Running a Chi-Square Test of Independence (Python)

Objective of the study
Testing the independence/dependence of "polityscore" (democracy level of a country) and "Internet use rate"

Explanation and Methodologies

The objective of this study is much more likely the one I did previously that investigating about the relationship between a country's democracy level and its internet use rate by ANOVA. Same as before, "Gapminder" dataset is the data source and we use two of the variables,"polityscore" and "internetuserate".

First of all, "polityscore" is going to be divided into 4 groups according to its values.

polityscore	group(level)
-10 to -5	No_democracy
-5 to 0	Unstatisfied_democracy
0 to 5	Satisfied_democracy
5 to 10	Good_democracy

Secondly, convert the "internetuserate" into 2 categorical groups by 50%-quantile.

Python Output1:

number of rows and columns of gapminder

213

16

a dataframe combining 2 chosen variables

    internetuserate_tile        polityscore_grp

0        userate=50%tile  Unsatisfied_democracy

1       userate=100%tile         Good_democracy

2        userate=50%tile    Satisfied_democracy

4        userate=50%tile  Unsatisfied_democracy

6       userate=100%tile         Good_democracy

7       userate=100%tile    Satisfied_democracy

9       userate=100%tile         Good_democracy

10      userate=100%tile         Good_democracy

11      userate=100%tile           No_democracy

13      userate=100%tile           No_democracy

14       userate=50%tile    Satisfied_democracy

16      userate=100%tile           No_democracy

17      userate=100%tile         Good_democracy

19       userate=50%tile         Good_democracy

21       userate=50%tile    Satisfied_democracy

22       userate=50%tile         Good_democracy

24       userate=50%tile         Good_democracy

25      userate=100%tile         Good_democracy

27      userate=100%tile         Good_democracy

28       userate=50%tile  Unsatisfied_democracy

29       userate=50%tile         Good_democracy

30       userate=50%tile    Satisfied_democracy

31       userate=50%tile  Unsatisfied_democracy

32      userate=100%tile         Good_democracy

35       userate=50%tile  Unsatisfied_democracy

36       userate=50%tile  Unsatisfied_democracy

37      userate=100%tile         Good_democracy

38      userate=100%tile           No_democracy

39      userate=100%tile         Good_democracy

40       userate=50%tile         Good_democracy

..                   ...                    ...

175     userate=100%tile         Good_democracy

176      userate=50%tile         Good_democracy

178      userate=50%tile         Good_democracy

179     userate=100%tile         Good_democracy

180      userate=50%tile         Good_democracy

183      userate=50%tile           No_democracy

184     userate=100%tile         Good_democracy

185     userate=100%tile         Good_democracy

186      userate=50%tile           No_democracy

188      userate=50%tile  Unsatisfied_democracy

189      userate=50%tile  Unsatisfied_democracy

190      userate=50%tile    Satisfied_democracy

191      userate=50%tile         Good_democracy

192      userate=50%tile  Unsatisfied_democracy

194     userate=100%tile         Good_democracy

195     userate=100%tile  Unsatisfied_democracy

196     userate=100%tile         Good_democracy

197      userate=50%tile           No_democracy

199      userate=50%tile  Unsatisfied_democracy

200     userate=100%tile         Good_democracy

201     userate=100%tile           No_democracy

202     userate=100%tile         Good_democracy

203     userate=100%tile         Good_democracy

204     userate=100%tile         Good_democracy

205      userate=50%tile           No_democracy

207     userate=100%tile  Unsatisfied_democracy

208      userate=50%tile           No_democracy

210      userate=50%tile  Unsatisfied_democracy

211      userate=50%tile         Good_democracy

212      userate=50%tile    Satisfied_democracy

-----

Now we have two categorical variables and a Chi Square Test is going to be applied.

Python Output2:

The contigency table of 'internetuserate_tile' and 'polityscore_grp' 

polityscore_grp       No_democracy  Unsatisfied_democracy  Satisfied_democracy  Good_democracy  

internetuserate_tile                                        

userate=50%tile                 12                     21                             17                         35  

userate=100%tile                11                      4                               2                         53  

The contigency table in percentage 

polityscore_grp       No_democracy  Unsatisfied_democracy  Satisfied_democracy  Good_democracy  

internetuserate_tile                                        

userate=50%tile           0.521739                   0.84                                0.894737        0.397727  

userate=100%tile          0.478261                   0.16                               0.105263        0.602273  

The Chi-Square test

(25.918522100123596, 9.9194759541075989e-06, 3, array([[ 12.61290323,  13.70967742,  10.41935484,  48.25806452],

       [ 10.38709677,  11.29032258,   8.58064516,  39.74193548]]))

-----

From the output, we can see that the p-value of the Chi-Square test is 9.9194759541075989e-06 which is lower than 0.05 significant level and reject the null hypothesis that democracy level and internet use rate has no dependence association (i.e.they have dependence association)

Python Output3:

.

-----

The charts shows that there is not a simple linear relationship between the two variables although they are dependent.

Now the followings are the post hoc tests for the Chi-Square test:

Python Output4:

Post hoc test for the Chi-Sqaure test

1,
NO_democracy vs Unsatisfied_democracy
NoVSUn                No_democracy  Unsatisfied_democracy
internetuserate_tile                                     
userate=50%tile                 12                     21
userate=100%tile                11                      4
(4.2634624505928862, 0.038940474422515331, 1, array([[ 15.8125,  17.1875],

       [  7.1875,   7.8125]]))


2,
No_democracy vs Satisfied_democracy
No_VS_Sat             No_democracy  Unsatisfied_democracy
internetuserate_tile                                     
userate=50%tile                 12                     21
userate=100%tile                11                      4
(4.2634624505928862, 0.038940474422515331, 1, array([[ 15.8125,  17.1875],
       [  7.1875,   7.8125]]))
C:\Users\user\Anaconda3\lib\site-packages\matplotlib\__init__.py:892: UserWarning: axes.color_cycle is deprecated and replaced with axes.prop_cycle; please use the latter.

  warnings.warn(self.msg_depr % (key, alt_key))

3,
No_democracy vs Good_democracy
No_VS_Go              Good_democracy_democracy  No_democracy
internetuserate_tile                                        
userate=50%tile                             35            12
userate=100%tile                            53            11
(0.69683212191731103, 0.4038501988882941, 1, array([[ 37.26126126,   9.73873874],

       [ 50.73873874,  13.26126126]]))

4,
Satisfied_democracy vs Unsatisfied_democracy
Sat_VS_Un             Satisfied_democracy  Unsatisfied_democracy
internetuserate_tile                                            
userate=50%tile                        17                     21
userate=100%tile                        2                      4
(0.0065004616805170515, 0.93573983334993682, 1, array([[ 16.40909091,  21.59090909],

       [  2.59090909,   3.40909091]]))

5,
Unsatisfied_democracy vs Good_democracy
Un_VS_Go              Good_democracy  Unsatisfied_democracy
internetuserate_tile                                       
userate=50%tile                   35                     21
userate=100%tile                  53                      4
(13.516299876110732, 0.00023650025871216763, 1, array([[ 43.61061947,  12.38938053],
       [ 44.38938053,  12.61061947]]))

6,
Satisfied_democracy vs Good_democracy
Sat_VS_Go             Good_democracy  Satisfied_democracy
internetuserate_tile                                     
userate=50%tile                   35                   17
userate=100%tile                  53                    2
(13.526401267691638, 0.00023523068534581418, 1, array([[ 42.76635514,   9.23364486],

       [ 45.23364486,   9.76635514]]))
-----
The post hoc test applies with Bonferroni Adjustnent. The significant level of rejecting null hypothesis is 0.05/(number of caparisons) =0.05/6=0.0083. From the output we can conclude that the 5th and 6th comparison reject the hypothesis and it means only can deduce the dependence association between democracy and internet use rate excluding the democracy level is "No_democracy".

Conclusion

The democracy level of a country and its internet use rate are dependent but not in a linear relationship. The post hoc test doesn't give the concrete and meaningful explanation or conclusion for the dependence association.A further research may be required 

Python Code
import pandas
import numpy
import scipy.stats as ss
import seaborn
import matplotlib.pyplot as p

gapminder = pandas.read_csv('gapminder.csv',low_memory=False)

print('number of rows and columns of gapminder')
print(len(gapminder))
print(len(gapminder.columns))


gapminder['internetuserate']=gapminder['internetuserate'].convert_objects(convert_numeric=True)
gapminder['polityscore']=gapminder['polityscore'].convert_objects(convert_numeric=True)


gapminder['polityscore_grp']=pandas.cut(gapminder.polityscore,4,labels=["No_democracy","Unsatisfied_democracy",
"Satisfied_democracy","Good_democracy"])

gapminder['internetuserate_tile']=pandas.qcut(gapminder.internetuserate,2,labels=["userate=50%tile",
"userate=100%tile"])


print('a dataframe combining 2 chosen variables')
sub1=gapminder[['internetuserate_tile','polityscore_grp']].dropna()
print(sub1)

print("The contigency table of 'internetuserate_tile' and 'polityscore_grp' ")
ct1=pandas.crosstab(sub1['internetuserate_tile'],sub1['polityscore_grp'])
print(ct1)

print("The contigency table in percentage ")
colsum=ct1.sum(axis=0)
colpct=ct1/colsum
print(colpct)


print("The Chi-Square test")
cs1=ss.chi2_contingency(ct1)
print(cs1)

sub1['internetuserate']=gapminder['internetuserate'].dropna()

sub1['internetuserate_tile']=sub1['internetuserate_tile'].convert_objects(convert_numeric=True)
sub1['polityscore_grp'].astype('category')

print("Bar chart of frequencies of 'internetuserate_tile'")
seaborn.factorplot(x='polityscore_grp',y='internetuserate',data=sub1,kind='bar',ci=None)
p.xlabel('polityscore_grp')
p.ylabel('internetuserate_tile')
print()

print("Post hoc test for the Chi-Sqaure test")
print("NO_democracy vs Unsatisfied_democracy")
recode1={"No_democracy":"No_democracy","Unsatisfied_democracy":"Unsatisfied_democracy"}
sub1["No_VS_Un"]=sub1['polityscore_grp'].map(recode1)
cc1=pandas.crosstab(sub1['internetuserate_tile'],sub1["No_VS_Un"])
print(cc1)
ccs1=ss.chi2_contingency(cc1)
print(ccs1)

print()
print("No_democracy vs Satisfied_democracy")
recode2={"No_democracy":"No_democracy","Satisfied_democracy":"Satisfied_democracy"}
sub1["No_VS_Sat"]=sub1['polityscore_grp'].map(recode1)
cc2=pandas.crosstab(sub1['internetuserate_tile'],sub1["No_VS_Sat"])
print(cc2)
ccs2=ss.chi2_contingency(cc2)
print(ccs2)


print()
print("No_democracy vs Good_democracy")
recode3={"No_democracy":"No_democracy","Good_democracy":"Good_democracy_democracy"}
sub1["No_VS_Go"]=sub1['polityscore_grp'].map(recode3)
cc3=pandas.crosstab(sub1['internetuserate_tile'],sub1["No_VS_Go"])
print(cc3)
ccs3=ss.chi2_contingency(cc3)
print(ccs3)

print()
print("Satisfied_democracy vs Unsatisfied_democracy")
recode4={"Satisfied_democracy":"Satisfied_democracy","Unsatisfied_democracy":"Unsatisfied_democracy"}
sub1["Sat_VS_Un"]=sub1['polityscore_grp'].map(recode4)
cc4=pandas.crosstab(sub1['internetuserate_tile'],sub1["Sat_VS_Un"])
print(cc4)
ccs4=ss.chi2_contingency(cc4)
print(ccs4)

print()
print("Unsatisfied_democracy vs Good_democracy")
recode5={"Unsatisfied_democracy":"Unsatisfied_democracy","Good_democracy":"Good_democracy"}
sub1["Un_VS_Go"]=sub1['polityscore_grp'].map(recode5)
cc5=pandas.crosstab(sub1['internetuserate_tile'],sub1["Un_VS_Go"])
print(cc5)
ccs5=ss.chi2_contingency(cc5)
print(ccs5)


print()
print("Satisfied_democracy vs Good_democracy")
recode6={"Satisfied_democracy":"Satisfied_democracy","Good_democracy":"Good_democracy"}
sub1["Sat_VS_Go"]=sub1['polityscore_grp'].map(recode6)
cc6=pandas.crosstab(sub1['internetuserate_tile'],sub1["Sat_VS_Go"])
print(cc6)
ccs6=ss.chi2_contingency(cc6)
print(ccs6)

Running an analysis of variance (Python)

Objective of the study:
Investigate the correlation between a country's democracy level and its internet use rate.

Explanation and Methodologies:

From the "Gapminder" dataset, two variables are extracted. They are "polityscore" and "internetuserate". The observations of the dataset are recorded among the countries around the world. The variable "politysocre" is measuring a country's democracy extent and the range of the score is -10 to 10. The key word "democracy level" will be used instead of "polity score" in the following explanation  for better understanding of the meaning of the variable.

First, "polityscore" is divided by four groups (4 levels) according to its value as follows:

polityscore	group(level)
-10 to -5	No_democracy
-5 to 0	Unstatisfied_democracy
0 to 5	Satisfied_democracy
5 to 10	Good_democracy

The "polityscore" will be an explanatory varibale and "internetuserate" will be a response variable in my study.The ANOVA analysis of four levels will  proceed.

The null hypothesis: The internet use rate has no mean difference among different levels of democracy.(i.e.  internet use rate's mean1=mean2=mean3=mean4)

Python Output 1:

 number of rows and columns of gapminder

213

16

a dataframe combining 4 chosen variables

     internetuserate        polityscore_grp

0           3.654122  Unsatisfied_democracy

1          44.989947         Good_democracy

2          12.500073    Satisfied_democracy

4           9.999954  Unsatisfied_democracy

6          36.000335         Good_democracy

7          44.001025    Satisfied_democracy

9          75.895654         Good_democracy

10         72.731576         Good_democracy

11         46.679702           No_democracy

13         54.992809           No_democracy

14          3.700003    Satisfied_democracy

16         32.052144           No_democracy

17         73.733934         Good_democracy

19          3.129962         Good_democracy

21         13.598876    Satisfied_democracy

22         20.001710         Good_democracy

24          5.999836         Good_democracy

25         40.650098         Good_democracy

27         45.986590         Good_democracy

28          1.400061  Unsatisfied_democracy

29          2.100213         Good_democracy

30          1.259934    Satisfied_democracy

31          3.999977  Unsatisfied_democracy

32         81.338393         Good_democracy

35          2.300027  Unsatisfied_democracy

36          1.700031  Unsatisfied_democracy

37         45.000000         Good_democracy

38         34.377790           No_democracy

39         36.499875         Good_democracy

40          5.098265         Good_democracy

..               ...                    ...

175        69.339971         Good_democracy

176         5.001375         Good_democracy

178        12.334893         Good_democracy

179        65.808554         Good_democracy

180        11.999971         Good_democracy

183         9.007736           No_democracy

184        90.016190         Good_democracy

185        82.166660         Good_democracy

186        20.663156           No_democracy

188        11.549391  Unsatisfied_democracy

189        11.000055  Unsatisfied_democracy

190        21.200072    Satisfied_democracy

191         0.210066         Good_democracy

192         5.379820  Unsatisfied_democracy

194        48.516818         Good_democracy

195        36.562553  Unsatisfied_democracy

196        39.820178         Good_democracy

197         2.199998           No_democracy

199        12.500255  Unsatisfied_democracy

200        44.585355         Good_democracy

201        77.996781           No_democracy

202        84.731705         Good_democracy

203        74.247572         Good_democracy

204        47.867469         Good_democracy

205        19.445021           No_democracy

207        35.850437  Unsatisfied_democracy

208        27.851822           No_democracy

210        12.349750  Unsatisfied_democracy

211        10.124986         Good_democracy

212        11.500415    Satisfied_democracy

This is the dataframe containing the two variables those are going to be analyzed.

Python Output 2:

-----

We can see the p-value of the F-statistics is  8.74e-08  lower than 0.05. The value 0.05 is the significant level to reject the null hypothesis statistically. Therefore, the null hypothesis of no mean difference of internet use rate among the democracy level groups is rejected.

Now we are going take a deeper look since the previous result does not tell us which groups are different from others. This means we don't know which pair of groups(democracy levels) have mean difference in internet use rate statistically. Therefore post hoc test for the ANOVA is going to be carried out. The next test is Tukey's Honestly Significant Different Test.

Python Out 3:

The values of the means and standard deviations of the internet use rate among the democracy level groups are given in the output for reference. It has a place drawing my attention that the mean value of "No_democracy" group has an unexpected high value since it is the lowest democracy level and expected that the mean difference between this group and "Good_democracy" should be largest among four groups according to my expectation. 

According the the Tukey's test, There is statistically mean difference of internet use rate between:

• "Good_democracy" and "Satisfied_democracy"
• "Good_democracy" and "Unsatisfied_democracy"

Conclusion

According to the ANOVA, democracy levels have significant effect on internet use rate but there is no verification on the causal relationship between the two variables in my study.

In fact, I have an expectation that the internet use rate positively relates to the democracy levels (one has a higher value corresponding to the increasing value or level of another variable.). However,Tukey's Honestly Significant Different Test gives an unexpected result. Therefore, there is not a simple positively linear relationship between internet use rate and democracy level and a further analysis may be required.

Python Code
import pandas
import numpy
import statsmodels.formula.api as smf
import statsmodels.stats.multicomp as multi

gapminder = pandas.read_csv('gapminder.csv',low_memory=False)

print('number of rows and columns of gapminder')
print(len(gapminder))
print(len(gapminder.columns))


gapminder['internetuserate']=gapminder['internetuserate'].convert_objects(convert_numeric=True)
gapminder['polityscore']=gapminder['polityscore'].convert_objects(convert_numeric=True)


gapminder['polityscore_grp']=pandas.cut(gapminder.polityscore,[-11,-5,0,5,10],labels=["No_democracy","Unsatisfied_democracy","Satisfied_democracy","Good_democracy"])



print('a dataframe combining 4 chosen variables')
sub1=gapminder[['internetuserate','polityscore_grp']].dropna()
print(sub1)


model1=smf.ols(formula='internetuserate~C(polityscore_grp)',data=sub1).fit()
print(model1.summary())

print('means of internetuserate among the four groups')
m2=sub1.groupby('polityscore_grp').mean()
print(m2)
print('standard deviatons of internetuserate among the four groups')
sd2=sub1.groupby('polityscore_grp').std()
print(sd2)

print()
print("Tukey's Honestly Significant Different Test")
model2=multi.MultiComparison(sub1['internetuserate'],sub1['polityscore_grp']).tukeyhsd()

print(model2.summary())