Dear Rafael, thank you for your comment which is indeed very interesting. In our project we have tested a number of methods to quantify the activity of difficult (DTM) and impossible (ITM) to measure radionuclides (according the IAEA definition). Among the methods suggested in the reference standards (ISO 21238 and IAEA NW-T-1.18) some use the geometric mean, other linear regression and some other regression of higher order. We decided to test the various methods and compare the results. According to our experience (over hundreds tons of radioactive waste already characterized and disposed in final disposal facilities) there is very little or no difference when using the geometric mean or the median of the ratios. We have also tested other average content statistics (such the mean or the mode) and various statistical learning approaches. When the waste population is fairly low heterogeneous there is little impact on the final characterization and quantification of DTMs and ITMs which, in any case, play a minor role when calculating hazard factors. In our case only a very limited number of gamma-emitters are of interest and we measure them via direct gamma-spectrometry on each waste package. Do not hesitate to write back to me if you have specific/technical questions. Regards, Biagio

Just another detail: in our papers and projects we discuss the characterization of legacy very-low-level radioactive waste so we do not refer to either clearance of material from regulatory control or high-level radioactive waste for which dedicated methods must be considered.

Thank you for your fast and clear answer

You are welcome. Regards, Biagio

Dear Rafael,

I am not sure of fully understand your question, but I will try to direct attention to some related issues. In fact, I know quite well the contribution of J. Aitchison and, nowadays, I am working on compositional data analysis.

With respect to the use of geometric means and medians, I would say that medians of a concentration or any other compositional variable is mainly reduced to be an univariate exploratory tool. The extension of medians to multivariate contexts is not easy. As compositional data are generally multivariate, the role of median is really unimportant. The geometric mean appears constantly in compositional data analysis (log-ratio approach) in several ways. However, it is not important by itself, but in the construction of the statistics which are used which main property is the scale invariance. The two main ways geometric means appear are: (1) when computing the compositional mean or center of a compositional sample (by columns); and (2) when representing compositions by ilr-coordinates or clr-coefficients where geometric means by rows play an important role.

Presently, we say that compositional data are arrays of positive components which carry relative information, i.e. information is in the ratios between components. In this sense, both concentration and activities are compositional. They are related by a perturbation, the group operation for compositional data, thus representing a shift of the compositions.

For an updated introduction to compositional data (log-ratio approach) you can have a look at

Pawlowsky-Glahn, Egozcue, Tolosana-Delgado: Modeling and Analysis of Compositional Data. Wiley (2015)

If you have any doubt, do not hesitate to contact me.