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How do I interpret factor loadings? A related question, I think, is how do I calculate factor scores?

In these forums, I've read that factor score = sum of loadings times standardized item scores. However, this paper http://pareonline.net/pdf/v14n20.pdf presents a number of ways of calculating factor scores, including sum of loadings times raw item scores, which is called "weighted sum scores". Because my items are all dummy variables, I think that the weighted sum scores are easier to interpret.

So let's say I want to go with this method of score computation. Here is where I start getting confused. I was initially coding my dummy variables as 0 = No / 1 = Yes. This means that factor score reduces to sum of loadings for the items where the answer is "Yes". However, then I thought, that maybe it would be more intuitive to code the dummy variables as -1 = No / 1 = Yes. This way, the factor score is the sum of loadings for the items where the answer is "Yes" minus the sum of loadings where the answer is "No".

Clearly, this cannot be correct. The correlation matrix of the data is the same, whether I code my dummy variables as -1/1 or 0/1. This means that the loadings are identical, regardless of the coding. However, the two codings lead to different factor scores. Not only are the factor scores different, you cannot get from one to the other through some simple transformation, such as multiplying by some constant.

My next thought is that maybe the loadings that the software (the unfortunately named factanal in R) is spitting out assume a particular kind of item score (standardized?) and a particular kind of factor score calculation (what kind?). Is this true? If that's the case, I need to adjust or recalculate the loadings to be consistent with my item score (dummy variables, not standardized) and factor score calculation (weighted sum scores). Is this correct? Do I need to adjust the loadings in some way? I don't think this is right, as I haven't seen any discussion of this anywhere.

Back to my original question, how do I interpret factor loadings? Do I need to adjust them in some way from what's reported by the software? Is the sum of loadings times raw item scores a valid way to calculate factor scores? Which coding for dummy variables would lead to a more intuitive loadings or scores?

Jessica
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    Jessica, the paper you link to is good one, I'm familiar with it. Please read it attentively. What I might recommend you though is not pay much attention to "non-refined methods" - they are very crude, were used mostly during pre-computer era. Read "refined methods". [This]http://stats.stackexchange.com/a/126985/3277) answer of mine presents formulas (in matrix algebra) and discusses factor scores very similarly to how that paper does it. – ttnphns Dec 02 '15 at 20:39
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    What, however, bothers in your question is that you seemingly do FA with dummy variables. Classic factor analysis [is not](http://stats.stackexchange.com/a/16335/3277) suited well - if at all - for binary data (and twice so with _dummy_ binary variables). – ttnphns Dec 02 '15 at 20:42
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    @ttnphns - excellent link to your other answer. So let me see if I get this right. 1. A loading is the correlation between an item and a factor. That's easy enough. 2. There is only one correct factor score, but many different estimates of it. Correct? – Jessica Dec 03 '15 at 13:31
  • Yes. I'd slightly changed your wording, though: "There is only one correct factor _value_ (which will eternally stay unknown to us), but many different ways to compute the approximate estimate of it - the factor _score_". – ttnphns Dec 03 '15 at 13:45
  • What's an appropriate data reduction technique for dummy variable data? – Jessica Dec 03 '15 at 13:54
  • Not quite get what you mean and want. Set of binary _dummy_ variables (aka indicator variables) is how a categorical variable is recoded in order to input it in some types of analysis. Try to read about Correspondence analysis and Multiple Correspondence analysis which are used to map such data in a low dimensional space. – ttnphns Dec 03 '15 at 14:01

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