Chi Square Hypothesis Testing

Chapter 10: Hypothesis Testing IV (Chi Square)

Chapter Outline

• Introduction
• Bivariate Table
• The Logic of Chi Square (\chi^2) test
• Chi Square Test for Independence
  • The Five-Step Model
  • Computation of Chi Square (\chi^2)

Bivariate Table

• Columns represent values of the independent variable.
• Rows represent values of the dependent variable.
• Cells represent the intersections of columns and rows.
• Each cell reports the number of times each combination of values occurred.

Bivariate Table Structure

Basic Logic of \chi^2 Test

• Chi-Square is a test of significance based on a bivariate table.
  • Most often, both variables are categorical (i.e., nominal or ordinal).
• We are looking for significant differences between the observed frequencies (fo) and the expected frequencies (fe) given two variables are independent.

Example of \chi^2 Test

• Is there any statistically significant relationship between gender and party identification?
• Data is based on the 1991 General Social Survey.

Example Data: Party Identification by Gender

Step 1: Assumptions and Test Requirements

• An independent random sample.
• Both variables are categorical (as is typical for the chi-square test).
• No assumption is made about the shape of the population distribution (the chi-square test is a non-parametric test).

Step 2: State the Null and Research Hypotheses (H0 and H1)

• H_0: Party identification and gender are independent. (Gender has no effect on party identification).
• H_1: Party identification and gender are dependent. (Gender has some effect on party identification).

Step 3: Select Sampling Distribution and Establish the Critical Region

• Sampling Distribution = \chi^2
• Use the table in Appendix C (“Distribution of Chi Square”) to find \chi^2 (critical):
  • df = (r-1)(c-1) = (3-1)(2-1) = 2
  • If we set \alpha = 0.05, \chi^2 (critical) = 5.991

Step 4: Compute the Test Statistic (\chi^2)

\chi^2 (obtained) = \sum \frac{(fo - fe)^2}{f_e}

• Where:

  • f_o represents the observed frequencies.
  • f_e represents the expected frequencies.
  • The expected frequencies (f_e) are computed as follows:

  f_e = \frac{\text{(RowMarginal)(ColumnMarginal)}}{N}

Step 4: Compute the Test Statistic (\chi^2) - Expected Frequencies

• Compute expected frequencies (f_e):
  • (444*577)/980 = 261.4
  • (444*403)/980 = 182.6
  • (120*577)/980 = 70.7
  • (120*403)/980 = 49.3
  • (416*577)/980 = 244.9
  • (416*403)/980 = 171.1

Step 4: Compute the Test Statistic (\chi^2): Party Identification by Gender (with Expected Frequencies)

Step 4: Compute the Test Statistic (\chi^2) - Calculation Table

Step 5: Interpret Results and Make a Decision

• \chi^2 (critical) = 5.991
• \chi^2 (obtained) = 7.0
• The test statistic is in the critical region. Therefore, we reject the H_0.
• There is a significant relationship between gender and party identification (or gender and party identification are dependent).

Interpreting Chi-Square Test

• The chi-square test tells us only if the variables are independent or not.
• It does not tell us the pattern or the nature of the relationship.
• To investigate the pattern, use observed frequencies to compute percentages within each column and compare across the columns.

Interpreting Chi-Square Test: Party Identification by Gender (Column Percentages)

Other Limitations of Chi-Square Test

• Similar to other types of hypothesis testing, chi-square is sensitive to sample size.
  • As N increases, obtained chi-square increases.
  • With large samples, trivial relationships may be significant.
• Remember: significance is not the same thing as importance.

Lab Exercise & Homework

• Lab Exercise 7 (ungraded)
  • 10.11 (p. 285)
  • Conduct the chi-square test and also compute the column percentage
• HW8 (graded)
  • 10.12 (p. 286)
    • Conduct the chi-square test and also compute the column percentage
    • Formula 10.4 in “The Limitations of the Chi Square Test” (p. 280) is not covered.