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.
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.
- 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.
Package is only compatible with Python 3.
You can install the package using pip (from PyPi):
pip install scikit-posthocsOr using conda (from conda-forge repo):
conda install -c conda-forge scikit-posthocsThe latest version from GitHub can be installed using:
pip install git+https://github.com/maximtrp/scikit-posthocs.gitHere 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 setosaNow, 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 NaNThe 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+00Remember 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.
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-14P 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+00Pairwise 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.
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 -1.000000This 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 11.51This 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 -1.000000Internally, 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.
>>> 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 -1.000000You 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')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.
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)
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}
}
Thorsten Pohlert, PMCMR author and maintainer