RE: Normalization of 5-Likert scale variables

[DY Kim]

My variable is not normally distributed. Skewness = -1.625266 and Kurtosis = 6.034496. My variable is 5-likert scale. Is anyway I can reduce the skewness and kurtosis problem. Does Stata help address the skewness of variables? Any help would be appreciated.

[Bruce Weaver]

Hello DY Kim. The FAQ includes this bit of advice for posters:

Asking about your real problem, not something else, may seem too obvious to mention, but do check http://xyproblem.info/.

I find myself wondering if you are "asking about your attempted solution rather than your actual problem." What is the actual problem? What are you trying to do? And why do you think normality is required? Thanks for clarifying.

[DY Kim]

Bruce, thank you for replying to my question!

Although people may have different criteria for skewness, some argued that if skewness in absolute value is greater than 1, the distribution is highly skewed. My variable (-1.625266) is greater than 1 in absolute value. As a rule of thumb, kurtoisis in absolute value should be greater than 3. My variable (6.034496) is greater than 3. I wanted to reduce the skewness and kurtoisis problem if possible. I hope my question is clear enough for you to answer. Thank you again.

[Bruce Weaver]

But what research question are you trying to answer? What type of analysis would you use if the data were "normally distributed"?

[DY Kim]

My research question is whether respondents' demographic and attitudinal characteristics influence their perceptions of job engagement. I plan to use ordinal regression given the variable was measured on a 5-point scale. Can I still use OLS if the variable is normally distributed? Thank you!

[Bruce Weaver]

I think you meant that you planned to use OLS linear regression. Is your outcome variable a single 5-point item? Or is it the mean (or sum) of several 5-point items intended to measure the same thing? (Only the latter is properly called a Likert-scale--see this page for further discussion). If it is a single 5-point item, then I would suggest that you forget about using OLS linear regression, and use the -ologit- command instead to estimate an ordinal logit model.

help ologit

HTH.

[DY Kim]

I have multiple items for my outcome variable. Since most items are skewed, I wanted to find a way to reduce the skewness problem. I created a composite scale, its distribution is still skewed. What should I do in this situation? Should I use ologit? or can I use OLS?

[Bruce Weaver]

Should I use ologit? or can I use OLS?

Answer: It depends very much on who you ask! If you ask one of my former bosses (Geoff Norman), I'm fairly certain he'd tell you to go ahead and use OLS regression. But if you asked John Kruschke, he would advise you to use an ordinal regression model despite having a mean (or sum) of several items. See the following articles.


Liddell, T. M., & Kruschke, J. K. (2018). Analyzing ordinal data with metric models: What could possibly go wrong?. Journal of Experimental Social Psychology, 79, 328-348. https://www.sciencedirect.com/scienc...KMicWputp5liE9

Norman, G. (2010). Likert scales, levels of measurement and the “laws” of statistics. Advances in health sciences education, 15(5), 625-632. https://link.springer.com/content/pd...010-9222-y.pdf

[Nick Cox]

Bruce Weaver makes excellent points. A little more can be said.

The point about ordinal scales is that they are arbitrary, other than being ordinal. That doesn't make skewness and kurtosis for such variables utterly meaningless, but it does make them highly arbitrary. Also, high skewness and/or kurtosis with measured variables may mean that something like a logarithmic transformation could be a good idea, to pull in a tail and subdue outliers, especially if it tackles nonlinearity too. But with a 5-point ordinal scale extreme outliers are impossible. Something like a U-shaped distribution with polarised attitudes or behaviour is more likely to be responsible here.

tab myvariable

to show us the distribution.

Whatever you do, 5 spikes in a frequency distribution can only be transformed into 5 different spikes. I don't think transforming to different scores is out of order but I wouldn't do that as a pursuit of marginal normality.

Most importantly, even plain linear regression (*) doesn't assume that any variable has a marginal normal distribution. What it does is assume, or imply, that looking at conditional means makes sense, or is pragmatic. (For a parable in favour of pragmatism, see the quotation below from a 2009 thread. Even for ordinal data means can be defensible: it's the principle of grade-point averages.)

Whatever you do here will be wrong or ill-advised from somebody's point of view. I don't find ordinal logit tremendously easy to work with, but that goes in a circle with only ever using it occasionally.

(*) I can't sign up to calling plain regression "OLS regression". That is not so much wrong, as putting emphasis in the wrong place, making too much of a method of estimation. The model is the message.

https://www.stata.com/statalist/arch.../msg00744.html

I don't vote for spurious precision, either. It's down there with mouldy apple pie. Gauss and Aristotle both said so too, but more elegantly.

But there is a still an issue between the purists and the pragmatists. Some of the purists were traumatised by a course in measurement theory in graduate school. That can have a lifelong effect, akin to the first really rigorous calculus course, which leaves students shaking and convinced that up to now they have just been waving arms and making coarse animal noises. Fortunately I did no such course (in either case). "You must pay attention to how the data were produced! These are ordinal scores and should be treated as such!" Well, yes, except that the pragmatists have a case too.

Consider these scores from recent Mata courses by different instructors:

Gould: 3 4 4 4 4 4 5 5
Cox: 3 3 3 4 4 4 4 5

Dr Gould gave a tough course. Dr Cox gave a tough course too. In fact, if we summarize these scores using "legitimate" summary measures, you can see that these instructors had the same median (4) and mode (4). Dr Foobar down the hall got all 5s and gets the promotion.

The pragmatist (meaning, in my book, the good data analyst) looks at these data and sees information there that should be dug out. There is a systematic difference between these score sets. The data analyst should want information-rich summaries! In fact, my university happily takes means of such data, and, whatever anyone says, that usually works fine. If you do that you get

Gould 4.125
Cox 3.75

Cox is the one to receive more counselling. What is the problem with his teaching?

Note: the allusion here to Gauss is to a saying attributed to him (e.g. by Oskar Morgenstern) about spurious precision. Like Einstein and Mark Twain, it seems that Gauss didn't say everything he is supposed to have said. At least no one can find that quotation in his work.. At this moment I cannot find the quotation or the debunking.

[Andrew Musau]

I much appreciate the history lessons in your posts Nick Cox (beyond the very enlightening content).