Chi-Square test for SEM in R

Question

I am working with the sem library in R for confirmatory factor analysis (CFA). As a part of the output, a chi-square test is returned.

I am totally unaware of what the null and alternative hypotheses are. I skimmed through the "help" section in R and couldn't spot the hypotheses being tested.

Does anyone know what the null hypothesis and alternative hypothesis are for running the CFA under the sem package in R?

Answer 1

Cited from Wikipedia (http://en.wikipedia.org/wiki/Confirmatory_factor_analysis#Chi-squared_test):

The chi-squared test indicates the difference between observed and expected covariance matrices. Values closer to zero indicate a better fit; smaller difference between expected and observed covariance matrices. Chi-squared statistics can also be used to directly compare the fit of nested models to the data. One difficulty with the chi-squared test of model fit, however, is that researchers may fail to reject an inappropriate model in small sample sizes and reject an appropriate model in large sample sizes. As a result, other measures of fit have been developed.

Or elsewere:

The Chi-Square value is the traditional measure for evaluating overall model fit and, assesses the magnitude of discrepancy between the sample and fitted covariances matrices.

You have to understand the basic principle of structural equation modeling, which is to try to reproduce the sample covariance matrix based on the model you've specified (e.g. your confirmatory factor analysis).

Thus, the null hypothesis states that the specified model holds in your sample data. The corresponding alternative hypothesis is that the reproduced covariance matrix is any symmetric positive definite matrix.

However, there has been an extensive discussions around this measure and a lot of simulations were done to illustrate that this measure is biased under various conditions. Most textbooks also wrap up this discussion and explain which alternative goodness of fit measures could and should be used. You can also refer to the aforementioned reference for further details (http://www.ejbrm.com/issue/download.html?idArticle=183).