Matrix-Based Introduction to Multivariate Data Analysis by Kohei Adachi

Author:Kohei Adachi
Language: eng
Format: epub, pdf
Publisher: Springer Singapore, Singapore

10.3 Distributional Assumptions for Factors

The factor vector f is assumed to have the multivariate normal (MVN) distribution whose average vector is 0 m and whose covariance matrix is Φ, respectively:

(10.4)

Here, the covariance matrix Φ (m × m) is constrained to be a correlation matrix with

(10.5)

It is equivalent to the assumption that factor scores are standard ones with their variances ones.

Let us consider the rationale of the above assumptions for averages and covariances. The average vector 0 m is matched by supposing that a data set to be analyzed contains centered scores. The reason for assuming the factor scores to be standard ones is that factors are unobserved latent variables ; thus, their variances can be freely determined; we may consider the values of a factor to be distributed over the range [−100, 90], [−50, 60], or [−0.01, 0.01]. For this reason, the variance is usually set to one, as it is a comprehensible value. This implies that the factor scores are standardized and the covariance matrix between factors is their correlation matrix. Thus, Φ in (10.5) is called a factor correlation matrix.

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