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https://github.com/maximtrp/scikit-posthocs

Multiple Pairwise Comparisons (Post Hoc) Tests in Python
https://github.com/maximtrp/scikit-posthocs

anova multiple-comparisons nonparametric-statistics post-hoc python statistical-analysis statistical-methods statistics stats

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Multiple Pairwise Comparisons (Post Hoc) Tests in Python

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README 

        
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===============



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**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

--------



.. image:: images/flowchart.png

  :alt: Tests Flowchart



- *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

------------



- `NumPy and SciPy packages `_

- `Statsmodels `_

- `Pandas `_

- `Matplotlib `_

- `Seaborn `_



Compatibility

-------------



Package is only compatible with Python 3.



Install

-------



You can install the package using ``pip`` (from PyPi):



.. code:: bash



  pip install scikit-posthocs



Or using ``conda`` (from conda-forge repo):



.. code:: bash



  conda install -c conda-forge scikit-posthocs



The latest version from GitHub can be installed using:



.. code:: bash



  pip install git+https://github.com/maximtrp/scikit-posthocs.git



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.



.. code:: python



  >>> 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.



.. code:: python



  >>> 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:



.. code:: python



  >>> 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.



.. code:: python



  >>> 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.



.. code:: python



  >>> 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.



.. code:: python



  >>> 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:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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

^^^^^^^^^^^^^^^^



.. code:: python



  >>> 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()``:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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:



.. code:: python



  >>> 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)



.. image:: images/plot-conover.png



Custom colormap applied to a plot:



.. code:: python



  >>> 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)



.. image:: images/plot-conover-custom-cmap.png



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



.. code::



  @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