Exploratory Factor Analysis

Initially, a measurement model can be evaluated using exploratory factor analysis (EFA); as it is beneficial to define the underlying structure among the items before further analysis is undertaken (Hair et al., 2006). It enables to check whether proposed dimensionality of the constructs examined is consistent with the data. Exploratory factor analysis is a suitable technique for studying the dimensionality of a scale (Bryant and Yarnold, 1995). By using inter-item correlation, exploratory factor analysis determines the underlying latent variables (‘factors’) responsible for the patterns of correlations observed in the data (Sharma, 1996). In factor analysis, variables are grouped together based on high correlations with each other. The total variance of each variable could be divided into common variance, specific variance (variance associated with only a specific variable) and error variance (variance unexplained by correlations with other variables) (Leandre and Duane, 2012). Exploratory factor analysis is primarily concerned with describing how much of an item’s variance is shared with other items (common variance). The higher the

correlation of an item with one or more other items, the higher the common variance (communality) of that item (Hair et al., 2006; Bryman and Cramer, 2009).

There are two main techniques available to evaluate the underlying dimensions of constructs, namely principal components analysis and common factor analysis (Dunteman, 1989). Both techniques explain part of the variation in a set of observed

variables. However, the aim of common factor analysis is to identify the least number of factors that account for the common (or shared) variance, whereas the goal of principal component analysis is to identify the number of factors that explain the total variation (common and unique/ ‘specific plus error’) variance in the observed

variables (Gorsuch, 1997).

In practice, the results obtained using principal component analysis are usually quite similar to the results obtained using common factor analysis (Stevens, 2009).

Nevertheless, it is important to understand the underlying differences between the techniques in order for the appropriate one to be chosen. The use of common factor analysis is warranted when the primary objective is to identify the latent dimensions and when a researcher wishes to eliminate specific and error variance due to little knowledge about its amount (Widaman, 1993). The use of principal component analysis is warranted when the main objective is data reduction (focus on minimum number of factors) and when a researcher has a prior knowledge that the amount of specific and error variance is relatively small (Diamantopoulos and

Schlegelmilch, 2000).

Despite the fact that it is rarely (if ever) possible to have a solid knowledge about the amount of specific and error variance, principal component analysis is more

commonly used (it is a default option in SPSS and other statistical packages within the factor analysis procedure) (Gorsuch, 1997). Moreover, there are major concerns associated with common factor analysis, including factor indeterminacy and non- estimation of commonalities. The former problem is concerned with the fact that for any individual respondent, several different factor scores can be calculated, whereas the latter means that sometimes the communalities are not estimated or can be invalid (e.g. greater than 1 or less than 0), which would lead to variable deletion (Hair et al., 2006). Thus, in the current study principal component analysis is used (see Chapter 6 for more details).

5.9.1.1 The Process of Factor Formation

The exploratory factor analysis consists of several iterative steps (Leandre and Duane, 2012). The first step includes the development of principal components, when one component (factor) is derived for each item being analysed. Each of these

initial factors is associated with the relative proportion of variance accounted for by each factor (eigenvalue) (Widaman, 1993). If the items do not correlate with one another and do not cluster into factors, the eigenvalues will be equal 1 as they reflect only the variance in the original items. When the inter-correlation of the items

increases, they will produce factors that contain more of the variance in the items, which will result in eigenvalues becoming larger than 1 (Hayton, Allen and

Scarpello, 2004).

5.9.1.2 Defining Number of Factors

In order to determine the appropriate number of factors, two main criteria are used. The first is known as Kaiser’s criterion (or latent root criterion) and is based on the selection of those factors which have an eigenvalue greater than 1. The logic behind that assumption is that each factor should explain the variance of at least a single variable. A factor with an eigenvalue less than 1 is contributing little to the

explanation of variance in the variables, so it is statistically insignificant (Norris et al., 2012). When a construct is unidimensional (only one factor is formed), it means that all items are perfectly correlated (Spector, 1992).

The second commonly used method was proposed by Cattell (1966), and is called a scree test. A scree test is a graph of the amount of variance explained by factors in the factor analysis. The point at which the plot slopes steeply downward (the ‘elbow’) indicates the start of non-significant factors (Bryman and Cramer, 2009).

5.9.1.3 Rotation

In order to increase the interpretability of the exploratory factor analysis, factors are usually rotated (Field, 2009). There are orthogonal (e.g. Varimax) and oblique (e.g. Oblimin) rotations. When orthogonal rotation is chosen, it is assumed that factors do not correlate with one another, so axes (factors) are maintained at 90 degree angles (Dunteman, 1989). The SPSS software package offers three types of orthogonal rotation: Quartimax, Varimax and Equamax (Sharma, 1996). When Quartimax rotation is used each variable loads high on one factor and as low as possible on other factors. That usually results in many variables loading on the same

variances of required loadings within the factor by loading a smaller number of variables highly onto a single factor, which provides a clearer separation of the factors compared to Quartimax. Equamax represents a compromise between Quartimax and Varimax rotation approaches (Kaiser, 1974).

In contrast to orthogonal rotation methods, oblique rotation methods allow factors to correlate instead of maintaining their independence. SPSS software package

provides only one oblique rotation method, namely Direct Oblimin (Bernaards and Jennrich, 2005).

In the current study oblique rotation (Oblimin) was used (see Chapter 6)

5.9.1.4 Factor Loadings

Rotations result in a loadings matrix that shows how strongly each item relates to each factor (Peterson, 2000). A factor loading represents the correlation between an item and a factor, whereas the squared loading is the amount of variable’s total variance explained by the factor. For example, a 0.20 loading would mean than only about 5% of the variable’s variance is explained, whereas 0.80 loading denotes that more than 60% of the variance is accounted for by the factor. Loadings of 0.4 are considered to be the minimal level for interpretation, and loadings 0.7 and above are considered to indicate well-defined structure (Hair et al., 2006).

However, aside from factor loadings, there are other important criteria which a researcher has to consider while performing a factor analysis.

5.9.1.5 Bartlett’s Test of Sphericity and Kaiser-Meyer-Olkin Test

Two statistical tests are usually used to evaluate the appropriateness of the results: the Bartlett’s test of sphericity and Kaiser-Meyer-Olkin (KMO). The Bartlett’s test of sphericity is a statistical test for the presence of correlations among the variables. It shows the statistical significance of the correlation matrix (that it has significant correlations among some of the variables) (Worthington and Whittaker, 2006). KMO can be calculated by dividing squared correlations between variables to the squared partial correlation between variables; it can vary between 0 and 1 (Kaiser, 1970). If the value of KMO is closer to 0, it indicates that the sum of partial correlations is very

large compared to sum of correlations, which means there is diffusion in the pattern of correlations and inappropriateness of factor analysis results. On the contrary, if the value of KMO is closer to 1, it indicates that patterns of correlations are relatively compact and it is likely that the results of factor analysis are appropriate and

meaningful (Field 2009). According to the rule of thumb if KMO value is 0.6 or higher the researcher can trust the results of exploratory factor analysis (Hair et al., 2006).