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- Abstract
- The Maximal Information Coefficient (MIC)
- Main Properties of MIC
- Comparisons to Other Methods
- An Expanded Toolkit for Exploration
- Application of MINE to real datasets
- Conclusion
- Supplementary Material
- References

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Science. Author manuscript; available in PMC 2012 June 16.

Published in final edited form as:

PMCID: PMC3325791

NIHMSID: NIHMS358982

David N. Reshef,^{1,}^{2,}^{3,}^{*}^{†} Yakir A. Reshef,^{2,}^{4,}^{*}^{†} Hilary K. Finucane,^{5} Sharon R. Grossman,^{2,}^{6} Gilean McVean,^{3,}^{7} Peter J. Turnbaugh,^{6} Eric S. Lander,^{2,}^{8,}^{9} Michael Mitzenmacher,^{10,}^{‡} and Pardis C. Sabeti^{2,}^{6,}^{‡}

Identifying interesting relationships between pairs of variables in large datasets is increasingly important. Here, we present a measure of dependence for two-variable relationships: the maximal information coefficient (MIC). MIC captures a wide range of associations both functional and not, and for functional relationships provides a score that roughly equals the coefficient of determination (R^{2}) of the data relative to the regression function. MIC belongs to a larger class of maximal information-based nonparametric exploration (MINE) statistics for identifying and classifying relationships. We apply MIC and MINE to datasets in global health, gene expression, major-league baseball, and the human gut microbiota, and identify known and novel relationships.

Imagine a dataset with hundreds of variables, which may contain important, undiscovered relationships. There are tens of thousands of variable pairs—far too many to examine manually. If you do not already know what kinds of relationships to search for, how do you efficiently identify the important ones? Datasets of this size are increasingly common in fields as varied as genomics, physics, political science, and economics, making this question an important and growing challenge (1, 2).

One way to begin exploring a large dataset is to search for pairs of variables that are closely associated. To do this, we could calculate some measure of dependence for each pair, rank the pairs by their scores, and examine the top-scoring pairs. For this strategy to work, the statistic we use to measure dependence should have two heuristic properties: generality and equitability.

By generality, we mean that with sufficient sample size the statistic should capture a wide range of interesting associations, not limited to specific function types (such as linear, exponential, or periodic), or even to all functional relationships (3). The latter condition is desirable because not only do relationships take many functional forms, but many important relationships—for example, a superposition of functions—are not well modeled by a function (4-7).

By equitability, we mean that the statistic should give similar scores to equally noisy relationships of different types. For example, we do not want noisy linear relationships to drive strong sinusoidal relationships from the top of the list. Equitability is difficult to formalize for associations in general but has a clear interpretation in the basic case of functional relationships: an equitable statistic should give similar scores to functional relationships with similar R^{2} values (given sufficient sample size).

In this paper, we describe an exploratory data analysis tool, the maximal information coefficient (MIC), that satisfies these two heuristic properties. We establish MIC's generality through proofs, show its equitability on functional relationships through simulations, and observe that this translates into intuitively equitable behavior on more general associations. Furthermore, we illustrate that MIC gives rise to a larger family of statistics, which we refer to as MINE, or maximal information-based nonparametric exploration. MINE statistics can be used not only to identify interesting associations, but also to characterize them according to properties such as non-linearity and monotonicity. We demonstrate the application of MIC and MINE to datasets in health, baseball, genomics, and the human microbiota.

Intuitively, MIC is based on the idea that if a relationship exists between two variables, then a grid can be drawn on the scatterplot of the two variables that partitions the data to encapsulate that relationship. Thus, to calculate the MIC of a set of two-variable data we explore all grids up to a maximal grid resolution, dependent on the sample size (Fig. 1A), computing for every pair of integers (*x*,*y*) the largest possible mutual information achievable by any *x*-by-*y* grid applied to the data. We then normalize these mutual information values to ensure a fair comparison between grids of different dimensions, and to obtain modified values between zero and one. We define the characteristic matrix *M* = (*m _{x,y}*), where

More formally, for a grid *G*, let *I _{G}* denote the mutual information of the probability distribution induced on the boxes of

Every entry of *M* falls between zero and one, and so MIC does as well. MIC is also symmetric (i.e. MIC(*X*, *Y*) = MIC(*Y*, *X*)) due to the symmetry of mutual information, and because *I _{G}* depends only on the rank order of the data, MIC is invariant under order-preserving transformations of the axes. Importantly, although mutual information is used to quantify the performance of each grid, MIC is not an estimate of mutual information (Section 2, SOM).

To calculate *M*, we would ideally optimize over all possible grids. For computational efficiency, we instead use a dynamic programming algorithm that optimizes over a subset of the possible grids and appears to approximate well the true value of MIC in practice (Section 3, SOM).

We have proven mathematically that MIC is general in the sense described above. Our proofs show that, with probability approaching 1 as sample size grows, (i) MIC assigns a perfect score of 1 to all never-constant noiseless functional relationships, (ii) MIC assigns scores that tend to 1 for a larger class of noiseless relationships (including superpositions of noiseless functional relationships), and (iii) MIC assigns a score of 0 to statistically independent variables.

Specifically, we have proven that for a pair of random variables *X* and *Y*, (i) if *Y* is a function of *X* that is not constant on any open interval, then data drawn from (*X*,*Y*) will receive an MIC tending to 1 with probability one as sample size grows; (ii) if the support of (*X*,*Y*) is described by a finite union of differentiable curves of the form *c*(*t*) = (*x*(*t*),*y*(*t*)) for *t* in [0,1], then data drawn from (*X*,*Y*) will receive an MIC tending to 1 with probability one as sample size grows, provided that *dx*/*dt* and *dy*/*dt* are each zero on finitely many points; (iii) the MIC of data drawn from (*X*,*Y*) converges to zero in probability as sample size grows if and only if *X* and *Y* are statistically independent. We have also proven that the MIC of a noisy functional relationship is bounded from below by a function of its R^{2}. (For proofs, see SOM.)

We tested MIC's equitability through simulations. These simulations confirm the mathematical result that noiseless functional relationships (i.e. R^{2} = 1.0) receive MIC scores of 1.0 (Fig. 2A). They also show that, for a large collection of test functions with varied sample sizes, noise levels, and noise models, MIC roughly equals the coefficient of determination R^{2} relative to each respective noiseless function. This makes it easy to interpret and compare scores across various function types (Fig. 2B, S4). For instance, at reasonable sample sizes, a sinusoidal relationship with a noise level of R^{2} = 0.80 and a linear relationship with the same R^{2} value receive nearly the same MIC score. For a wide range of associations that are not well modeled by a function, we also show that MIC scores degrade in an intuitive manner as noise is added (Fig. 2G, Figs. S5-6).

We compared MIC to a wide range of methods – including methods formulated around the axiomatic framework for measures of dependence developed by Rényi (8), other state-of-the-art measures of dependence, and several nonparametric curve estimation techniques that can be used to score pairs of variables based on how well they fit the estimated curve.

Methods such as splines (1) and regression estimators (1, 9, 10) tend to be equitable across functional relationships (11), but are not general: they fail to find many simple and important types of relationships that are not functional. (Figs. S5 and S6 depict examples of relationships of this type from existing literature, and compare these methods to MIC on such relationships.) Although these methods are not intended to provide generality, the failure to assign high scores in such cases makes them unsuitable for identifying all potentially interesting relationships in a dataset.

Other methods such as mutual information estimators (12-14), maximal correlation (8, 15), principal curve-based methods (16-19)(20), distance correlation (21), and the Spearman rank correlation coefficient all detect broader classes of relationships. However, they are not equitable even in the basic case of functional relationships: they show a strong preference for some types of functions, even at identical noise levels (Fig. 2A,C-F). For example, at a sample size of 250, the Kraskov *et al.* mutual information estimator (14) assigns a score of 3.65 to a noiseless line but only 0.59 to a noiseless sinusoid, and it gives equivalent scores to a very noisy line (R^{2} = 0.35) and to a much cleaner sinusoid (R^{2} = 0.80) (Fig. 2D). Again, these results are not surprising—they correctly reflect the properties of mutual information. But this behavior makes these methods less practical for data exploration.

The basic approach of MIC can be extended to define a broader class of MINE statistics based on both MIC and the characteristic matrix *M*. These statistics can be used to rapidly characterize relationships that may then be studied with more specialized or computationally intensive techniques.

Some statistics are derived, like MIC, from the spectrum of grid resolutions contained in *M*. Different relationship types correspond to different characteristic matrices (Fig. 3). For example, just as a characteristic matrix with a high maximum indicates a strong relationship, a symmetric characteristic matrix indicates a monotonic relationship. We can thus detect deviation from monotonicity with the Maximum Asymmetry Score (MAS), defined as the maximum over *M* of |*m _{x,y}* −

Because MIC is general and roughly equal to R^{2} on functional relationships, we can also define a natural measure of non-linearity by MIC − *ρ ^{2}*, where

Similar MINE statistics can be defined to detect properties that we refer to as “complexity” and “closeness to being a function.” We provide formal definitions and a performance summary of these two statistics (Section 2.3, SOM, Table S1). Finally, MINE statistics can also be used in cluster analysis to observe the higher-order structure of datasets (Section 4.9, SOM).

We used MINE to explore four high-dimensional datasets from diverse fields. Three datasets have previously been analyzed and contain many well-understood relationships. These datasets are (i) social, economic, health, and political indicators from the World Health Organization (WHO) and its partners (7, 25); (ii) yeast gene expression profiles from a classic paper reporting genes whose transcript levels vary periodically with the cell cycle (26); and (iii) performance statistics from the 2008 Major League Baseball (MLB) season (27, 28). For our fourth analysis, we applied MINE to a dataset that has not yet been exhaustively analyzed: a set of bacterial abundance levels in the human gut microbiota (29). All relationships discussed in this section are significant at a false discovery rate of 5%; p-values and q-values are listed in the SOM.

We explored the WHO dataset (357 variables, 63,546 variable pairs) with MIC, the commonly used Pearson correlation coefficient (*ρ*), and Kraskov's mutual information estimator (Fig. 4, Table S9). All three statistics detected many linear relationships. However, mutual information gave low ranks to many non-linear relationships that were highly ranked by MIC (Figs. 4A-B). Two-thirds of the top 150 relationships found by mutual information were strongly linear (|*ρ*| ≥ 0.97), whereas most of the top 150 relationships found by MIC had |*ρ*| below this threshold. Further, although equitability is difficult to assess for general associations, the results on some specific relationships suggest that MIC comes closer than mutual information to this goal (Fig. 4I). Using the non-linearity measure MIC − *ρ ^{2}*, we found several interesting relationships (Figs. 4E-G), many of which are confirmed by existing literature (30-32). For example, we identified a superposition of two functional associations between female obesity and income per person, one from the Pacific Islands, where female obesity is a sign of status, (33) and one from the rest of the world, where weight and status do not appear to be linked in this way (Fig. 4F).

We next explored a yeast gene expression dataset (6,223 genes) that was previously analyzed with a special-purpose statistic developed by Spellman *et al.* to identify genes whose transcript levels oscillate during the cell cycle (26). Of the genes identified by Spellman *et al.* and MIC, 70% and 69%, respectively, were also identified in a later study with more time points conducted by Tu *et al.* (22). However, MIC identified genes at a wider range of frequencies than Spellman *et al.*, and MAS sorted those genes by frequency (Fig. 5). Of the genes identified by MINE as having high frequency (MAS > 75^{th} percentile), 80% were identified by Spellman *et al.*, while of the low-frequency genes (MAS < 25^{th} percentile) Spellman *et al.* identified only 20% (Fig. 5B). For example, although both methods found the well-known cell-cycle regulator HTB1 (Fig. 5G) required for chromatin assembly, only MIC detected the heat-shock protein HSP12 (Fig. 5E), which Tu *et al.* confirmed to be in the top 4% of periodic genes in yeast. HSP12, along with 43% of the genes identified by MINE but not Spellman *et al.*, was also in the top third of statistically significant periodic genes in yeast according to the more sophisticated specialty statistic of Ahdesmaki *et al*., which was specifically designed for finding periodic relationships without a pre-specified frequency in biological systems (24). Due to MIC's generality and the small size of this dataset (*n*=24), relatively few of the genes analyzed (5%) had significant MIC scores after multiple testing correction at a false discovery rate of 5%. However, using a less conservative false discovery rate of 15% yielded a larger list of significant genes (16% of all genes analyzed) and this larger list still attained a 68% confirmation rate by Tu *et al.*.

In the MLB dataset (131 variables), MIC and *ρ* both identified many linear relationships, but interesting differences emerged. On the basis of *ρ*, the strongest three correlates with player salary are walks, intentional walks, and runs batted in. In contrast, the strongest three associations according to MIC are hits, total bases, and a popular aggregate offensive statistic called Replacement Level Marginal Lineup Value (27)(34) (Fig. S12, Table S12). We leave it to baseball enthusiasts to decide which of these statistics are (or should be!) more strongly tied to salary.

Our analysis of gut microbiota focused on the relationships between prevalence levels of the trillions of bacterial species that colonize the gut of humans and other mammals (35, 36). The dataset consisted of large-scale sequencing of 16S ribosomal RNA from the distal gut microbiota of mice colonized with a human fecal sample (29). After successful colonization, a subset of the mice was shifted from a low-fat/plant-polysaccharide-rich (LF/PP) diet to a high-fat/high-sugar ‘Western’ diet. Our initial analysis identified 9,472 significant relationships (out of 22,414,860) between ‘species’-level groups called operational taxonomic units (OTUs); significantly more of these relationships occurred between OTUs in the same bacterial family than expected by chance (30% vs. 24±0.6%).

Examining the 1,001 top-scoring non-linear relationships (MIC-*ρ*^{2}>0.2), we observed that a common association type was ‘non-coexistence’: when one species is abundant the other is less abundant than expected by chance, and *vice versa* (Fig. 6A-B,D). Additionally, we found that 312 of the top 500 non-linear relationships were affected by one or more factors for which data were available (host diet, host sex, identity of human donor, collection method, and location in the gastrointestinal tract; SOM, Section 4.7). Many are non-coexistence relationships that are explained by diet (Fig. 6A, Table S13). These diet-explained non-coexistence relationships occur at a range of taxonomic depths—inter-phylum, inter-family, and intra-family—and form a highly interconnected network of non-linear relationships (Fig. 6E).

The remaining 188 of the 500 highly ranked non-linear relationships were not affected by any of the factors in the dataset, and included many non-coexistence relationships (Table S14, Fig. 6D). These unexplained non-coexistence relationships may suggest interspecies competition and/or additional selective factors that shape gut microbial ecology, and therefore represent promising directions for future study.

Given the ever-growing, technology-driven data stream in today's scientific world, there is an increasing need for tools to make sense of complex datasets in diverse fields. The ability to examine all potentially interesting relationships in a dataset—independent of their functional form—allows tremendous versatility in the search for meaningful insights. On the basis of our tests, MINE is useful for identifying and characterizing structure in data.

The authors thank C. Blättler, B. Eidelson, M.D. Finucane, M.M. Finucane, M. Fujihara, T. Gingrich, E. Goldstein, R. Gupta, R. Hahne, T. Jaakkola, N. Laird, M. Lipsitch, S. Manber, G. Nicholls, A. Papageorge, N. Patterson, E. Phelan, J.Rinn, B. Ripley, I. Shylakhter, and R. Tibshirani for their invaluable support and critical discussions throughout; and O. Derby, M. Fitzgerald, S. Hart, M. Huang, E. Karlsson, S. Schaffner, C. Edwards and D. Yamins for assistance. PCS and this work are supported by the Packard Foundation, MM by NSF grant 0915922, HKF by ERC grant no. 239985, SRG by MSTP, and PJT by NIH P50 GM068763. Data and software are available online at http://exploredata.net.

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