Exploratory Factor Analysis

Exploratory factor analysis (EFA) is a set of statistical procedures used to determine the number and nature of constructs required to account for the pattern of correlations among a set of measures. EFA is used when there is little theoretical and/or empirical basis to generate specific predictions regarding the underlying structure of correlations among the measures. This entry further describes EFA and its purpose and then discusses how to conduct an EFA.

EFA is generally conducted for purposes of theory development (i.e., identifying fundamental constructs in a domain of interest) or measure development (i.e., determining specific measures that effectively represent constructs). For example, a researcher might be presented with several academic tests (e.g., verbal reasoning, vocabulary, numerical reasoning, and arithmetic skills) and unsure of the academic abilities (e.g., verbal and mathematical ability) underlying these tests. Although a researcher might speculate which tests [Page 649]reflect common underlying abilities, EFA provides a more formal method of assessing which measures reflect the same constructs.

To understand EFA, it is useful to have some insight into the common factor model (CFM), which is a mathematic framework upon which EFA is based. The CFM can be illustrated via a path diagram using the academic tests example in the previous paragraph (see Figure 1). Common factors, also called latent variables, are hypothetical constructs that cannot be directly measured and influence more than one measured variable. In the academic tests example, the common factors are verbal and mathematical ability. Verbal ability influences verbal reasoning, vocabulary, and numerical reasoning, whereas mathematical ability influences numerical reasoning and arithmetic skills.

Figure 1 Path diagram for academic tests example

Measured variables, also called manifest or surface variables, are observed scores that can be directly computed and are drawn from the domain under investigation. In the academic tests example, the measured variables are verbal reasoning, vocabulary, numerical reasoning, and arithmetic skills tests. Unique factors are unobservable variables that account for the variance in measured variables that is unaccounted for by the common factors. Each unique factor only influences one measured variable and includes two components: the specific factor and measurement error. The specific factor consists of systematic sources of influence on a measured variable that are specific to that variable, while the measurement error is random influences on a measured variable.

Another way to represent the CFM is through its formal matrix algebra mathematical expression:

$$P=\lambda \Phi {\lambda }^T+D\psi ,$$

where P is the measured variable correlation matrix in the population and λ is the factor loadings matrix that contains numerical values representing the strength and direction of common factors’ influence on the measured variables. In this matrix, columns represent common factors and rows represent measured variables. The elements comprising the matrix reflect the influence of each common factor on each measured variable.

For example, as seen in Table 1, for every 1 unit of increase in verbal ability, there is a corresponding 0.8 unit increase in vocabulary. Φ is the matrix of correlations among the common factors. λT is the factor loadings matrix transposed (a reexpression of the columns of a matrix as rows). Dψ is the unique factors covariance matrix. In this matrix, the diagonal elements are the unique variances associated with each measured variable, and the off-diagonal elements (covariances among unique factors) are assumed to be zero because unique factors are assumed to be independent of one another.

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