Dimensionality Analysis Via EFA

Test developers typically aim to construct uni-dimensional scales or subscales containing items that all to some degree reflect a single common underlying latent variable or latent dimension. If

67 this is the case all the items of such a scale of subscale should have reasonably high factor loadings on a single underlying factor. Dimensionality analysis was consequently conducted via exploratory factor analysis (EFA) on each of the two subscales of the SD scale of the SAPI. The objective of the EFA analysis was to explore the number of factors underlying each of the two subscales of the Social Desirability scale. Exploratory factor analysis analyses the shared or common variance amongst the items in order to determine the number of common factors that need to be assumed to explain the observed inter-item correlation matrix. There are a variety of factor extraction methods, and the two most commonly used methods are, principal-components analysis (PCA) and common-factor analysis (FA) (Worthington & Whittaker, 2006). The purpose of PCA, according to these researchers, is to reduce the number of items whilst still maintaining as much of the original item variance as possible (Worthington & Whittaker, 2006). With this SAPI project the aim is to not reduce the number of items measuring the two hypothesised latent dimensions of social desirability but to rather understand the latent factors or constructs that account for the shared variance among items. Therefore, FA is more closely aligned to the objective of the current study and Worthington and Whittaker (2006) recommend that this approach be implemented in the case of the development of new scales.

The unidimensionality assumption was tested for each of the social desirability subscales by subjecting each of the subscales to a principal axis factor analysis(PAF) with oblique rotation using direct oblimin rotation. The statistical technique of PAF was chosen in preference to principal component analysis because the reasons outlined earlier. The decision as to the appropriate number of factors to extract was based on the rule of thumb known as the ‘eigenvalue-greater-than-one’ criterion. If there are two eigenvalues greater than one this suggest the presence of two factors. Kaiser (1960) recommended that all factors with eigenvalues greater than one should be retained. This criterion is based on the argument that in the standardised item data set the variance of a single item is one, and since eigenvalues represent the amount of variation explained by a factor, only factors that explain more variance than a single item should be considered for retention (Field, 2013). The number of factors to extract was also determined through the use of a scree plot. Using this criterion the number of factors to the left of the elbow indicates the number of factors that should be extracted. It is also possible to set the factor extraction to a specific number. Parallel analysis (Horn, 1965), is another procedure that can be employed to decide how many factors to retain. This analysis is done when a random data set of

the same order as the raw data set (i.e. an equal number of observations and items) is created and factor analysis is then conducted on both the original data set and the random data set. The number of factors to retain is determined by looking at the eigenvalues in both data sets and a factor is retained if the eigenvalue for the original data is larger than the eigenvalue from the random data (Worthington & Whittaker, 2006). The eigenvalues-greater-than-unity rule, the scree plot and parallel analysis were employed in this study to determine the number of factors that should be extracted to explain the observed inter-item correlation matrix.

To examine the factor analysability of the two subscales the Kaiser-Meyer-Olkin (KMO) (Kaiser, 1960) measure of sampling adequacy was examined. When the KMO value approaches one, or is at least greater than .60, the inter-item correlation matrix is considered to be suitable for factor analysis. A KMO value close to one indicates that patterns of correlations are relatively compact and so factor analysis should yield distinct and reliable factors (Field, 2013). If in the parameter the variables in the correlation matrix do not correlate with each other than the matrix would be an identity matrix. The Bartlett test tests the null hypothesis that the correlation matrix in the parameter is an identity matrix meaning that every item correlates only with itself and nothing else. The identity matrix implies that there are no common factors and that every item is unique with no evidence of common variance. Low factor analysability would constitute comment negatively on the subscales of the SD scale as a common factor should emerge if a subscale is successfully measuring a specific latent trait. In factor analysis, when the sample size is large, the Bartlett’s test will nearly always be significant (Field, 2013).

Once one or more factors have been extracted it is possible to calculate the degree to which variables load on the factor (Field, 2013). In most rotated factor analytic solutions the variables being factor analysed have high loadings on a single factor and low loadings on the remaining factors. In a single-factor solution and in an orthogonal rotated factor solution the factor loadings can be interpreted a correlation coefficients. The square of the factor loadings therefore reflect the proportion of variance in the item that is explained by the underlying factor. In an oblique solution the interpretation is less straight-forward (Tabachnick & Fidell, 2007). An oblique rotation method assumes that the factors that have being extracted are related to each other and are not independent (Field, 2013). The factors therefore share variance. When utilising oblique rotation the pattern matrix reflects the partial regression coefficients that express the

69 slope of the regression of the items on each extracted factor when controlling for the other factors in the solution (Tabachnick & Fidell, 2007). Direct oblimin rotation was used in this study.

The decision to perform the dimensionality analysis on the two subscales of the SAPI SD scale again made good sense in the case of operational research hypothesis 1, but not when viewed from the perspectives of operational research hypotheses 2, 3 and 4. The same argument that applied to the item analysis also applies to the use of EFA to evaluate the assumption that each subscale measures a single, undifferentiated latent social desirability dimension.