Chi-squared test in Data Analysis Python - Time & Space Complexity

Time Complexity: Chi-squared test

O(n x m)

Understanding Time Complexity

We want to understand how the time it takes to run a Chi-squared test changes as the size of the data grows.

Specifically, how does the number of operations increase when we have more categories or observations?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy.stats import chi2_contingency

# Create a contingency table with n rows and m columns
n, m = 100, 5
data = np.random.randint(1, 10, size=(n, m))

# Perform Chi-squared test
chi2, p, dof, expected = chi2_contingency(data)

This code creates a table of counts and runs the Chi-squared test to check if rows and columns are independent.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Calculating expected counts and summing squared differences for each cell in the table.
• How many times: Once for each cell in the contingency table, so n x m times.

How Execution Grows With Input

As the number of rows or columns increases, the number of cells grows, so the work grows with the total number of cells.

Input Size (n x m)	Approx. Operations
10 x 5 = 50	About 50 operations
100 x 5 = 500	About 500 operations
1000 x 5 = 5000	About 5000 operations

Pattern observation: The operations grow roughly in direct proportion to the number of cells in the table.

Final Time Complexity

Time Complexity: O(n x m)

This means the time to run the test grows linearly with the number of rows times columns in the data table.

Common Mistake

[X] Wrong: "The Chi-squared test runs in constant time no matter the data size."

[OK] Correct: The test must look at every cell to calculate expected counts and differences, so more data means more work.

Interview Connect

Understanding how the Chi-squared test scales helps you explain performance when working with larger datasets in real projects.

Self-Check

"What if we changed the data to a 3D contingency table? How would the time complexity change?"

Practice

1. What does the Chi-squared test primarily check in data analysis?

easy

A. If two numerical variables have a linear relationship

B. If two categorical variables are related or independent

C. If a dataset follows a normal distribution

D. If the mean of a dataset is greater than zero

Solution

1. Step 1: Understand the purpose of the Chi-squared test

   The Chi-squared test is used to check relationships between categorical variables, not numerical data.

2. Step 2: Identify the correct relationship tested

   It tests if two categories are independent or related based on observed counts.

3. Final Answer:

   If two categorical variables are related or independent -> Option B

4. Quick Check:

   Chi-squared tests category relationships [OK]

Hint: Chi-squared tests category links, not means or correlations [OK]

Common Mistakes:

• Confusing Chi-squared with correlation tests
• Thinking it tests numerical data relationships
• Assuming it checks data distribution shape

2. Which Python library and function is commonly used to perform a Chi-squared test on a contingency table?

easy

A. scipy.stats.chi2_contingency

B. numpy.chisquare_test

C. pandas.chi_squared

D. matplotlib.chi2_test

Solution

1. Step 1: Recall common Python libraries for statistics

   Scipy is the standard library for statistical tests in Python.

2. Step 2: Identify the correct function for Chi-squared test

   The function to perform Chi-squared test on contingency tables is chi2_contingency in scipy.stats.

3. Final Answer:

   scipy.stats.chi2_contingency -> Option A

4. Quick Check:

   Use scipy.stats.chi2_contingency for Chi-squared [OK]

Hint: Remember: scipy.stats has chi2_contingency for Chi-squared [OK]

Common Mistakes:

• Using numpy or pandas which don't have this function
• Confusing function names or libraries
• Trying to use plotting libraries for tests

3. What will be the output of this Python code snippet?

import numpy as np
from scipy.stats import chi2_contingency

observed = np.array([[10, 20], [20, 40]])
chi2, p, dof, expected = chi2_contingency(observed, correction=False)
print('{:.3f}'.format(p))

medium

A. 1.000

B. 0.000

C. 0.500

D. 0.250

Solution

1. Step 1: Understand the observed data and test

   The observed table has proportional counts (10:20 and 20:40), suggesting independence.

2. Step 2: Interpret the p-value from chi2_contingency

   The p-value is approximately 0.5, indicating no significant relationship; the test finds data consistent with independence.

3. Final Answer:

   0.500 -> Option C

4. Quick Check:

   Proportional counts give moderate p-value around 0.5 [OK]

Hint: Proportional tables yield moderate p-values near 0.5 [OK]

Common Mistakes:

• Assuming low p-value means independence
• Confusing chi2 statistic with p-value
• Rounding errors changing p-value interpretation

4. Identify the error in this code that tries to perform a Chi-squared test:

import scipy.stats as stats
observed = [[15, 25], [35, 25]]
chi2, p = stats.chi2_contingency(observed)
print(p)

medium

A. Incorrect unpacking of chi2_contingency output

B. Missing import of numpy for array input

C. Observed data must be a pandas DataFrame

D. No error; code runs correctly

Solution

1. Step 1: Check the output of chi2_contingency

   The function returns four values: chi2, p, dof, expected.

2. Step 2: Verify unpacking matches returned values

   The code unpacks only two values, causing a ValueError due to mismatch.

3. Final Answer:

   Incorrect unpacking of chi2_contingency output -> Option A

4. Quick Check:

   chi2_contingency returns 4 values, unpack all [OK]

Hint: Unpack all 4 outputs from chi2_contingency [OK]

Common Mistakes:

• Unpacking fewer than 4 values
• Assuming input must be numpy array
• Thinking pandas DataFrame is required

5. You have a 3x3 contingency table of observed counts for three categories of two variables. After running chi2_contingency, you get a p-value of 0.03. What is the best interpretation if your significance level is 0.05?

hard

A. The variables are independent

B. You need a larger sample size to decide

C. The test is inconclusive; p-value is too high

D. There is a significant relationship between the variables

Solution

1. Step 1: Compare p-value with significance level

   The p-value 0.03 is less than 0.05, indicating statistical significance.

2. Step 2: Interpret significance in context of independence

   A significant result means we reject the null hypothesis of independence; variables are related.

3. Final Answer:

   There is a significant relationship between the variables -> Option D

4. Quick Check:

   p-value < 0.05 means reject independence [OK]

Hint: p-value < alpha means variables related, not independent [OK]

Common Mistakes:

• Thinking p-value < 0.05 means independence
• Ignoring significance level in decision
• Assuming inconclusive without checking p-value