Ch. 9Study Notes on Chi-square Tests and Categorical Variables

Overview of Categorical Variables

• Categorical variables: limited values (e.g., gender, favorite color)

• Participants assigned to one category, count frequency of values.

Contingency Tables

• Summarize associations between two categorical variables.

• Show cell frequencies and row/column totals.

Chi-square (c2) Test

• Assess observed vs. expected frequencies.

• If null hypothesis (H0) is true, expected frequencies calculated as:
  $(Row\ total \times Column\ total) / Total\ participants$

• Comparison formula:
  $(obs\ freq - exp\ freq)^2 / exp\ freq$ summed across cells.

Hypotheses for Chi-square Independence Test

• H0: No association between variables.

• H1: There is an association.

• Decision based on p-value relative to significance level (α).

SPSS Chi-square Test Process

1. Data entry for each cell as rows.

2. Weight cases using frequency column.

3. Analyze -> Descriptive Statistics -> Crosstabs.

4. Select required statistics (Chi-square, Frequencies).

Assumptions of the Test

• Categories must be mutually exclusive.

• At least one observation per cell.

• Expected frequencies > 5 in > 20% of cells.
  (Use Fisher’s Exact test if violated)

Goodness-of-Fit Test

• One categorical variable to check if observed frequencies are as expected.

• Null hypothesis expects even distribution.

• Calculate c2 similarly as above.

Goodness-of-Fit Test in SPSS

• Analyze -> Nonparametric Tests -> Chi-square.

• Enter expected counts manually or select equal.

Interpretation of Results

• Significant p-value (< 0.05) leads to rejection of H0.

• Indicates significant association or preference among variables.

Summary of c2 Applications

• Used for observing distribution across values or association between two categorical variables.

• Can be extended to larger tables, logistic regression may be applicable.