Estimation technique

The method most widely used in SEM to estimate the model is maximum likelihood estimation (ML). It is the default estimation method in most available programs (Diamantopoulos and Siguaw, 2000). A key assumption of this method is multivariate normality for the exogenous variables. Although ML has appeared in the past to demonstrate sensitivity to data which is not normally distributed, it is now believed to be robust in cases where non-normality exists, and has produced reliable results in a variety of circumstances (Hair, Black, Babin, Anderson and Tatham, 2006). The ML method is adopted for the town centre image study reported in this thesis.

5.4.4 Identification

Before analysis can take place, the researcher must determine whether there is sufficient information in the model to obtain a unique solution for the parameters to be estimated (Diamantopoulos and Siguaw, 2000; Hair, Black, Babin, Anderson and Tatham, 2006). In other words, model identification must be assessed. If there are more parameters than there are item variances, there is insufficient

information for estimation, the model cannot be calculated and is therefore underidentified. If there are exactly equal numbers of parameters and item variances, the model is just-identified and will produce one perfect solution, and theoretical testing of the model becomes unrealistic. Hence the researcher is seeking a model which is overidentified, where more than one solution can be calculated, so that the statistical analysis can seek the best possible solution to explain the observed data (Kelloway, 1998). Identification is measured by the formula 1/2 [p (p+1)] - k, where p is the number of measured items and k is the number of parameters to be estimated, and should always be greater than 1 (Bentler and Chou, 1987).

157 5.4.5 Model fit

The validity of the measurement model is assessed by goodness-of-fit measures.

Goodness-of-fit measures assess the extent to which the hypothesised model fits the data by calculating how closely the observed sample matrix fits the estimated population covariance matrix. The fundamental measure of fit is the

chi-square (χ ²) statistic. This test assumes a null hypothesis that there is no difference between the observed matrix and the estimated matrix, chi-square is not significant and the model fits the data perfectly. If chi-square is large and statistically significant, the null hypothesis is rejected and the model is assumed to be a poor fit. In contrast to conventional hypothesis testing, the aim in SEM is not to reject the null hypothesis (Diamantopoulos and Siguaw, 2000). However, the chi-square test is highly sensitive to sample size (Anderson and Gerbing, 1988;

Bentler and Chou, 1987; Fornell and Larcker, 1981). As discussed above, SEM requires a substantial sample for analysis to be performed, yet it is unlikely that a non-significant chi-square can be achieved in a sample size which is sufficiently large for analysis to proceed (Kelloway, 1998). In addition, the chi-square test measures perfect fit, which is an unlikely circumstance in real life research situations, since researchers generally test models which are approximations of reality at best (Diamantopoulos and Siguaw, 2000). The chi-square test is also sensitive to degrees of freedom (df), which is the number of observed variables and parameters in the model (Hair, Black, Babin, Anderson and Tatham, 2006).

Hence, an alternative and more pragmatic measure of fit calculates the ratio between chi-square and the number of degrees of freedom (χ ² /df) (Byrne, 2010).

Although values of under 5 have been supported (Diamantopulos and Siguaw, 2000), a model with good fit is generally considered to have been achieved when this ratio is less than 2 (Tabachnick and Fidell, 2007).

Hence, although the chi-square test is useful as a guide to model fit, in that a small chi-square indicates good fit and a large chi-square indicates poor fit, alternative measures of goodness-of-fit have subsequently been developed. These can be grouped into three categories: indices which measure absolute fit, indices which measure incremental fit, and indices of parsimonious fit (Hu and Bentler, 1999).

158 5.4.5.1 Absolute fit indices

Absolute fit indices measure how well the model specified by the researcher reproduces the observed data (Hu and Bentler, 1999). The model is evaluated directly, without comparison to other models. One of the most common absolute indices reported is the Goodness-of-Fit Index (GFI), which assesses the relative amount of variances and covariances accounted for in the model and how closely the model comes to perfectly reproducing the observed covariance matrix (Byrne, 2010; Diamantopoulos and Siguaw, 2000). An Adjusted Goodness-of-Fit index (AGFI) adjusts the GFI values in relation to the complexity of the model and hence accounts for degrees of freedom. Both GFI and AGFI values should exceed 0.90 if the model fits well (Diamantopoulos and Siguaw, 2000).

The Root Mean Square Residual (RMSR), Standardised Root Mean Residual (SRMR) and Root Mean Square Error of Approximation (RMSEA) are all measures of fit based on the values of the error terms in the covariances, or

residuals. However, unlike the other two measures, the RMSEA assesses not only how well the model fits the population, but also considers model complexity and sample size; hence it is a more informative measure of model fit (Diamantopoulos and Siguaw, 2000; Hair, Black, Babin, Anderson and Tatham, 2006). Lower RMSEA values represent better fit: values below 0.05 indicate good fit, between 0.05 and 0.08 suggest reasonable fit, 0.08 and 0.10 mediocre fit, and above 0.10 poor fit (Diamantopoulos and Siguaw, 2000; Hu and Bentler, 1999).

5.4.5.2 Incremental fit indices

Incremental fit indices, in contrast, compare alternative models. Initially the specified model is compared relative to an alternative baseline or null model, which assumes that all variables are unspecified and uncorrelated and which does not result in good fit (Hu and Bentler, 1999; Kelloway, 1998). Incremental indices assess the improvement in fit when parameters are specified as in the model. The Normed Fit Index (NFI) compares the chi-square value of the null model with the specified model, so that values range between 0 and 1, with perfect fit values at 1.

The Non Normed Fit Index (NNFI) adjusts the NFI for model complexity by including the number of degrees of freedom (Kelloway, 1998). The Tucker-Lewis

159 Index (TLI) is a similar index to the NNFI, falling between 0 and 1 with values close to 0.95 indicating good fit (Byrne, 2010; Hu and Bentler, 1999). The Comparative Fit Index (CFI) is less sensitive to model complexity and is thus more desirable (Hair, Black, Babin, Anderson and Tatham, 2006). The CFI again ranges between 0 and 1, with values greater than 0.95 assumed to represent good fit (Hu and Bentler, 1999; Kelloway, 1998).

5.4.5.3 Parsimony fit indices

Parsimony fit indices assess the fit of competing models to determine if a better fit can be achieved by a simpler model, i.e. one with fewer specified parameter

paths. They may be seen as combining absolute and comparative measures when comparing two models. The basis for parsimony fit measures is the ratio of

degrees of freedom in the model as specified to the total degrees of freedom available. The most widely used index is the Parsimony Normed Fit Index (PNFI) which again falls between 0 and 1. However even with a model which fits well, this index is unlikely to reach as high a cut off point as 0.90 and is most useful when assessing the relative fit of two competing models (Hair, Black, Babin, Anderson and Tatham, 2006; Kelloway, 1998).

5.4.5.4 Reporting goodness-of-fit indices

Since these measures of fit are all considered a guide to the plausibility of a model rather than evidence of perfect fit, no one measure can determine a correct model, and researchers are recommended to report more than one measure from each of at least the absolute fit and incremental fit indices, together with the chi-square statistic, when assessing a model‟s fit (Bentler and Chou, 1987; Diamantopoulos and Siguaw, 2000; Hair, Black, Babin, Anderson and Tatham, 2006).