Correlation Coefficient
Example Data  244  238 
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 = "everything", method = c("pearson", "kendall", "spearman")) cor(x, y = NULL, use = "everything", method = c("pearson", "kendall", "spearman")) cov2cor(V)
Arguments
x 
a numeric vector, matrix or data frame. 
y 

na.rm 
logical. Should missing values be removed? 
use 
an optional character string giving a
method for computing covariances in the presence
of missing values. This must be (an abbreviation of) one of the strings

method 
a character string indicating which correlation
coefficient (or covariance) is to be computed. One of

V 
symmetric numeric matrix, usually positive definite such as a covariance matrix. 
Details
For cov
and cor
one must either give a matrix or
data frame for x
or give both x
and y
.
The inputs must be numeric (as determined by is.numeric
:
logical values are also allowed for historical compatibility): the
"kendall"
and "spearman"
methods make sense for ordered
inputs but xtfrm
can be used to find a suitable prior
transformation to numbers.
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 = "na.or.complete"
) to
compute the variance. Otherwise, by default use = "everything"
.
If use
is "everything"
, NA
s will
propagate conceptually, i.e., a resulting value will be NA
whenever one of its contributing observations is NA
.
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 (and if there are no complete cases, that gives an error).
"na.or.complete"
is the same unless there are no complete
cases, that gives NA
.
Finally, if use
has the value "pairwise.complete.obs"
then the correlation or covariance 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, as well as NA
entries if there are no complete
pairs for that pair of variables. For cov
and var
,
"pairwise.complete.obs"
only works with the "pearson"
method.
Note that (the equivalent of) var(double(0), use = *)
gives
NA
for use = "everything"
and "na.or.complete"
,
and gives an error in the other cases.
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 SPLUS 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 rankbased 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 nonPearson 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")
. In the case of missing values, the
ranks are calculated depending on the value of use
, either
based on complete observations, or based on pairwise completeness with
reranking for each pair.
When there are ties, Kendall's tau_b is computed, as proposed by Kendall (1945).
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(abs(r) <= 1)
.
Note
Some people have noted that the code for Kendall's tau is slow for
very large datasets (many more than 1000 cases). It rarely makes
sense to do such a computation, but see function
cor.fk
in package pcaPP.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Kendall, M. G. (1938) A new measure of rank correlation, Biometrika 30, 81â€“93. https://dx.doi.org/10.1093/biomet/30.12.81
Kendall, M. G. (1945) The treatment of ties in rank problems. Biometrika 33 239â€“251. https://dx.doi.org/10.1093/biomet/33.3.239
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.values = TRUE)$values) # 6.19 1921 ## swM := "swiss" with 3 "missing"s : swM < swiss colnames(swM) < abbreviate(colnames(swiss), min=6) swM[1,2] < swM[7,3] < swM[25,5] < NA # create 3 "missing" ## Consider all 5 "use" cases : (C. < cov(swM)) # use="everything" quite a few NA's in cov.matrix try(cov(swM, use = "all")) # Error: missing obs... C2 < cov(swM, use = "complete") stopifnot(identical(C2, cov(swM, use = "na.or.complete"))) range(eigen(C2, only.values = TRUE)$values) # 6.46 1930 C3 < cov(swM, use = "pairwise") range(eigen(C3, only.values = TRUE)$values) # 6.19 1938 ## Kendall's tau doesn't change much: symnum(Rc < cor(swM, method = "kendall", use = "complete")) symnum(Rp < cor(swM, method = "kendall", use = "pairwise")) symnum(R. < cor(swiss, method = "kendall")) ## "pairwise" is closer componentwise, summary(abs(c(1  Rp/R.))) summary(abs(c(1  Rc/R.))) ## but "complete" is closer in Eigen space: EV < function(m) eigen(m, only.values=TRUE)$values summary(abs(1  EV(Rp)/EV(R.)) / abs(1  EV(Rc)/EV(R.)))
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