RV function

Three measures of the correlations between sets of variables

How to measure the the correlation between two clusters or groups of variables x and y from the same data set is a recurring problem. Perhaps the most obvious is simply the unweighted correlation Ru.

Consider the matrix M composed of four submatrices

	Rx	Rxy
M =	Rxy	Ry

The unit weighted correlation, Ru is merely

$Ru = \Sigma{r_{xy}/\sqrt{\Sigma{r_x}}\Sigma{r_y}}$

A second is the Set correlation (also found in lmCor) by Cohen 1982) which is

$Rset = 1- \frac{det(m)}{det(x)* det(y)}$

Where m is the full matrix (x+y)by (x+y). and det represents the determinant.

A third approach (the RV coeffiecent) was introduced by Escoufier (1970) and Robert and Escoufier (1976).

$RV= tr( xy$.

Where tr is the trace operator. (The sum of the diagonals).

The analysis can be done from the raw data or from correlation or covariance matrices. From the raw data, just specify the x and y variables. If using correlation/covariance matrixes, the xy matrix must be specified as well.

If using raw data, just specify the x and y columns and the data file.

RV(x, y, xy = NULL, data=NULL, cor = "cor",correct=0)

Arguments

• x: Columns of the data matrix of n rows and p columns, (if data is specified) or a p x p correlation matrix.
• y: Columns of a raw data matrix of n rows and q columns, or a q * q correlation matrix.
• xy: A p x q correlation or covariance matrix, if not using the raw data.
• data: A matrix or data frame containing the raw data.
• cor: If xy is NULL, find the p x p correlations or covariances from x, and the q x q correlations from y as well as the p x q covariance/correlation matrix.. Options are "cor" (for Pearson), "spearman" , "cov" for covariances, "tet" for tetrachoric, or "poly" for polychoric correlation.
• correct: The correction for continuity if desired.

Details

If using raw data, just specify the columns in x and y. If using a correlation matrix or covariance matrix, the inter corrlations/covariances) are specified in xy. The results match those of the RV function from matrixCalculations and the coeffRV function from factoMineR.

Returns

• RV: The RV statistic

• Rset: Cohen's set correlation

• Ru: The unit weighted correlation between x and y.

• Rx: The correlation matrix of the x variables.

• Ry: The correlation matrix of the y variables.

• Rxy: The intercorrelations of x and y.

References

P. Robert and Y. Escoufier, 1976, A Unifying Tool for Linear Multivariate Statistical Methods: The RV- Coefficient. Journal of the Royal Statistical Society. Series C (Applied Statistics), Volume 25, pp. 257-265.

J. Cohen (1982) Set correlation as a general multivariate data-analytic method. Multivariate Behavioral Research, 17(3):301-341.

Author(s)

William Revelle

See Also

lmCor for unit weighted correlations.

Examples

#from raw data
RV (attitude[1:3],attitude[4:7])  #find the correlations
RV (attitude[1:3],attitude[4:7],cor="cov")

#find the correlations
R <- cor(attitude)
r1 <- R[1:3,1:3]
r2 <- R[4:7,4:7]
r12 <- R[1:3,4:7]
RV(r1,r2,r12)

#or find the covariances
C <- cov(attitude)
c1 <- C[1:3,1:3]
c2 <- C[4:7,4:7]
c12 <- C[1:3,4:7]
RV(c1, c2, c12)