Performing the Factor rotation is the next step after identifyng the number of factors to be extracted. The objective of factor rotation is to facilitate a clear understanding and interpretation of the data (Walker and Maddan, 2012). The reference axes (coordinate plane) of the original factor solution are rotated in this process to simplify the factor structure (i.e., placing all of the factors in the same quadrant) (e.g., Figure 5.8). This makes the geometric location of the factors more meaningful and creates an interpretable solution simply by creating the rotated Pattern Matrix from the original Factor Matrix.
Figure 5.8 An example of factor rotation (ucla: Statistical Consulting Group., 2007)
There are two basis approaches in SPSS to rotate the factors. They are known as orthogonal (uncorrelated) and oblique (correlated) approaches (Abdi, 2004; Field, 2005; Walker and Maddan, 2012). Orthogonal rotation (Kaiser, 1958) (i.e., varimax, quartermax and equamax) is used for the conditions where the reference axes are uncorrelated and the angle between the reference axes of factors are maintained at 90 degrees (ucla: Statistical Consulting Group., 2007; Walker and Maddan, 2012). Oblique rotation (Carroll, 1953) (i.e., direct oblimin and promax) is applied where the axes are correlated and the angle between the reference axes are not at right angles (Field, 2005; Pallant, 2007; Walker and Maddan, 2012). Either of the approaches often result in very similar results, particularly when the pattern of correlations amongst the items is clear (Tabachnick and Fidell, 2007). Nevertheless, the orthogonal rotations should be applied if the factors are uncorrelated, and oblique rotations for the cases where the factors are correlated (ucla: Statistical Consulting Group., 2007).
Use of factors in other analyses
The last step is ‘Use of Factors’, in which the results of factor analysis can be used in other analyses, for example, structural equations modelling (Walker and Maddan, 2012).
Structural Equation Modelling
Structural equation modelling, commonly known as SEM, is a second-generation multivariate statistical analytical tool (Suhr, 2012) which is used to identify and depict relationships among variables (Kline, 2005). It can be defined as “a hypothesis of a specific pattern of relations among a set of measured variables and latent variables” (Shah and Goldstein, 2006, p166).
An overview structural equation modelling and path analysis
SEM constitutes a family of statistical methods (i.e., multiple regression, variances and covariance) (Valluzzi et al., 2003) to test in tandem the causal processes denoted by a series of regression equations (Byrne, 2010). It has been used widely for complex hypothesis testing (Grace, 2012) and confirmatory study (Wu, 2009). Apart from this, SEM is also a form of graphical modelling that intends to provide an effective quantitative test (Kline, 2005). Hence, in SEM, the causal processes can be modelled pictorially to generate a clearer theoretical framework. The graphical symbols used for this purpose are shown below in Figure 5.9:
Figure 5.9 SEM diagram symbols, adopted from Schumacker and Lomax (2004, p153)
There are two types of variables included in SEM (McDonald and Ho, 2002): observed variables and latent variables (Bentler and Weeks, 1980; Kline, 2005). The observed (measured) variables are ones which can be directly observed or measured (Streiner, 2006). The latent variables are ones which are not directly observable; SEM additionally depicts the measurement error concerned with each observed variable (Byrne, 2010). Latent variables are inferred constructs based on the observed variables. Hence, SEM makes use of correlation coefficients and regression analysis to establish causal relationships among observed/latent variables (Ullman, 2007). It uses causal arrows to denote the causes and effects between exogenous (independent or source) variables and endogenous (dependent or downstream) variables based upon some underlying theory (Loehlin, 2004; Byrne, 2010).
Figure 5.10 Illusions of Regression Model and Factor Model based on Shah and Goldstein (2006, p150)
The three types of basic models are tested by SEM (Schumacker and Lomax, 2004), namely,
(1) Regression Models (Figure 5.10 (a)) to test hypothesised causal relationships between latent variables (Pearson, 1936);
(2) Factor Models (Figure 5.10 (b)) to test relationships amongst observed and latent variables (Spearman, 1904b; Spearman, 1927); and
(3) Path Models (Figure 5.11) to structure complex causal relationship amongst observed/latent variables as well as between latent variables (Wright, 1918; Wright, 1921; Wright, 1934; Wright, 1960).
Figure 5.11 A general structural equation model demarcated into measurement (Byrne, 2010, p13)