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Several articles I've read stated that MCA and PCA both work as "reducing dimensionality" tools but MCA is used for categorical variables and PCA is used for numerical variables. But is there any difference with the result both methods produce? I have never used MCA nor PCA before so my knowledge is limited to only this.

Let's say I have 9 variables, all of them are about power over deciding different types of expenditure in the household but with the same 5-point likert scale (1: least power to 5: most power). Then I run both MCA and PCA (PCA passed the Bartlett Sphericity Test & KMO Measure) and obtain the predictor variables. If I generate a variable that is the sum of all responses from 9 variables (that should amount to maximum of 45) and run a twoway scatterplot with the MCA predictor and PCA predictor separately, I got a graph that looks like this: the sum of response variable and PCA graph has positive relationship, while sum of response variable and MCA has negative relationship.

My question is: is this normal? Should MCA and PCA results be similar? How do I interpret both results?

Lily RR
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    This is an interesting topic and having been touched occasionally on this site. One answer of mine discusses affinities and differences between PCA, biplot and simple (2-way) CA. – ttnphns Apr 02 '23 at 19:14
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    Now, regarding Multiple CA. It is CA with more than 2 categorical variables. MCA is the dim. reduction analysis for nominal categorical variables. PCA is the linear dim. reduction analysis for scale (interval) variables. There exist the more general method Categorical PCA (CATPCA). Now, MCA can be seen a special case of it: "CATPCA with nominal level of quantification applied to all the (nominal) variables". If all the variables are dichotomous (such as YES/NO) MCA and PCA become essentially identical, though their customary results output may differ in form. – ttnphns Apr 02 '23 at 19:28

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