Correlation, Variance and Covariance (Matrices)

Description

var, cov and cor compute the variance of x and the covariance or correlation of x and y if these are vectors. If x and y are matrices then the covariances (or correlations) between the columns of x and the columns of y are computed.

cov2cor scales a covariance matrix into the corresponding correlation matrix efficiently.

Usage

var(x, y = NULL, na.rm = FALSE, use)

cov(x, y = NULL, use = "all.obs",
    method = c("pearson", "kendall", "spearman"))

cor(x, y = NULL, use = "all.obs",
     method = c("pearson", "kendall", "spearman"))

cov2cor(V)

Arguments

Details

For cov and cor one must either give a matrix or data frame for x or give both x and y.

var is just another interface to cov, where na.rm is used to determine the default for use when that is unspecified. If na.rm is TRUE then the complete observations (rows) are used (use = "complete") to compute the variance. Otherwise (use = "all"), var will give an error if there are missing values.

If use is "all.obs", then the presence of missing observations will produce an error. If use is "complete.obs" then missing values are handled by casewise deletion. Finally, if use has the value "pairwise.complete.obs" then the correlation between each pair of variables is computed using all complete pairs of observations on those variables. This can result in covariance or correlation matrices which are not positive semidefinite.

The denominator n - 1 is used which gives an unbiased estimator of the (co)variance for i.i.d. observations. These functions return NA when there is only one observation (whereas S-PLUS has been returning NaN), and fail if x has length zero.

For cor(), if method is "kendall" or "spearman", Kendall's \tau or Spearman's \rho statistic is used to estimate a rank-based measure of association. These are more robust and have been recommended if the data do not necessarily come from a bivariate normal distribution.
For cov(), a non-Pearson method is unusual but available for the sake of completeness. Note that "spearman" basically computes cor(R(x), R(y)) (or cov(.,.)) where R(u) := rank(u, na.last="keep"). Notice also that the ranking is (currently) done removing only cases that are missing on the variable itself, which may not be what you expect if you let use be "complete.obs" or "pairwise.complete.obs".

Scaling a covariance matrix into a correlation one can be achieved in many ways, mathematically most appealing by multiplication with a diagonal matrix from left and right, or more efficiently by using sweep(.., FUN = "/") twice. The cov2cor function is even a bit more efficient, and provided mostly for didactical reasons.

Value

For r <- cor(*, use = "all.obs"), it is now guaranteed that all(r <= 1).

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth \& Brooks/Cole.

See Also

cor.test for confidence intervals (and tests).

cov.wt for weighted covariance computation.

sd for standard deviation (vectors).

Examples

var(1:10)# 9.166667

var(1:5,1:5)# 2.5

## Two simple vectors
cor(1:10,2:11)# == 1

## Correlation Matrix of Multivariate sample:
(Cl <- cor(longley))
## Graphical Correlation Matrix:
symnum(Cl) # highly correlated

## Spearman's rho  and  Kendall's tau
symnum(clS <- cor(longley, method = "spearman"))
symnum(clK <- cor(longley, method = "kendall"))
## How much do they differ?
i <- lower.tri(Cl)
cor(cbind(P = Cl[i], S = clS[i], K = clK[i]))


## cov2cor() scales a covariance matrix by its diagonal
##	     to become the correlation matrix.
cov2cor # see the function definition {and learn ..}
stopifnot(all.equal(Cl, cov2cor(cov(longley))),
          all.equal(cor(longley, method="kendall"),
            cov2cor(cov(longley, method="kendall"))))

##--- Missing value treatment:
C1 <- cov(swiss)
range(eigen(C1, only=TRUE)$val) # 6.19	1921
swM <- swiss
swM[1,2] <- swM[7,3] <- swM[25,5] <- NA # create 3 "missing"
try(cov(swM)) # Error: missing obs...
C2 <- cov(swM, use = "complete")
range(eigen(C2, only=TRUE)$val) # 6.46	1930
C3 <- cov(swM, use = "pairwise")
range(eigen(C3, only=TRUE)$val) # 6.19	1938

(scM <- symnum(cor(swM, method = "kendall", use = "complete")))
## Kendall's tau doesn't change much: identical symnum codings!
identical(scM, symnum(cor(swiss, method = "kendall")))

all.equal(cov2cor(cov(swM, method = "kendall", use = "pairwise")),
                  cor(swM, method = "kendall", use = "pairwise"))