Several weeks ago I had to compare three machine learning algorithm implementations and decide if one of them performed significantly better than the other two. The output I had from the algorithms was in the form of series of accuracy scores. Turns out that an easy way to compare two or more data sets is to use analysis of variance (ANOVA).
The analysis of variance statistical models were developed by the English statistician Sir R. A. Fisher and are commonly used to determine if there is a significant difference between the means of two or more data sets. In this post we will be focusing exclusively on one-way ANOVA, which means that we'll be examining the influence of one independent on one dependent variable (i.e. we are measuring the significant effects of one factor).
Another way of putting this is that we can use ANOVA for testing hypothesis. A common approach is to assume that the data sets are samples of the same distribution (i.e. the null hypothesis is that their means are equal). Rejecting the null hypothesis would imply that at least one of the means is different.
One-way ANOVA in Python
Let's look at a fictitious problem and see how we can solve it using one-way ANOVA in Python. Three archers - Pat, Jack, and Alex are participating in an archery contest. They are shooting at targets with 10 evenly spaced concentric rings. The rings have score values from 1 through 10 assigned to them, with 10 being the highest. Each participant shoots 6 arrow, scoring the following points:
Pat - 5, 4, 4, 3, 9, 4
Jack - 4, 8, 7, 5, 1, 5
Alex - 9, 9, 8, 10, 4, 10
Based on the above results we would like to know who the best archer is. In other words our null hypothesis is that the means of all populations are equal.
Rejecting the null hypothesis would mean that there is a significant difference between at least two of the archers.
The decision to reject the null hypothesis and accept the alternative hypothesis is based on the significance level of the test () and the probability of observing the effect given that the null hypothesis is true (-value). If the null hypothesis is ruled out. We typically use a value of , which corresponds to 95% confidence.
Using one-way ANOVA in Python is quite straightforward - the f_oneway function from SciPy performs a one-way ANOVA and returns the F and p values from the test. We can use the following code to run the analysis against the data sets from our example.
import numpy as np from scipy import stats data = np.rec.array([ ('Pat', 5), ('Pat', 4), ('Pat', 4), ('Pat', 3), ('Pat', 9), ('Pat', 4), ('Jack', 4), ('Jack', 8), ('Jack', 7), ('Jack', 5), ('Jack', 1), ('Jack', 5), ('Alex', 9), ('Alex', 8), ('Alex', 8), ('Alex', 10), ('Alex', 5), ('Alex', 10)], dtype = [('Archer','|U5'),('Score', '<i8')]) f, p = stats.f_oneway(data[data['Archer'] == 'Pat'].Score, data[data['Archer'] == 'Jack'].Score, data[data['Archer'] == 'Alex'].Score) print ('One-way ANOVA') print ('=============') print ('F value:', f) print ('P value:', p, '\n')
Running the above produces the following output:
One-way ANOVA ============= F value: 5.0 P value: 0.0216837493201
As we reject the null hypothesis and we conclude that at least one of the means is different from at least one other population mean (i.e. not all archers perform equally).
The thing with one-way ANOVA is that although we now know that there is difference in the performance of the archers, we do not know know exactly who performs best or worst. This is why the analysis of variance is often followed by a post hoc analysis.
Tukey's range test
Tukey's range test, named after the American mathematician John Tukey, is a common method used as post hoc analysis after one-way ANOVA. This test compares all possible pairs and we can use it to precisely identify difference between two means that's greater than the expected standard error.
The statsmodels library provides an easy to use implementation of Tukey's range test. First, we have to modify our code to import the required classes:
from statsmodels.stats.multicomp import pairwise_tukeyhsd from statsmodels.stats.multicomp import MultiComparison
We can then append the following to the code to run the actual test.
mc = MultiComparison(data['Score'], data['Archer']) result = mc.tukeyhsd() print(result) print(mc.groupsunique)
Note the last line in the snippet. We need this to see the group IDs assigned to the archers, as the algorithm won't necessarily follow the group order from the array. Also note that the tukeyhsd() function has a parameter named alpha, which we are not setting explicitly as we are happy with its default value ().
You can get the complete code of archers.py from GitHub.
Running the code produces the following output:
$ ./archers.py One-way ANOVA ============= F value: 5.0 P value: 0.0216837493201 Multiple Comparison of Means - Tukey HSD,FWER=0.05 ============================================= group1 group2 meandiff lower upper reject --------------------------------------------- 0 1 -3.3333 -6.5755 -0.0911 True 0 2 -3.5 -6.7422 -0.2578 True 1 2 -0.1667 -3.4089 3.0755 False --------------------------------------------- ['Alex' 'Jack' 'Pat'] $
The results above reveal that Alex (group 0) significantly differs from the other two archers. The third column tells us that there is significant evidence to reject the null hypothesis for the groups Alex-Jack (0-1) and Alex-Pat(0-2).
The test also shows the difference between the group means (the meandiff column).
This leads to the conclusion that Alex is the best archer in the group.