I admit that I have been converting raw data into normal scores or normal quantiles for further analysis. There are many supporters of such a ranking methodology and there seem to exist many more statisticians or users of statistics who seem to be appalled by transforming data into ranks. What is really happening in a statistical analysis in which the raw data have been transformed into z scores? Is it valid? Is it defensible? Is it clearly incorrect? Are there superior methods for dealing with non-normal data?
What is your stand on it?
It depends why you are doing the ranking. If the data have some underlying metric, ranking data is a odd transformation (so if the data are 1,2,3,4 you are saying they are treating them as if interval, but 1,2,3,6 you are saying the distance from 3->6 is the same as 1->2). If you are doing it so outliers don't have as much affect there are lots of other methods (e.g., trimmed means, M-estimators). But is the data don't have an underlying metric (i.e., qualitative) and each possible value is populated, this may be useful. Further, there are a lot of research hypotheses that are qualitative (are more of the data in this group higher than in this group) and ranking can be appropriate. It might help if you provide more context about the situations that you are asking about.
Hello Daniel.
Consider analyzing cancer rates (continuous data) for thousands of geographic locations. We want to do a cluster analysis on the cancer rates. This is when I may consider transforming data into normal scores. The results are robust to occasional outliers, and the interpretation of the cluster values becomes easier to do.
Another application was estimating the distances at which sharks approach divers. The distances are estimated based on videos, and everything is an approximation there.
We also used such transformations with large data sets on air pollution for billions of data values.
Then there are cases when I would like to be able to use a nonparametric test, but it is unavailable. For example, we have a three-factor factorial with interactions (ANOVA or MANOVA) with non-normal data. Using normal scores reduces the impact of non-normality.
Raid, Nice question, but harder to answer. When you have data, you need to know which transformations are admissible. With rank data any transformation is admissible that does not change the ranking. But your analysis should only use the ranking to be analyzed and not take the numbers as being equidistant.
I do not see cancer rates as 'continuous data'. There are absolute lower and upper bounds (0 and 100%). When you transform this data to Z-scores, it seems that 100 standard deviation above the mean would theoretically be possible, but in most cases this will not be true. 100% would be at a specific value for the standard deviation and although very high values are rare, they are admissible. This restriction in range is no problem as long as the analysis does not result in predictions outside the permissible range. With binary data people think you need to perform logistic regression, but that need not be. You could use normal regression analysis (see Dodge (2008) Concise Encyclopedia of Statistics, pag 4).
Within the GLM framework, you could perform a MANOVA with a non-normal distribution (as long as the distribution is from the exponential family, like normal, Poisson, Gamma, Chi-Square (see https://en.wikipedia.org/wiki/Exponential_family)). However to choose another distribution might require strong theoretical considerations or a risk of getting your paper turned down. Why not perform a regression analysis with a Chi-Squire distribution instead of a normal distribution? Possible, but which Chi-Squire to use might be a difficult choice to make.
Hi Peter,
Yes, if we had cancer counts and matching population counts, then we do not use the normal model. It would be most likely the Poisson model, and there would be no need for such a transformation. On the other hand, if counts are unavailable, and only (say) age adjusted cancer rates are available, we may then use such rank transformations.
I first got exposed to such ideas when I was a graduate student many eyars ago. We used a text by W. J. Conover "PRACTICAL NONPARAMETRIC STATISTICS" . In Section 5.10 (Tests with A.R.E. of 1 or More", the author discussed the ranking based test by van der Waerden, and he compared it with the Kruskal-Wallis Test. Once you have the normal scores, you could use the new data in a variety of statistical tests.
http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/vanderwa.htm
Thanks for your thoughts. I am going over your feedback.
Here is an article where the normal scores tests are criticized.
Article Rank-Based Inverse Normal Transformations are Increasingly U...
arcgis.com discusses such normal scores:
http://pro.arcgis.com/en/pro-app/help/analysis/geostatistical-analyst/normal-score-transformation.htm
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