Nonparametric Test Statistics

## Project description

# PyNonpar

Test statistics based on ranks may lead to paradoxical results. A solution are so-called pseudo-ranks. This package provides a function to calculate pseudo-ranks which are used in nonparametric statistics for rank tests. For a definition and discussion of pseudo-ranks, see for example [1].

To install the package from PyPI, simply type

```
pip install PyNonpar
```

## Table of Contents

**Two-Sample Tests**

**Paired Two-Sample Tests**

**Multi-Sample Tests**

## Pseudo-Ranks

If there are ties (i.e., observations with the same value) in the data, then the pseudo-ranks have to be adjusted. There are the options 'minimum', 'maximum' and 'average'. It is recommended to use 'average' as for this adjusmtent, normalized empirical distribution functions are used. See the example for details on the usage of the function 'psrank'.

```
import PyNonpar
from PyNonpar import*
# some artificial data
x = [1, 1, 1, 1, 2, 3, 4, 5, 6]
group = ['C', 'C', 'B', 'B', 'B', 'A', 'C', 'A', 'C']
PyNonpar.pseudorank.psrank(x, group, ties_method = "average")
```

## Nonparametric Test Statistics

### Two-Sample Tests

- Asymptotic Wilcoxon-Mann-Whitney test: wilcoxon_mann_whitney_test()
- Brunner-Munzel test (Generalized Wilcoxon test): brunner_munzel_test()

The Hodges-Lehmann estimator can be calculated in a location shift model: hodges_lehmann(). The confidence interval for this estimator is only asymptotic and assumes continuous distributions.

#### 1. Asymptotic Wilcoxon-Mann-Whitney test

```
import PyNonpar
from PyNonpar import*
x = [8,4,10,4,9,1,3,3,4,8]
y = [10,5,11,6,11,2,4,5,5,10]
PyNonpar.twosample.wilcoxon_mann_whitney_test(x, y, alternative="less", alpha = 0.05)
```

#### 2. Brunner-Munzel test

```
import PyNonpar
from PyNonpar import*
x = [8,4,10,4,9,1,3,3,4,8]
y = [10,5,11,6,11,2,4,5,5,10]
PyNonpar.twosample.brunner_munzel_test(x, y, alternative="less", quantile = "t")
PyNonpar.twosample.brunner_munzel_test(x, y, alternative="less", quantile = "normal")
```

### Paired Two-Sample Tests

#### 1. Paired ranks test

The paired ranks test compares the marginal distributions F1 and F2. The Null hypothesis is H0: F1 = F2 (var_equal = True) or H0: p = 1/2 (var_equal = False). The two sided alternative is for both cases p != 1/2.

p = Probability(X_i < Y_j) + 1/2 * Probability(X_i = Y_j) for i != j where (X_i, Y_i), (X_j, Y_j) are paired observations.

```
import PyNonpar
from PyNonpar import*
x = [1, 2, 3, 4, 5, 7, 1, 1, 1]
y = [4, 6, 8, 7, 6, 5, 9, 1, 1]
PyNonpar.twosample_paired.paired_ranks_test(x, y, alternative="two.sided", var_equal=False, quantile="normal")
```

### Multi-Sample Tests

- The Hettmansperger-Norton Test for Patterned Alternatives: hettmansperger_norton_test()
- Kruskal-Wallis test: kruskal_wallis_test()

#### 1. The Hettmansperger-Norton Test for Patterned Alternatives

This package provides a function to calculate the Hettmansperger-Norton test for patterned alternatives using pseudo-ranks. Originally, this test was developed for ranks but this version was adapted to pseudo-ranks.

For the alternative, it is possible to use 'increasing' (i.e., trend = [1, 2, 3, ..., g]), 'decreasing' (i.e., trend = [g, g-1, g-2, ..., 1]) or 'custom' where the trend has to be specified manually. Note, that the trend is a list of length g where g is the number of groups.

```
import PyNonpar
from PyNonpar import*
# some artificial data
x = [1, 1, 1, 1, 2, 3, 4, 5, 6]
group = ['C', 'C', 'B', 'B', 'B', 'A', 'C', 'A', 'C']
PyNonpar.hettmansperger.hettmansperger_norton_test(x, group, alternative = "custom", trend = [1,3,2])
```

#### 2. Kruskal-Wallis Test

```
import PyNonpar
from PyNonpar import*
# some artificial data
x = [1, 1, 1, 1, 2, 3, 4, 5, 6]
group = ['C', 'C', 'B', 'B', 'B', 'A', 'C', 'A', 'C']
# Using pseudo-ranks
PyNonpar.multisample.kruskal_wallis_test(x, group, pseudoranks = True)
# Using ranks
PyNonpar.multisample.kruskal_wallis_test(x, group, pseudoranks = False)
```

## References

[1] Brunner, E., Bathke A. C. and Konietschke, F: Rank- and Pseudo-Rank Procedures in Factorial Designs - Using R and SAS, Springer Verlag, to appear.

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