RV coefficient

RV coefficients that measure the proximity between two data matrices. Compute them on your data in Excel using the XLSTAT statistical software.

What is the RV coefficient?

The RV coefficient depicts the similarity between two matrices of quantitative variables or two configurations resulting from multivariate analysis.

RV coefficient: definition

This tool allows computing the RV coefficient between two matrices of quantitative variables. The RV coefficient is defined as (Robert and Escoufier, 1976; Schlich, 1996):

RV(Wi,Wj) = trace(Wi,Wj) / [trace(Wi,Wi).trace(Wj,Wj)]1/2

Where trace(Wi,Wj) = Σl,mwil,mwjl,m is a generalized covariance coefficient between matrices between matrices Wi and Wj, trace(Wi,Wi) = Σl,mwil,m2 is a generalized variance of matrix Wi and wil,m is the (l,m) element of matrix Wi.

The RV coefficient is a generalization of the squared Pearson correlation coefficient. The RV coeffcient lays between 0 and 1. The closer to 1 the RV is, the more similar the two matrices Wi and Wj are. XLSTAT offers the possibility:

  • To compute the RV coefficient between two matrices, including all variables form both matrices;
  • To choose the k first variables from both matrices and compute the RV coefficient between the two resulting matrices.

XLSTAT allows testing if the obtained RV coefficient is significantly different from 0 or not.

Two methods to compute the p-value are proposed by XLSTAT. The user can choose between a p-value computed using on an approximation of the exact distribution of the RV statistic with the Pearson type III approximation (Kazi-Aoual et al., 1995), and a p-value computed usingMonte Carlo resamplings.

Results

RV coefficients: A table including the RV coefficient(s), standardized RV coefficient(s), and mean(s) and variance(s) of the RV coefficient distribution; and the adjusted RV coefficient(s) and p-value(s) if requested by the user.

RV bar chart: A bar chart showing the RV coefficient(s) (with color codes to show significance of the associated p-value(s) if requested).

References

Kazi-Aoual F., Hitier S., Sabatier R., Lebreton J.-D., (1995) Refined approximations to permutations tests for multivariate inference. Computational Statistics and Data Analysis, 20, 643–656.

Robert P. and Escoufier Y. (1976) A unifying tool for linear multivariate statistical methods: the RV-coefficient. Applied Statistics, 25, 257–265.

Schlich P. (1996). Defining and validating assossor compromises about product distances and attribute correlations. In T, Næs, & E. Risvik (Eds.), Multivariate analysis of data in sensory sciences. New York: Elsevier.