It is not a must to transform the data for analyzing by the common statistical methods.

It is always ideal if the data, as obtained in raw form or primary form, can be analyzed (or are analyzed) by the use of common statistical methods.

However, if it is not convenient to apply the common statistical methods to the raw data directly but convenient to apply on transformed data for the purpose, only then it is advisable to transform the data.

Regarding my question, Prof. Juan José Egozcue sent me the following comments by e-mail. Many thanks to him.

Let me comment on your questions. But my question is about the papers how take logarithm to reduce the skewness of the data like elements concentration and use common statistical methods to analyze the data in log scale and their results seems reasonable to them and put some interpretation on their results and publish it.

Although reducing skewness is a common practice in statistics, I think it is not a good idea. The actual problem is that of the sample space. For instance, for positive variables the sample space is R+ and many times the zero is not attainable, i.e. it is placed at infinity. In these cases, it is reasonable to take logarithms. Importantly, a central question is which is the addition in this context. It is no longer the sum which implies a subtraction which easily leads to negative values. A better strategy is to think that addition in R+ is a multiplication. For instance, the variable X>0 has incremented 5\%, meaning that it changed from X to 1.05X; or it decreased 5\% meaning that it attained the value 0.95X. In these conditions taking logarithms everything is transformed into the standard R with addition as additive group and absolute scale. So, to conduct a compositional analysis by simply taking logs is assuming that differences between X and Y are measured by log(X/Y), that log(X) has absolute scale and that the central limit distribution is the log-normal; the average is now the geometric mean... This is acceptable for a single concentration or single proportion but not applicable for a composition where concentrations or proportions are linked together. In fact, a proportion is never isolated from other proportions. Imagine the X is a proportion. This means that, although not considered, 1-X is also another proportion strongly linked to X. Traditionally this case of X and 1-X y reduced to absolute scale by the logit-transform log(X/(1-X)). Now the proper addition in the logit space is log(X/(1-X)) + log(Y/(1-Y)); the central limit distribution is the logistic-normal. The summary is that the analysis of compositions by simply taking logs is not appropriate although some results can be similar or even acceptable.

My question is: Regarding ALR, CLR, and ILR transformations, is it really a MUST to transform compositional data by, say, CLR or ILR transformations before applying statistical methods to the data such as factor analysis?

No, it is not compulsory but it is comfortable and reduces the compositional data to real data (absolute scale, sum as proper addition, normal distribution as central limit, average is an appropriate central point,...). ALR, CLR, ILR represent compositions in coordinates in a real space, but they have different properties and uses. ILR(X) represent a composition X in orthogonal Cartesian coordinates, whereas ALR does it in an oblique system of coordinates, what makes the computation of distances and norms a complicated computation. CLR(X) is a representation of X in a generating system (the clr-coefficients are contrained to add to zero). CLR provides straight-forward computational algorithms but, as coordinates, they are also oblique. Summarizing there is not a MUST in transforming the data, but then the operations get more involved and the interpretation of results is cumbersome. For instance, it is known that the information from a composition is contained in the ratios between its components (called parts). Therefore, the interpretation should be based on the behavior of the ratios of parts and not on single parts. To end, note that the above mentioned logit transform is both an ILR and a ALR. Also, is an scale invariant function. This is the main requirement on compositions. For instance, the proportions (0.1,0.3,0.6) provide the same information as in the percentages (10,30,60).

Both participants Hamid Ghorbani and Dhritikesh Chakrabarty, agree in that transforming Compositional Data (CoDa) by ILR or CLR is not a MUST. I agree with their opinion. However, in order to get insight on this point it is necessary to clarify which are the assumptions underlying the use of such transformations.

First, the main assumption on the log-ratio approach to CoDa is the "scale invariance", that is, a composition X=(x1,x2,...,xD), xi>0, conveys the same information as Y=aX=(ax1,ax2,...,axD), a>0. Consequently, they are equal as compositions. A second assumption is the so called "subcompositional coherence": when considering a subcomposition of X, say Y=(y1,y2,...,yd), d