# Confirmatory Factor Analysis

Historically, factor analysis has been the prime statistical technique for the development of structural theories in social science, such as the hierarchical factor model of human cognitive abilities, or the Five Factor Model of personality. In confirmatory factor analysis the researcher specifies the number of factors which underlie a set of observed variables, together with the relationships between the observed variables and the factors. In the example below, three observed variables (Conscientiousness, Emotional Stability and Agreeableness) load on the factor of Stability, and two observed variables (Extraversion and Openness to Experience) load on Plasticity. This model is hypothesized in advance and the data to test the model comprises the correlations between the observed variables, known as the sample correlation matrix. The parameters of the model (factor loadings and error variances) are estimated so that the factor model generates a correlation matrix which is as close an approximation to the sample correlation matrix as possible. The preferred method of estimation for continuous data is maximum likelihood. The discrepancies between the sample correlation matrix and the model generated correlation matrix are used to assess the fit of the model. Factor models which fit the data well are judged to be plausible, while those which do not are rejected.

Figure 1. The General Factor of Personality (GFP) going to the Big Two to the Big Five using the medians from Digman’s (1997) 14 samples. From Rushton and Irwing (2008).

Suitable sources in progressive order from basic to advanced:

Kline, R. B. (2005). Principles and Practice of Structural Equation Modelling. London: The Guilford Press.

Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York, N.J.: Guiford Press.

Bollen, K. A. (1989). Structural equations with latent variables. New York, NJ: Wiley.

Mulaik, S. A. (2010). Foundations of factor analysis. London: Chapman & Hall.

Useful website: Mplus

Recommended computer package: Muthén, L. K. and Muthén, B. O. (2010). Mplus Version 6. Los Angeles, CA: Muthén & Muthén