In Depth Notes on Chi-Square Tests

Module 5: Chi-Square Tests

Comparing Counts
  • Covers 2 types of tests:
    • (a) Goodness-of-fit tests for experiments with more than 2 categories.
    • (b) Homogeneity and independence tests for contingency tables.
Goodness-of-Fit Test (Univariate Chi-Squared Test)
  • Purpose: Checks if the frequency distribution of a categorical variable from a sample matches the model's expected distribution.
  • Key Focus: Measures how well our data fits a predetermined model.
Hypothesis & Assumption (Goodness-of-Fit Test)
  • Distribution: $p1, p2, …, p_k$.
  • Hypotheses:
    • $H0: p1 = p{01}, p2 = p{02}, …, pk = p_{0k}$
    • $Ha: H0$ is not true.
  • Assumptions and Conditions:
    • (a) Counted Data Condition: Data must be counts for the categories of a categorical variable.
    • (b) Independence Assumption: Counts in cells must be independent.
    • (i) Randomization Condition: The sample must be a random selection from the population.
    • (c) Sample Size Assumption:
    • (i) Expected Cell Frequency Condition: Expect at least 5 counts in each cell.
Test Statistic (Goodness-of-Fit Test)
  • Chi-square statistic ($ ext{χ}^2$):
    extχ2<em>0=extsumofthesquaresofobserved−expectedcountsextexpectedcounts=extΣ</em>extallcategories(O−E)2Eext{χ}^2<em>0 = \frac{ ext{sum of the squares of observed - expected counts}}{ ext{expected counts}} = ext{Σ}</em>{ ext{all categories}} \frac{(O - E)^2}{E}
  • Note on Interpretation:
    • If $O ext{ ≈ } E$, then $ ext{χ}^20$ is small, do not reject $H0$.
    • If $O ext{ is large or significantly deviates from } E$, reject $H_0$.
Idea Behind the Test Statistic
  • The chi-square statistic is intended for hypothesis testing, not for confidence intervals.
  • Large statistics indicate the observed counts diverge significantly from the expected.
  • p-value:
    • pext−value=P(extχ2>extχ02)p ext{-value} = P( ext{χ}^2 > ext{χ}^2_0) is the upper tail area for a distribution with degrees of freedom $c-1$.
  • Decision Rule:
    • If $p ext{-value} ext{ ≤ } ext{α}$, reject $H0$; if $p ext{-value} ext{ > } ext{α}$, do not reject $H0$.
Distribution of $ ext{χ}^2$
  • The chi-square value follows a distribution characterized by degrees of freedom (df).
  • For goodness-of-fit test, degrees of freedom are $c - 1$, where $c$ is the number of categories.
Example of Goodness-of-Fit Test
  • Scenario: Proportion of beans in groups A, B, C, D should be $9:3:3:1$ based on a sample of 1600 beans: 882, 313, 287, and 118.
  • Calculations:
    • Null Hypothesis: Theory fits experiment well.
    • Expected Counts:
    • A: $ rac{9}{16} imes 1600 = 900$; B: $ rac{3}{16} imes 1600 = 300$; C: $ rac{3}{16} imes 1600 = 300$; D: $ rac{1}{16} imes 1600 = 100$.
    • Calculate $ ext{χ}^2_0$:
      =(882−900)2900+(313−300)2300+(287−300)2300+(118−100)2100=4.73= \frac{(882-900)^2}{900} + \frac{(313-300)^2}{300} + \frac{(287-300)^2}{300} + \frac{(118-100)^2}{100} = 4.73
  • Conclusion: $p ext{-value} > 0.05$, do not reject $H_0$.
Test of Homogeneity
  • Compares distributions across two or more groups on the same categorical variable.
  • Essentially generalizes the two-proportion z-test.
  • Conditions:
    • Similar assumptions as goodness-of-fit test: counted data condition, randomization condition, expected cell frequency condition.
Test Statistic and p-value for Homogeneity
  • Calculate the $ ext{χ}^2_0$ as in goodness-of-fit test.
  • Degrees of freedom: $(r-1)(c-1)$, where $r$ is the number of rows, $c$ is the number of columns.
Example of Homogeneity Test
  • Scenario: Study on children's TV program preferences among 200 first graders (80 boys, 120 girls).
  • Hypotheses:
    • Null Hypothesis: Distribution of preferences is the same across boys and girls.
    • Alternative Hypothesis: At least one preference distribution differs.
  • Expected Counts Calculation: Based on row and column totals.
  • Results: $p ext{-value} > 0.05$, do not reject $H_0$.
Chi-Square Test for Independence
  • Evaluates the association between two categorical variables.
  • Uses same calculations as homogeneity test—interpretation differs.
  • Implies independence if the counts of one variable do not depend on the other.
Assumptions for Independence Test
  • Need counts in each cell, expect counts to be at least 5.
  • Sample must be a representative random sample.
Example: Independence Test
  • Scenario: Study on hand vs foot lengths; right-handed vs left-handed participants.
  • Set up null and alternative hypotheses regarding independence.
  • Calculate $ ext{χ}^2_0$ and derive $p ext{-value}$.
Conclusion for Independence Test
  • If $p ext{-value} < 0.05$, reject $H_0$, suggesting a relationship exists between the variables.