11. Chi-Square Statistics

Chi-Square Statistics

Overview

  • Prepared by: Jhunar John M. Tauy, MAPsy, RPsy, RPM

Learning Objectives

  • At the end of this topic, students should be able to:

    • Learn the basic logic behind hypothesis testing using chi-square statistic with nominal data.

    • Show competency in calculating, analyzing, and interpreting results from a chi-square statistic.

    • Recognize when to use and when not to use chi-square statistic.

Parametric and Nonparametric Tests

  • Parametric Test

    • A statistical test that makes assumptions about the population parameters and the distributions of the data (e.g., normality and homogeneity of variance).

  • Nonparametric Test

    • Also known as distribution-free tests or rank-order tests.

    • Statistical tests that do not assume anything about population parameters.

Chi-Square Tests

  • Types of Chi-Square Tests

    • Goodness of Fit Test

    • Test for Independence

  • Chi-Square Statistic

    • Hypothesis testing procedure for nominal variables.

Chi-Square Goodness of Fit Test

  • Utilizes sample data to test hypotheses about population proportions.

  • Determines how well the obtained sample proportions fit population proportions specified by the null hypothesis.

  • Formula:

    [ \chi^2 = \sum \frac{(f_o - f_e)^2}{f_e} ]

Example Case Study

  • Color Association and Hunger

    • Participants choose colors associated with hunger: Red, Yellow, Green, Blue.

    • Observed frequencies:

      • Red: 19

      • Yellow: 16

      • Green: 10

      • Blue: 5

Observed and Expected Frequencies

  • Observed Frequency (f_o)

    • Actual number of individuals found in a study category.

  • Expected Frequency (f_e)

    • Number expected in a category if the null hypothesis is true.

Steps to Calculate Chi-Square Statistic

  1. Determine actual observed frequencies in each category.

  2. Determine expected frequencies in each category.

  3. For each category, calculate observed minus expected frequencies.

  4. Square each of the differences.

  5. Divide each squared difference by its expected frequency.

  6. Sum results of step 5 for all categories.

Chi-Square Distribution

  • Mathematically defined curve used as the comparison distribution in chi-square tests.

  • Reflects the distribution of the chi-square statistic.

Chi-Square Table

  • Table of cutoff scores on the chi-square distribution for various degrees of freedom and significance levels.

Degrees of Freedom for Goodness of Fit Test

  • Formula:

    [ df = C - 1 ]

    • Where C = number of categories.

Reporting Chi-Square Goodness of Fit Test Results

  • Example report:

    • Significant differences in color preference proportions.

    • Conclusion: Certain colors more likely associated with hunger (e.g., Red: n = 19, Yellow: n = 16).

Assumptions of Chi-Square Test for Goodness of Fit

  1. One categorical variable (dichotomous, nominal, ordinal).

  2. Independence of observations.

  3. Groups of categorical variables must be mutually exclusive.

  4. At least 5 expected frequencies in each group of categorical variables.

Chi-Square Test of Independence

  • Examines if the distribution of frequencies over categories of one nominal variable is unrelated to the distribution of frequencies over another nominal variable.

Example of Chi-Square Test of Independence

  • Study on the relationship between personality (introvert/extrovert) and color preference (red/yellow/green/blue) among 200 students.

Contingency Table

  • Two-dimensional chart showing frequencies in each combination of categories of two nominal variables.

Expected Frequencies for Each Cell

  • Formula:

    [ f_e = \frac{R \times n}{C} ]

    • R = row total, C = column total, n = total respondents.

Degrees of Freedom for Chi-Square Test of Independence

  • Formula:

    [ df = (N_C - 1)(N_R - 1) ]

    • Where N_C = number of columns, N_R = number of rows.

Assumptions of Chi-Square Test for Independence

  1. Two variables measured at ordinal or nominal level.

  2. Variables consist of two or more categorical independent groups.

  3. Less than 20% of the cells should have an expected frequency of less than 5.

Effect Size for Chi-Square Test of Independence

  • Phi Coefficient (φ)

    • Effect-size measure for a 2x2 contingency table.

    • Formula:

      [ \phi = \sqrt{\frac{\chi^2}{n}} ]

  • Cramer's Phi (φ)

    • For contingency tables larger than 2x2.

    • Formula:

      [ V = \frac{\chi^2}{n \cdot df} ]

Reporting Chi-Square Test for Independence Results

  • Example report:

    • Significant relationship between personality and color preference, with details of specific frequencies.

Chi-Square Goodness of Fit Test in Software (Jamovi & JASP)

  • Procedures outlined for conducting goodness of fit tests using various statistical software including steps for setting up, running tests, and interpreting results.

References

  • Aron, A., Coups, E., & Aron, E. (2013). Statistics for psychology (6th ed.). Pearson Education Inc.

  • Goss-Sampson, M. A. (2024). Statistical Analysis in JASP 0.16.1: A Guide for Students.

  • Gravetter, F.J., Wallnau, L. B., & Forzano, L.B. (2020). Essentials of statistics for the behavioral science (9th ed.). Cengage Learning.

  • Navarro, D. J., & Foxcroft, D. R. (2019). Learning statistics with jamovi: A tutorial for psychology students and other beginners (Version 0.70).

  • Statistics How To (2018). Parametric statistics, tests and data.

  • Sharpe, Donald (2015). Your Chi-Square Test is Statistically Significant: Now What? Practical Assessment, Research & Evaluation.