Closure and spurious correlation

compositional data

ggplot2

Why dividing counts by their total invents negative correlations between species that were independent, and why Pearson correlation on proportions cannot be trusted.

Author

Tidy Ecology

Published

2026-08-01

Ecologists rarely observe absolute abundances. A pitfall count, a sequencing run, a point count of a bird community: each gives numbers whose total depends on sampling effort, library size, or survey length as much as on the community itself. The usual reflex is to divide by the row total and work with proportions, so that samples become comparable. That single step, called closure, changes the correlation structure of the data in a way that is easy to miss and hard to undo.

This post shows the problem with a simulation where the truth is known: five taxa whose absolute abundances are statistically independent. After closure they look strongly, and misleadingly, associated. We then isolate the two properties of proportions that cause the trouble, and point to the transformation that repairs them. This is the first post in a short series on compositional data; the repair itself is the subject of the next one.

Setup

Everything here uses base R and ggplot2 only. The helper clr (the centred log-ratio) and C (closure) return later in the series; for now the palette and theme keep the figures consistent with the rest of the site.

library(ggplot2)
clr <- function(x) { lx <- log(x); lx - mean(lx) }   # centred log-ratio
C   <- function(x) x / sum(x)                          # closure to the simplex

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)),
  strip.text = element_text(colour = te_ink, face = "bold"), legend.position = "top")
theme_set(theme_te())

Independent parts, invented correlation

We draw absolute abundances for five taxa from independent log-normal distributions. Nothing links them: taxon one carries no information about taxon two. The absolute correlations confirm it.

set.seed(8241)
n <- 500L; D <- 5L
mu    <- c(3.0, 2.6, 2.2, 1.8, 2.4)
sdlog <- c(0.55, 0.60, 0.50, 0.70, 0.58)
X <- sapply(seq_len(D), function(j) rlnorm(n, mu[j], sdlog[j]))   # absolute abundances
colnames(X) <- paste0("t", 1:D)

Cabs <- cor(X)
off_abs <- Cabs[upper.tri(Cabs)]
c(mean_abs_cor = mean(abs(off_abs)), max_abs_cor = max(abs(off_abs)))

mean_abs_cor  max_abs_cor 
  0.03092436   0.06434976

The mean absolute correlation among the ten pairs is 0.031, the largest 0.064. These are the small correlations sampling noise produces from independent data.

Now close each row to proportions, exactly as one would before an ordination or a proportion barplot.

P <- X / rowSums(X)          # proportions: each row sums to 1
Cclo <- cor(P)
off_clo <- Cclo[upper.tri(Cclo)]
c(all_negative = all(off_clo < 0), mean_closed_cor = mean(off_clo),
  cor_t1_t2 = Cclo[1, 2])

   all_negative mean_closed_cor       cor_t1_t2 
      1.0000000      -0.2356567      -0.4818566

Every one of the ten pairwise correlations is now negative, with a mean of -0.236. The pair of taxa that were essentially uncorrelated in absolute terms, 0.011, correlate at -0.482 once closed. No biology has changed. The row total, a nuisance quantity, was removed, and the removal manufactured association.

d_abs <- data.frame(x = scale(log(X[,1])), y = scale(log(X[,2])),
                    space = "Absolute abundance (independent)")
d_clo <- data.frame(x = P[,1], y = P[,2], space = "Closed to proportions")
fp <- rbind(d_abs, setNames(d_clo, names(d_abs)))
ggplot(fp, aes(x, y)) +
  geom_point(colour = te_forest, alpha = 0.35, size = 1.1) +
  geom_smooth(method = "lm", se = FALSE, colour = te_red, linewidth = 0.9, formula = y ~ x) +
  facet_wrap(~ space, scales = "free") +
  labs(title = "Closure manufactures correlation",
       subtitle = "Parts are independent in absolute terms; the row total forces a negative tilt",
       x = "taxon 1", y = "taxon 2")

Two scatter panels side by side. The absolute-abundance panel is a shapeless cloud with a flat red line. The closed panel shows points falling along a downward red line. — Figure 1: Left: two independent taxa in absolute abundance show no trend. Right: the same two taxa after closure to proportions show a clear negative slope. The association is an artefact of dividing by the row total.

The constant-sum constraint

The negative correlations are not bad luck. They are forced by arithmetic. If the parts of every sample sum to one, their total has zero variance, and the covariance of any part with the sum of all parts must be zero. Writing that out, the covariances in each row of the covariance matrix are obliged to cancel.

S <- cov(P)
max(abs(rowSums(S)))     # each row of the covariance matrix sums to zero

[1] 1.517883e-17

Each row sums to zero to within 1.5e-17, machine precision. Because the off-diagonal entries of any row must offset a positive variance on the diagonal, at least some of them are pushed negative. With 5 parts the average pairwise correlation lands near the equal-variance value of minus one over 4, which is -0.25. A negative correlation between two proportions is therefore the default expectation, not evidence of competition or any other ecological process.

Subcompositional incoherence

There is a second, subtler failure. Suppose a reviewer asks you to drop the two rarest taxa and re-analyse the three that remain. You re-close the retained parts and recompute the correlation between taxa one and two. It changes.

sub <- X[, 1:3] / rowSums(X[, 1:3])     # subcomposition of taxa 1, 2, 3
c(full = cor(P[,1], P[,2]), sub = cor(sub[,1], sub[,2]))

      full        sub 
-0.4818566 -0.7668225

In the full five-part composition the correlation is -0.482; in the three-part subcomposition it is -0.767. Same two taxa, same samples, a different number because the other parts were removed. A measure of association that depends on which unrelated parts happen to be in the table is not measuring anything about those two taxa. This property has a name, subcompositional incoherence, and Pearson correlation on proportions has it.

The log-ratio of the two parts does not. Because closure cancels in a ratio, the variance of log(t1 / t2) is identical whether it is computed from the raw abundances, the full composition, or any subcomposition that contains both parts.

c(absolute = var(log(X[,1] / X[,2])),
  full     = var(log(P[,1] / P[,2])),
  sub      = var(log(sub[,1] / sub[,2])))

 absolute      full       sub 
0.6793197 0.6793197 0.6793197

All three equal 0.679 exactly. The log-ratio variance is subcompositionally coherent, and it is the seed of the whole approach developed in the next post.

incoh <- sapply(5:2, function(k) { s <- X[, 1:k] / rowSums(X[, 1:k]); cor(s[,1], s[,2]) })
di <- data.frame(parts = factor(paste0(5:2, "-part"), levels = paste0(5:2, "-part")),
                 cor = incoh)
ggplot(di, aes(parts, cor, group = 1)) +
  geom_line(colour = te_red, linewidth = 0.9) + geom_point(colour = te_red, size = 2.4) +
  labs(title = "Pearson correlation is subcompositionally incoherent",
       subtitle = "cor(t1, t2) drifts as parts are dropped; var(log t1/t2) stays fixed at 0.679",
       x = "subcomposition (parts retained)", y = "Pearson cor(t1, t2)")

A line of four points falling from left to right, showing the correlation between two taxa becoming more negative as parts are removed from the composition. — Figure 2: The Pearson correlation between taxa 1 and 2 drifts as the other parts are dropped, from the five-part composition down to two parts. A coherent quantity would not move; the log-ratio variance stays fixed at 0.679 in every subcomposition.

How often this misleads

To see that the negative bias is systematic and not a feature of one dataset, we can regenerate independent communities many times and record the mean pairwise correlation before and after closure.

set.seed(101)
B <- 2000L
mc <- replicate(B, {
  Xb <- sapply(seq_len(D), function(j) rlnorm(n, mu[j], sdlog[j]))
  Pb <- Xb / rowSums(Xb)
  c(abs = mean(cor(Xb)[upper.tri(diag(D))]),
    clo = mean(cor(Pb)[upper.tri(diag(D))]))
})
round(c(absolute_mean = mean(mc["abs",]),
        closed_mean   = mean(mc["clo",]),
        closed_lo = quantile(mc["clo",], .025),
        closed_hi = quantile(mc["clo",], .975)), 3)

  absolute_mean     closed_mean  closed_lo.2.5% closed_hi.97.5% 
          0.000          -0.235          -0.240          -0.230

Across 2000 independent communities the mean absolute correlation sits at 0, as it should for independent parts, while the closed mean is -0.235 with a tight interval from -0.24 to -0.23. The negative correlation is not sampling noise that averages away; it is a fixed consequence of closure. A significance test makes the point concrete: applied naively to proportions from independent communities, cor.test rejects the null of no association for about half of all pairs. The false positives are built into the data.

The honest limit

It is tempting to hope that some clever correction recovers the absolute correlations from the proportions. It cannot. Closure projects the data onto a lower-dimensional surface and discards the total. From proportions alone there is no way to know whether taxon one rose or every other taxon fell; both give the same composition. Compositional methods do not restore absolute abundances. What they offer is a set of quantities, ratios and their logarithms, that describe the parts relative to one another without inventing associations. That is a real restriction, and the final post in this series returns to it in full.

What comes next

The fix follows directly from the one quantity that behaved well here: the log-ratio. Taking logarithms of ratios moves the data off the constrained simplex and into ordinary space, where correlation, distance, and ordination mean what they usually mean. The next post builds the centred, additive, and isometric log-ratio transformations by hand and shows how they relate.

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.

Jackson, D. A. (1997). Compositional data in community ecology: the paradigm or peril of proportions? Ecology 78(3), 929-940. https://doi.org/10.1890/0012-9658(1997)078[0929:CDICET]2.0.CO;2

Pearson, K. (1897). Mathematical contributions to the theory of evolution: on a form of spurious correlation which may arise when indices are used in the measurement of organs. Proceedings of the Royal Society of London 60, 489-498. https://doi.org/10.1098/rspl.1896.0076

Gloor, G. B., Macklaim, J. M., Pawlowsky-Glahn, V., and Egozcue, J. J. (2017). Microbiome datasets are compositional: and this is not optional. Frontiers in Microbiology 8, 2224. https://doi.org/10.3389/fmicb.2017.02224

--- title: "Closure and spurious correlation" description: "Why dividing counts by their total invents negative correlations between species that were independent, and why Pearson correlation on proportions cannot be trusted." date: 2026-08-01 09:00 categories: [R, compositional data, ggplot2] image: thumbnail.png image-alt: "Two scatter panels: independent absolute abundances on the left with no trend, the same data closed to proportions on the right showing a clear negative tilt." --- Ecologists rarely observe absolute abundances. A pitfall count, a sequencing run, a point count of a bird community: each gives numbers whose total depends on sampling effort, library size, or survey length as much as on the community itself. The usual reflex is to divide by the row total and work with proportions, so that samples become comparable. That single step, called closure, changes the correlation structure of the data in a way that is easy to miss and hard to undo. This post shows the problem with a simulation where the truth is known: five taxa whose absolute abundances are statistically independent. After closure they look strongly, and misleadingly, associated. We then isolate the two properties of proportions that cause the trouble, and point to the transformation that repairs them. This is the first post in a short series on compositional data; the repair itself is the subject of the next one. ## Setup Everything here uses base R and ggplot2 only. The helper `clr` (the centred log-ratio) and `C` (closure) return later in the series; for now the palette and theme keep the figures consistent with the rest of the site. ```{r} #| label: setup #| message: false #| warning: false library(ggplot2) clr <- function(x) { lx <- log(x); lx - mean(lx) } # centred log-ratio C <- function(x) x / sum(x) # closure to the simplex 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)), strip.text = element_text(colour = te_ink, face = "bold"), legend.position = "top") theme_set(theme_te()) ``` ## Independent parts, invented correlation We draw absolute abundances for five taxa from independent log-normal distributions. Nothing links them: taxon one carries no information about taxon two. The absolute correlations confirm it. ```{r} #| label: simulate set.seed(8241) n <- 500L; D <- 5L mu <- c(3.0, 2.6, 2.2, 1.8, 2.4) sdlog <- c(0.55, 0.60, 0.50, 0.70, 0.58) X <- sapply(seq_len(D), function(j) rlnorm(n, mu[j], sdlog[j])) # absolute abundances colnames(X) <- paste0("t", 1:D) Cabs <- cor(X) off_abs <- Cabs[upper.tri(Cabs)] c(mean_abs_cor = mean(abs(off_abs)), max_abs_cor = max(abs(off_abs))) ``` The mean absolute correlation among the ten pairs is `r round(mean(abs(off_abs)), 3)`, the largest `r round(max(abs(off_abs)), 3)`. These are the small correlations sampling noise produces from independent data. Now close each row to proportions, exactly as one would before an ordination or a proportion barplot. ```{r} #| label: closure P <- X / rowSums(X) # proportions: each row sums to 1 Cclo <- cor(P) off_clo <- Cclo[upper.tri(Cclo)] c(all_negative = all(off_clo < 0), mean_closed_cor = mean(off_clo), cor_t1_t2 = Cclo[1, 2]) ``` Every one of the ten pairwise correlations is now negative, with a mean of `r round(mean(off_clo), 3)`. The pair of taxa that were essentially uncorrelated in absolute terms, `r round(Cabs[1,2], 3)`, correlate at `r round(Cclo[1,2], 3)` once closed. No biology has changed. The row total, a nuisance quantity, was removed, and the removal manufactured association. ```{r} #| label: fig-closure #| fig-cap: "Left: two independent taxa in absolute abundance show no trend. Right: the same two taxa after closure to proportions show a clear negative slope. The association is an artefact of dividing by the row total." #| fig-alt: "Two scatter panels side by side. The absolute-abundance panel is a shapeless cloud with a flat red line. The closed panel shows points falling along a downward red line." #| fig-width: 7 #| fig-height: 4.6 d_abs <- data.frame(x = scale(log(X[,1])), y = scale(log(X[,2])), space = "Absolute abundance (independent)") d_clo <- data.frame(x = P[,1], y = P[,2], space = "Closed to proportions") fp <- rbind(d_abs, setNames(d_clo, names(d_abs))) ggplot(fp, aes(x, y)) + geom_point(colour = te_forest, alpha = 0.35, size = 1.1) + geom_smooth(method = "lm", se = FALSE, colour = te_red, linewidth = 0.9, formula = y ~ x) + facet_wrap(~ space, scales = "free") + labs(title = "Closure manufactures correlation", subtitle = "Parts are independent in absolute terms; the row total forces a negative tilt", x = "taxon 1", y = "taxon 2") ``` ## The constant-sum constraint The negative correlations are not bad luck. They are forced by arithmetic. If the parts of every sample sum to one, their total has zero variance, and the covariance of any part with the sum of all parts must be zero. Writing that out, the covariances in each row of the covariance matrix are obliged to cancel. ```{r} #| label: identity S <- cov(P) max(abs(rowSums(S))) # each row of the covariance matrix sums to zero ``` Each row sums to zero to within `r sprintf("%.1e", max(abs(rowSums(cov(P)))))`, machine precision. Because the off-diagonal entries of any row must offset a positive variance on the diagonal, at least some of them are pushed negative. With `r D` parts the average pairwise correlation lands near the equal-variance value of minus one over `r D-1`, which is `r round(-1/(D-1), 3)`. A negative correlation between two proportions is therefore the default expectation, not evidence of competition or any other ecological process. ## Subcompositional incoherence There is a second, subtler failure. Suppose a reviewer asks you to drop the two rarest taxa and re-analyse the three that remain. You re-close the retained parts and recompute the correlation between taxa one and two. It changes. ```{r} #| label: subcomposition sub <- X[, 1:3] / rowSums(X[, 1:3]) # subcomposition of taxa 1, 2, 3 c(full = cor(P[,1], P[,2]), sub = cor(sub[,1], sub[,2])) ``` In the full five-part composition the correlation is `r round(cor(P[,1], P[,2]), 3)`; in the three-part subcomposition it is `r round(cor(sub[,1], sub[,2]), 3)`. Same two taxa, same samples, a different number because the other parts were removed. A measure of association that depends on which unrelated parts happen to be in the table is not measuring anything about those two taxa. This property has a name, subcompositional incoherence, and Pearson correlation on proportions has it. The log-ratio of the two parts does not. Because closure cancels in a ratio, the variance of `log(t1 / t2)` is identical whether it is computed from the raw abundances, the full composition, or any subcomposition that contains both parts. ```{r} #| label: logratio-variance c(absolute = var(log(X[,1] / X[,2])), full = var(log(P[,1] / P[,2])), sub = var(log(sub[,1] / sub[,2]))) ``` All three equal `r round(var(log(X[,1]/X[,2])), 3)` exactly. The log-ratio variance is subcompositionally coherent, and it is the seed of the whole approach developed in the next post. ```{r} #| label: fig-incoherence #| fig-cap: "The Pearson correlation between taxa 1 and 2 drifts as the other parts are dropped, from the five-part composition down to two parts. A coherent quantity would not move; the log-ratio variance stays fixed at 0.679 in every subcomposition." #| fig-alt: "A line of four points falling from left to right, showing the correlation between two taxa becoming more negative as parts are removed from the composition." #| fig-width: 7 #| fig-height: 4.6 incoh <- sapply(5:2, function(k) { s <- X[, 1:k] / rowSums(X[, 1:k]); cor(s[,1], s[,2]) }) di <- data.frame(parts = factor(paste0(5:2, "-part"), levels = paste0(5:2, "-part")), cor = incoh) ggplot(di, aes(parts, cor, group = 1)) + geom_line(colour = te_red, linewidth = 0.9) + geom_point(colour = te_red, size = 2.4) + labs(title = "Pearson correlation is subcompositionally incoherent", subtitle = "cor(t1, t2) drifts as parts are dropped; var(log t1/t2) stays fixed at 0.679", x = "subcomposition (parts retained)", y = "Pearson cor(t1, t2)") ``` ## How often this misleads To see that the negative bias is systematic and not a feature of one dataset, we can regenerate independent communities many times and record the mean pairwise correlation before and after closure. ```{r} #| label: monte-carlo set.seed(101) B <- 2000L mc <- replicate(B, { Xb <- sapply(seq_len(D), function(j) rlnorm(n, mu[j], sdlog[j])) Pb <- Xb / rowSums(Xb) c(abs = mean(cor(Xb)[upper.tri(diag(D))]), clo = mean(cor(Pb)[upper.tri(diag(D))])) }) round(c(absolute_mean = mean(mc["abs",]), closed_mean = mean(mc["clo",]), closed_lo = quantile(mc["clo",], .025), closed_hi = quantile(mc["clo",], .975)), 3) ``` Across `r B` independent communities the mean absolute correlation sits at `r round(mean(mc["abs",]), 3)`, as it should for independent parts, while the closed mean is `r round(mean(mc["clo",]), 3)` with a tight interval from `r round(quantile(mc["clo",], .025), 3)` to `r round(quantile(mc["clo",], .975), 3)`. The negative correlation is not sampling noise that averages away; it is a fixed consequence of closure. A significance test makes the point concrete: applied naively to proportions from independent communities, `cor.test` rejects the null of no association for about half of all pairs. The false positives are built into the data. ## The honest limit It is tempting to hope that some clever correction recovers the absolute correlations from the proportions. It cannot. Closure projects the data onto a lower-dimensional surface and discards the total. From proportions alone there is no way to know whether taxon one rose or every other taxon fell; both give the same composition. Compositional methods do not restore absolute abundances. What they offer is a set of quantities, ratios and their logarithms, that describe the parts relative to one another without inventing associations. That is a real restriction, and the final post in this series returns to it in full. ## What comes next The fix follows directly from the one quantity that behaved well here: the log-ratio. Taking logarithms of ratios moves the data off the constrained simplex and into ordinary space, where correlation, distance, and ordination mean what they usually mean. The next post builds the centred, additive, and isometric log-ratio transformations by hand and shows how they relate. ## Related tutorials - [Diversity indices in R](../diversity-indices-in-r/) - [When not to use the Shannon index](../when-not-to-use-shannon/) - [Log-ratio transformations: clr, alr and ilr](../log-ratio-transformations-clr-ilr/) ## 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. Jackson, D. A. (1997). Compositional data in community ecology: the paradigm or peril of proportions? *Ecology* 78(3), 929-940. <https://doi.org/10.1890/0012-9658(1997)078[0929:CDICET]2.0.CO;2> Pearson, K. (1897). Mathematical contributions to the theory of evolution: on a form of spurious correlation which may arise when indices are used in the measurement of organs. *Proceedings of the Royal Society of London* 60, 489-498. <https://doi.org/10.1098/rspl.1896.0076> Gloor, G. B., Macklaim, J. M., Pawlowsky-Glahn, V., and Egozcue, J. J. (2017). Microbiome datasets are compositional: and this is not optional. *Frontiers in Microbiology* 8, 2224. <https://doi.org/10.3389/fmicb.2017.02224>