A one-way ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups.
If the overall p-value from the ANOVA table is less than some significance level, then we have sufficient evidence to say that at least one of the means of the groups is different from the others.
However, this doesn’t tell us which groups are different from each other. It simply tells us that not all of the group means are equal. In order to find out exactly which groups are different from each other, we must conduct a post hoc test.
One of the most commonly used post hoc tests is Tukey’s Test, which allows us to make pairwise comparisons between the means of each group while controlling for the family-wise error rate.
This tutorial provides a step-by-step example of how to perform Tukey’s Test in Python.
Step 1: Load Necessary Packages and Functions
First, we’ll load the necessary packages and functions in Python:
import pandas as pd
import numpy as np
from scipy.stats import f_oneway
from statsmodels.stats.multicomp import pairwise_tukeyhsd Step 2: Fit the ANOVA Model
The following code shows how to create a fake dataset with three groups (A, B, and C) and fit a one-way ANOVA model to the data to determine if the mean values for each group are equal:
#enter data for three groups a = [85, 86, 88, 75, 78, 94, 98, 79, 71, 80] b = [91, 92, 93, 90, 97, 94, 82, 88, 95, 96] c = [79, 78, 88, 94, 92, 85, 83, 85, 82, 81] #perform one-way ANOVA f_oneway(a, b, c) F_onewayResult(statistic=5.167774552944481, pvalue=0.012582197136592609)
We can see that the overall p-value from the ANOVA table is 0.01258.
Since this is less than .05, we have sufficient evidence to say that the mean values across each group are not equal.
Thus, we can proceed to perform Tukey’s Test to determine exactly which group means are different.
Step 3: Perform Tukey’s Test
To perform Tukey’s test in Python, we can use the pairwise_tukeyhsd() function from the statsmodels library:
#create DataFrame to hold data df = pd.DataFrame({'score': [85, 86, 88, 75, 78, 94, 98, 79, 71, 80, 91, 92, 93, 90, 97, 94, 82, 88, 95, 96, 79, 78, 88, 94, 92, 85, 83, 85, 82, 81], 'group': np.repeat(['a', 'b', 'c'], repeats=10)}) # perform Tukey's test tukey = pairwise_tukeyhsd(endog=df['score'], groups=df['group'], alpha=0.05) #display results print(tukey) Multiple Comparison of Means - Tukey HSD, FWER=0.05 ===================================================== group1 group2 meandiff p-adj lower upper reject ----------------------------------------------------- a b 8.4 0.0158 1.4272 15.3728 True a c 1.3 0.8864 -5.6728 8.2728 False b c -7.1 0.0453 -14.0728 -0.1272 True -----------------------------------------------------
Here’s how to interpret the output:
- P-value for the difference in means between a and b: .0158
- P-value for the difference in means between a and c: .8864
- P-value for the difference in means between b and c: .0453
Thus, we would conclude that there is a statistically significant difference between the means of groups a and b and groups b and c, but not a statistically significant difference between the means of groups a and c.
Additional Resources
How to Perform a One-Way ANOVA in Python
How to Perform a Two-Way ANOVA in Python
How to Perform a Repeated Measures ANOVA in Python