Statistical post-hoc analysis and outlier detection algorithms

## Project description

**scikit-posthocs** is a Python package that provides post hoc tests for
pairwise multiple comparisons that are usually performed in statistical
data analysis to assess the differences between group levels if a statistically
significant result of ANOVA test has been obtained.

**scikit-posthocs** is tightly integrated with Pandas DataFrames and NumPy
arrays to ensure fast computations and convenient data import and storage.

This package will be useful for statisticians, data analysts, and researchers who use Python in their work.

## Background

Python statistical ecosystem comprises multiple packages. However, it still has numerous gaps and is surpassed by R packages and capabilities.

SciPy (version 1.2.0) offers *Student*, *Wilcoxon*,
and *Mann-Whitney* tests that are not adapted to multiple pairwise
comparisons. Statsmodels (version 0.9.0)
features *TukeyHSD* test that needs some extra actions to be fluently
integrated into a data analysis pipeline.
Statsmodels also has good helper
methods: `allpairtest` (adapts an external function such as
`scipy.stats.ttest_ind` to multiple pairwise comparisons) and
`multipletests` (adjusts *p* values to minimize type I and II errors).
PMCMRplus is a very good R package that
has no rivals in Python as it offers more than 40 various tests (including
post hoc tests) for factorial and block design data. PMCMRplus was an
inspiration and a reference for *scikit-posthocs*.

**scikit-posthocs** attempts to improve Python statistical capabilities by
offering a lot of parametric and nonparametric post hoc tests along with
outliers detection and basic plotting methods.

## Features

*Omnibus*tests:- Durbin test (for balanced incomplete block design).
- Mack-Wolfe test.
- Hayter (OSRT) test.

*Parametric*pairwise multiple comparisons tests:- Scheffe test.
- Student T test.
- Tamhane T2 test.
- TukeyHSD test.

*Non-parametric*tests for factorial design:- Conover test.
- Dunn test.
- Dwass, Steel, Critchlow, and Fligner test.
- Mann-Whitney test.
- Nashimoto and Wright (NPM) test.
- Nemenyi test.
- van Waerden test.
- Wilcoxon test.

*Non-parametric*tests for block design:- Conover test.
- Durbin and Conover test.
- Miller test.
- Nemenyi test.
- Quade test.
- Siegel test.

- Outliers detection tests:
- Simple test based on interquartile range (IQR).
- Grubbs test.
- Tietjen-Moore test.
- Generalized Extreme Studentized Deviate test (ESD test).

- Other tests:
- Anderson-Darling test.

- Global null hypothesis tests:
- Fisher’s combination test.
- Simes test.

- Plotting functionality (e.g. significance plots).

All post hoc tests are capable of p adjustments for multiple pairwise comparisons.

## Dependencies

## Compatibility

Package is compatible with only Python 3.

## Install

You can install the package from PyPi:

pip install scikit-posthocs

## Examples

### Parametric ANOVA with post hoc tests

Here is a simple example of the one-way analysis of variance (ANOVA)
with post hoc tests used to compare *sepal width* means of three
groups (three iris species) in *iris* dataset.

To begin, we will import the dataset using statsmodels
`get_rdataset()` method.

>>> import statsmodels.api as sa >>> import statsmodels.formula.api as sfa >>> import scikit_posthocs as sp >>> df = sa.datasets.get_rdataset('iris').data >>> df.columns = df.columns.str.replace('.', '') >>> df.head() SepalLength SepalWidth PetalLength PetalWidth Species 0 5.1 3.5 1.4 0.2 setosa 1 4.9 3.0 1.4 0.2 setosa 2 4.7 3.2 1.3 0.2 setosa 3 4.6 3.1 1.5 0.2 setosa 4 5.0 3.6 1.4 0.2 setosa

Now, we will build a model and run ANOVA using statsmodels `ols()`
and `anova_lm()` methods. Columns `Species` and `SepalWidth`
contain independent (predictor) and dependent (response) variable
values, correspondingly.

>>> lm = sfa.ols('SepalWidth ~ C(Species)', data=df).fit() >>> anova = sa.stats.anova_lm(lm) >>> print(anova) df sum_sq mean_sq F PR(>F) C(Species) 2.0 11.344933 5.672467 49.16004 4.492017e-17 Residual 147.0 16.962000 0.115388 NaN NaN

The results tell us that there is a significant difference between
groups means (p = 4.49e-17), but does not tell us the exact group pairs which
are different in means. To obtain pairwise group differences, we will carry
out a posteriori (post hoc) analysis using `scikits-posthocs` package.
Student T test applied pairwisely gives us the following p values:

>>> sp.posthoc_ttest(df, val_col='SepalWidth', group_col='Species', p_adjust='holm') setosa versicolor virginica setosa -1.000000e+00 5.535780e-15 8.492711e-09 versicolor 5.535780e-15 -1.000000e+00 1.819100e-03 virginica 8.492711e-09 1.819100e-03 -1.000000e+00

Remember to use a FWER controlling procedure, such as Holm procedure, when making multiple comparisons. As seen from this table, significant differences in group means are obtained for all group pairs.

### Non-parametric ANOVA with post hoc tests

If normality and other assumptions are violated, one can use a non-parametric Kruskal-Wallis H test (one-way non-parametric ANOVA) to test if samples came from the same distribution.

Let’s use the same dataset just to demonstrate the procedure. Kruskal-Wallis
test is implemented in SciPy package. `scipy.stats.kruskal` method
accepts array-like structures, but not DataFrames.

>>> import scipy.stats as ss >>> import statsmodels.api as sa >>> import scikit_posthocs as sp >>> df = sa.datasets.get_rdataset('iris').data >>> df.columns = df.columns.str.replace('.', '') >>> data = [df.loc[ids, 'SepalWidth'].values for ids in df.groupby('Species').groups.values()]

`data` is a list of 1D arrays containing *sepal width* values, one array per
each species. Now we can run Kruskal-Wallis analysis of variance.

>>> H, p = ss.kruskal(*data) >>> p 1.5692820940316782e-14

P value tells us we may reject the null hypothesis that the population medians
of all of the groups are equal. To learn what groups (species) differ in their
medians we need to run post hoc tests. `scikit-posthocs` provides a lot of
non-parametric tests mentioned above. Let’s choose Conover’s test.

>>> sp.posthoc_conover(df, val_col='SepalWidth', group_col='Species', p_adjust = 'holm') setosa versicolor virginica setosa -1.000000e+00 2.278515e-18 1.293888e-10 versicolor 2.278515e-18 -1.000000e+00 1.881294e-03 virginica 1.293888e-10 1.881294e-03 -1.000000e+00

Pairwise comparisons show that we may reject the null hypothesis (p < 0.01) for each pair of species and conclude that all groups (species) differ in their sepal widths.

### Block design

In block design case, we have a primary factor (e.g. treatment) and a blocking
factor (e.g. age or gender). A blocking factor is also called a *nuisance*
factor, and it is usually a source of variability that needs to be accounted
for.

An example scenario is testing the effect of four fertilizers on crop yield in four cornfields. We can represent the results with a matrix in which rows correspond to the blocking factor (field) and columns correspond to the primary factor (yield).

The following dataset is artificial and created just for demonstration of the procedure:

>>> data = np.array([[ 8.82, 11.8 , 10.37, 12.08], [ 8.92, 9.58, 10.59, 11.89], [ 8.27, 11.46, 10.24, 11.6 ], [ 8.83, 13.25, 8.33, 11.51]])

First, we need to perform an omnibus test — Friedman rank sum test. It is
implemented in `scipy.stats` subpackage:

>>> import scipy.stats as ss >>> ss.friedmanchisquare(*data.T) FriedmanchisquareResult(statistic=8.700000000000003, pvalue=0.03355726870553798)

We can reject the null hypothesis that our treatments have the same
distribution, because p value is less than 0.05. A number of post hoc tests are
available in `scikit-posthocs` package for unreplicated block design data.
In the following example, Nemenyi’s test is used:

>>> import scikit_posthocs as sp >>> sp.posthoc_nemenyi_friedman(data) 0 1 2 3 0 -1.000000 0.220908 0.823993 0.031375 1 0.220908 -1.000000 0.670273 0.823993 2 0.823993 0.670273 -1.000000 0.220908 3 0.031375 0.823993 0.220908 -1.000000

This function returns a DataFrame with p values obtained in pairwise comparisons between all treatments. One can also pass a DataFrame and specify the names of columns containing dependent variable values, blocking and primary factor values. The following code creates a DataFrame with the same data:

>>> data = pd.DataFrame.from_dict({'blocks': {0: 0, 1: 1, 2: 2, 3: 3, 4: 0, 5: 1, 6: 2, 7: 3, 8: 0, 9: 1, 10: 2, 11: 3, 12: 0, 13: 1, 14: 2, 15: 3}, 'groups': {0: 0, 1: 0, 2: 0, 3: 0, 4: 1, 5: 1, 6: 1, 7: 1, 8: 2, 9: 2, 10: 2, 11: 2, 12: 3, 13: 3, 14: 3, 15: 3}, 'y': {0: 8.82, 1: 8.92, 2: 8.27, 3: 8.83, 4: 11.8, 5: 9.58, 6: 11.46, 7: 13.25, 8: 10.37, 9: 10.59, 10: 10.24, 11: 8.33, 12: 12.08, 13: 11.89, 14: 11.6, 15: 11.51}}) >>> data blocks groups y 0 0 0 8.82 1 1 0 8.92 2 2 0 8.27 3 3 0 8.83 4 0 1 11.80 5 1 1 9.58 6 2 1 11.46 7 3 1 13.25 8 0 2 10.37 9 1 2 10.59 10 2 2 10.24 11 3 2 8.33 12 0 3 12.08 13 1 3 11.89 14 2 3 11.60 15 3 3 11.51

This is a *melted* and ready-to-use DataFrame. Do not forget to pass `melted`
argument:

>>> sp.posthoc_nemenyi_friedman(data, y_col='y', block_col='blocks', group_col='groups', melted=True) 0 1 2 3 0 -1.000000 0.220908 0.823993 0.031375 1 0.220908 -1.000000 0.670273 0.823993 2 0.823993 0.670273 -1.000000 0.220908 3 0.031375 0.823993 0.220908 -1.000000

### Data types

Internally, `scikit-posthocs` uses NumPy ndarrays and pandas DataFrames to
store and process data. Python lists, NumPy ndarrays, and pandas DataFrames
are supported as *input* data types. Below are usage examples of various
input data structures.

#### Lists and arrays

>>> x = [[1,2,1,3,1,4], [12,3,11,9,3,8,1], [10,22,12,9,8,3]] >>> # or >>> x = np.array([[1,2,1,3,1,4], [12,3,11,9,3,8,1], [10,22,12,9,8,3]]) >>> sp.posthoc_conover(x, p_adjust='holm') 1 2 3 1 -1.000000 0.057606 0.007888 2 0.057606 -1.000000 0.215761 3 0.007888 0.215761 -1.000000

You can check how it is processed with a hidden function `__convert_to_df()`:

>>> sp.__convert_to_df(x) ( vals groups 0 1 1 1 2 1 2 1 1 3 3 1 4 1 1 5 4 1 6 12 2 7 3 2 8 11 2 9 9 2 10 3 2 11 8 2 12 1 2 13 10 3 14 22 3 15 12 3 16 9 3 17 8 3 18 3 3, 'vals', 'groups')

It returns a tuple of a DataFrame representation and names of the columns
containing dependent (`vals`) and independent (`groups`) variable values.

*Block design* matrix passed as a NumPy ndarray is processed with a hidden
`__convert_to_block_df()` function:

>>> data = np.array([[ 8.82, 11.8 , 10.37, 12.08], [ 8.92, 9.58, 10.59, 11.89], [ 8.27, 11.46, 10.24, 11.6 ], [ 8.83, 13.25, 8.33, 11.51]]) >>> sp.__convert_to_block_df(data) ( blocks groups y 0 0 0 8.82 1 1 0 8.92 2 2 0 8.27 3 3 0 8.83 4 0 1 11.80 5 1 1 9.58 6 2 1 11.46 7 3 1 13.25 8 0 2 10.37 9 1 2 10.59 10 2 2 10.24 11 3 2 8.33 12 0 3 12.08 13 1 3 11.89 14 2 3 11.60 15 3 3 11.51, 'y', 'groups', 'blocks')

#### DataFrames

If you are using DataFrames, you need to pass column names containing variable values to a post hoc function:

>>> import statsmodels.api as sa >>> import scikit_posthocs as sp >>> df = sa.datasets.get_rdataset('iris').data >>> df.columns = df.columns.str.replace('.', '') >>> sp.posthoc_conover(df, val_col='SepalWidth', group_col='Species', p_adjust = 'holm')

`val_col` and `group_col` arguments specify the names of the columns
containing dependent (response) and independent (grouping) variable values.

## Significance plots

P values can be plotted using a heatmap:

>>> pc = sp.posthoc_conover(x, val_col='values', group_col='groups') >>> heatmap_args = {'linewidths': 0.25, 'linecolor': '0.5', 'clip_on': False, 'square': True, 'cbar_ax_bbox': [0.80, 0.35, 0.04, 0.3]} >>> sp.sign_plot(pc, **heatmap_args)

Custom colormap applied to a plot:

>>> pc = sp.posthoc_conover(x, val_col='values', group_col='groups') >>> # Format: diagonal, non-significant, p<0.001, p<0.01, p<0.05 >>> cmap = ['1', '#fb6a4a', '#08306b', '#4292c6', '#c6dbef'] >>> heatmap_args = {'cmap': cmap, 'linewidths': 0.25, 'linecolor': '0.5', 'clip_on': False, 'square': True, 'cbar_ax_bbox': [0.80, 0.35, 0.04, 0.3]} >>> sp.sign_plot(pc, **heatmap_args)

## Citing

If you want to cite *scikit-posthocs*, please refer to the publication in
the Journal of Open Source Software:

Terpilowski, M. (2019). scikit-posthocs: Pairwise multiple comparison tests in Python. Journal of Open Source Software, 4(36), 1169, https://doi.org/10.21105/joss.01169

@ARTICLE{Terpilowski2019, title = {scikit-posthocs: Pairwise multiple comparison tests in Python}, author = {Terpilowski, Maksim}, journal = {The Journal of Open Source Software}, volume = {4}, number = {36}, pages = {1169}, year = {2019}, doi = {10.21105/joss.01169} }

## Acknowledgement

Thorsten Pohlert, PMCMR author and maintainer

Copyright (c) 2020 Maksim Terpilowski

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

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