Chi-squared test in R Programming - Time & Space Complexity

Time Complexity: Chi-squared test

O(n^2)

Understanding Time Complexity

When running a chi-squared test in R, it is helpful to know how the time it takes grows as the data size increases.

We want to understand how the test's execution time changes when we have more data.

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

# Create a contingency table
observed <- matrix(c(30, 10, 20, 40), nrow=2)

# Perform chi-squared test
result <- chisq.test(observed)
print(result)
    

This code creates a simple table of counts and runs a chi-squared test to check for independence.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Summing and comparing counts across the contingency table cells.
• How many times: The operations repeat once for each cell in the table.

How Execution Grows With Input

As the table size grows, the number of cells increases, so the test does more work.

Input Size (n x n table)	Approx. Operations
2 x 2	4
10 x 10	100
100 x 100	10,000

Pattern observation: The work grows roughly with the number of cells, which is the square of the table dimension.

Final Time Complexity

Time Complexity: O(n^2)

This means the time to run the test grows roughly with the square of the table size.

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 in the table, so more data means more work.

Interview Connect

Understanding how statistical tests scale helps you write efficient code and explain performance clearly.

Self-Check

"What if we changed the table from square (n x n) to rectangular (n x m)? How would the time complexity change?"

Practice

1. What does the Chi-squared test in R primarily check for?

easy

A. Whether two categorical variables are related

B. Whether two numerical variables have the same mean

C. Whether a variable follows a normal distribution

D. Whether the data contains missing values

Solution

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

   The Chi-squared test is used to check if there is an association between two categorical variables.

2. Step 2: Compare with other options

   Options A and B relate to numerical data tests, and D is about data quality, not relationships.

3. Final Answer:

   Whether two categorical variables are related -> Option A

4. Quick Check:

   Chi-squared test checks categorical variable relationship = C [OK]

Hint: Chi-squared tests relationships between categories, not numbers [OK]

Common Mistakes:

• Confusing Chi-squared with t-test or ANOVA
• Thinking it tests numerical data means
• Assuming it checks data normality

2. Which of the following is the correct way to perform a Chi-squared test on a table named tbl in R?

easy

A. chisq(tbl)

B. chisq.test(tbl)

C. chi.test(tbl)

D. test.chisq(tbl)

Solution

1. Step 1: Recall the correct function name in R

   The function to perform a Chi-squared test in R is chisq.test().

2. Step 2: Check the syntax

   The correct syntax is calling chisq.test() with the table as the argument.

3. Final Answer:

   chisq.test(tbl) -> Option B

4. Quick Check:

   Correct function is chisq.test() = B [OK]

Hint: Use chisq.test() function for Chi-squared test in R [OK]

Common Mistakes:

• Using incorrect function names like chisq() or chi.test()
• Placing the function name after the table
• Forgetting parentheses after function name

3. What will be the output of the following R code?

tbl <- matrix(c(10, 20, 30, 40), nrow=2)
result <- chisq.test(tbl)
round(result$p.value, 3)

medium

A. 0.504

B. 0.025

C. 0.000

D. Error: object 'result' not found

Solution

1. Step 1: Understand the input table

   The matrix has counts: 10, 20 in first row and 30, 40 in second row.

2. Step 2: Perform Chi-squared test and check p-value

   Running chisq.test on this table gives a p-value around 0.504, meaning no strong association.

3. Final Answer:

   0.504 -> Option A

4. Quick Check:

   Chi-squared p-value rounded = 0.504 [OK]

Hint: Check p-value from chisq.test() to judge significance [OK]

Common Mistakes:

• Confusing p-value with test statistic
• Expecting zero p-value for small counts
• Mistaking error message for output

4. Identify the error in this R code snippet for Chi-squared test:

data <- matrix(c(5, 15, 20, 10), ncol=2)
chisq.test(data)

medium

A. Matrix should have rows and columns swapped

B. chisq.test() requires a data frame, not a matrix

C. No error, code runs correctly

D. Matrix must be named 'tbl' to work

Solution

1. Step 1: Check matrix creation

   The matrix is created with 4 values and 2 columns, which is valid.

2. Step 2: Check chisq.test() input type

   chisq.test() accepts a matrix of counts, so this input is valid and runs without error.

3. Final Answer:

   No error, code runs correctly -> Option C

4. Quick Check:

   chisq.test() accepts matrix input = A [OK]

Hint: chisq.test() works with matrices or tables of counts [OK]

Common Mistakes:

• Thinking chisq.test() only accepts data frames
• Assuming matrix dimensions must be square
• Believing variable name affects function

5. You have survey data of 3 groups choosing among 3 products. How do you apply a Chi-squared test in R to check if product preference depends on group?

group <- c('A', 'A', 'B', 'B', 'C', 'C')
product <- c('X', 'Y', 'X', 'Z', 'Y', 'Z')
# What next?

hard

A. Convert group and product to numeric and run chisq.test()

B. Run chisq.test(group, product) directly without table

C. Use t.test() on group and product vectors

D. Create a table with table(group, product) then run chisq.test() on it

Solution

1. Step 1: Create a contingency table

   Use table(group, product) to get counts of each group-product combination.

2. Step 2: Apply Chi-squared test on the table

   Run chisq.test() on the table to test independence between group and product preference.

3. Final Answer:

   Create a table with table(group, product) then run chisq.test() on it -> Option D

4. Quick Check:

   Table then chisq.test() = A [OK]

Hint: Always make a table before chisq.test() for categorical data [OK]

Common Mistakes:

• Trying to run chisq.test() on raw vectors
• Using t.test() instead of chisq.test() for categories
• Converting categories to numbers incorrectly