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I am developing a questionnaire to measure four factors which constitute spirituality, and I would like to ask the following question:

Are data transformations on non-normal data necessary for an exploratory factor analysis when using the principal axis factoring extraction method?

I finished screening my data yesterday, and I found that 3 out of 20 questions are positively skewed while 1 out of 20 is negatively skewed (Question 6 = 4.88, Question 9 = 7.22, Question 12 = 11.11, Question 16 = -6.26). I also found that 1 of the questions (out of 20) is leptokurtic (Question 12 = 12.21).

I chose the principal axis factoring extraction method because I read that it is used on "severely non-normal data" while maximum-likelihood is used on normal data, but:

  1. How would I know if my data is "severely" non-normal?

  2. If my data is "severely non-normal", does this mean that I can leave the data as it is now (not transform it) and analyze it using the principal axis factoring extraction method? Or do I need to transform the data before proceeding with the EFA?

  3. If I do need to transform the data, what transformations would I use for positively skewed, negatively skewed, and leptokurtic items?

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Factor analysis is essentially a (constrained) linear regression model. In this model, each analyzed variable is the dependent variable, common factors are the IVs, and the implied unique factor serve as the error term. (The constant term is set to zero due to centering or standardizing which are implied in computation of covariances or correlations.) So, exactly like in linear regression, there could exist "strong" assumption of normality - IVs (common factors) are multivariate normal and errors (unique factor) are normal, which automatically leads to that the DV is normal; and "weak" assumption of normality - errors (unique factor) are normal only, therefore the DV needs not to be normal. Both in regression and FA we usually admit "weak" assumption because it is more realistic.

Among classic FA extraction methods only the maximum likelihood method, because it departs from the characteristics of population, states that the analyzed variables be multivariate normal. Methods like principal axes or minimal residuals do not require this "strong" assumption (albeit you can make it anyway).

Please remember that even if your variables are normal separately, it doesn't necessarily guarantee that your data are multivariate normal.

Let us accept "weak" assumption of normality. What is the potential threat coming from strongly skewed data, like your, then? It is outliers. If the distribution of a variable is strongly asymmetric the longer tail becomes extra influential in computing correlations or covariances, and simultaneously it provokes apprehension about whether it still measures the same psychological construct (the factor) as the shorter tail does. It might be cautious to compare whether correlation matrices built on the lower half and the upper half of the rating scale are similar or not. If they are similar enough, you may conclude that both tails measure the same thing and do not transform your variables. Otherwise you should consider transforming or some other action to neutralize the effect of "outlier" long tail.

Transformations are plenty. For example, raising to a power>1 or exponentiation are used for left-skewed data, and power<1 or logarithm - for right-skewed. My own experience says that so called optimal transformation via Categorical PCA performed prior FA is almost always beneficial, for it usually leads to more clear, interpretable factors in FA; under the assumption that the number of factors is known, it transforms your data nonlinearly so as to maximize the overall variance accounted by that number of factors.

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  • $\begingroup$ Thanks for your answer! And would you recommend removing question 12 since it is both positively skewed and leptokurtic, or would it be better to leave it in and transform it? $\endgroup$ – Madeline Aug 21 '12 at 8:33
  • $\begingroup$ I'd refrain from direct advice, being not in your shoes. Speaking most generally, chronically very skewed (= too difficult or too easy) or leptokurtic (insensitive to differences among typical respondents) questions are psychometrically "bad" ones, and are worth to revise. $\endgroup$ – ttnphns Aug 21 '12 at 8:49
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I just post what I learned from Yong and Pearce (2013).

To perform a factor analysis, there has to be univariate and multivariate normality within the data (Child, 2006)

Yong, A. G., & Pearce, S. (2013). A beginner’s guide to factor analysis: Focusing on exploratory factor analysis. Tutorials in quantitative methods for psychology, 9(2), 79-94. DOI:10.20982/tqmp.09.2.p079

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