Once the constructs have been defined and their observed or indicator variables measured, we are in a position to specify the measurement model. This involves the assignment of the relevant measured variables to each latent construct. The measurement model is usually represented by a
diagram, as indicated in Figure 22.3. Figure 22.3 represents a simple measurement model having two correlated constructs with each construct being represented by three indicator or measured variables. The assignment of measured variables to each latent construct is graphically equivalent to drawing arrows from each construct to the measured variables that represent that construct.
The degree to which each measured variable is related to its construct is represented by that variable’s loading, also shown in Figure 22.3. Only the loading linking each measured variable to its latent construct as specified by the arrows are estimated; all other loadings are set to zero. Also, since a latent factor does not explain a measured variable perfectly, an error term is added. In a measurement model, we do not distinguish between exogenous and endogenous constructs; they are all treated as being of the same type, similar to that in factor analysis
In a measurement model, it is common to represent constructs by Greek characters and measured variables by alphabets. The common notations used are
Assess Measurement Model Reliability and Validity
The validity of the measurement model depends on the goodness-of-fit results, reliability, and evidence of construct validity, especially convergent and discriminant validity.
Assess Measurement Model Fit
As stated earlier, goodness of fit means how well the specified model reproduces the covariance matrix among the indicator items. Thai is, how similar is the estimated covariance of the indicator variables (rk) to the observed covariance in the sample data (S). The closer the values of the two matrices are to each other, the better the model is said to fit. As shown in Figure 22.4, the various measures designed to assess fit consist of absolute fit, incremental fit, and parsimony fit indices.
In absolute fit indices, each model is evaluated independently of other possible models. These indices directly measure how well the specified model reproduces the observed or sample data. Absolute fit indices may measure either goodness of fit or badness of fit. Goodness-of-fit indices indicate how well the specified model fits the observed or sample data, and so higher values of these measures are desirable. Measures that are commonly used are the goodness-of-fit index (GFI) and the adjusted goodness-of-fit index (AGA). On the other hand, badness-of-fit indices measure error or deviation in some form and so lower values on these indices are desirable. The commonly used
In contrast to the absolute fit indices, the incremental fit indices evaluate how well the specified model fits the sample data relative to some alternative model that is treated as a baseline model. The baseline model that is commonly used is the null model that is based on the assumption that the observed variables are uncorrelated. These are goodness-of-fit measures, and the commonly used incremental fit indices include the normed fit index (NFl), non-normcd fit index (NNFI) , comparative fit index (CFI) , the Tucker Lewis Index (TLI). and the relative non centrality index (RNI).
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. These indices are based on the parsimony ratio that is calculated as the ratio of degrees of freedom used by the model to the total degrees of freedom available. The commonly used parsimony fit indices are the parsimony goodness-of-fit index (PGFI) and the parsimony normed fit Index (PNFI). We discuss these indices briefly and provide guidelines for their use. Given its foundational nature, chi-square (Xl) is discussed first, followed by other indices
OTHER ABSOLUTE FIT INDICES: BADNESS OF FIT The notion of a residual was discussed earlier. The root mean square residual (RMSR) is the square root of the mean of these squared residuals. Thus, RMSR is an average residual covariance that is a function of the units used to measure the observed variables. Therefore, it is problematic to compare RMSR across models unless standardization is done. Standardized root mean residual (SRMR) is the standardized value of the root mean square residual and helps in comparing fit across models. Like RMSR, lower values of SRMR indicate better model fit, and values of 0.08 or less are desirable
PARSIMONY FIT INDICES It should be emphasized that parsimony fit indices are not appropriate for evaluating the fit of a single model but are useful in comparing models of different complexities. The parsimony goodness-of-fit index (PGFI) adjusts the goodness-of-fit index by using the parsimony ratio that was defined earlier. The values of PGFI range between °and I. A model with a higher PGFI is preferred based on fit and complexity. The parsimony normed fit index (PNFI) adjusts the normed fit index (Nfl) by multiplication with the parsimony ratio. Like PGFI, higher values of PNFI also indicate better models in terms of fit and parsimony. Both PGFI and PNFI should be used only in a relative sense, i.e., in comparing models. PNFI is used to a greater extent as compared to PGFI . Of the measures we have considered, CFI and RMSEA are among the measures least affected by sample size and quite popular in use. It is highly desirable that we use multiple (at least three) indices of different types. It is a good practice to always report the X2 value with the associated degrees of freedom. In addition, use at least one absolute goodness-of-fit, one absolute badness-of-fit, and one incremental fit measure. If models of different complexities are being compared, one parsimony fit index should also be considered.
Lack of Validity: Diagnosing Problems
If the validity of the proposed measurement model is not satisfactory, then you can malee use of the diagnostic information provided by CFA to malee appropriate modifications. The diagnostic
cues that can be used to malee appropriate modifications include (I) the path estimates or loadings, (2) standardized residual, (3) modification indices, and (4) specification search.
As noted earlier, residuals refer to the differences between the observed covariances (i.e the sample data) and the estimated covariance terms. A standardized residual is the residual divided by its standard error. The following guidelines are observed with respect to the absolute values of the standardized residuals. Absolute values of standardized residuals exceeding 4.0 are problematic, while those between 2.5 and 4.0 should also be examined carefully but may not suggest any changes to the model if no other problems are associated with the corresponding indicators or observed variables.
A specification search is an empirical approach that uses the model diagnostics and trial and error to find a better-fitting model. It can be easily implemented using SEM software. In spite of this, the approach should not be used without caution because there are problems associated with determining a better-fitting model simply based on empirical data. We do not recommend this approach for the non expert user.
It should be noted that all such adjustments, whether based on path estimates, standardized residuals, modification indices, or specification searches, are against the intrinsic nature of CFA, which is a confirmatory technique. In fact, such adjustments are more in keeping with exploratory factor analysis (EFA). However, if the modifications are minor (e.g., deleting less than 10 percent of the . observed variables), you may be able to proceed with the prescribed model and data after the suggested changes. However, if the modifications are substantial then you must modify the measurement theory, specify a new measurement model, and collect new data to test the new model
Specify the Structural Model
Once the validity of the measurement model has been established, you can proceed with the specification of the structural model. In moving from the measurement model to the structural model, the emphasis shifts from the relationships between latent constructs and the observed variables to the nature and magnitude of the relationships between constructs. Thus, the measurement model is altered based on the relationships among the latent constructs. Because the measurement model is changed, the estimated covariance matrix based on the let of relationships examined will also change. However, the observed covariance matrix, based on the sample data, does not change as the same data are used to estimate the structural model. Thus, in general, the fits statistics will also change, indicating that the fit of the structural model is different from the fit of the measurement model
Figure 225 shows the structural model that is based on the measurement model of Figure 22.3. While the constructs C1 and C2 were correlated in Figure 22.3, there is now a dependence relationship, with C2 being dependent on C1. Note that the two-headed curved arrow in Figure 22.3 is now replaced with a one-headed straight arrow representing the path from C I to C2. There are also some changes in the notations and symbols. The construct C2 is now represented by “Ii’ This change helps us to distinguish an endogenous construct (C2) from an exogenous construct (CI). Also note that only the observed variables for the exogenous construct CI are represented by X (Xi to X3). On the other hand, the observed variables for the endogenous construct (C2) are represented by Y(Y1 to Y3). The error variance terms for the Yvariables are denoted bye, rather than by 5. The loadings also reflect the endogenous and exogenous distinction. Loadings for the exogenous construct, as before, are still represented by Ax. However, the loadings for the endogenous construct are represented by Ay• The graphical representation of a structural model, such as in Figure 225, is called a path diagram. The relationships among the latent constructs, shown with one-headed straight arrows in a path diagram, are examined by estimating the structural parameters, such as in Figure 225. Note that only the free parameters are shown with one-headed straight arrows in the path diagram; fixed parameters, typically set at zero, are not shown. The structural parameters fall into two groups. Parameters representing relationships from exogenous constructs (g) to endogenous constructs (“I) are denoted by the symbol ‘Y(gamma), as shown in Figure 225. Parameters representing relationships from endogenous constructs to endogenous constructs are denoted by the symbol f3 (beta)
In specifying the structural model, it is desirable to also estimate the factor loadings and the error variances along with the structural parameters. These standardized estimates from the structural model can then be compared with the corresponding estimates from the measurement model to identify any inconsistencies (differences larger than 0.05). This approach also allows us to use the measurement model fit as a basis of evaluating the fit of the structural model. An alternative approach that uses the estimates of factor loadings and error variances obtained in the measurement model as fixed parameters in the structural model is not recommended. The reason is that the change in fit between the measurement model and the structural model may be due to problems with the measurement theory instead of problems with the structural theory
Assessing the validity of the structural model involves (1) examining the fit, (2) comparing the proposed structural model with competing models, and (3) testing structural relationships and hypotheses .
The fit of a structural model is examined along the same lines as that for the measurement model discussed earlier. As explained earlier. generally a recursive structural model has less relationships than a measurement model from which it is derived. At most, the number of relationships in a structural model can equal those in a measurement model. This means that comparatively less
parameters are estimated in the structural model. Therefore, the value of X2 in a recursive structural model cannot be lower than that in the corresponding measurement model. In other words, a recursive structural model cannot have a better fit. Thus, the fit of the measurement model provides an upper bound to the goodness of fit of a structural model. The closer the fit of a structural model is to the fit of a measurement model, the better. The other statistics and guidelines for assessing the fit of a structural model are similar to those discussed earlier for the measurement model and the same fit indices are used.