Copyright | (c) 2010 Neil Brown |
---|---|

License | BSD3 |

Maintainer | bos@serpentine.com |

Stability | experimental |

Portability | portable |

Safe Haskell | None |

Language | Haskell98 |

Mann-Whitney U test (also know as Mann-Whitney-Wilcoxon and Wilcoxon rank sum test) is a non-parametric test for assesing whether two samples of independent observations have different mean.

- mannWhitneyUtest :: TestType -> Double -> Sample -> Sample -> Maybe TestResult
- mannWhitneyU :: Sample -> Sample -> (Double, Double)
- mannWhitneyUCriticalValue :: (Int, Int) -> Double -> Maybe Int
- mannWhitneyUSignificant :: TestType -> (Int, Int) -> Double -> (Double, Double) -> Maybe TestResult
- wilcoxonRankSums :: Sample -> Sample -> (Double, Double)
- data TestType
- data TestResult

# Mann-Whitney U test

:: TestType | Perform one-tailed test (see description above). |

-> Double | The p-value at which to test (e.g. 0.05) |

-> Sample | First sample |

-> Sample | Second sample |

-> Maybe TestResult | Return |

Perform Mann-Whitney U Test for two samples and required
significance. For additional information check documentation of
`mannWhitneyU`

and `mannWhitneyUSignificant`

. This is just a helper
function.

One-tailed test checks whether first sample is significantly larger than second. Two-tailed whether they are significantly different.

mannWhitneyU :: Sample -> Sample -> (Double, Double) #

The Mann-Whitney U Test.

This is sometimes known as the Mann-Whitney-Wilcoxon U test, and
confusingly many sources state that the Mann-Whitney U test is the same as
the Wilcoxon's rank sum test (which is provided as `wilcoxonRankSums`

).
The Mann-Whitney U is a simple transform of Wilcoxon's rank sum test.

Again confusingly, different sources state reversed definitions for U₁
and U₂, so it is worth being explicit about what this function returns.
Given two samples, the first, xs₁, of size n₁ and the second, xs₂,
of size n₂, this function returns (U₁, U₂)
where U₁ = W₁ - (n₁(n₁+1))/2
and U₂ = W₂ - (n₂(n₂+1))/2,
where (W₁, W₂) is the return value of `wilcoxonRankSums xs1 xs2`

.

Some sources instead state that U₁ and U₂ should be the other way round, often expressing this using U₁' = n₁n₂ - U₁ (since U₁ + U₂ = n₁n₂).

All of which you probably don't care about if you just feed this into `mannWhitneyUSignificant`

.

:: (Int, Int) | The sample size |

-> Double | The p-value (e.g. 0.05) for which you want the critical value. |

-> Maybe Int | The critical value (of U). |

Calculates the critical value of Mann-Whitney U for the given sample sizes and significance level.

This function returns the exact calculated value of U for all sample sizes; it does not use the normal approximation at all. Above sample size 20 it is generally recommended to use the normal approximation instead, but this function will calculate the higher critical values if you need them.

The algorithm to generate these values is a faster, memoised version of the simple unoptimised generating function given in section 2 of "The Mann Whitney Wilcoxon Distribution Using Linked Lists"

:: TestType | Perform one-tailed test (see description above). |

-> (Int, Int) | The samples' size from which the (U₁,U₂) values were derived. |

-> Double | The p-value at which to test (e.g. 0.05) |

-> (Double, Double) | The (U₁, U₂) values from |

-> Maybe TestResult | Return |

Calculates whether the Mann Whitney U test is significant.

If both sample sizes are less than or equal to 20, the exact U critical value
(as calculated by `mannWhitneyUCriticalValue`

) is used. If either sample is
larger than 20, the normal approximation is used instead.

If you use a one-tailed test, the test indicates whether the first sample is significantly larger than the second. If you want the opposite, simply reverse the order in both the sample size and the (U₁, U₂) pairs.

## Wilcoxon rank sum test

wilcoxonRankSums :: Sample -> Sample -> (Double, Double) #

The Wilcoxon Rank Sums Test.

This test calculates the sum of ranks for the given two samples. The samples are ordered, and assigned ranks (ties are given their average rank), then these ranks are summed for each sample.

The return value is (W₁, W₂) where W₁ is the sum of ranks of the first sample
and W₂ is the sum of ranks of the second sample. This test is trivially transformed
into the Mann-Whitney U test. You will probably want to use `mannWhitneyU`

and the related functions for testing significance, but this function is exposed
for completeness.

# Data types

data TestResult #

Result of hypothesis testing

Significant | Null hypothesis should be rejected |

NotSignificant | Data is compatible with hypothesis |

# References

- Cheung, Y.K.; Klotz, J.H. (1997) The Mann Whitney Wilcoxon
distribution using linked lists.
*Statistica Sinica*7:805–813. http://www3.stat.sinica.edu.tw/statistica/oldpdf/A7n316.pdf.