Test for Association/Correlation Between Paired Samples
Test for association between paired samples, using one of Pearson's product moment correlation coefficient, Kendall's $\tau$ or Spearman's $\rho$.
## S3 method for class 'default': cor.test(x, y, alternative = c("two.sided", "less", "greater"), method = c("pearson", "kendall", "spearman"), exact = NULL, conf.level = 0.95, continuity = FALSE, ...)
## S3 method for class 'formula': cor.test(formula, data, subset, na.action, \dots)
- x, y
- numeric vectors of data values.
ymust have the same length.
- indicates the alternative hypothesis and must be
"less". You can specify just the initial letter.
"greater"corresponds to positive association,
"less"to negative association.
- a character string indicating which correlation
coefficient is to be used for the test. One of
"spearman", can be abbreviated.
- a logical indicating whether an exact p-value should be
computed. Used for Kendall's $\tau$ and
Detailsfor the meaning of
- confidence level for the returned confidence interval. Currently only used for the Pearson product moment correlation coefficient if there are at least 4 complete pairs of observations.
- logical: if true, a continuity correction is used for Kendall's $\tau$ and Spearman's $\rho$ when not computed exactly.
- a formula of the form
~ u + v, where each of
vare numeric variables giving the data values for one sample. The samples must be of the same length.
- an optional matrix or data frame (or similar: see
model.frame) containing the variables in the formula
formula. By default the variables are taken from
- an optional vector specifying a subset of observations to be used.
- a function which indicates what should happen when
the data contain
NAs. Defaults to
- further arguments to be passed to or from methods.
The three methods each estimate the association between paired samples and compute a test of the value being zero. They use different measures of association, all in the range $[-1, 1]$ with $0$ indicating no association. These are sometimes referred to as tests of no correlation, but that term is often confined to the default method.
"pearson", the test statistic is based on
Pearson's product moment correlation coefficient
cor(x, y) and
follows a t distribution with
length(x)-2 degrees of freedom
if the samples follow independent normal distributions. If there are
at least 4 complete pairs of observation, an asymptotic confidence
interval is given based on Fisher's Z transform.
$\tau$ or Spearman's $\rho$ statistic is used to
estimate a rank-based measure of association. These tests may be used
if the data do not necessarily come from a bivariate normal
For Kendall's test, by default (if
exact is NULL), an exact
p-value is computed if there are less than 50 paired samples containing
finite values and there are no ties. Otherwise, the test statistic is
the estimate scaled to zero mean and unit variance, and is approximately
For Spearman's test, p-values are computed using algorithm AS 89 for
$n < 1290$ and
exact = TRUE, otherwise via the asymptotic
$t$ approximation. Note that these are
- A list with class
"htest"containing the following components:
statistic the value of the test statistic. parameter the degrees of freedom of the test statistic in the case that it follows a t distribution. p.value the p-value of the test. estimate the estimated measure of association, with name
"rho"corresponding to the method employed.
null.value the value of the association measure under the null hypothesis, always
alternative a character string describing the alternative hypothesis. method a character string indicating how the association was measured. data.name a character string giving the names of the data. conf.int a confidence interval for the measure of association. Currently only given for Pearson's product moment correlation coefficient in case of at least 4 complete pairs of observations.
- Kendall correlation coefficient
- Kendall's tau
- Pearson correlation coefficient
- Spearman correlation coefficient
- Spearman's rho
D. J. Best & D. E. Roberts (1975), Algorithm AS 89: The Upper Tail Probabilities of Spearman's $\rho$. Applied Statistics, 24, 377--379.
Myles Hollander & Douglas A. Wolfe (1973), Nonparametric Statistical Methods. New York: John Wiley & Sons. Pages 185--194 (Kendall and Spearman tests).
Kendall in package
## Hollander & Wolfe (1973), p. 187f. ## Assessment of tuna quality. We compare the Hunter L measure of ## lightness to the averages of consumer panel scores (recoded as ## integer values from 1 to 6 and averaged over 80 such values) in ## 9 lots of canned tuna. x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1) y <- c( 2.6, 3.1, 2.5, 5.0, 3.6, 4.0, 5.2, 2.8, 3.8) ## The alternative hypothesis of interest is that the ## Hunter L value is positively associated with the panel score. cor.test(x, y, method = "kendall", alternative = "greater") ## => p=0.05972 cor.test(x, y, method = "kendall", alternative = "greater", exact = FALSE) # using large sample approximation ## => p=0.04765 ## Compare this to cor.test(x, y, method = "spearm", alternative = "g") cor.test(x, y, alternative = "g") ## Formula interface. require(graphics) pairs(USJudgeRatings) cor.test(~ CONT + INTG, data = USJudgeRatings)