable model
fit, for indices such as GFI, CFI, NFI, etc.
 Typically cite Bentler & Bonett (1980) for this assertation
• Basis for this is rather thin (Lance et al., 2006)
• What Bentler and Bonett (1980) actually said:
 “experience will be required to establish values of the indices that are
associated with various degrees of meaningfulness of results. In our
experience, models with overall fit indices of less than 0.90 can usually
be improved substantially” (Bentler & Bonett, 1980, p. 600).

11. Step 5: Model Re-specification/Modification
• Goal is to improve model fit – changing the model to fit the data
• Caveats
 Modifications are post hoc & capitalize on chance!
• General guidelines
 Must be theoretically consistent
 Must be replicated with new data

12. Evaluating Your Model
• Theoretical/clinical meaning
 Guiding principle
• Residuals and implied correlations
 Discrepancies between sample covariance matrix and those implied
by the model
• Tests of path coefficients
 Direction, magnitude
• Absence of numerical problems
 Direction and magnitude of residuals
 Pattern of standardized residuals (z-scores)

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