<i>R</i>
            <sup>2</sup>
            -equitability is satisfiable

Murrell, Ben; Murrell, Daniel S.; Murrell, Hugh

doi:10.1073/pnas.1403623111

Cited by 14 publications

(7 citation statements)

References 2 publications

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“…Kinney and Atwal [ 16 ] have recently asserted that “No nontrivial dependence measure can satisfy R 2 -equitability”, providing a theorem to support this. However, as we show [ 17 ], their result hinges on a peculiar definition of “noise”, which allows a trend to be embedded in the noise term, effectively introducing an un-identifiability in their definition which they then exploit to prove the notion incoherent. When you take the trend out of the noise term, you are left with a perfectly sensible notion of “ R 2 -equitability”.…”

Section: Discussionmentioning

confidence: 99%

Discovering General Multidimensional Associations

Murrell

2016

PLoS ONE

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When two variables are related by a known function, the coefficient of determination (denoted R2) measures the proportion of the total variance in the observations explained by that function. For linear relationships, this is equal to the square of the correlation coefficient, ρ. When the parametric form of the relationship is unknown, however, it is unclear how to estimate the proportion of explained variance equitably—assigning similar values to equally noisy relationships. Here we demonstrate how to directly estimate a generalised R2 when the form of the relationship is unknown, and we consider the performance of the Maximal Information Coefficient (MIC)—a recently proposed information theoretic measure of dependence. We show that our approach behaves equitably, has more power than MIC to detect association between variables, and converges faster with increasing sample size. Most importantly, our approach generalises to higher dimensions, estimating the strength of multivariate relationships (Y against A, B, …) as well as measuring association while controlling for covariates (Y against X controlling for C). An R package named matie (“Measuring Association and Testing Independence Efficiently”) is available (http://cran.r-project.org/web/packages/matie/).

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Section: Discussionmentioning

confidence: 99%

Discovering General Multidimensional Associations

Murrell

2016

PLoS ONE

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“…Nevertheless, Justin B. Kinney & Gurinder S. Atwal indicate that MIC does not have the property of “equitability”, and the reported simulation evidences contain artifacts [ 42 ]. However, Reshef et al [ 43 ] and Murrell et al [ 44 ] have called Kinney and Atwal’s methodology into question. Their work led to the better understanding of equitability and MIC and allowed researchers in the area to move forward.…”

Section: Discussionmentioning

confidence: 99%

Efficient test for nonlinear dependence of two continuous variables

Wang

Cao

et al. 2015

BMC Bioinformatics

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BackgroundTesting dependence/correlation of two variables is one of the fundamental tasks in statistics. In this work, we proposed a new way of testing nonlinear dependence between two continuous variables (X and Y).ResultsWe addressed this research question by using CANOVA (continuous analysis of variance, software available at https://sourceforge.net/projects/canova/). In the CANOVA framework, we first defined a neighborhood for each data point related to its X value, and then calculated the variance of the Y value within the neighborhood. Finally, we performed permutations to evaluate the significance of the observed values within the neighborhood variance. To evaluate the strength of CANOVA compared to six other methods, we performed extensive simulations to explore the relationship between methods and compared the false positive rates and statistical power using both simulated and real datasets (kidney cancer RNA-seq dataset).ConclusionsWe concluded that CANOVA is an efficient method for testing nonlinear correlation with several advantages in real data applications.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0697-7) contains supplementary material, which is available to authorized users.

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“…Since its publication, Kinney and Atwal (2014) has been the subject of two published technical comments [Murrell, Murrell and Murrell (2014), Reshef et al (2014)] describing its main limitations, which are threefold. First, the central proof of the impossibility of equitability with respect to R 2 in Kinney and Atwal (2014) applies only to perfect equitability, and says nothing about the achievability of the more general (approximate) notion with which we are primarily interested and regarding which we have previously made claims about MIC.…”

Section: Relationship To Equitability Analysis Frommentioning

confidence: 99%

“…That is, even if no method is perfectly equitable with respect to R 2 , some methods can be more equitable with respect to R 2 than others, and the question remains which methods come meaningfully close to the ideal [Reshef et al (2014)]. Second, the impossibility result relies crucially on a nonidentifiable noise model Q in which, for example, a noiseless parabola can be obtained as a "noisy" linear relationship [Murrell, Murrell and Murrell (2014)]. Third, though mutual information indeed outperforms MIC under the specific sample size and noise model chosen in Kinney and Atwal (2014), this is not the case in general [Reshef et al (2014)].…”

Section: Relationship To Equitability Analysis Frommentioning

confidence: 99%

An empirical study of the maximal and total information coefficients and leading measures of dependence

Reshef¹,

Reshef²,

Sabeti³

et al. 2018

Ann. Appl. Stat.

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In exploratory data analysis, we are often interested in identifying promising pairwise associations for further analysis while filtering out weaker ones. This can be accomplished by computing a measure of dependence on all variable pairs and examining the highest-scoring pairs, provided the measure of dependence used assigns similar scores to equally noisy relationships of different types. This property, called equitability and previously formalized, can be used to assess measures of dependence along with the power of their corresponding independence tests and their runtime. Here we present an empirical evaluation of the equitability, power against independence, and runtime of several leading measures of dependence. These include the two recently introduced and simultaneously computable statistics MIC e , whose goal is equitability, and TIC e , whose goal is power against independence. Regarding equitability, our analysis finds that MIC e is the most equitable method on functional relationships in most of the settings we considered. Regarding power against independence, we find that TIC e and Heller and Gorfine's S DDP share state-of-the-art performance, with several other methods achieving excellent power as well. Our analyses also show evidence for a trade-off between power against independence and equitability consistent with recent theoretical work. Our results suggest that a fast and useful strategy for achieving a combination of power against independence and equitability is to filter relationships by TIC e and then to rank the remaining ones using MIC e. We confirm our findings on a set of data collected by the World Health Organization.

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R ² -equitability is satisfiable

Cited by 14 publications

References 2 publications

Discovering General Multidimensional Associations

Discovering General Multidimensional Associations

Efficient test for nonlinear dependence of two continuous variables

An empirical study of the maximal and total information coefficients and leading measures of dependence

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R 2 -equitability is satisfiable

Cited by 14 publications

References 2 publications

Discovering General Multidimensional Associations

Discovering General Multidimensional Associations

Efficient test for nonlinear dependence of two continuous variables

An empirical study of the maximal and total information coefficients and leading measures of dependence

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Product

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R ² -equitability is satisfiable