Bartels Rank Test of Randomness for Detecting Non-Random Order in Continuous Sequences

A nonparametric test for randomness of a sequence (so it tests whether data is sampled randomly from an underlying population), based on the ratio of the mean square successive difference of ranks to the rank variance.

Usage

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

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 "auto" (default), "normal" or "beta". With "auto" the beta approximation is used for sample sizes up to 100 and the normal approximation otherwise.

Value

A list with class "htest" containing the following components:

statistic

the value of the normalized test statistic.

parameter, n

the sample size, after the removal of missing 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 shown on screen).

nm

the value of the NM statistic, the numerator of RVN (not shown on screen).

mu

the mean value of the RVN statistic (not shown on screen).

var

the variance of the RVN statistic (not shown on screen).

Details

Data must at least be measured on an ordinal scale. The RVN test statistic is $$RVN=\frac{\sum_{i=1}^{n-1}(R_i-R_{i+1})^2}{\sum_{i=1}^{n}\left(R_i-(n+1)/2\right)^2}$$ where \(R_i=rank(X_i), i=1,\dots, n\). It is known that \((RVN-2)/\sigma\) is asymptotically standard normal, where \(\sigma^2=\frac{4(n-2)(5n^2-2n-9)}{5n(n+1)(n-1)^2}\).

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, see vonNeumannTest().

Note

Based on code by Frederico Caeiro previously published in the randtests package, adapted to conform to package standards.

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., New York: Marcel Dekker (pp. 97-98).

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

Other test.timeseries: adfTest(), kpssTest(), runsTest(), vonNeumannTest()

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 rank test of randomness (beta approximation)
#> 
#> data:  tourists
#> z = -3.6453, n = 13, p-value = 1.21e-08
#> alternative hypothesis: trend
#> 

#  Bartels rank test of randomness (beta approximation)
#
# data:  tourists
# z = -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")
#> 
#> 	Bartels rank test of randomness (beta approximation)
#> 
#> data:  x
#> z = 0.083357, n = 10, p-value = 0.9379
#> alternative hypothesis: nonrandomness
#>