Bartels Rank Test of Randomness
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.
BartelsRankTest(x, alternative = c("two.sided", "trend", "oscillation"),
method = c("normal", "beta", "auto"))x |
a numeric vector containing the observations |
alternative |
a character string specifying the alternative hypothesis, must be one of " |
method |
a character string specifying the method used to compute the p-value. Must be one of |
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.
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). |
Frederico Caeiro <fac@fct.unl.pt>
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.
## 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")Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.