# Chi-square (χ2) test in Python (Pearson Chi-square test)

## Chi-square (χ2) test for independence (Pearson Chi-square test)

- Chi-square test is a non-parametric (distribution-free) method used to compare the relationship between the two categorical (nominal) variables in a contingency table.
- For example, we have different treatments (treated and nontreated) and treatment outcomes (cured and noncured), here we could use the chi-square test for independence to check whether treatments are related to treatment outcomes.
- Chi-square test relies on approximation (gives approximate
*p*value) and hence require larger sample size. The expected frequency count should not be < 5 for more than 20% of cells. If the sample size is small, the chi-square test is not accurate, and you should use Fisher’s exact test. - Note: Chi-square test for independence is different than the chi-square goodness of fit test

### Formula

### Hypotheses for Chi-square test for independence

*Null hypothesis*: The two categorical variables are independent (no association between the two variables) ( H_{0}: O_{i}= E_{i})*Alternative hypothesis*: The two categorical variables are dependent (there is an association between the two variables) ( H_{a}: O_{i}≠ E_{i})- Note: There are no
one
or two-tailed
*p*value. Rejection region of the chi-square test is always on the right side of the distribution.

Learn more about hypothesis testing and interpretation

### Chi-square test assumptions

- The two variables are categorical (nominal) and data is randomly sampled
- The levels of variables are mutually exclusive
- The expected frequency count for at least 80% of the cell in a contingency table is at least 5. Fisher’s exact test is appropriate for small frequency counts.
- The expected frequency count should not be less than 1
- Observations should be independent of each other
- Observation data should be frequency counts and not percentages, proportions or transformed data

### Calculate a chi-square test for independence in Python

- We will use
`bioinfokit`

v0.9.5 or later and scipy python packages - Check bioinfokit documentation for installation and documentation
- Download a hypothetical dataset for chi-square test for independence

Note: If you have your own dataset, you should import it as pandas dataframe. Learn how to import data using pandas

chi-square test for independence using bioinfokit,

```
from bioinfokit.analys import stat, get_data
# load example dataset
df = get_data('drugdata').data
df.head()
# output
treatments cured noncured
0 treated 60 10
1 nontreated 30 25
# set treatments column as index
df = df.set_index('treatments')
# output
df.head()
cured noncured
treatments
treated 60 10
nontreated 30 25
# run chi-square test for independence
res = stat()
res.chisq(df=df)
# output
print(res.summary)
# corrected for the Yates’ continuity
Chi-squared test for independence
Test Df Chi-square P-value
-------------- ---- ------------ -----------
Pearson 1 13.3365 0.000260291
Log-likelihood 1 13.4687 0.000242574
print(res.expected_df)
Expected frequency counts
cured noncured
-- ------- ----------
0 50.4 19.6
1 39.6 15.4
```

chi-square test for independence using `chi2_contingency`

function from scipy package,

```
import numpy as np
from scipy.stats import chi2_contingency
# using Pearson’s chi-squared statistic
# corrected for the Yates’ continuity
observed = np.array([[60, 10], [30, 25]])
chi_val, p_val, dof, expected = chi2_contingency(observed)
chi_val, p_val, dof, expected
# output
(13.3364898989899, 0.0002602911116400899, 1, array([[50.4, 19.6],
[39.6, 15.4]]))
# without Yates’ correction for continuity
chi_val, p_val, dof, expected = chi2_contingency(observed, correction=False)
chi_val, p_val, dof, expected
# output
(14.842300556586274, 0.00011688424010613195, 1, array([[50.4, 19.6],
[39.6, 15.4]]))
# for log-likelihood method run command as below
chi_val, p_val, dof, expected = chi2_contingency(observed, lambda_="log-likelihood")
```

### Yates’ correction for continuity

- In the χ2 test, the discrete probabilities of observed counts can be approximated by the continuous chi-squared probability distribution. This can cause errors and needs to be corrected using continuity correction.
- Yates’ correction for continuity modifies the 2x2 contingency table and adjust the difference of observed and expected counts by subtracting the value of 0.5 (see formula).
- Yates’ correction for continuity increases the
*p*value by reducing the χ2 value. The corrected*p*value is close to exact tests such as the Fisher exact test. Sometimes, Yates’ correction may give an overcorrected*p*value. - χ2 and Yates’ corrected χ2 produce similar results on large samples, but Yates’ corrected χ2 can be conservative on
smaller samples and gives a higher
*p*value.

### Interpretation

The *p* value obtained from chi-square test for independence is significant (*p* < 0.05), and therefore, we
conclude that there is a significant association between treatments (treated and nontreated) with treatment outcome
(cured and noncured)

## Related reading

### References

- Virtanen P, Gommers R, Oliphant TE, Haberland M, Reddy T, Cournapeau D, Burovski E, Peterson P, Weckesser W, Bright J, van der Walt SJ. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature methods. 2020 Mar;17(3):261-72.
- Bewick V, Cheek L, Ball J. Statistics review 8: Qualitative data–tests of association. Critical care. 2003 Feb 1;8(1):46.
- Serra N, Rea T, Di Carlo P, Sergi C. Continuity correction of Pearson’s chi-square test in 2x2 Contingency Tables: A mini-review on recent development. Epidemiology, Biostatistics and Public Health. 2019 Jun 21;16(2).

If you have any questions, comments or recommendations, please email me at
**reneshbe@gmail.com**

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