# Model estimation In this step suitable fitting procedure is selected to estimate the free parameters of the model

Model estimation

In this step suitable fitting procedure is selected to estimate the free parameters of the model. Many procedures, like maximum likelihood, LISREL’s initial estimates (Jöreskog and Sörbom, 2001), EQS’s distribution-specific and distribution-free estimates (Bentler, 1988), and many others (see Hair et al., 2010, p638) may be deployed in SEM for parameter estimations. The objective of these procedures is to estimates a best- fitting solution and to evaluate the model fit in accordance with the size, distributional assumptions and scale dependency of the data samples (Wu, 2009). The changes in observed variable scale yield different solutions or sets of estimates. AMOS facilitates five SEM procedures (Table 5.15) for parameters estimations: Maximum likelihood (ML), Generalised least squares (GLS), Unweighted-least squares (ULS), Scale-free least squares (e.g., Weighted-least squares or WLS) and Asymptotic-distribution free (ADF) (Anderson and Gerbing, 1988; Barndorff-Nielsen and Cox, 1989; Schumacker and Lomax, 2004, p66).

Estimation procedures Normality assumption Sample size Scale Dependent

Maximum likelihood (ML) Yes the more the better No

Generalised least squares (GLS) Yes the more the better No

Unweighted-least squares (ULS) No more than 1000 Yes

Scale-free least squares (WLS) No more than 1000 No

Asymptotic-distribution free (ADF) No more than 1000 No

Table 5.15 The SEM estimation procedures in AMOS, adopted from Byrne (2010) and Wu (2009)

Since the ULS, WLS and ADF (i.e. three of above five) have no distributional assumptions or associated statistical tests, the multivariate normality of the data is not required (Rong, 2009). Nevertheless, these carry the desirable asymptotic (large sample) properties, that is, the random sample size must be over 1000 (Wu, 2009). Also, the ULS method is a scale dependent method. Hence, the changes in observed variable scale yield different estimates (Schumacker and Lomax, 2004).

In contrast, the ML and GLS are more robust techniques in many cases (McDonald and Ho, 2002). These are scale free (Schumacker and Lomax, 2004) and require lower sample size, not to forget the more the better. However with decrease in the size non- normality increases (Jöreskog and Sörbom, 2001; Byrne, 2010) and the statistical power diminishes (Saris and Satorra, 1993; McQuitty, 2004; Fadlelmula, 2011). These two aforesaid methods may yield very similar results (Muthen, 1973; Wu, 2009, p. 25), specifically, in the cases in which the model is correctly specified (Olsson et al., 2000), and having similar implementation requirements (Olsson et al., 2000; Byrne, 2010). Each one of these methods needs the data to be continuous, with minimum variance, unbiased, and multivariate normal (Schumacker and Lomax, 2004; Byrne, 2010). The violation of these assumptions could significantly influencethe estimated results (Kline, 2005). The GLS could be preferred over the ML in cases where the assumptions are not too seriously violated (Wu, 2009), since it produces a better empirical fit (Browne, 1973; Ding et al., 1995). Nevertheless, Olsson et al. (2000) demonstrated that this superiority is compromised at the cost of lower theoretical fit (supported by the theory). Olsson et al. (2000, p560) elaborated that “parameter estimates for correctly specified paths within a partly incorrectly specified model, i.e. not fully supported by the theory, were found to be significantly more biased for GLS than for ML.”