Thank you for the paper recommendation Eduard. I see many options in that paper and I decided to go with:

- the basic select07 (Any correlation between two variables that is higher than .7 will lead to one variable being excluded)

- principal component analysis implemented in lm

I have two additional questions relating to my analysis:

- How do I interpret the contribution of principal components on the regression model? Can I look at the coefficients and infer directly on how large of an effect my original variables have on the model?

- In my dataset, I have data that include all 0 percentages. Is it okay if I incorporate them like that in the dataset? (look at the dataset that i provide below, my 5th row is all zeros because there were no data to calculate the fractions for aa,bb,cc)

a = data.frame(score = c(12,321,411,511,19),yapa = c(1,2,1,1,2),ran=c(3,4,5,6,8),aa=c(0.1,0.4,0.7,0.8,0),bb=c(0.2,0.2,0.2,0.1,0),cc=c(0.7,0.4,0.1,0.1,0))

pc

There is a macro published by SAS Institute, that might be adaptable to your needs in R. It's called SumtoOne.sas, which I've attached. It's a text file, even though the suffix is *.sas. This macro addresses situations where the predictors are proportions that sum to 1. Good luck!

Thanks for the file Mary. It looks promising although I have to translate it to R and see how it works. My only issue is that it assumes that all variables sum to 1, however I have cases where variables don't sum up to 1 but to 0. These are cases that I could not calculate the ratios for and therefore I marked them all as zeros. How should I deal with them? Leaving them as 0 is okay or should I NA them?

The clr transformed scores add up to zero, thus yu wo't get rid of collinearity with it.

If the idea of working with logratios fits the scale of your problem, then the only thing you need to avoid collinearity is to take a set of one-to-one logratios. If you have 3 components, that means you have enough with 2 logratios, like e.g.

x1 = log(aa/cc)

x2 = log(bb/cc)

and then

lm(score~x1+x2)

You can even take the coefficients and convert them back to a composition,

compcoef = c( exp(x1), exp(x2), 1)

compcoef = compcoef/sum(compcoef)

which coefficients must be interpreted as the gradient of "scores" in the simplex. The final estimates do not depend on which denominator you took, which set of logratios you used, etc. As long as it is a one-to-one transformation, regression results after back-transformation are the same. The package "compositions" does actually include these transformations as functions alr and clr, so that your code would also be fine with

cmp = a[,4:6]

dd = ilr(cmp) #centered log ratio transform

mdl = lm(score~dd,a)

summary(mdl)

ilrInv(mdl$coefficients[-1], orig=cmp)

You can check what I say trying once with the alr and once with the ilr ;).

With regard to the issue of zeros, you better treat them as NAs.

Raimon sorry for the delayed response. I just check your code and I would like to thank you. Your answer is exactly what I needed it and works like a charm. Interpretation of coefficients becomes harder but as a whole I can say that these compositional data as a whole matter.

Glad to read that!

Yes, the issue with the interpretation of the coefficients is one of the hardest aspects of the log-ratio approach to compositional data analysis. And in a regression framework with compositional input that is even more complex... In a given sense, you can see it as the gradient of "scores" with respect to "dd". And as a gradient, it shows the "compositional direction" in which "scores" increases fastest. To get a better idea of what does that mean you can compute all possible pairwise logratios of your coefficient

compcoef = ilrInv(mdl$coefficients[-1], orig=cmp)

D = length(compcoef)

z = log(outer(compcoef, compcoef, "/")) / (4*D^2)

If the element z[i,j] of this matrix is positive this means that "scores" increases z[i,j] units for each time that the ratio of variables i and j is increased by a factor of e=2.71~3. The same applies to negative elements, meaning a decrease, of course. This may help in deciding which components are relevant to predict "scores", and of course, in interpreting the whole thing. The (2D)^2 constant appears because of the own Euclidean structure of compositions. Perhaps a look at

https://www.researchgate.net/publication/230492190_Linear_Models_with_Compositions_in_R?ev=prf_pub

can help you in understanding the meaning of these coefficients.

Chapter Linear Models with Compositions in R

I would like to insist that the logratio approach delivers exactly the same result in a least-squares regression problem, whichever logratio transformation is actually used, provided that it is one-to-one transform. Both alr and ilr are so, thus it doesn't matter which one is used in regression. Actually even the clr could be used, provided that one is able to work with generalized inverses. That invariance is quite a general property, occurring to all statistical methods based on second-order moments (regression and ANOVA, discriminant analysis, linear geostatistics and time series analysis, etc.), but for instance, in general not to a cluster analysis. The authors of robCompositions, Filzmoser and Hron, proved that in a couple of papers in Mathematical Geosciences.

anybody have R code for isometric log transformation of compositional data and correlation analysis?