cor.test                package:stats                R Documentation

Test for Association/Correlation Between Paired Samples

Description:

     Test for association between paired samples, using one of
     Pearson's product moment correlation coefficient, Kendall's tau or
     Spearman's rho.

Usage:

     cor.test(x, ...)

     ## Default S3 method:
     cor.test(x, y,
              alternative = c("two.sided", "less", "greater"),
              method = c("pearson", "kendall", "spearman"),
              exact = NULL, conf.level = 0.95, ...)

     ## S3 method for class 'formula':
     cor.test(formula, data, subset, na.action, ...)

Arguments:

    x, y: numeric vectors of data values.  'x' and 'y' must have the
          same length.

alternative: indicates the alternative hypothesis and must be one of
          '"two.sided"', '"greater"' or '"less"'.  You can specify just
          the initial letter.  '"greater"' corresponds to positive
          association, '"less"' to negative association.

  method: a character string indicating which correlation coefficient
          is to be  used for the test.  One of '"pearson"',
          '"kendall"', or '"spearman"', can be abbreviated.

   exact: a logical indicating whether an exact p-value should be
          computed.  Only used for Kendall's tau. See the Details for
          the meaning of 'NULL' (the default).

conf.level: 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.

 formula: a formula of the form '~ u + v', where each of 'u' and 'v'
          are numeric variables giving the data values for one sample. 
          The samples must be of the same length.

    data: an optional data frame containing the variables in the model
          formula.

  subset: an optional vector specifying a subset of observations to be
          used.

na.action: a function which indicates what should happen when the data
          contain 'NA's.  Defaults to 'getOption("na.action")'.

     ...: further arguments to be passed to or from methods.

Details:

     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.

     If 'method' is '"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.

     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 tests may be used if the data do not
     necessarily come from a bivariate normal distribution.

     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 normally distributed.

     For Spearman's test, p-values are computed using algorithm AS 89.

Value:

     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 '"cor"',
          '"tau"', or '"rho"' correspoding to the method employed.

null.value: the value of the association measure under the null
          hypothesis, always '0'.

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.

References:

     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 inference_. New York: John Wiley & Sons. Pages 185-194
     (Kendall and Spearman tests).

Examples:

     ## 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.
     data(USJudgeRatings)
     pairs(USJudgeRatings)
     cor.test(~ CONT + INTG, data = USJudgeRatings)

