R: Test for Association Between Paired Samples

cor.test {ctest} | R Documentation

## Test for Association 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, y,
             alternative = c("two.sided", "less", "greater"),
             method = c("pearson", "kendall", "spearman"),
             exact = NULL, conf.level = 0.95, ...)
    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)

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