Proportionality: A Valid Alternative to Correlation for Relative Data

1This is a PLOS Computational Biology Methods paper.

2Relative abundance measurements are common in molecular biology: nucleic acids typically have to be provided at a set concentration for sequencing or microarray analysis; sequencing methods report a large but finite total of reads, of which any particular sequence is a proportion. Sometimes, researchers are interested in the relative abundance of different components. Other times, they have to make do with relative abundance to gain insight into the system under study. Whatever the case, data that carry only relative information need special treatment.

3Awareness is growing but it is not yet widely appreciated that common analysis methods—including correlation—can be very misleading for data carrying only relative information. Compositional data analysis (CoDA) is a valid alternative that harks back to Pearson’s observation of ‘spurious correlation’, i.e., while statistically independent variables X, Y, and Z are not correlated, their ratios X/Z and Y/Z must be, because of their common divisor. (Note: this differs from the logical fallacy that “correlation implies causation”.)

4Proportions, percentages and parts per million are familiar examples of compositional data; the fact that the representation of their components is constrained to sum to a constant (i.e., 1, 100, 106) emphasizes that the data carry only relative information. Note that compositional data do not necessarily have to sum to a constant; what is essential is that only the ratios of the different components are regarded as informative.

5Correlation—Pearson, Spearman or other—leads to meaningless conclusions if applied to compositional data because its value depends on which components are analyzed. Problems with correlation can also be demonstrated geometrically (Fig. 1): the bivariate joint distribution of relative abundances says nothing about the distribution of absolute abundances that gave rise to them. Thus, relative data is also problematic for mutual information and other distributional measures of association. To further illustrate how correlation can be misleading we applied it to absolute and relative gene expression data in fission yeast cells deprived of a key nutrient.

6How then can we make sound inferences from relative data? We show how proportionality provides a valid alternative to correlation and can be used as the basis of familiar analyses and visualizations. We conclude by putting this analysis strategy in perspective, discussing challenges, caveats and issues for further work, as well as the biological questions raised in this study.

7Our results are based on data from Marguerat et al. on the absolute levels of gene expression (i.e., mRNA copies per cell) in fission yeast after cells were deprived of a key nutrient (Fig. 2). Unlike many experiments where researchers ensure (or assume) cells produce similar amounts of mRNA across conditions, this experiment ensured cells produced very different amounts so as to illustrate the merits of absolute quantification (S1 Fig.). Total abundance may vary dramatically in other experimental settings—such as in comparing diseased and normal tissues, tissues at different stages of development, or microbial communities in different environments.

8To illustrate the key points of this paper, we worked with positive data only (i.e., we excluded records with any zero or NA values): measurements of 3031 components (i.e., mRNAs) at 16 time points. Furthermore, we applied analysis methods (specifically, correlation) to the absolute abundance data without transformation (e.g., taking logarithms) because we believe this approach yields useful insights and simplifies the presentation of the central ideas of this paper (see and S1 Supporting Information).

9Before looking at issues with pairs of components, it is important to note that interpreting differences in the relative abundance of a single component can be challenging.

10Tests for differential expression are popular for analyzing relative data in bioscience. Much attention has been given to dealing with small numbers of observations and large numbers of tests, but comparatively little to “…the commonly believed, though rarely stated, assumption that the absolute amount of total mRNA in each cell is similar across different cell types or experimental perturbations”.

11The relationship between the relative and absolute abundance of a component can be understood in terms of fold change over time. When total absolute abundance of mRNA stays constant, fold changes in both absolute and relative abundance of each mRNA are equal. When total absolute abundance varies, fold changes in absolute and relative abundances of each mRNA are no longer equal and can change in different directions. Between 0 and 3 hours there were 1399 yeast mRNAs whose absolute abundance decreased, and whose relative abundance increased. Clearly, mRNAs are being expressed differently, but to describe them as “under- or over-expressed” is too simplistic—here lies the interpretation challenge (see S1 Supporting Information).

12While “differential expression” of relative abundances is challenging to interpret, in the absence of any other information or assumptions, correlation of relative abundances is just wrong. We stress in the absence of any other information or assumptions to highlight the common assumption of constant absolute abundance of total mRNA across all experimental conditions. If this assumption holds, and all the mRNAs comprising that total are considered, the relative abundance of each kind of mRNA will be proportional to its absolute abundance, and analyses of correlation or “differential expression” of the relative values will have clear interpretations. The revisitation of this assumption should raise alarm bells about the inferences drawn from many gene expression studies.

13Fig. 1(a) shows why correlation between relative abundances tells us nothing about the relationship between the absolute abundances that gave rise to them: the perfectly correlated relative abundances could come from any set of absolute abundance pairs that lie on the rays from the origin. This many-to-one mapping means that other measures of statistical association (e.g., rank correlations or mutual information) will not tell us anything either when applied to purely relative data.

14But is this problem just a theoretical construct? A rare issue? Consider the red mRNA pair in Fig. 2: while their absolute abundances over time are strongly positively correlated, if someone (inappropriately) used correlation to measure the association between the relative abundances of these two mRNAs they would form the opposite view (Fig. 3(a)); correlation between the blue mRNA pair in Fig. 2 is similarly misleading (S2 Fig.). What of the other 4.5 million pairs of mRNAs? Fig. 3(b) summarizes all discrepancies between correlations of absolute abundance, and correlations of relative abundance, showing clearly that the apparent correlations of relative abundances tell a very different story from those of the absolute data. So how should we go about analyzing these relative data?

15CoDA theory provides three principles: Scale invariance: analyses must treat vectors with proportional positive components as representing the same composition (e.g., (2, 3, 4) is equivalent to (20, 30, 40))Subcompositional coherence: inferences about subcompositions (subsets of components) should be consistent, regardless of whether the inference is based on the subcomposition or the full composition.Permutation invariance: the conclusions of analyses must not depend on the order of the components. Correlation is not subcompositionally coherent: its value depends on which components are considered in the analysis, e.g., if you deplete the most abundant RNAs from a sample and use correlation to measure association between relative abundances, you get different correlations to the undepleted sample (S3 Fig.).

16Scale invariance: analyses must treat vectors with proportional positive components as representing the same composition (e.g., (2, 3, 4) is equivalent to (20, 30, 40))

17Subcompositional coherence: inferences about subcompositions (subsets of components) should be consistent, regardless of whether the inference is based on the subcomposition or the full composition.

18Permutation invariance: the conclusions of analyses must not depend on the order of the components.

19Proportionality obeys all three principles for analyzing relative data. If relative abundances x and y are proportional across experimental conditions i, their absolute abundances must be in proportion: where t i is the total abundance in condition i (Fig. 1(b)).

20We proposed a “goodness-of-fit to proportionality” statistic ϕ to assess the extent to which a pair of random variables (x, y) are proportional. ϕ is related to logratio variance, var(log(x/y)), and is zero when x and y behave perfectly proportionally. However, when x and y are not proportional, ϕ has both a clear geometric interpretation and a meaningful scale, addressing concerns raised about logratio variance: the closer ϕ is to zero, the stronger the proportionality. We consider “strength” of proportionality (goodness-of-fit) rather than testing the hypothesis of proportionality because it allows us to compare relationships between different pairs of mRNAs (S1 Supporting Information).

21We calculated ϕ for the relative abundances of all pairs of mRNAs and compared it to the correlations between their absolute abundances (S4 Fig.): clearly, the absolute abundances of most mRNA pairs are strongly positively correlated; far fewer are also strongly proportional. Focusing on these strongly proportional mRNAs, we extracted the 424 pairs with ϕ < 0.05. We graphed the network of relationships between these mRNAs (S5 Fig.), an approach similar to gene co-expression network or weighted gene co-expression analysis but founded on proportionality and therefore valid for relative data. The network revealed one cluster of 96, and many other smaller clusters of mRNAs behaving proportionally across conditions. Using ϕ as a dissimilarity measure, we formed heatmaps of the three largest clusters (S6 and S7 Figs.) similar to the method of Eisen et al. but, again, using proportionality not correlation.

22This paper does not deny pairwise statistical associations between absolute abundances. What it does say is that once all the absolute information has been removed, only a subset of those associations will reliably endure in the remaining relative data, specifically, associations where values behave proportionally across observations.

23Other researchers have recognized the compositional nature of molecular bioscience data, including as discussed in. Strategies have been proposed to ameliorate spurious correlation in the analysis of relative abundances. We contend that there is no way to salvage a coherent interpretation of correlations from relative abundances without additional information or assumptions; our argument is based on Fig. 1.

24ReBoot attempts to establish a null distribution of correlations against which bootstrapped estimates of correlations can be compared. Aitchison articulates problems with this approach [, p.56–58]. SparCC injects additional information by assuming the number of different components is large and the true correlation network is sparse. This equates to assuming “that the average correlations [between absolute abundances] are small, rather than requiring that any particular correlation be small” [, Eq.14]. This means the expected value of the total absolute abundance will be constant (as the sum of many independently distributed amounts). We are concerned with situations where that assumption cannot be made, or where the aim is to describe associations between relative amounts.

25We are also keen to raise awareness that correlation (and other statistical methods that assume measurements come from real coordinate space) should not be applied to relative abundances. This is highly relevant to gene coexpression networks. Correlation is at the heart of methods like Weighted Gene Co-expression Network Analysis and heatmap visualization. These methods are potentially misleading if applied to relative data. This concern extends to methods based on mutual information (e.g., relevance networks) since, as Fig. 1 shows, the bivariate joint distribution of relative abundances (from which mutual information is estimated) can be quite different from the bivariate joint distribution of the absolute abundances that gave rise to them.

26Measures of association produce results regardless of the data they are applied to—it is up to the analyst to ensure that the measures are appropriate to the data. Currently, there are many gene co-expression databases available that provide correlation coefficients for the relative expression levels of different genes, generally from multiple experiments with different experimental conditions (see e.g.,). As far as we are aware, none of the database providers explicitly address whether absolute levels of gene expression were constant across experimental conditions. If the answer to this question is “no”, we would not recommend these correlations be used for the reasons demonstrated in this paper. If the answer is “yes” we still advocate caution in applying correlation to absolute abundances for reasons discussed in S1 Supporting Information.

27While the main aim of this study is to present and illustrate principles for analyzing relative abundances, it has also uncovered intriguing biological insight with respect to gene regulation.

28The largest cluster of proportionally regulated mRNAs (96 genes, S1 Supporting Information) was highly enriched for mRNAs down-regulated as part of the core environmental stress response, including 66 mRNAs that encode ribosomal proteins, and the remaining mRNAs also associated with roles in protein translation, such as ribosome biogenesis, rRNA processing, tRNA methyltransferases and translation elongation factors. The absolute levels of these mRNAs decrease after removal of nitrogen. The notable coherence in biological function among the mRNAs in this cluster is higher than typically seen when correlative similarity metrics for clustering are applied (e.g.,). These 96 mRNAs show remarkable proportionality to each other over the entire timecourse (S8 Fig.), and maintain near constant ratios across all conditions (S9 Fig.). Given the huge energy invested by yeast cells for protein translation (most notably ribosome biogenesis, it certainly makes sense for cells to synchronize the expression of relevant genes such that translation is finely tuned to nutritional conditions.

29Evidently, numerous ribosomal proteins and RNAs function together in the ribosome, demanding their coordinated expression; more surprisingly, multiple other genes, with diverse functions in translation, show equally pronounced proportional regulation across the timecourse. These findings raise intriguing questions as to the molecular mechanisms underlying this proportional regulation, suggesting sophisticated, coordinated control of numerous mRNAs at both transcriptional and post-transcriptional levels of gene expression.

30While proportionality and the ϕ-statistic provide a valid alternative to correlation for relative data, there are still some challenges in their application. First is the treatment of zeroes, for which there is currently no simple general remedy. Second, and related, is the fact that “many things that we measure and treat as if they are continuous are really discrete count data, even if only at the molecular extremes” and count data is not purely relative—the count pair (1, 2) carries different information than counts of (1000, 2000) even though the relative amounts of the two components are the same. Correspondence analysis, or methods based on count distributions (e.g., logistic regression and other generalized linear models) may provide ways forwards.

31All data and code needed to reproduce the analyses and visualizations set out in this paper are contained in the Supporting Information, along with additional illustrations and detailed explanations.

32The “goodness-of-fit to proportionality” statistic ϕ can be used to assess the extent to which a pair of random variables (x, y) are proportional. Aitchison proposed logratio variance, var(log(x/y)), as a measure of association for variables that carry only relative information. When x and y are exactly proportional var(log(x/y)) = 0, but when x and y are not exactly proportional, “it is hard to interpret as it lacks a scale. That is, it is unclear what constitutes a large or small value… (does a value of 0.1 indicate strong dependence, weak dependence, or no dependence?)”. Logratio variance can be factored into two more interpretable terms: where β is the standardized major axis estimate of slope of random variables log y on log x, and r the correlation between those variables. The first term in Equation 2, var(log x), is solely about the magnitude of variation at play and has nothing to do with y. The second term, ϕ, describes the degree of proportionality between x and y, and forms the basis of our analysis of the relationships between relative values. Other non-negative functions of β and r that are zero when x and y are perfectly proportional could be formed; this is described in more detail in S1 Supporting Information, as well as why ϕ is preferable to an hypothesis testing approach. There is no need to calculate β or r to assess strength of proportionality; they simply provide a clear geometric interpretation of ϕ; in practice, one can use the relationship ϕ(log x, log y) = var(log(x/y))/var(log x).

33The ϕ statistic is a measure of goodness-of-fit to proportionality that combines two quantities of interest: β, the slope of the line best describing the relationship between random variables log x and log y; and r, whose magnitude estimates the strength of the linear relationship between log x and log y. “Goodness-of-fit” describes how well a statistical model fits a set of observations and is a familiar concept in regression, including linear and generalised linear models, but note that ϕ—specifically the slope (β) of the standardized major axis—is motivated by allometry rather than regression modeling. We are interested in assessing whether two variables are directly proportional, rather than predicting one from the other: “use of regression would often lead to an incorrect conclusion about whether two variables are isometric or not” [, p.265]. Note also that ordinary least squares regression fits are not symmetric: in general, the slope of y regressed on x is different to the slope of x regressed on y.

34While goodness-of-fit measures for regression may not generally be appropriate for assessing proportionality, Zheng explores the concordance correlation coefficient ρ c which could be modified to provide an alternative measure of proportionality defined as and related to var(log(x/y)) by the terms in Equation 1. This “proportionality correlation coefficient” ranges from −1 (perfect reciprocality) to +1 (perfect proportionality) and lacks the clear geometric interpretation of ϕ.

35We have used ϕ(log x, log y) to emphasize the relationship between ϕ and logratio variance. However to ensure that the ϕ values for component pair (i, j) are on the same scale (i.e., comparable to) the ϕ values for component pair (m, n), it is necessary to use the centered logratio (clr) transformation instead of just the logarithm (S1 Supporting Information). The clr representation of composition x = (x 1, …, x i, …, x D) is the logarithm of the components after dividing by the geometric mean of x: ensuring that the sum of the elements of clr(x) is zero. Note that dividing all components in a composition by a constant (i.e., the geometric mean gm(x)) does not alter the ratios of components.

36Gene co-expression networks are generally based on a pairwise distance or dissimilarity matrix which is often a function of correlation and thus not appropriate for relative data. Proportionality is appropriate, but ϕ does not satisfy the properties of a distance—most obviously, it is not symmetric unless β = 1: We are most interested in pairs of variables where β and r are near 1 and want to preserve the link between ϕ(log x, log y), β and r. Hence, our approach to forming a dissimilarity matrix is simply to work with ϕ(log x i, log x j) where i < j, in effect, the lower triangle of the matrix of ϕ values between all pairs of components. This symmetrised form of ϕ was then used to lay out a network of the 145 mRNAs that were involved in 424 pairwise relationships with ϕ < 0.05. We used the symmetrised form of ϕ as the basis of the cluster analysis and heatmap expression pattern display (e.g., S10 Fig.) described by Eisen et al..