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 " |

`method` |
a character string specifying the method used to compute the p-value. Must be one of |

### Details

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.

### 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

### 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")
```

*DescTools*version 0.99.55 Index]