Discovering General Multidimensional Associations

Abstract

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/).

Global Gene Expression Profiling of Body-Mass Index in Middle-Aged Danish Twins

“…We then applied a generalized measure of association, the generalized correlation coefficient (GCC), to investigate the association between intra-pair difference of BMI and intra-pair difference of expression to control the genetic and shared environmental effects, as proposed by Tan et al [13]. GCC was computed using a ratio of maximum likelihoods for the marginal distribution and maximum weighted likelihoods for the joint distribution using the R package matie [14]. The mRNA probes with p<0.05 were used for gene set enrichment analysis (GSEA) to detect biological pathways over-represented by the listed probes for functional interpretation [15].…”

Generalized Correlation Coefficient for Non-Parametric Analysis of Microarray Time-Course Data

“…By providing the amount of information one variable reveals about another, MI measures the dependency between two variables of any type. Based on the concept of rank correlation, Murrell et al [15] very recently proposed a generalized correlation coefficient for non-parametric measurement of association between variables. The association score, A, ranges from 0 (when the variables are independent) to 1 (when they are perfectly associated).…”

“…Under the normal assumption in least squares regression, the squared deviations can be expressed as probability density so that . The last part of the formula (the proportion of unexplained variance) is the geometric mean of the squared ratio of the probability of observing a data point under the null model over the probability of that data point under the alternative model [15]. Here, the formulation of correlation depends only on the ratio of the probability density between two models which do not necessarily require normality assumption.…”

“…Here, the formulation of correlation depends only on the ratio of the probability density between two models which do not necessarily require normality assumption. A generalized version of R 2 , or the so called association score A, can be calculated as long as the probability distributions for the null and alternative models can be evaluated using a kernel density approach as described by Murrell et al [15]. The generalization enables direct estimation of the association score A when the form of correlation is unknown.…”

“…The generalization enables direct estimation of the association score A when the form of correlation is unknown. The generalized association approach has been shown to have more power than MIC and fast convergence [15]. We apply the generalized association to microarray time-course data to detect the dynamic patterns of gene expression over the experiment time and across genes without any assumption for the heterogeneous relationship.…”