Tukey’s Test For Post-Hoc Analysis Using Python

by Amrita Mitra | Feb 9, 2023 | Statistics For Machine Learning

$F-statistic=\frac{Mean\ Square\ Due\ To\ Treatment(MST)}{Mean\ Square\ Due\ To\ Error(MSE)}$

$MST=\frac{Sum\ Of\ Squares\ Due\ To\ Treatment(SST)}{df_{k}}\\ df_{k}=degrees\ of\ freedom=(the\ total\ number\ of\ groups-1)$

$SST=n_{1}(\bar{X}_{1}-\bar{X})^{2}+n_{2}(\bar{X}_{2}-\bar{X})^{2}+n_{3}(\bar{X}_{3}-\bar{X})^{2}\\ n_{1}=the\ total\ number\ of\ observations\ in\ the\ first\ group\\ n_{2}=the\ total\ number\ of\ observations\ in\ the\ second\ group\\ n_{3}=the\ total\ number\ of\ observations\ in\ the\ third\ group\\ \bar{X}=the\ mean\ marks\ across\ all\ groups\\ \bar{X}_{1}=the\ mean\ marks\ obtained\ by\ the\ first\ group\\ \bar{X}_{2}=the\ mean\ marks\ obtained\ by\ the\ second\ group\\ \bar{X}_{3}=the\ mean\ marks\ obtained\ by\ the\ third\ group\\$

$MSE=\frac{Sum\ Of\ Squares\ Due\ To\ Error(SSE)}{df}\\ df=the\ total\ degrees\ of\ freedom\\ =(the\ total\ number\ of\ observations-the\ total\ number\ of\ groups)$

$SSE=(n_{1}-1)\sigma_{1}^{2}+(n_{2}-1)\sigma_{2}^{2}+(n_{3}-1)\sigma_{3}^{2}\\ \sigma_{1}^{2}=the\ sample\ variance\ in\ the\ first\ group\\ \sigma_{2}^{2}=the\ sample\ variance\ in\ the\ second\ group\\ \sigma_{3}^{2}=the\ sample\ variance\ in\ the\ third\ group\\ n_{1}=the\ total\ number\ of\ observations\ in\ the\ first\ group\\ n_{2}=the\ total\ number\ of\ observations\ in\ the\ second\ group\\ n_{3}=the\ total\ number\ of\ observations\ in\ the\ third\ group\\$

While calculating the Tukey statistic, we use the MSE calculated in the one-way ANOVA test.

Tukey’s Test For Post-Hoc Analysis Using Python

We can use the following Python code to calculate the statistics in Tukey’s test for post-hoc analysis.

import numpy
import pandas
from statsmodels.stats.multicomp import pairwise_tukeyhsd
 
# Perform Tukey's test
df = pandas.DataFrame({'marks': [75, 50, 67, 59, 73, 82, 63, 69, 57, 81, 80, 92, 69, 69, 82, 57, 59, 82, 76, 66,
                                 84, 92, 81, 63, 75, 80, 68, 86, 86, 77, 70, 69, 84, 88, 66, 67, 85, 60, 69, 64,
                                 73, 73, 69, 71, 85, 86, 75, 77, 90, 70, 95, 84, 88, 85, 90, 91, 85, 74, 91, 65],
                        'group': numpy.repeat(['I', 'II', 'III'], repeats=20)})
 
result = pairwise_tukeyhsd(endog=df['marks'], groups=df['group'], alpha=0.05)
print(result)

Here, we are using the pairwise_tukeyhsd() function to calculate the test statistics in the Tukey’s test for post-hoc analysis. The output of the above program will be like the following: …

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