10.4 Practice Problems
Problem 10.1. State two differences between principal component analysis and factor analysis, and give a situation in which each would be the appropriate choice.
Problem 10.2. For a one-factor model with \(p=3\), write \(\Sigma =LL'+\Psi \) in full and count the free parameters against the entries of \(\Sigma \). Is the model testable? Where to start: \(\Sigma \) has \(\tfrac 12 p(p+1)=6\) distinct entries; the model has \(pm+p=6\) parameters.
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Solution. With \(m=1\), \(L=(\ell _1,\ell _2,\ell _3)'\) and \(\Psi =\operatorname {diag}(\psi _1,\psi _2,\psi _3)\), so \[\Sigma = \begin {pmatrix} \ell _1^{2}+\psi _1 & \ell _1\ell _2 & \ell _1\ell _3\\ \ell _1\ell _2 & \ell _2^{2}+\psi _2 & \ell _2\ell _3\\ \ell _1\ell _3 & \ell _2\ell _3 & \ell _3^{2}+\psi _3 \end {pmatrix}.\] There are \(6\) parameters and \(6\) distinct entries, and with \(m=1\) the rotational constraint count \(\tfrac 12 m(m-1)\) is zero. The degrees of freedom are \(6-6=0\): the model is exactly determined, fits any \(\Sigma \) with the right sign pattern, and there is nothing to test. Testability needs \(\tfrac 12\left [(p-m)^{2}-p-m\right ]>0\), which for \(m=1\) requires \(p\geq 4\).
Problem 10.3. Prove that if \(T\) is orthogonal then \(L^{*}=LT\) reproduces the same \(\Sigma \), and explain why this means the factors cannot be identified from the covariance matrix alone.
Problem 10.4. Define the communality of a variable and explain what a low communality indicates about that variable’s place in the analysis.
Problem 10.5. A maximum likelihood factor analysis returns an estimated specific variance of \(-0.03\). What is this called, and what does it tell you about the model?
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