Good nonparametric measures of skewness are ofered by so called data depth concept. It could be for example scaled difference between multivariate median and the mean vector - for details see Robert Serfling' pages (paper nonparametric descriptive measures...), within R you can simply use 2d boxplot (bagplot within aplpack) for a visual iinspection.

Daniel Kosiorowski

Department of Statistics

Cracow University of Economics

This should be extensible to bivariate case

https://en.wikipedia.org/wiki/Skewness#Sample_skewness

However, why not to fit a model distribution to your data and estimate the skewness as the third moment:

https://en.wikipedia.org/wiki/Skew_normal_distribution

https://en.wikipedia.org/wiki/Multivariate_stable_distribution

HTH,

Peter.

One more link about how to calculate moments:

https://en.wikipedia.org/wiki/Moment_%28mathematics%29

(and sorry for the link to Skew Normal, it is univariate)

Hmmmm. I would look for the linear combination of your two variables that maximized the skew. What I might consider is rotating around the bivariate mean, and measuring the skew at each slice. Then you would have a maximum and minimum skew. To do this for two variables X and Y define Z = X sin(theta) + Y cos(theta), and then vary theta from 0 to 90 degrees. This would give you the minimum and maximum skew for the data set. If you wanted to put statistics on it you would probably have to do some sort of bootstrap or randomization.

Actually make that 0 to 180 degrees (oops)

Phenotypic traits are seldom real (real numbers, including positive and negative). They are frequently positive (real), or compositional (given in proportions). The skewness is normally thought in a real space (absolute scale) and it changes when it is computed in a different scale.

For a study of skewness of traits, a first step should be to decide which is the sample space of the traits and the corresponding scale. A second step is to transform quantitative traits into a real-absolute scale. And then proceed to skewess analysis as recommended in previous answers.

Let me put a couple of examples. Trait X: size of the body. We can accept it is in a ratio scale. A diference of 5cm tallness is not the same on an individual of 150cm as on an individual of 180cm. An individual 0cm tall is not an individual... In this cases it is advisable to take logarithms, log X, and then perform the skewness analysis on log X.

The compositional example appears when the trait is a proportion of a total, like weight of head over total weight. This proportion has a symmetrical ratio scale. To reduce it to an absolute scale log-ratio analysis should be used. In this simple case, I would recommend a logit transformation. If X is the proportion, the skewness should be studied on log( X /(T-X)), where T is the total an X is the proportion measured. More intricated cases arise when the treats are several proportions of the same total. This leads to a full compositional analysis.

See for instance skew-normal distribution in the simplex.

I saw in the answers some comments on the skew-normal (univariate, multivariate, in the simplex for compositional data). Here some references for multivariate cases bib-Tex format:

@ARTICLE{AC99,

author = {Azzalini, A. and Capitanio, A.},

title = {Statistical applications of the multivariate skew-normal distribution},

journal = {Journal of the Royal Statistical Society, Series B (Statistical Methodology)},

year = {1999},

volume = {61},

number = {3},

pages = {579-602},

note = { },

}

@ARTICLE{AD96,

author = {Azzalini, A. and {Dalla Valle} , A.},

title = {The multivariate skew-normal distribution},

journal = {Biometrika},

year = {1996},

volume = {83},

number = {4},

pages = {715-726},

note = { },

}

@ARTICLE{MPB05,

author = {Mateu-Figueras, G. and Pawlowsky-Glahn, V. and Barcel{\'o}-Vidal, C.},

title = {The additive logistic skew-normal distribution on the simplex},

journal = {Stochastic Environmental Research and Risk Assessment (SERRA)},

year = {2005},

volume = {19},

number = {3},

pages = {205-214},

month = {August},

}

Thanks for the article reference. And, to everyone else, thanks for the many excellent suggestions. It will, of course, take me a little while to figure out what will be the best application to my data (and how I will execute this), but in the meantime I wanted to say thanks to everyone. These comments and suggestions were very helpful in getting my head around this problem and identifying its solution. (So, thanks everyone for taking all those math courses....!!)

@Juan, if I'm understanding your comment about the trait space correctly, I should note that this work examines multiple traits (some positive, continuous and other Gaussian) that have been scale to mean=0 and unity variance. My goal is to understand how relative fitness varies with standardized unit changes in each trait (in this case, units=standard deviations). There are some disadvantages of this, but it's the best framework we have right now. You're absolutely right that the original units play a huge role in model inference, but I hope this addresses some of your concerns. You got me thinking more about this, for sure...

Dear Patrick,

I had no time to read your last post. I would add a comment on Gaussian and standardized traits.

I think that a trait is seldom real (unlimited positive and/or negative). Most times, they are qualitative, positive, compositional (proportions) or directional (directions). For a trait being Gaussian it should be real. Standardization requires a real trait, etc. Then, what is called a "Gaussian trait", a "standardized trait" are transformations of original measurements or characteristics. The question is whether the applied transformation is consistent with the scale and sample space of the original measurement. For instance, one proportion X is not real; it lives in the simplex; it cannot be Gaussian. When it is transformed as logit(X), it is a real variable and its distribution can be Gaussian or skewed.

However, I know that what I am trying to explain (structure of sample space, scale of measurements, transformations into absolute scale,...) are not of common use in applied sciences although the associated problems are known for more than a century (K. Pearson 1897). I would recommend to have a look to the J. Aitchison (1986) book on compositional data analysis.

I wish you a successful and encouraging research!