Chi-squared Test Overview

Introduction

• Chi-squared test: A statistical hypothesis test used to analyze contingency tables with large sample sizes.
• Examines the independence of two categorical variables by analyzing test statistic values.

Types of Chi-squared Tests

• Pearson's chi-squared test: Determines statistical significance between expected and observed frequencies in contingency tables.
• Fisher's exact test: Used for contingency tables with small sample sizes.

Applications

• Observations are classified into mutually exclusive classes.
• Under the null hypothesis of no differences, the test statistic follows a chi-squared distribution.
• Used to test independence of random variables.

Mathematical History

• Developed in the late 19th and early 20th century by Karl Pearson.
• Pearson introduced the Pearson distribution to model various data distributions, challenging normal distribution assumptions.

Pearson's Chi-squared Test

• Published by Karl Pearson in 1900, it forms the basis of modern statistics.
• Involves classifying n observations into k classes with expected numbers mi = npi.
• Test statistic (X²) follows a chi-squared distribution with k - 1 degrees of freedom in large samples.

Additional Chi-squared Tests

• Test for variance: Tests if the variance of a normally distributed population has a specific value.
• Other Tests:
  • Cochran-Mantel-Haenszel test
  • McNemar's test
  • Tukey's test of additivity
  • Portmanteau test in time-series
  • Likelihood-ratio tests

Yates's Correction for Continuity

• Suggested by Frank Yates to reduce error in chi-squared approximations.
• Adjusts Pearson’s chi-squared test by subtracting 0.5 from the observed vs. expected value difference in 2x2 tables.

Chi-squared Test for Variance in a Normal Population

• Applicable when sample from a normal distribution tests if population variance holds a pre-determined value.

Applied Example

• Example given of city residents and occupational categories to test the independence of variables.
• The test statistic is calculated, with rejection of the null hypothesis if statistic is improbably large.

Uses in Various Fields

• Cryptanalysis: Used to compare plaintext and ciphertext distributions.
• Bioinformatics: Compares properties of genes across different categories.