Computational Statistics and Data Analysis 55 (2011) 1357–1366 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Partial correlation with copula modeling Jong-Min Kim a,∗ , Yoon-Sung Jung b , Taeryon Choi c , Engin A. Sungur a a Statistics Discipline, Division of Science and Mathematics, University of Minnesota at Morris, Morris, MN, 56267, USA b Office of Research, Alcorn State University, Alcorn State, MS, 39096, USA c Department of Statistics, Korea University, Seoul, 136-701, South Korea article info Article history: Received 26 January 2010 Received in revised form 21 September 2010 Accepted 22 September 2010 Available online 16 October 2010 Keywords: Partial correlation Gaussian copula Gene network abstract We propose a new partial correlation approach using Gaussian copula. Our empirical study found that the Gaussian copula partial correlation has the same value as that which is obtained by performing a Pearson’s partial correlation. With the proposed method, based on canonical vine and d-vine, we captured direct interactions among eight histone genes. © 2010 Elsevier B.V. All rights reserved. 1. Introduction The current Pearson partial correlation approach is popular because of the simple computation advantage it confers. But the current approach has many drawbacks: for example, it does not exist if the first or second moments do not exist. Possible values depend on the marginal distributions which are not invariant under non-linear strictly increasing transformations (Kurowicka and Cooke, 2006). This was our motivation to propose a new approach to partial correlation using copula, specifically a Gaussian copula. Since Sklar (1959) proposed the theorem of the copula, numerous copula functions have been introduced in the last five decades. Recently, Nelson (2006) summarized the theories of numerous copula functions and Yan (2007) developed the R package of multivariate dependence with copulas. But most copulas have a limitation which fails to satisfy the copula properties when extended from bivariate to multivariate cases. To overcome this limitation, Aasa et al. (2009) proposed pair-copula constructions of multiple dependence, based on the work of Bedford and Cooke (2002). Since model construction is hierarchical, it is not simple to incorporate more variables in the conditioning sets with pair-copula which uses the inverse of the conditional bivariate distribution function, h-function inverse. But pair-copula constructions by Aasa et al. (2009) are promising way to derive a partial correlation, so we adopted a Gaussian bivariate copula by using the conditional distributions to find a partial correlation. To find a partial correlation, we derive a conditional standard normal distribution by using multivariate normal distribution properties and estimate the partial correlation coefficient by the Gaussian copula. In the general theory of partial correlation, the partial correlation coefficient is a measure of the strength of the linear relationship between two variables after we control for the effects of other variables. If the two variables of interest are Y and X , and the control variables are Z 1 , Z 2 ,..., Z n , then we denote the corresponding partial correlation coefficient by ρ YX |Z 1 ,Z 2 ,...,Z n . ∗ Corresponding author. E-mail address: jongmink@morris.umn.edu (J.-M. Kim). 0167-9473/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2010.09.025