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Under sufficiently large $n$ the test statistic has approximately chi-square distribution with df $/2$, and you can obtain p-value ( $m$ thus must be small enough to give positive df according to the formula). That roughly means all positive eigenvalues of $\bf R-U^2$ except first $m$ ones are close to zero if the $m$-factor model fits. to $\bf \hat$ residuals are random noise, sliding to $0$ as the sample size $n$ grows to infinity. This chi-square tests the null hypothesis that the observed data correlation matrix p x p $\bf R$ is a random sample realization from population having correlation matrix equal to the one returned by the extracted m factors, i.e. The test assumes that the data comes from multivariate normal population. This chi-square goodness-of-fit test which SPSS outputs under Maximum likelihood or Generalized least squares methods of factor extraction is one of the many methods to estimate the "best" number of factors to extract from the data. Why does is the test only available with ML and GLS and not with other methods also offered, e.g. Is it legitimate/useful to convert this particular Chi Square test to RMSEA? The Chi Square test is very sensitive to sample size. If so, is such a method any good? I've heard of all sorts of other ways of deciding the number of factors (scree plots, Kaiser-Guttman rule, MAP test, parallel test) but had never heard of this one before and it seems very problematic. Is it true that what they 'want' me to do is to run this test with increasing numbers of factors selected for extraction, and then to select the number of factors when the test is no longer statistically significant? I also found the formula used, which is as follows Would not get a test of whether the factor loading matrix conformed to Were adequate to explain the covariances among your variables. Goodness of fit, which is a test of the null hypothesis that 3 factors (GLS) as your extraction method, you would get a chi-square measure of If you choose maximum likelihood (ML) or generalized least squares The SPSS documentation seems to suggest that it's a way of deciding how many factors to select (number of factors in factor analysis problem). I was not used to seeing goodness-of-fit tests in the context of EFA (as opposed to CFA), and wondered what the point of it was. I ran an Exploratory Factor Analysis in SPSS recently with ML as the extraction method, and got the following table in my output: