# 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(W_{i},W_{j}) = trace(W_{i},W_{j}) / [trace(W_{i},W_{i}).trace(W_{j},W_{j})]^{1/2}

Where trace(W_{i},W_{j}) = Σ_{l,m}w^{i}_{l,m}w^{j}_{l,m} is a generalized covariance coefficient between matrices between matrices W_{i} and W_{j}, trace(W_{i},W_{i}) = Σ_{l,m}w^{i}_{l,m}^{2} is a generalized variance of matrix W_{i} and w^{i}_{l,m} is the (l,m) element of matrix W_{i}.

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 W_{i} and W_{j} 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.