10.3 Estimation
Two methods are standard.
The principal factor method estimates the communalities, subtracts the resulting \(\widehat {\Psi }\) from \(S\) or \(R\), and takes the spectral decomposition of the remainder, setting \(\widehat {L} = \left (\sqrt {\widehat {\lambda }_1}\widehat {\underline {e}}_1, \dots ,\sqrt {\widehat {\lambda }_m}\widehat {\underline {e}}_m\right )\). It requires no distributional assumption and always produces an answer.
The maximum likelihood method assumes \(\underline {X}\) multivariate normal and maximises the likelihood over \(L\) and \(\Psi \) subject to a uniqueness condition that removes the rotational freedom. It is iterative, may fail to converge, and can return a Heywood case — an estimated specific variance that is zero or negative, which is impossible and signals that the model with that \(m\) does not fit. Against these drawbacks it offers what the other method cannot: a likelihood ratio test of the adequacy of \(m\) factors, \[-2\ln \Lambda = \left (n-1-\tfrac {2p+4m+5}{6}\right ) \ln \frac {\left |\widehat {L}\widehat {L}'+\widehat {\Psi }\right |}{\left |S\right |} \ \overset {\cdot }{\sim }\ \chi ^{2}_{\left [(p-m)^{2}-p-m\right ]/2}.\]
Note 10.7. The degrees of freedom are worth reading rather than copying. They count \(\tfrac 12 p(p+1)\) free entries in \(\Sigma \) against \(pm+p\) parameters, less the \(\tfrac 12 m(m-1)\) constraints imposed to fix the rotation. When the count is negative the model has more parameters than the covariance matrix has entries and cannot be tested at all — which places a hard upper limit on \(m\) for a given \(p\), independent of any data.
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