Suppose I have several compositions of 4 parts $A_i, B_i, C_i,$ and $D_i$ with $i = 1, \ldots , n$ taken from two locations ($X$ and $Y$), which are coordinates in the simplex $S^4$. I apply the isometric log-transformation to each composition to obtain the following balances based on the sequential binary partition $\{A ||| B || C | D\}$ (selected because of some nice interpretations)
$$ \begin{split} &b1 = \frac{1}{\sqrt2} \log\left( \frac{A}{g(B, C, D)} \right) \\ &b2 = \sqrt{\frac{2}{3}} \log\left( \frac{B}{g(C, D)} \right) \\ &b3 = \log\left( \frac{C}{D}\right) \end{split} $$
where $g(\cdot)$ is the geometric mean (center) of the subcomposition. These balances are defined according to Pawlowsky-Glahn and Egozcue in "Exploring Compositional Data with the CoDa-Dendrogram" (p. 107-108). I can't find a working DOI, but here is a link to it https://stat.tugraz.at/AJS/ausg111+2/111+2Pawlowsky.pdf. I have performed t-tests on the following differences between means of the balances $b1_Y - b1_X$, $b2_Y - b2_X$, and $b3_Y - b3_X$.
Would it be reasonable to scale these differences in means by the inverse of the balance normalization coefficients (the square-root terms in front of the balances)?
For example, let $\Delta_1 = b1_Y - b1_X$ (details below). Then, can I interpret the quantity $\frac{g(R_Y)}{g(R_X)}$ as "The ratio of the proportion of $A$ compared to the center of the proportions of the subcomposition $\{B, C, D\}$ (increases or decreases) by $\frac{g(R_Y)}{g(R_X)}$ fold in location Y compared to location X"?
$$ \begin{align} \overline{b1}_Y - \overline{b1}_X &= \frac{1}{\sqrt2} \overline{\log\left( \frac{A}{g(B, C, D)} \right)}_Y - \frac{1}{\sqrt2} \overline{\log\left( \frac{A}{g(B, C, D)} \right)}_X \\ &= \frac{1}{\sqrt2} \overline{\log(R_Y)} - \frac{1}{\sqrt2} \overline{\log(R_X)} \\ &= \frac{1}{\sqrt2} \left[ \overline{\log(R_Y)} - \overline{\log(R_X)} \right] \\ &= \frac{1}{\sqrt2} \log\left(\frac{g(R_Y)}{g(R_X)} \right) \\ &= \Delta_1 \end{align} $$
Therefore, I should be able to convert this to a ratio between the two groups, like so
$$ \exp\left(\Delta_1 \frac{1}{1/\sqrt2}\right) = \exp(\Delta_1\sqrt2) = \frac{g(R_Y)}{g(R_X)} $$
Edit: based on some simulations, it appears that
$$ g(R) = g\left( \frac{A}{g(B, C, D)} \right) = \frac{g(A)}{g(B, C, D)} $$
That is, $g(R)$ is the ratio of the geometric mean of all $A$ proportions in location $X$ or $Y$ to the geometric mean of all non-$A$ proportions from the same location. So I guess that mostly answers my question. If anyone has a more intuitive interpretation, please let me know.
R code:
# Geometric mean
gm <- function(x) {
prod(x) ^ (1 / length(x))
}
# Toy example: 3-part compositions, each element
# is from a different sample
A <- c(0.3, 0.35, 0.2, 0.25)
B <- c(0.5, 0.5, 0.55, 0.45)
C <- 1 - (A + B)
gm(A) / gm(c(B, C)) # 0.8167283
gm(A / gm(c(B, C))) # 0.8167283
# A bit weird, but equivalent!
gm(c(A, A) / c(B, C)) # 0.8167283
gm(c(gm(A / B), gm(A / C))) # 0.8167283
```
