Structural Equation Modelling(SEM)

An Introduction (Part 3)

CFA Models: Important Steps

• Model Specification

• Model Identification

• Model Estimation

• Assessment of Model Fit

• Model Re-specification

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)

Step 1: Model Specification

• Exogenous variables

• Variables whose causes are unknown and/or not included in themodel

• Variables that explain other variables in the model (i.e. independentvariables (IVs))

• Endogenous variables

• Variables that serve as DVs in a model

• May also serve as IVs

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 is100, 1000, 10,000, etc.

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 covariancematrices.

• Process begins with initial estimates- start values.

• Is an iterative process – will stop when a minimum fitting criterion isreached.

When the difference between the observed and implied covariancematrices are minimized

Step 4: Assessing Model Fit

• Absolute fit

• Relative (Comparative) fit

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

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

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).

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

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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