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I received the following email:

My project involves rewording a questionnaire and seeing if the internal reliability and CFA results improve. There are 30 items that are meant to form 4 subscales. Each item is scored always, sometimes, never. I have heard that I should check that each of the items is normally distributed before putting them into CFA. I've found that all items are normally distributed except 2 items. I was then going to transform these two items using a log transformation, but it seems odd to me to transform individual items, especially when the subscales themselves are normally distributed. I guess I just want to check: do I need each item to be normally distributed or not, and if so, is a log transformation okay if I recode 0-1-2 to 1-2-3?

So in summary the questions are:

  • Do you need to check whether individual test items are normally distributed when performing CFA?
  • If individual items are not normally distributed, should you transform them before performing CFA?
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  • $\begingroup$ Just to be clear from the start - none of the items are actually normally distributed, even the ones stated in the question to be normal. $\endgroup$
    – Glen_b
    Oct 3 '13 at 9:20
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Ordered categorical items and normality:

First, ordered categorical items are discrete and lumpy. In particular, 3-point response scales lack the granularity required to even provide a rudimentary approximation of normality. When you have more response options in your ordered categorical variable, the item has more potential to approximate a normally distributed variable.

Ordered categorical items have ceilings and floors. If the mean is close the the ceiling or floor, then they are almost always skewed with the tail pointing away from the floor or ceiling as the case may be.

There is a model of responding to ordered categorical items, which suggests that a latent continuous numeric variable underlies responses. But that is different to examining the raw variable.

Whether to look at normality of items when doing CFA: When looking at typical self-report scales on 3,4,5 point scales, I don't think it makes much sense to look at normality.

Generally, you find that the size of the skew is related the degree to which the item deviates from the scale mid-point, and moves towards either the minimum or maximum possible value. Thus, I imagine your most skewed items would be those on a 0-1-2 scales that have means either close to 0 or close to 2. That said, there are exceptions where people can focus their responses towards either extreme (e.g., see some contaminated Amazon 5-star ratings).

So in summary, I think it is useful to get a sense of how people have used the response scale. Is it consistent with the most common responses being around the mean for the item or are scores around the extremes, or is there something else guiding responses. The aim here is just to build an understanding of how people have used the scale.

In most psychological contexts, I find that the mean captures most of what is going on with the distribution. The main preliminary step I like to do is to see whether any items suffer from severe floor or ceiling effects. So for example, if you had an item with a mean below 0.4 or above 1.6 on a 0-1-2 scale, you might want to think about whether the item is adequately discriminating people. I wouldn't automatically drop such an item, but I would think about what it is contributing.

Should non-normal items be transformed before CFA? As already mentioned, the items are not normally distributed anyway. Furthermore, the natural scale of ordered categorical items, prevents extreme outliers, and limits extreme skew. Furthermore, items will typically be scored using their original scaling, so it is best to leave them as is. For this reason, I tend not to transform individual items when performing CFA.

A second point relates generally to how to model ordered categorical items. While you can do CFA on individual items, there are also a range of more advanced, and arguably better, alternative approaches that are designed to explicitly model ordered categorical items:

  • MPlus has various models for ordinal items with thresholds between items
  • Amos has some models for modelling ordinal items
  • Optimal scaling PCA
  • Factor analysis on polychoric correlations

You might want to read this presentation by Bowen and Wegmann where they discuss solutions.

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  • $\begingroup$ Always answering your own questions... $\endgroup$
    – Taal
    Oct 3 '13 at 6:59
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    $\begingroup$ @Taal Yep. I get lots of questions from students and random people on the internet. I'm going to answer them anyway. So I find it more motivating to integrate it into the stack exchange workflow. Answers benefit the googling masses, and I can see what others have to say :-) I prefer it if the person posts the question themselves as per this post, but if it is self-contained, it is often easier if I just post it. $\endgroup$ Oct 3 '13 at 7:02
  • $\begingroup$ Why are you speaking of ordinal variables at all? Did your email correspondent stated that he sees his variables ordinal? It would be foolish to speak about normality or non-normality of an ordinal variable. $\endgroup$
    – ttnphns
    Oct 3 '13 at 7:56
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    $\begingroup$ @ttnphns In the post, it says how each item was scored: "Always", "sometimes", "Never". That sounds like ordinal data to me! $\endgroup$
    – Peter Flom
    Oct 3 '13 at 10:48
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    $\begingroup$ Regarding the question, I tried to convey the question as it was sent to me, embodying the problematic assumption that you could even check for normality for a 3-point variable. I felt that any answer should address that assumption. I figure that if one student has made this assumption other people will make the same assumption in the future and may stumble upon this question in their google searches. $\endgroup$ Oct 3 '13 at 23:29

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