# Do data transformations before factor analysis need to be consistent across different variables?

(This question continues the previous one)

I am creating a questionnaire, and I have identified 3 questions which are skewed (2 positively skewed & 1 negatively skewed). I successfully transformed two of the questions using Lg10 and inverse of Lg10 on SPSS, but the second positively skewed question is still positively skewed even after the Lg10 transformation. My questions are the following:

1. Is it "okay" for the question to still be positively skewed after the transformation? Is any further action needed (any further transformation(s))?
2. Can I use a different transformation on this specific question (the remaining positively skewed one) or do I have to use the same transformation for all of the skewed questions?
3. What is the next "strongest" transformation after Lg10 on SPSS? How would it be entered on SPSS? (e.x. negatively skewed question = Lg10((Max. Score + 1) - Question))

Regarding 1) Factor analysis is based on correlations/covariances. When a highly skewed variable is part of a correlation, the correlation can be affected by the extreme points. This will affect the factor analysis, although I do not know of literature on the extent of the effect (it's probably been studied, though).

Regarding 2) You do not need to use the same transformation on each variable. But transforming variables in different ways and then doing factor analysis can lead to factors that are somewhat hard to interpret.

Regarding 3) I don't know SPSS, sorry.

More generally, what is the nature of these questions? Are they Likert-type scales? Physical measurements? Or what? Ideally, you could tell us what they actually mean.

• Thanks for your response! The questions are Likert-type scales. Example: "I do not have a reason for living" with answer choices 1 - 6 (1 = strongly disagree/ 6 = strongly agree). – Madeline Aug 21 '12 at 11:52
• I would NOT transform Likert type items in any way. Items that are very highly skewed, with only 7 levels, are items where nearly all gave a response at the end. These items may not be useful at all; they add little information. I would first look at Cronbach alpha for items that don't correlate well with the others. I'd consider deleting the highly skewed items. However, all this depends on SUBSTANCE and what the actual questions are. It may well be that factor analysis is not the best tool. – Peter Flom - Reinstate Monica Aug 21 '12 at 12:15

I agree with Peter that scores on 1-6-range items should not be transformed - not because it's theoretically wrong but because it's unlikely to help. One option is, as I think Peter implies, to find sound ways to combine such items into scales, which may have more interval-level properties than do the items themselves. These scales could then be inputs for further procedures that address your research questions.

A second option is actually to use the original items in a factor analysis. This goes against common wisdom, and it won't help you if you are using maximum likelihood factor extraction, which depends on normally distributed items. But I am often surprised at how useful principal axis factor analytic results turn out even when based on skewed distributions from what were originally considered ordinal-level items. Commonalities, variances explained, and factor loadings often come out satisfyingly high despite this apparent violation of best practices. If you try this, you may find it's helpful, but don't take the loadings and other results too literally: they'll be approximations. E.g., I wouldn't report loadings to more than 1 decimal place.

• I'm not to say I disagree with you, but I (a kind of) feel lack of clearness in your response. For example, you don't explain why transformation of a 6-point rating scale is unlikely to help and what you mean under "help" here. Also skewed distributions from ordinal-level items looks like farrago. Linear FA don't have ordinal input, it treats everything as interval. – ttnphns Aug 21 '12 at 16:44
• @rolando2 - Thank you for your response, but as ttnphns said, I would appreciate it if you could clarify your answer. The reason why I am doing an exploratory factor analysis on the individual questions is so that I can identify factors and then compare these factors to other mental health constructs such as depression, anxiety, and stress. Within psychological research, the steps for questionnaire development are principal components analysis (which I already did), EFA (to develop factors), then confirmatory factor analysis (to confirm factors). – Madeline Aug 22 '12 at 5:40
• I already tried doing the EFA but backtracked upon realizing that I didn't check for normality in my data. When I checked for skew and kurtosis, I realized that there were issues with 4 out of 20 questions. I don't know what you mean by "don't take the loadings and other results too literally" - how would it be possible to report results when they are not exact? In journals which report psychological research, I have only seen loadings reported to 2 decimal places. – Madeline Aug 22 '12 at 5:45
• Think about what it means to estimate a quantity or parameter, such as a factor loading. Then think about what it means for a parameter to have uncertainty, as quantified by a standard error. This is always the case even under random sampling from a population. Now add to that idealized level of uncertainty an additional element... – rolando2 Aug 22 '12 at 11:29
• ...you are not only dealing with sampling error but also inexactness that comes out of trying to approximate, with non-interval-level data, what you would ideally learn through interval-level data. Now consider the fact that the data are skewed, taking you further away from your idealized situation. All of this adds up to a considerable level of uncertainty. It's highly debatable whether confirmatory factor analysis would be appropriate under these conditions. – rolando2 Aug 22 '12 at 11:29

In my experience, transforming data using any combination of techniques (Square-root, Log, Inverse, etc.) is entirely valid from a statistical point-of-view. The only issue is that interpretation becomes increasingly difficult. For example, if two items load on the same factor and one has a log transformation, then you must interpret the two as related in some nonlinear fashion.

Moreover, Norris & Aroian (2004) indicate transformations may not even be necessary: http://www.ncbi.nlm.nih.gov/pubmed/14726780