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quick question regarding principal component analysis (PCA) and multiple correspondence analysis (MCA).

I am trying to extract scores to create index variables from ordinal data. I first used PCA, but then have since used MCA. The results are nearly identical, except the scores for one factor/index variable is negative in the MCA compared to the the PCA. Essentially the scores are multiple by -1 for this index variable. This then means when I go to use these in a Cox regression, I get opposite hazard ratio for this index variable (reflected across HR=1).

Any idea to why this is? Or which one is correct?

Doing a bit more reading on it, I see that the assignment of positive and negative signs can be arbitrary. But since I need to use these scores to create an index variable, whether it is negative or positive has an impact on the direction of the results. How do I know which it is?

capmo
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  • You really need to give more details for this to be answerable! Did you use pca/mca on the predictor variables for the regression? How? What are those variables, what do they measure? Can you give details of the regression you fitted, for both cases? – kjetil b halvorsen Apr 25 '22 at 17:33
  • Even when you run a PCA on the same data with different software you often find some scores are negated. That's because the PCs determine subspaces, not specific directions within them. All are equally correct. The duplicate explains and illustrates this. We have other threads on PCA that make this point as well. – whuber Apr 25 '22 at 17:34
  • I ran the PCA and MCA on SES variables to create 2 factors (they seem to load on well). I then extracted the scores and used and added them to my original dataframe. I then used these in the regression model. I am then using these as predictor variables in a regression. Hope. this is a bit more clear. – capmo Apr 25 '22 at 17:35
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    A nice analogy occurs to me. PCA is like identifying a street on a map. In one calculation you have found that street and seen a sign that says it is northbound. Another calculation showed you the street from the other side, where there's a sign that says it's southbound. Which direction is the correct one? – whuber Apr 25 '22 at 17:37
  • Doing a bit more reading on it, I see that the assignment of positive and negative signs can be arbitrary. But since I need to use these scores to create an index variable, whether it is negative or positive has an impact on the direction of the results. How do I know which it is? – capmo Apr 25 '22 at 17:55
  • You have to study the coefficients to see how this PC is related to the original variables. There's no general answer, because increases in some of the variables might need to be associated with larger values of the index while decreases in other variables might need to be associated with larger index values. – whuber Apr 26 '22 at 13:01

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