# What are the metrics to assess the quality of a multiple correspondence analysis (MCA) model?

We are trying to implement a multiple correspondence analysis (MCA) model. I was looking for metrics to assess the quality of an MCA to evaluate our model. Sadly, I didn’t find much literature about the topic (how to implement an MCA) but I found these 3 papers about factor analysis:

This paper recommends 3 metrics:

• The “factor determinacy index”
• Test for construct replicability
• Explained common variance

This paper gives a more detailed approach on how they implemented their exploratory factor analysis :

• First, it tests the adequacy of the sample and data for factor analysis using the Kaiser-Meyer-Olkin (KMO), Bartlett’s Test of Sphericity, communality, testing for ceiling and floor effects, Test of Sampling Adequacy and more…
• Then it proceeds to test the model using robust diagonally weighted least squares (RDWLS), the Kaiser’s criteria (Eigenvalue ≥1), the scree test, the cumulative variance explained rule (>40%), the robust parallel analysis, the Comparative Fit Index (CFI), the goodness of fit index (GFI), the weighted root mean square residual (WRMR), the root mean square error of approximation (RMSEA), the non-normed fit index (NNFI), the χ2…

This paper distinguishes between

• Exploratory factor analysis that requires: Kaiser’s criterion, scree test the cumulative variance
• Confirmatory Factor Analysis that requires: The chi-square test, Comparative Fit Index, Root Mean Square Error of Approximation

My first question is; are these metrics correct? These papers all relate to factor analysis, so can I really use the metrics they recommend for an MCA? If yes, which metrics, because these papers differ quite a bit?

Second, the SAS paper recommends the use of only 3 metrics for an EFA. But the second paper used many more metrics, even though it implements an EFA. Which one should I trust? Are 3 metrics enough or should I also check the CFI, RMSEA, WRMR, the Chi-square etc…

My final question is about the Chi-square test. My intuition about the Chi-square test is that when we reject the null hypothesis, it means that the variables are not independent. Does this implicitly mean there is a common factor (hence its use in MCA for testing for common factors)? So if I am correct, we need to reject the null hypothesis to test for the existence of at least one factor; is my understanding correct ?

Thank you very much,

• (Multiple) Correspondence analysis is a way to visually represent, in a low dimensional space, chi-square distances between categories forming a p-way frequency table. It is like PCA, only for nominal, categorical data. Strictly, it is not like Factor analysis which is a latent variable modelling technique. Generally, it is not a good idea to compare it with FA. Jul 20, 2021 at 14:45
• Why is it not a good idea to compare MCA with FA? Aren't both PCA and MCA types of FA? Jul 20, 2021 at 15:46
• Strictly speaking - not. FA is a modelling of latent variables. PCA (and MCA) are only dimensionality reductions which yield dimensions that could be tentatively interpreted as latent variables. Here on this site is a huge collection of posts discussing PCA vs FA topic. Jul 20, 2021 at 16:08
• For MCA, like for PCA, I would recommend just look only on the overall data variability explained, plus interpretability of categories' juxtapositions on the map produced. The majority of approaches/indices/tests you are mentioning may have sense in FA and have no sense or applicability for MCA. Jul 20, 2021 at 16:20
• Oh all right, thank you. So you're saying these metrics are good for FA but for MCA the ones you said are enough. Btw, could you post the link to the site discussing PCA vs FA? Thank you ! Jul 21, 2021 at 8:54