Structural Equation Modeling

Structural equation modeling is a multivariate method that can be used to examine a set of regression equations simultaneously (Bollen 1989, Hair et al. 1998:584).

Structural equation modeling may be used as a more powerful alternative for instance to multiple regression, path analysis, factor analysis, time series analysis, and analysis of covariance. These procedures can be viewed as special cases of structural equation modeling which is an extension of the general linear model.

Structural equation modeling has some advantages compared to multiple regression including for instance more flexible assumptions, use of confirmatory factor analysis to reduce measurement error by having multiple indicators per latent variable, overall testing of the model fit rather than coefficients individually, the ability to test models with multiple dependent variables, the ability to model mediating variables, the ability to model error terms.

Structural equation modeling is normally viewed as a confirmatory rather than exploratory procedure (Byrne 2001:3). Structural equation modeling uses goodness-of-fit tests to determine if the pattern of variances and covariances in the data is consistent with the hypothesized structural model specified a priori. Structural equation modeling can also be used to test two or more causal models to determine which has the best fit (Loehlin 1987). Because structural equation modeling cannot itself draw causal arrows in models or resolve causal ambiguities, theoretical insight and judgment by the researcher is critically important.

Interpretation of the Results in Structural Equation Modeling

The fit of a structural equation model is a multidimensional concept and should therefore be examined from a variety of perspectives. The examination of model fit includes the assessment of the parameter estimates and the model as a whole. The procedures used to examine the model fit in the present study are discussed below in more detail.

Parameter estimates. Byrne (2001:75) summarizes three areas of assessment on parameter estimates: (1) the feasibility of the parameter estimates, (2) the appropriateness of standard errors, and (3) the statistical significance of the parameter estimates.

The first step when assessing the model fit on the parameter estimate level is the examination of the feasibility of the parameter estimates. Parameters should have the

correct sign and size according to the underlying theory. Clear examples of unreasonable estimates include correlations >1.00, negative variances, and covariance or correlation matrices that are not positively definite (Byrne 2001).

The second step in the determination of the model fit on the parameter estimate level is the assessment of the appropriateness of the standard errors. Standard errors that are either excessively large or small are indicative of poor model fit (Byrne 2001).

However, this assessment is subjective because the magnitude of standard errors is dependent on the unit of measurement and the parameter estimates.

The third step in the assessment of the model fit on the parameter estimate level is the examination of the statistical significance of the parameter estimates. Non-significant parameter estimates, with the exception of error variances, can be considered unimportant for the model. However, it should be noted that sample size influences the significance of the parameters (Byrne 2001).

These three steps were followed in the analyses carried out in the present study.

The model as a whole. When examining the fit of the model as a whole, multiple indices are typically used to determine the model fit. Table 4-9 describes the goodness-of-fit measures used in this study.

Table 4-9 Goodness-of-fit criteria in structural equation modeling used in this study

Criterion Description Interpretation

Chi-square Calculation of difference between observe and estimated covariance matrices

p>.05 for model to be acceptable; sensitivity to sample size

Normed Chi-square Chi-square adjusted for degrees of freedom

Recommendation between 1.0 and 2.0

Goodness of fit index (GFI)

Predicted squared residuals compared with obtained residuals, not adjusted by degrees of freedom

Range between 0 (no fit) to 1.0

Proposed model compared with the null model, adjusted by degrees of freedom

Proposed model compared with the null model, adjusted by degrees of freedom

Values closer to zero indicate better fit and greater parsimony Root mean square error of

approximation (RMSEA)

Discrepancy per degree of freedom Values below .08 are acceptable

One of the commonly used measures of model fit is the chi-square test where the predicted covariance matrix is tested for statistical difference from the original covariance matrix. If the difference is statistically insignificant, the model fit is considered to be good.

Besides the chi-square test, there are many other indices used in the testing of model fit. Normed chi-square adjusts the chi-square by the degrees of freedom. Values between 1.0-2.0 are considered to indicate a good fit (Hair et al. 1998).

Goodness of fit index (GFI) is calculated by comparing the predicted squared residuals with the obtained residuals. This measure is for absolute fit, and not adjusted by degrees of freedom. The range of this index is between 0 (no fit) and 1.0 (perfect fit). Models with GFI is above .90 are considered to have a good fit. This index has been argued to be insufficient because, for example, it is overly influenced by sample size (Fan et al. 1999).

Non-normed fit index (NNFI) compares the proposed model with a null model. This index is also called Tucker and Lewis’ index (TLI). NNFI is adjusted by degrees of freedom and ranges between 0 (no fit) and 1.0 (perfect fit). Models with NNFI above .90 have traditionally been considered to have a good fit. However, it should be noted that when the sample size is small, the NNFI tends to reject correct models too easily (Hu and Bentler 1999).

Comparative fit index (CFI) compares the proposed model to the null model. This index is also adjusted by the degrees of freedom. Also CFI ranges between 0 (no fit) and 1.0 (perfect fit) Models with CFI above .90 are considered to have a good fit (Bentler 1992). However, the recent research recommends higher cut-off value close to .95 (Hu & Bentler 1999).

Akaike information criterion (AIC) compares models with different number of constructs. AIC is based on information theory. Values closer to zero indicate better fit and greater parsimony. When comparing different models, the model with the lowest AIC is considered to have the best fit (Akaike 1987).

Root mean square error of approximation (RMSEA) measures the discrepancy per degree of freedom. Values less than .05 are considered to be good and values ranging from .05-.08 are considered to be acceptable (Browne & Cudeck 1993:144, MacCallum et al. 1996). However, Hu and Bentler (1999) cautioned that when the sample size is small, the RMSEA tends to reject correct models too easily.

Model misspecification. Finally, after the assessment of the model on the parameter estimate level and the model as a whole, the potential model misspecification is examined. The residual covariance matrix is the discrepancy between the restricted hypothesized model and the sample covariance matrix. Each residual represents the difference between the observed and hypothesized parameter estimate. Large residuals indicate potential misfit in the model. Because the magnitude of residuals is dependent on the measurement units, standardized residuals are typically used in this analysis.

Standardized residuals, being defined as fitted residuals divided by their asymptotical standard errors, are analogous to Z scores (Byrne 2001). Jöreskog and Sorbom (1988) suggested a cut-off value of 2.58 residuals to be considered large. In order to identify signs of potential misspecification, residuals are examined in the present study following the above guideline.

Structural equation model is often used to combine confirmatory factor analysis and path analysis. Various processes have been proposed for doing this (Anderson &

Gerbing 1988, Mulaik & Millsap 2000). In the present study, a relatively small sample

size prevents the use of these techniques. Instead, path analysis was carried out using separately validated summated scales employing structural equation modeling.

In line with some other recent studies (e.g. Zahra et al. 2000a), the present study used multiple regression analysis with summated scales as the primary analytical method. However, the multiple regression analyses were supplemented with path analyses carried out using structural equation modeling in order to test all the hypotheses simultaneously and to test that there are no other important paths in the model. Path analysis using structural equation model thereby enables a test of the whole model and thereby adds to the multiple regression analyses. The use of two methods to carry out the analyses increases the robustness of the results and conclusions.