I have a basic question about confirmatory factor analysis. When performing CFA, Chi-square test is often used. I read from the book that after using MLE for factor extraction (computing covariance matrix), the test statistic $v=(N-1)F_{ml}$ is distributed as a chi-squared distribution. For example, on Pg. 63 of Statistical Analysis of Management Data by Hubert Gatignon:

Based on large-sample distribution theory, $v=(N-1)\hat{F}$ (where $N$ is the sample size used to generate the covariance matrix of the observed variables and $\hat{F}$ is the minimum value of the expression $F$ as defined by Equation 4.15) is distributed as a chi-squared with number of degrees of freedom...

However, I only know by Pearson's theory, the test statistic $T=\sum_{j=1}^r \frac{(v_j-\mathbb{E}v_j)^2}{\mathbb{E}v_j}\rightarrow\chi^2_{r-1}$ where $v_j$ is the observation in category $j$. The test statistic is clearly different from $v=(N-1)\hat{F}$. Could someone explain why $v$ can be interpreted as a test statistic distributed as $\chi^2$? I searched online but couldn't find much explanation.

In addition, I understand we are trying to test the covariance matrix, but I fail to see how it is connected to $v$ and it would be great if someone can explain this too!



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