Chi-Square Test in Python

GOAL

To write program of chi-square test using python. 

What is chi-square test?

Chi-square test which means “Pearson’s chi-square test” here, is a method of statistical hypothesis testing for goodness-of-fit and independence.

Goodness-of-fit test is the testing to determine whether the observed frequency distribution is the same as the theoretical distribution.
Independence test is the testing to determine whether 2 observations that is represented by 2*2 table, on 2 variables are independent of each other.

Details will be longer. Please see the following sites and document.

• Explainer video on youtube
• Chi-Square Test (Math is fun)
• The Chi square test: an introduction

Implementation

The following is implementation for chi-square test.

Import libraries

import numpy as np
import pandas as pd
import scipy as sp
from scipy import stats

Data preparing

chi_square_data.csv

A,B,C
23,65,158
100,44,119

Read and Set Data

csv_line = []
with open('chi_square_data.csv', ) as f:
    for i in f:
        items = i.split(',')
        for j in range(len(items)):
            if '\n' in items[j]:
                items[j] =float(items[j][:-1])
            else:
                items[j] =float(items[j])
        csv_line.append(items)

group = csv_line[0]
success = [int(n) for n in csv_line[1]]
failure = [int(n) for n in csv_line[2]]

groups = [] 
result =[]
count = []
for i in range(len(group)):
    groups += [group[i], group[i]] #['A','A', 'B', 'B', 'C', 'C']
    result += ['success', 'failure'] #['success', 'failure', 'success', 'failure', 'success', 'failure']
    count += [success[i], failure[i]] #[23, 100, 65, 44, 158, 119]
    
data =  pd.DataFrame({
    'groups' : groups,
    'result' : result,
    'count' : count
})

cross_data = pd.pivot_table(
    data = data,
    values ='count',
    aggfunc = 'sum',
    index = 'groups',
    columns = 'result'
)
print(cross_data)
>>result  failure  success
groups                  
A           100       23
B            44       65
C           119      158

Chi-square test

print(stats.chi2_contingency(cross_data, correction=False))
>> (57.23616422920877, 3.726703617716424e-13, 2, array([[ 63.554,  59.446],
       [ 56.32 ,  52.68 ],
       [143.126, 133.874]]))

• chi2 : 57.23616422920877
  • The test statistic
• p : 3.726703617716424e-13
  • The p-value of the test
• dof : 2
  • Degrees of freedom
• expected : array
  • The expected frequencies, based on the marginal sums of the table.

The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
When statistically significant, that is, p-value is less than 0.05 (typically ≤ 0.05), the difference between groups is significant.

Paired One-way ANOVA And Multiple Comparisons In Python

GOAL

To write program of paired one-way ANOVA(analysis of variance) and multiple comparisons using python. Please refer another article “Unpaired One-way ANOVA And Multiple Comparisons In Python” for unpaired one-way ANOVA.

What is ANOVA

ANOVA(analysis of variance) is a method of statistical hypothesis testing that determines the effects of factors and interactions, which analyzes the differences between group means within a sample.
Details will be longer. Please see the following site.

• Explainer video on Youtube
• A Simple Introduction to ANOVA (Analyticsvidhya)
• Analysis of Variance (Online Statistics Education)

One-way ANOVA is ANOVA test that compares the means of three or more samples. Null hypothesis is that samples in groups were taken from populations with the same mean.

Implementation

The following is implementation example of paired one-way ANOVA.

Import Libraries

Import libraries below for ANOVA test.

import statsmodels.api as sm
from statsmodels.formula.api import ols
import pandas as pd
import numpy as np
import statsmodels.stats.anova as anova

Data Preparing

test_data.csv

85, 90, 88, 69, 78, 98, 87
55, 82, 67, 64, 78, 54, 49
46, 95, 59, 80, 52, 73, 70

Read and Set Data

csv_line = []
with open('test_data.csv', ) as f:
    for i in f:
        items = i.split(',')
        for j in range(len(items)):
            if '\n' in items[j]:
                items[j] =float(items[j][:-1])
            else:
                items[j] =float(items[j])
        print(items)
        csv_line.append(items)

groupA = csv_line [0]
groupB = csv_line [1]
groupC = csv_line [2]
tdata = pd.DataFrame({'A':groupA, 'B':groupB, 'C':groupC})
tdata.index = range(1,10)
tdata 

If you want to display data summary, use DataFrame.describe().

tdata.describe()

ANOVA

subjects=['id1','id2','id3','id4','id5','id6','id7']
points = np.array(groupA +groupB + groupC)
conditions = np.repeat(['A','B','C'],len(group0))
subjects = np.array(subjects+subjects+subjects)
df = pd.DataFrame({'Point':points,'Conditions':conditions,'Subjects':subjects})

aov=anova.AnovaRM(df, 'Point','Subjects',['Conditions'])
result=aov.fit()

print(result)

>>                 Anova
========================================
           F Value Num DF  Den DF Pr > F
----------------------------------------
Conditions  5.4182 2.0000 12.0000 0.0211
========================================

The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
When statistically significant, that is, p-value is less than 0.05 (typically ≤ 0.05), perform a multiple comparison. This p value is different between paired ANOVA and unpaired ANOVA.

Tukey’s multiple comparisons

Use pairwise_tukeyhsd(endog, groups, alpha=0.05) for tuky’s HSD(honestly significant difference) test. Argument endog is response variable, array of data (A[0] A[1]… A[6] B[1] … B[6] C[1] … C[6]). Argument groups is list of names(A, A…A, B…B, C…C) that corresponds to response variable. Alpha is significance level.

def tukey_hsd(group_names , *args ):
    endog = np.hstack(args)
    groups_list = []
    for i in range(len(args)):
        for j in range(len(args[i])):
            groups_list.append(group_names[i])
    groups = np.array(groups_list)
    res = pairwise_tukeyhsd(endog, groups)
    print (res.pvalues) #print only p-value
    print(res) #print result

print(tukey_hsd(['A', 'B', 'C'], tdata['A'], tdata['B'],tdata['C']))
>> [0.02259466 0.06511251 0.85313142]
 Multiple Comparison of Means - Tukey HSD, FWER=0.05 
=====================================================
group1 group2 meandiff p-adj   lower    upper  reject
-----------------------------------------------------
     A      B -20.8571 0.0226 -38.9533 -2.7609   True
     A      C -17.1429 0.0651 -35.2391  0.9533  False
     B      C   3.7143 0.8531 -14.3819 21.8105  False
-----------------------------------------------------
None