2 Building a Phenotypic Factor Model Mx Script

The factor model may be written as

$$Y_{ij} = b_i X_j + E_{ij}$$

with

$$\begin{eqnarray*} i & = & 1, \cdots, p \mbox{ (variables)}\\ j & = & 1, \cdots, n \mbox{ (subjects)} \end{eqnarray*}$$

and where the measured variables $Y$ are a function of a subject's value on the underlying factor $X$ (henceforth the $j$ subscript indicating subjects in $Y$ will be omitted). These subject values are called factor scores. Although the use of factor scores is always implicit in the application of factor analysis, they cannot be determined precisely but must be estimated, since the number of common and unique factors always exceeds the number of observed variables. In addition, there is a specific part ($E$) to each variable. The $b$'s are the $p$-variate factor loadings of measured variables on the latent factors. To estimate these loadings we do not need to know the individual factor scores, as the expectation for the $p\times p$ covariance matrix ($\Sigma_{Y,Y}$) consists only of a $p\times m$ matrix of factor loadings (B) ($m$ equals the number of latent factors), a $m \times m$ correlation matrix of factor scores (P), and a $p\times p$ diagonal matrix of specific variances (E) :

$$\Sigma_{Y,Y} = {\bf B \bf P \bf B'} + {\bf E}.$$ (59)

In problems with uncorrelated latent factors, P is an identity matrix, so equation 10.1 reduces to

$$\Sigma_{Y,Y} = {\bf B \bf B'} + {\bf E}.$$ (60)

Thus, the parameters in the model consist of factor loadings and specific variances (sometimes also referred to as error variances).