Log-ratio transformations: clr, alr and ilr

compositional data

ggplot2

Building the centred, additive and isometric log-ratio transformations by hand, with Aitchison distance, perturbation and powering, using base R only.

Author

Tidy Ecology

Published

2026-08-01

The previous post showed that proportions carry a constant-sum constraint that manufactures negative correlations and makes ordinary correlation subcompositionally incoherent. The one quantity that behaved well was the log-ratio. This post takes that hint seriously and builds the three standard log-ratio transformations from first principles: the centred log-ratio, the additive log-ratio, and the isometric log-ratio. Each maps a composition off the constrained simplex and into ordinary Euclidean space, where distance, correlation and ordination recover their usual meaning. All three are a few lines of base R.

Setup

library(ggplot2)
clr <- function(x) { lx <- log(x); lx - mean(lx) }          # centred log-ratio
alr <- function(x, ref = length(x)) log(x[-ref] / x[ref])   # additive log-ratio
C   <- function(x) x / sum(x)                                # closure
ilr_basis <- function(D) {                                  # orthonormal contrast matrix
  V <- matrix(0, D, D - 1)
  for (i in seq_len(D - 1)) {
    a <- sqrt(1 / (i * (i + 1))); b <- sqrt(i / (i + 1))
    V[1:i, i] <- a; V[i + 1, i] <- -b
  }
  V
}
te_ink <- "#16241d"; te_forest <- "#275139"; te_faint <- "#5d6b61"
te_paper <- "#f5f4ee"; te_line <- "#dad9ca"; te_red <- "#b5534e"
theme_te <- function(base = 12) theme_minimal(base_size = base) + theme(
  plot.background = element_rect(fill = te_paper, colour = NA),
  panel.background = element_rect(fill = te_paper, colour = NA),
  panel.grid.major = element_line(colour = te_line, linewidth = 0.3),
  panel.grid.minor = element_blank(),
  axis.text = element_text(colour = "#2c3a31"), axis.title = element_text(colour = te_ink),
  plot.title = element_text(colour = te_ink, face = "bold"),
  plot.subtitle = element_text(colour = te_faint, size = rel(0.9)), legend.position = "top")
theme_set(theme_te())

set.seed(3390)
D <- 5L; n <- 300L
mu <- c(2.8, 2.4, 2.0, 1.6, 2.2); sdlog <- c(0.5, 0.6, 0.55, 0.7, 0.5)
X <- sapply(1:D, function(j) rlnorm(n, mu[j], sdlog[j]))
P <- X / rowSums(X)
V <- ilr_basis(D)

The centred log-ratio

The centred log-ratio takes logarithms of the parts and subtracts their mean, which is the same as dividing every part by the geometric mean of the composition. The result is a vector that sums to zero.

CLR <- t(apply(P, 1, clr))
max(abs(rowSums(CLR)))       # rows sum to zero

[1] 1.110223e-15

The rows sum to zero to within 1.1e-15. That zero-sum property is exactly what makes the clr convenient and what makes it awkward. Convenient, because clr coordinates live in ordinary space and each one is interpretable as a part against the average part. Awkward, because the zero-sum means the coordinates are linearly dependent: their covariance matrix is singular and cannot be inverted.

ev <- eigen(cov(CLR), only.values = TRUE)$values
c(smallest_eigenvalue = min(abs(ev)), rank = qr(cov(CLR))$rank, parts = D)

smallest_eigenvalue                rank               parts 
       1.440082e-16        4.000000e+00        5.000000e+00

The smallest eigenvalue is 1.4e-16 and the matrix has rank 4 for 5 parts. Any method that needs a full-rank covariance, discriminant analysis or a multivariate normal likelihood among them, cannot take clr coordinates directly. That is what the isometric log-ratio fixes.

A picture of the simplex

Before the algebra, a picture helps. A three-part composition is a point in a triangle: the three corners are the pure compositions, and every interior point is a mixture. Distances and directions in this triangle are not the ordinary ones. The natural operations are perturbation, which plays the role of a translation, and powering, which plays the role of scaling. Perturbation of one composition by another is the closed elementwise product; powering is the closed elementwise power.

set.seed(11)
A3 <- sapply(1:3, function(j) rlnorm(9, c(2.2, 2.0, 1.6)[j], 0.5)); T3 <- A3 / rowSums(A3)
tern <- function(m) data.frame(tx = m[,2] + 0.5 * m[,3], ty = (sqrt(3)/2) * m[,3])
pts <- tern(T3)
vert <- data.frame(tx = c(0, 1, 0.5), ty = c(0, 0, sqrt(3)/2), lab = c("part A", "part B", "part C"))
edge <- data.frame(x = c(0, 1, 0.5), y = c(0, 0, sqrt(3)/2),
                   xend = c(1, 0.5, 0), yend = c(0, sqrt(3)/2, 0))
xstar <- C(T3[1,]); yv <- C(c(1.8, 1.0, 1.0)); xp <- C(xstar * yv)
pth <- tern(rbind(xstar, xp))
ggplot() +
  geom_segment(data = edge, aes(x, y, xend = xend, yend = yend), colour = te_line, linewidth = 0.6) +
  geom_point(data = pts, aes(tx, ty), colour = te_forest, size = 2.2, alpha = 0.8) +
  geom_segment(data = data.frame(x = pth$tx[1], y = pth$ty[1], xend = pth$tx[2], yend = pth$ty[2]),
               aes(x, y, xend = xend, yend = yend), colour = te_red, linewidth = 0.9,
               arrow = arrow(length = unit(0.18, "cm"), type = "closed")) +
  geom_text(data = vert, aes(tx, ty, label = lab), vjust = c(1.6, 1.6, -0.8), colour = te_ink, size = 3.4) +
  coord_equal() + theme_void() +
  theme(plot.background = element_rect(fill = te_paper, colour = NA),
        plot.title = element_text(colour = te_ink, face = "bold"),
        plot.subtitle = element_text(colour = te_faint, size = rel(0.9))) +
  labs(title = "The simplex: sample space of compositions",
       subtitle = "Points are three-part compositions; the arrow is a perturbation")

An upward triangle labelled part A, part B and part C at its corners, with green points scattered inside and a red arrow between two of them. — Figure 1: Nine three-part compositions plotted in the simplex. The red arrow is a perturbation, the compositional analogue of adding a fixed vector: it moves a point by multiplying its parts and re-closing.

These operations are not exotic once viewed through the clr. Perturbation becomes ordinary addition, and powering becomes ordinary multiplication, in clr space.

y <- P[3, ]; a <- 2.3; xa <- P[1, ]
c(perturbation = max(abs(clr(C(xa * y)) - (clr(xa) + clr(y)))),
  powering     = max(abs(clr(C(xa^a)) - a * clr(xa))))

perturbation     powering 
2.220446e-16 4.440892e-16

Both identities hold to machine precision. The simplex, with perturbation and powering, is a vector space, and the clr is the map that turns its geometry into the geometry we already know.

Distance that respects ratios

The distance attached to this geometry is the Aitchison distance. It can be written three ways that all agree: as the ordinary distance between clr vectors, as the ordinary distance between isometric log-ratio vectors, and as a sum over all pairwise log-ratios. Computing one composition against another confirms the identity.

xa <- P[1, ]; xb <- P[2, ]
d_clr <- sqrt(sum((clr(xa) - clr(xb))^2))                 # Euclidean on clr
d_ilr <- sqrt(sum(((clr(xa) - clr(xb)) %*% V)^2))         # Euclidean on ilr
Lr <- function(x) outer(1:D, 1:D, function(i, j) log(x[i] / x[j]))
d_lr <- sqrt(sum((Lr(xa) - Lr(xb))^2) / (2 * D))          # pairwise log-ratio form
round(c(clr = d_clr, ilr = d_ilr, pairwise = d_lr), 5)

     clr      ilr pairwise 
 1.35279  1.35279  1.35279

All three give 1.353. This distance ranks pairs of samples very differently from ordinary Euclidean distance on the raw proportions. Euclidean distance is dominated by the abundant parts, so two samples that differ only in a rare taxon look almost identical even when that taxon changed by an order of magnitude. Aitchison distance weighs the ratios and sees the change.

CLRfull <- t(apply(P, 1, clr))
Draw <- as.matrix(dist(P)); Dait <- as.matrix(dist(CLRfull)); ut <- upper.tri(Draw)
rho <- cor(Draw[ut], Dait[ut], method = "spearman")
set.seed(9); s <- sample(which(ut), 3000)
ggplot(data.frame(euc = Draw[s], ait = Dait[s]), aes(euc, ait)) +
  geom_point(colour = te_forest, alpha = 0.25, size = 0.9) +
  labs(title = "Naive Euclidean distance is not Aitchison distance",
       subtitle = sprintf("Spearman rank agreement %.2f over all pairs", rho),
       x = "Euclidean distance on raw proportions", y = "Aitchison distance (Euclidean on clr)")

A scatter of many points comparing two distance measures. The cloud is broadly increasing but wide, with clear disagreement between the two orderings. — Figure 2: Naive Euclidean distance on raw proportions against Aitchison distance for every pair of samples. The rank agreement is only 0.80: many pairs that sit close on proportions are far apart on log-ratios, and the reverse.

The Spearman rank agreement between the two distance matrices is 0.798. As one example, among the 44,850 pairs there is one that ranks 2161 of 44,850 by Euclidean distance, near the closest, yet ranks 4.1119^{4} by Aitchison distance, near the farthest. The two samples look almost the same in proportions but have very different internal ratios.

Additive and isometric coordinates

The additive log-ratio divides every part by a chosen reference part and takes logarithms. It gives 4 coordinates and is simple, but it is not isometric: distances measured on alr coordinates do not equal Aitchison distance, and they depend on which part was chosen as the reference.

d_alr <- sqrt(sum((alr(xa, 5) - alr(xb, 5))^2))
round(c(alr_euclidean = d_alr, aitchison = d_clr), 5)

alr_euclidean     aitchison 
      1.92256       1.35279

The alr distance here is 1.923 against an Aitchison distance of 1.353. They differ, and a different reference would give a different alr distance. For interpreting a single contrast against a fixed baseline the alr is fine; for anything that relies on distances or angles it distorts them.

The isometric log-ratio removes that distortion. It projects the clr onto an orthonormal basis of the zero-sum space, giving 4 coordinates that are uncorrelated by construction and preserve Aitchison distance exactly. The basis used here contrasts the geometric mean of the first i parts against part i + 1, scaled so that the columns are orthonormal.

c(orthonormal = max(abs(crossprod(V) - diag(D - 1))),
  reconstruct = max(abs(CLRfull - (CLRfull %*% V) %*% t(V))))

 orthonormal  reconstruct 
2.220446e-16 6.661338e-16

The basis satisfies V'V = I to machine precision, and applying it and its transpose recovers the clr exactly. The ilr coordinates are the ones to feed to a discriminant analysis, a distance-based ordination, or a multivariate normal model, because they are full rank and isometric.

The honest limit

Each transformation trades one inconvenience for another. The clr is interpretable part by part but singular, so it cannot go into a method that inverts a covariance. The alr is simple but reference-dependent and non-isometric. The ilr is full rank and isometric but its coordinates are not individually interpretable: only the whole geometry means something, and a different orthonormal basis gives different coordinate values for the same distances. None of this recovers absolute abundance. Every log-ratio describes parts relative to one another, and the total that closure removed stays gone. The three transformations agree on distances and disagree on interpretation, so the right choice depends on the question, not on which is best in the abstract.

What comes next

With coordinates that behave, the standard toolkit opens up: principal components on clr, clustering on Aitchison distance, and tests on balances. The next post applies these to a two-group comparison, shows how a single real change becomes many spurious ones under naive proportion tests, and confronts the practical obstacle that log-ratios cannot handle a zero.

References

Aitchison, J. (1982). The statistical analysis of compositional data. Journal of the Royal Statistical Society: Series B (Methodological) 44(2), 139-160. https://doi.org/10.1111/j.2517-6161.1982.tb01195.x

Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Monographs on Statistics and Applied Probability. Chapman and Hall, London. ISBN 978-0-412-28060-3.

Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., and Barcelo-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology 35(3), 279-300. https://doi.org/10.1023/A:1023818214614

Egozcue, J. J., and Pawlowsky-Glahn, V. (2005). Groups of parts and their balances in compositional data analysis. Mathematical Geology 37(7), 795-828. https://doi.org/10.1007/s11004-005-7381-9

Pawlowsky-Glahn, V., Egozcue, J. J., and Tolosana-Delgado, R. (2015). Modeling and Analysis of Compositional Data. Wiley, Chichester. ISBN 978-1-118-44306-4.

Filzmoser, P., Hron, K., and Templ, M. (2018). Applied Compositional Data Analysis: With Worked Examples in R. Springer, Cham. ISBN 978-3-319-96420-1.

--- title: "Log-ratio transformations: clr, alr and ilr" description: "Building the centred, additive and isometric log-ratio transformations by hand, with Aitchison distance, perturbation and powering, using base R only." date: 2026-08-01 10:00 categories: [R, compositional data, ggplot2] image: thumbnail.png image-alt: "A triangular simplex with sample points and a red arrow showing a perturbation, the compositional analogue of a translation." --- The previous post showed that proportions carry a constant-sum constraint that manufactures negative correlations and makes ordinary correlation subcompositionally incoherent. The one quantity that behaved well was the log-ratio. This post takes that hint seriously and builds the three standard log-ratio transformations from first principles: the centred log-ratio, the additive log-ratio, and the isometric log-ratio. Each maps a composition off the constrained simplex and into ordinary Euclidean space, where distance, correlation and ordination recover their usual meaning. All three are a few lines of base R. ## Setup ```{r} #| label: setup #| message: false #| warning: false library(ggplot2) clr <- function(x) { lx <- log(x); lx - mean(lx) } # centred log-ratio alr <- function(x, ref = length(x)) log(x[-ref] / x[ref]) # additive log-ratio C <- function(x) x / sum(x) # closure ilr_basis <- function(D) { # orthonormal contrast matrix V <- matrix(0, D, D - 1) for (i in seq_len(D - 1)) { a <- sqrt(1 / (i * (i + 1))); b <- sqrt(i / (i + 1)) V[1:i, i] <- a; V[i + 1, i] <- -b } V } te_ink <- "#16241d"; te_forest <- "#275139"; te_faint <- "#5d6b61" te_paper <- "#f5f4ee"; te_line <- "#dad9ca"; te_red <- "#b5534e" theme_te <- function(base = 12) theme_minimal(base_size = base) + theme( plot.background = element_rect(fill = te_paper, colour = NA), panel.background = element_rect(fill = te_paper, colour = NA), panel.grid.major = element_line(colour = te_line, linewidth = 0.3), panel.grid.minor = element_blank(), axis.text = element_text(colour = "#2c3a31"), axis.title = element_text(colour = te_ink), plot.title = element_text(colour = te_ink, face = "bold"), plot.subtitle = element_text(colour = te_faint, size = rel(0.9)), legend.position = "top") theme_set(theme_te()) set.seed(3390) D <- 5L; n <- 300L mu <- c(2.8, 2.4, 2.0, 1.6, 2.2); sdlog <- c(0.5, 0.6, 0.55, 0.7, 0.5) X <- sapply(1:D, function(j) rlnorm(n, mu[j], sdlog[j])) P <- X / rowSums(X) V <- ilr_basis(D) ``` ## The centred log-ratio The centred log-ratio takes logarithms of the parts and subtracts their mean, which is the same as dividing every part by the geometric mean of the composition. The result is a vector that sums to zero. ```{r} #| label: clr CLR <- t(apply(P, 1, clr)) max(abs(rowSums(CLR))) # rows sum to zero ``` The rows sum to zero to within `r sprintf("%.1e", max(abs(rowSums(CLR))))`. That zero-sum property is exactly what makes the clr convenient and what makes it awkward. Convenient, because clr coordinates live in ordinary space and each one is interpretable as a part against the average part. Awkward, because the zero-sum means the coordinates are linearly dependent: their covariance matrix is singular and cannot be inverted. ```{r} #| label: clr-singular ev <- eigen(cov(CLR), only.values = TRUE)$values c(smallest_eigenvalue = min(abs(ev)), rank = qr(cov(CLR))$rank, parts = D) ``` The smallest eigenvalue is `r sprintf("%.1e", min(abs(ev)))` and the matrix has rank `r qr(cov(CLR))$rank` for `r D` parts. Any method that needs a full-rank covariance, discriminant analysis or a multivariate normal likelihood among them, cannot take clr coordinates directly. That is what the isometric log-ratio fixes. ## A picture of the simplex Before the algebra, a picture helps. A three-part composition is a point in a triangle: the three corners are the pure compositions, and every interior point is a mixture. Distances and directions in this triangle are not the ordinary ones. The natural operations are perturbation, which plays the role of a translation, and powering, which plays the role of scaling. Perturbation of one composition by another is the closed elementwise product; powering is the closed elementwise power. ```{r} #| label: fig-simplex #| fig-cap: "Nine three-part compositions plotted in the simplex. The red arrow is a perturbation, the compositional analogue of adding a fixed vector: it moves a point by multiplying its parts and re-closing." #| fig-alt: "An upward triangle labelled part A, part B and part C at its corners, with green points scattered inside and a red arrow between two of them." #| fig-width: 6.6 #| fig-height: 5.2 set.seed(11) A3 <- sapply(1:3, function(j) rlnorm(9, c(2.2, 2.0, 1.6)[j], 0.5)); T3 <- A3 / rowSums(A3) tern <- function(m) data.frame(tx = m[,2] + 0.5 * m[,3], ty = (sqrt(3)/2) * m[,3]) pts <- tern(T3) vert <- data.frame(tx = c(0, 1, 0.5), ty = c(0, 0, sqrt(3)/2), lab = c("part A", "part B", "part C")) edge <- data.frame(x = c(0, 1, 0.5), y = c(0, 0, sqrt(3)/2), xend = c(1, 0.5, 0), yend = c(0, sqrt(3)/2, 0)) xstar <- C(T3[1,]); yv <- C(c(1.8, 1.0, 1.0)); xp <- C(xstar * yv) pth <- tern(rbind(xstar, xp)) ggplot() + geom_segment(data = edge, aes(x, y, xend = xend, yend = yend), colour = te_line, linewidth = 0.6) + geom_point(data = pts, aes(tx, ty), colour = te_forest, size = 2.2, alpha = 0.8) + geom_segment(data = data.frame(x = pth$tx[1], y = pth$ty[1], xend = pth$tx[2], yend = pth$ty[2]), aes(x, y, xend = xend, yend = yend), colour = te_red, linewidth = 0.9, arrow = arrow(length = unit(0.18, "cm"), type = "closed")) + geom_text(data = vert, aes(tx, ty, label = lab), vjust = c(1.6, 1.6, -0.8), colour = te_ink, size = 3.4) + coord_equal() + theme_void() + theme(plot.background = element_rect(fill = te_paper, colour = NA), plot.title = element_text(colour = te_ink, face = "bold"), plot.subtitle = element_text(colour = te_faint, size = rel(0.9))) + labs(title = "The simplex: sample space of compositions", subtitle = "Points are three-part compositions; the arrow is a perturbation") ``` These operations are not exotic once viewed through the clr. Perturbation becomes ordinary addition, and powering becomes ordinary multiplication, in clr space. ```{r} #| label: geometry y <- P[3, ]; a <- 2.3; xa <- P[1, ] c(perturbation = max(abs(clr(C(xa * y)) - (clr(xa) + clr(y)))), powering = max(abs(clr(C(xa^a)) - a * clr(xa)))) ``` Both identities hold to machine precision. The simplex, with perturbation and powering, is a vector space, and the clr is the map that turns its geometry into the geometry we already know. ## Distance that respects ratios The distance attached to this geometry is the Aitchison distance. It can be written three ways that all agree: as the ordinary distance between clr vectors, as the ordinary distance between isometric log-ratio vectors, and as a sum over all pairwise log-ratios. Computing one composition against another confirms the identity. ```{r} #| label: distance xa <- P[1, ]; xb <- P[2, ] d_clr <- sqrt(sum((clr(xa) - clr(xb))^2)) # Euclidean on clr d_ilr <- sqrt(sum(((clr(xa) - clr(xb)) %*% V)^2)) # Euclidean on ilr Lr <- function(x) outer(1:D, 1:D, function(i, j) log(x[i] / x[j])) d_lr <- sqrt(sum((Lr(xa) - Lr(xb))^2) / (2 * D)) # pairwise log-ratio form round(c(clr = d_clr, ilr = d_ilr, pairwise = d_lr), 5) ``` All three give `r round(d_clr, 3)`. This distance ranks pairs of samples very differently from ordinary Euclidean distance on the raw proportions. Euclidean distance is dominated by the abundant parts, so two samples that differ only in a rare taxon look almost identical even when that taxon changed by an order of magnitude. Aitchison distance weighs the ratios and sees the change. ```{r} #| label: fig-distance #| fig-cap: "Naive Euclidean distance on raw proportions against Aitchison distance for every pair of samples. The rank agreement is only 0.80: many pairs that sit close on proportions are far apart on log-ratios, and the reverse." #| fig-alt: "A scatter of many points comparing two distance measures. The cloud is broadly increasing but wide, with clear disagreement between the two orderings." #| fig-width: 7 #| fig-height: 4.6 CLRfull <- t(apply(P, 1, clr)) Draw <- as.matrix(dist(P)); Dait <- as.matrix(dist(CLRfull)); ut <- upper.tri(Draw) rho <- cor(Draw[ut], Dait[ut], method = "spearman") set.seed(9); s <- sample(which(ut), 3000) ggplot(data.frame(euc = Draw[s], ait = Dait[s]), aes(euc, ait)) + geom_point(colour = te_forest, alpha = 0.25, size = 0.9) + labs(title = "Naive Euclidean distance is not Aitchison distance", subtitle = sprintf("Spearman rank agreement %.2f over all pairs", rho), x = "Euclidean distance on raw proportions", y = "Aitchison distance (Euclidean on clr)") ``` The Spearman rank agreement between the two distance matrices is `r round(cor(Draw[ut], Dait[ut], method = "spearman"), 3)`. As one example, among the `r format(sum(ut), big.mark = ",")` pairs there is one that ranks `r rank(Draw[ut])[which.max(rank(Dait[ut]) - rank(Draw[ut]))]` of `r format(sum(ut), big.mark = ",")` by Euclidean distance, near the closest, yet ranks `r rank(Dait[ut])[which.max(rank(Dait[ut]) - rank(Draw[ut]))]` by Aitchison distance, near the farthest. The two samples look almost the same in proportions but have very different internal ratios. ## Additive and isometric coordinates The additive log-ratio divides every part by a chosen reference part and takes logarithms. It gives `r D - 1` coordinates and is simple, but it is not isometric: distances measured on alr coordinates do not equal Aitchison distance, and they depend on which part was chosen as the reference. ```{r} #| label: alr d_alr <- sqrt(sum((alr(xa, 5) - alr(xb, 5))^2)) round(c(alr_euclidean = d_alr, aitchison = d_clr), 5) ``` The alr distance here is `r round(d_alr, 3)` against an Aitchison distance of `r round(d_clr, 3)`. They differ, and a different reference would give a different alr distance. For interpreting a single contrast against a fixed baseline the alr is fine; for anything that relies on distances or angles it distorts them. The isometric log-ratio removes that distortion. It projects the clr onto an orthonormal basis of the zero-sum space, giving `r D - 1` coordinates that are uncorrelated by construction and preserve Aitchison distance exactly. The basis used here contrasts the geometric mean of the first `i` parts against part `i + 1`, scaled so that the columns are orthonormal. ```{r} #| label: ilr c(orthonormal = max(abs(crossprod(V) - diag(D - 1))), reconstruct = max(abs(CLRfull - (CLRfull %*% V) %*% t(V)))) ``` The basis satisfies `V'V = I` to machine precision, and applying it and its transpose recovers the clr exactly. The ilr coordinates are the ones to feed to a discriminant analysis, a distance-based ordination, or a multivariate normal model, because they are full rank and isometric. ## The honest limit Each transformation trades one inconvenience for another. The clr is interpretable part by part but singular, so it cannot go into a method that inverts a covariance. The alr is simple but reference-dependent and non-isometric. The ilr is full rank and isometric but its coordinates are not individually interpretable: only the whole geometry means something, and a different orthonormal basis gives different coordinate values for the same distances. None of this recovers absolute abundance. Every log-ratio describes parts relative to one another, and the total that closure removed stays gone. The three transformations agree on distances and disagree on interpretation, so the right choice depends on the question, not on which is best in the abstract. ## What comes next With coordinates that behave, the standard toolkit opens up: principal components on clr, clustering on Aitchison distance, and tests on balances. The next post applies these to a two-group comparison, shows how a single real change becomes many spurious ones under naive proportion tests, and confronts the practical obstacle that log-ratios cannot handle a zero. ## Related tutorials - [Closure and spurious correlation](../closure-and-spurious-correlation/) - [NMDS ordination](../nmds-ordination/) - [Compositional analysis and the zero problem](../compositional-analysis-and-zeros/) ## References Aitchison, J. (1982). The statistical analysis of compositional data. *Journal of the Royal Statistical Society: Series B (Methodological)* 44(2), 139-160. <https://doi.org/10.1111/j.2517-6161.1982.tb01195.x> Aitchison, J. (1986). *The Statistical Analysis of Compositional Data*. Monographs on Statistics and Applied Probability. Chapman and Hall, London. ISBN 978-0-412-28060-3. Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., and Barcelo-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. *Mathematical Geology* 35(3), 279-300. <https://doi.org/10.1023/A:1023818214614> Egozcue, J. J., and Pawlowsky-Glahn, V. (2005). Groups of parts and their balances in compositional data analysis. *Mathematical Geology* 37(7), 795-828. <https://doi.org/10.1007/s11004-005-7381-9> Pawlowsky-Glahn, V., Egozcue, J. J., and Tolosana-Delgado, R. (2015). *Modeling and Analysis of Compositional Data*. Wiley, Chichester. ISBN 978-1-118-44306-4. Filzmoser, P., Hron, K., and Templ, M. (2018). *Applied Compositional Data Analysis: With Worked Examples in R*. Springer, Cham. ISBN 978-3-319-96420-1.