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# Confirmatory Factor Analysis

Part 2. Marketing Research

1. Richard J. Fox

Published Online: 15 DEC 2010

DOI: 10.1002/9781444316568.wiem02060

## Wiley International Encyclopedia of Marketing

#### How to Cite

Fox, R. J. 2010. Confirmatory Factor Analysis. Wiley International Encyclopedia of Marketing. 2.

#### Author Information

1. University of Georgia, Athens, GA, USA

#### Publication History

1. Published Online: 15 DEC 2010

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### Abstract

Confirmatory factor analysis is a statistical technique used when investigating the structure of multivariate data. Each of a set of n observed variables is represented as a linear combination of m (m < n) unobserved latent variables or factors, plus an independent error term. The n × n covariance matrix corresponding to the n-dimensional observed variables contains the individual variable variances on the diagonal, and the covariances for all pairs of the individual observed variables in the corresponding off-diagonal positions. A matrix equation, called the covariance equation, relates this n × n covariance matrix to the matrix whose n rows correspond to the coefficients of the m latent factors referred to as factor loadings, and the covariance matrices of the unobserved latent factors and the errors.

Confirmatory factor analysis is used to test whether a hypothesized structure is appropriate for multivariate data. The hypothesized structure constrains the matrices appearing in the covariance equation. Individual covariances among the latent factors or among the error terms can be assumed equal or set to zero. Likewise, selected variances (diagonal entries) may be presumed to be equal within each of these matrices. Also, selected factor loadings may be set to zero. A random sample of multivariate observations is used to estimate the corresponding sample covariance matrix with and without the constraints imposed by the hypothesized structure. The “constraint free” covariance matrix is the matrix containing the typical sample descriptive statistics. Maximum likelihood estimation is typically used to determine estimates, with the constraints imposed, of the covariance matrices of the latent factors and errors, and the matrix of factor loadings. A statistical test is conducted to determine whether the hypothesized structure fits the data by comparing the sample covariance matrix to its counterpart produced from the covariance equation using the matrices estimated by maximum likelihood (constraints imposed).

### Keywords:

• confirmatory factor analysis;
• covariance;
• maximum likelihood estimation;
• goodness of fit;
• latent factors