R: Bartels Rank Test of Randomness

BartelsRankTest {DescTools} | R Documentation

## Bartels Rank Test of Randomness

### Description

Performs the Bartels rank test of randomness, which tests if a sample is sampled randomly from an underlying population. Data must at least be measured on an ordinal scale.

### Usage

    BartelsRankTest(x, alternative = c("two.sided", "trend", "oscillation"),
                    method = c("normal", "beta", "auto"))

### Arguments

`x` | a numeric vector containing the observations
`alternative` | a character string specifying the alternative hypothesis, must be one of "`two.sided`" (default), "`trend`" or "`oscillation`".
`method` | a character string specifying the method used to compute the p-value. Must be one of `normal` (default), `beta` or `auto`.

### Details

The RVN test statistic is

RVN=∑(R_i-R_{i+1})^2 / ∑(R_i-(n+1)/2)^2

where R_i=rank(X_i), i=1,...,n. It is known that (RVN-2)/σ is asymptotically standard normal, where σ^2=[4(n-2)(5n^2-2n-9)]/[5n(n+1)(n-1)^2].

By using the alternative "`trend`" the null hypothesis of randomness is tested against a trend. By using the alternative "`oscillation`" the null hypothesis of randomness is tested against a systematic oscillation.

Missing values are silently removed.

Bartels test is a rank version of von Neumann's test.

### Value

A list with class "htest" containing the components:

`statistic` | the value of the normalized statistic test.
`parameter, n` | the size of the data, after the remotion of consecutive duplicate values.
`p.value` | the p-value of the test.
`alternative` | a character string describing the alternative hypothesis.
`method` | a character string indicating the test performed.
`data.name` | a character string giving the name of the data.
`rvn` | the value of the RVN statistic (not show on screen).
`nm` | the value of the NM statistic, the numerator of RVN (not show on screen).
`mu` | the mean value of the RVN statistic (not show on screen).
`var` | the variance of the RVN statistic (not show on screen).

### Author(s)

Frederico Caeiro <fac@fct.unl.pt>

### References

Bartels, R. (1982) The Rank Version of von Neumann's Ratio Test for Randomness, Journal of the American Statistical Association, 77 (377), 40-46.

Gibbons, J.D. and Chakraborti, S. (2003) Nonparametric Statistical Inference, 4th ed. (pp. 97-98). URL: http://books.google.pt/books?id=dPhtioXwI9cC&lpg=PA97&ots=ZGaQCmuEUq

von Neumann, J. (1941) Distribution of the ratio of the mean square successive difference to the variance. Annals of Mathematical Statistics 12, 367-395.

### See Also

`rank.test`, `RunsTest`

### Examples

    ## Example 5.1 in Gibbons and Chakraborti (2003), p.98.
    ## Annual data on total number of tourists to the United States for 1970-1982.

    years <- 1970:1982
    tourists <- c(12362, 12739, 13057, 13955, 14123,  15698, 17523, 18610, 19842,
          20310, 22500, 23080, 21916)
    plot(years, tourists, pch=20)

    BartelsRankTest(tourists, alternative="trend", method="beta")

    #  Bartels Ratio Test
    #
    # data:  tourists
    # statistic = -3.6453, n = 13, p-value = 1.21e-08
    # alternative hypothesis: trend

    ## Example in Bartels (1982).
    ## Changes in stock levels for 1968-1969 to 1977-1978 (in $A million), deflated by the
    ## Australian gross domestic product (GDP) price index (base 1966-1967).
    x <- c(528, 348, 264, -20, - 167, 575, 410, -4, 430, - 122)

    BartelsRankTest(x, method="beta")

[Package DescTools version 0.99.28 Index]