Hello!
I am currently analyzing data regarding the validation of a greek translation of the anxiety sensitivity index 3. I have a sample of around 200. Can I transform my variables (e.g. log or Z-scores) to approach normality, or does this render the convergent and discriminant evidence meaningless?
Thank you!
Anxiety scores will not be normally distributed for a number of reasons.
1. They are bounded at both ends of the scale
2. They will be skewed
3. They will take on a limited number of integer values
Even if you could force them into a normal distribution, it would not affect validity.
I agree with Ronán Michael Conroy that anxiety scores are not generally normally distributed. If, as expected, your scores on the anxiety sensitivity index are not normally distributed, it is advisable to transform them into something that approaches a normal distribution for parametric analyses because most parametric analyses assume normality. Choose your transformation carefully and examine what it does to the distribution. If you find a transformation that brings the distribution into something that approaches normality, then you can proceed. Transforming variables does make interpretation somewhat more difficult, but is be okay for hypothesis-testing.
If you cannot transform the distribution into something approximately normal, then use a nonparametric analysis that does not assume normality.
Rhianon Allen Ronán Michael Conroy
I have ran both parametric and non parametric and they do not differ much. What bugs me however (and makes me search for normality) is that I intend to use the Steiger's test (William's method) to compare correlation coeficients. I am sure I can run it on parametric correlations, but I am not sure if I can run it on non parametric correlations (sprearman, kendal-t) and I can't find any information online. Do you know about that?
Thank you!
Rhianon Allen wrote: "...most parametric analyses assume normality."
Normality of what, though? There is much confusion about this!
Better textbooks clarify that for OLS models (including t-tests and ANOVA), it is the errors (i.e., deviation of actual values from fitted values using the true, population regression expression) that are assumed to be normally distributed. But sadly, not many books point out that normality of the errors is a sufficient condition, but not a necessary condition.
The necessary condition is that the sampling distributions of the parameter estimates (i.e., the constant & the regression coefficients) be approximately* normal. As was implied above, if the errors are normally distributed, the sampling distributions of the parameter estimates will also be normal. But the latter sampling distributions approach normality as n increases, even if the error distribution is not normal. If you need published references to support what I've said here, see the textbooks by Wooldridge (2021) and Vittinghoff et al (2012) that are mentioned in the slides I have attached.
* Approximate normality is the best one can hope for when working with real world data. See the famous statement by George Box that I have attached below. The implication is that the t- and F-tests for OLS models are really approximate tests when one is using real world data, and the question is not whether we have exact normality or not, but whether the approximation is good enough to make the model "useful" (to borrow a word from another famous statement by Box!).
HTH.
Bruce Weaver I think I have a déjà vu (or many) and I love how you calmly repeat the same sermon over and over again. ;-) Thumbs up that you do not get tired mentioning it. Recommendation earned for the Box quote ;-)
Bruce Weaver – I have to agree with Rainer Duesing : you are dismantling misconceptions one person at a time. Where did this misconception come from? Along with the idea that data can be parametric or nonparametric, it must be at the top of the list of statistics myths.
Thanks Ronán Michael Conroy. To really drive home the point, I suppose I could have added that the normality assumption for binary logistic regression is exactly the same as the normality assumption for OLS linear regression: In both cases, the sampling distributions of the parameter estimates have to be approximately normal! ;-)
Nikos Makrygiorgos , I have never compared nonparametric correlations, and am drawing a blank on how one might do that. Do you have a statistics expert at your organisation? See whether you can get confidence intervals for the correlations.
Rhianon Allen Yes, but I have not extended this question outside of my immidiate circle in my department yet. First thing that crossed my mind was to open a question here on research gate to tell you the truth :)
Rhianon Allen Bruce Weaver Ronán Michael Conroy Thanks for the prompt replies!
getting z scores brings your mean to 0 and std to 1. that is all. it doesn't make your variable normally distributed. In other words, if your X variable doesn't follow a normal distribution, so as Z won't.