Explaining degrees of freedom when testing the number of common factors

Question

To test the adequacy of the $m$ common factor model, the null hypothesis is $H_0: \Sigma = LL'+\Psi$, where $\Sigma_{p\times p}$, $L_{p\times m}$ is the loading factors matrix, and $\Psi_{p\times p}$ is a diagonal matrix with variances of the specific factors in the main diagonal.

When using the MLR statistic, under $H_0$, we get an approximate chi-squared distribution, with degrees of freedom $$dim(\Theta)-dim(\Theta_0)=\frac{1}{2}p(p+1)-[p(m+1)-\frac{1}{2}m(m-1)]$$

I can explain all terms, except the last one, $\frac{1}{2}m(m-1)$. Where does it come from?

For example, the $1/2 p(p+1)$ is the number of parameters in $\Sigma$ (symmetric), $pm$ is for $L$, $p$ is for $\Psi$

Answer 1

By chance, I stumbled upon the answer to my question.

For the matrices $L, \Psi$ to be uniquely determined there is an extra condition which is imposed when finding the ML estimators: $$L'\Psi^{-1}L=\Delta$$ where $\Delta$ is a diagonal matrix. Well, the zeros in the off the main diagonal impose $\frac{1}{2}m(m-1)$ conditions. And these need to be subtracted to $dim(\Theta_0)$.

Answer 2

$\frac{1}{2}p(p+1)-[p(m+1)-\frac{1}{2}m(m-1)] = \frac{(p-m)^2-(p+m)}{2}$

is the $df$ in the likelihood ratio chi-square test statistic. The test is used in Exploratory Factor analysis (mostly ML or GLS methods of extraction, with normal population assumed) as well as in Confirmatory Factor analysis, to test if the $m$ factors is a good fit. The null hypothesis of the test is that the considered $p$-variable correlation (or covariance) matrix is reasonably a random sample realization of the population matrix defined by the factors (factor loadings). Or in other words, that the residual correlations between the two matrices are random noise.

$\frac{1}{2}p(p+1)$ is the the number of correlations, including diagonal, too.

The quantity in brackets is the number of free parameters implied in the model. It is comprised of: (i) the number of pattern loadings of common factors, $pm$; (ii) the number of unique factors loadings, $p$; the number of correlations between the common factors, $\frac{1}{2}m(m-1)$ (subtracts because, to test for the number of factors, we fix factor correlations to be zero).

In confirmatory factor analysis, the number of free parameter may be less than that maximal shown in brackets, because you might wish to constrain the model by setting some parameters (loadings or factor correlations) to specific values, for example zero.