In exploratory factor analysis, how much variance would a good model be likely to explain

Exploratory Factor Analysis

Exploratory factor analysis (EFA) is a classical formal measurement model that is used when both observed and latent variables are assumed to be measured at the interval level. Characteristic of EFA is that the observed variables are first standardized (mean of zero and standard deviation of 1). EFA is executed on the correlation matrix between the items. In EFA, a latent variable is called a factor and the associations between latent and observed variables are called factor loadings. Factor loadings are standardized regression weights. Since EFA is an exploratory technique, there is no expected distribution of loadings; hence, it is not possible to test statistically whether or not factor loadings are the same across cultural groups. However, congruence measures, such as Tucker's ϕ, have been developed to indicate whether the pattern of factor loadings across items on a factor is the same across cultural groups. Sufficient congruence for structural equivalence is usually taken to be found if Tucker's ϕ exceeds 0.95. Values below 0.90 are taken to indicate that one or more items show deviant factor loadings and thus show bias. Bootstrap procedures have been developed to test the identity of factor loadings in EFA.

EFA is used to investigate structural equivalence. However, since it works on standardized variables (mean of zero and standard deviation of 1), this model is not suited to detect nonuniform and especially uniform item bias.

EFA is often used in the multidimensional situation where more than one latent variable is measured at the same time. Before evaluating congruence in this case, the factor structures should be rotated toward a target structure.

Construct/Factor Analytic

An exploratory-factor analysis (maximum-likelihood method, varimax rotation) on the data from a sample of 189 undergraduate students indicated a clear four-factor structure with the selected 16-items; the average factor loading of these items on their respective WLEIS dimensions was .80. The four-factor solution explained 71.5% of the total variance and fit the data reasonably well in a confirmatory-factor analysis (N=72) from the first cross-validation study, χ2(98)=132.41, RMR=.08, CFI=.95, TLI=.93. The second study (N=146) also showed that the four-factor model fit the data reasonably well, χ2(98)=179.33, RMR=.07, CFI=.91, TLI=.89 (Wong & Law, 2002).

2.4 Statistical analyses

First, Exploratory factor analysis (EFA) was performed to investigate the factor structure of the questionnaire and reveal the latent factors underlying the attitude of clinical professionals toward the use of tES in children. Assumptions of sphericity (Barlett's test of Sphericity: P < 0.001) and sample adequacy (overall KMO = 0.68; all single-item KMOs greater than 0.50) were satisfied. The minimum residuals extraction method was used in combination with a direct Oblimin rotation. The number of factors to be extracted was estimated using parallel analysis and the examination of the scree plot.

Second, descriptive statistics have been computed for the propensity to use tES in children ratings and the reported theoretical and practical knowledge about tES. For the following analyses, a tES knowledge composite variable was computed by averaging the ratings to the tES theoretical knowledge and tES practical knowledge items, which were highly correlated with each other (r = 0.88; P < 0.001).

Descriptive statistics has been performed for each questionnaire item (Table 1). After that, gender differences were tested on all predictors. Given that gender groups had unequal sample sizes, Welch's t-test was used. Factorial 2 × 3 ANCOVAs were tested to assess the main and interaction effects of the clinical field (i.e., healthcare vs. psychology) and practice type (i.e., public vs. private vs. freelance) on all questionnaire factors, controlling for years of clinical experience. Post-hoc comparisons have been performed using Tukey's HSD test.

Finally, two linear regression models were tested to measure the contributions of participants' professional characteristics (i.e., years of experience, clinical field) and questionnaire factors to (a) propensity to use tES in children and (b) ethical concerns. The assumption of no multicollinearity was confirmed in all multiple regression models (all VIF values ranged between 1.00 and 1.09). The predictors were entered in each model using the forward method and they were selected by comparing the models' goodness of fit (F-tests) and considering AIC values.

Effect sizes have been reported as Cohen's d.

Construct/Factor Analytic

Structural validity was analyzed by both exploratory (EFA) and confirmatory factor analyses (CFA). Five factors were explicitly extracted in the EFA calibration sample using a principal components method with Varimax rotation (N=4621). The five factors accounted for 25.69% of the variance. A CFA was then performed on the 50 items (χ2=5664.66; d.f.=1165; χ2/d.f.=4.86; SMSR=0.01; CFI=0.78; GFI=0.90; RMSEA=0.04). Correlations between the latent variables for oblique models were: ImpSS/N-Anx: −0.11/−0.04; ImpSS/Agg-Host: 0.31/0.34, ImpSS/Act: 0.19/0.14, ImpSS/Sy: −0.40/−0.36, N-Anx/Agg-Host: 0.28/0.24, N-Anx/Act: −0.05/−0.08, N-Anx/Sy: 0.17/0.19, Agg-Host/Act: 0.04/0.03, Agg-Host/Sy: −0.08/−0.09 and Act-Sy: −0.18/−0.19.

Construct/Factor Analytic

Triandis and Gelfand (1998) performed an exploratory factor analysis on the responses to the 27-item scale (N=326). They reported that, ‘The 27×27 matrix of correlations among the items was subjected to an exploratory factor analysis. Bentler and Wu’s (1995) equal prior instant communalities method was used.’ (Triandis & Gelfand, 1998). They found that the items measuring VC, HC, VI, and HI had significant loadings on the respective factors, providing evidence for the factor analytic validity of the measure. For the 32-item scale, confirmatory factor analysis using LISREL 7 showed that the 4-factor model (GFI=.79, AGFI=.75, RMR=.09) had a superior fit than either the 1-factor or 2-factor models (Triandis, 1995; Singelis et al., 1995).

There is also confirmatory factor analytic evidence for structural equivalence of the 16-item scale (Soh & Leong, 2002). In a factor analytic study of 180 American undergraduates and 184 Singaporean graduates, Soh and Leong (2002) found that factor loadings were invariant in the two validation samples (RMSEA=.05, SRMR=.10, NFI=.81, CFI=.83, IFI=.84). However, one VC item (‘It is important to me that I respect decisions by my groups’) showed higher loading on HC than VC for both samples. In addition, the means were not invariant across two samples (Δχ2(12)=65.0, p<.05), and neither were the variances (Δχ2(16)=29.4, p<.05).

Exploratory Factor Analysis

Of the two types of factor analytic techniques, exploratory factor analysis is the most commonly used. These techniques consist of methods such as principal-components analysis and principal-axes analysis. These approaches have in common that the researcher does not pre-specify the nature of the latent variables, that is, which items they define. Instead, he or she determines the number of factors believed to be sufficient to explain the intercorrelations among variables, extracts these factors, and then interprets factors based on how strongly items reflect or load on these factors. This last procedure often first involves rotating the factors, which serves to rescale the loadings so that high loadings are as close to 1 or −1 as possible and low loadings are as close to 0 as possible.

Three abductive research methods are described: (1) The multivariate statistical method of exploratory factor analysis is presented as an abductive method of theory generation that exploits an important principle of scientific inference known as the principle of the common cause. (2) The theory of explanatory coherence is an abductive method for evaluating the explanatory worth of competing theories. (3) Grounded theory method promotes the inductive generation of theories grounded in qualitative data. However, it can be plausibly reconstructed as an abductive account of scientific method. It is recommended that these methods should be part of the methodological armamentarium of educational and social science researchers.

Conceptual/theoretical considerations

Factor analysts should choose an appropriate factor model, usually component analysis (CA) versus EFA, in accordance with the purpose of the analysis. If the purpose of factor analysis is essentially data reduction, then CA will yield a fewer number of components which represent the original set of variables. The resulting component scores are used in follow-up analyses.

If the goal of the researcher is to interpret the correlations among variables as arising from a smaller set of latent variables/factors, EFA is the method of choice. The latter model recognizes that variables are measured with error and yield coefficients which are less biased.

Construct/Factor Analytic

The sample (N=677) was split into two random, independent samples. An exploratory factor analysis with oblimin rotation, resulted in a three-dimensional model with factors labeled: Public Identification as Gay (5 items), Sexual Comfort with Gay Men (4 items), and Social comfort with gay men (4 items; Currie et al., 2004). Subsequent confirmatory factor analysis indicated adequate to good fit for all three factors after removing one redundant item from the Public Identification subscale (leaving a total of 12 items; Currie et al., 2004). This model fit the data well (χ2 (51, N=335)=66.67, p=.069; CFI=.98). Intercorrelations between subscales ranged from .42 to .48 (Currie et al., 2004).

Construct/Factor Analytic

As indicated above, in constructing the original AAS, Collins and Read (1990) conducted an exploratory factor analysis with oblique rotation (N=406) based on the 21×21 item intercorrelation matrix and extracted three factors that clearly defined the AAS structure (see Collins & Read, Table 2, p. 647, for the factor loadings on each of the original 198 items). ‘Factor 1 (Depend) and Factor 3 (Close) were moderately correlated (.41)’ suggesting some measurement overlap between these two AAS subscales. Subsequently, Collins (1996, pp. 814–815, N=295) confirmed the tripartite structure of the AAS (based on a factor analysis of the revised items).