Calculating Chi-Square

Practice questions covering the conceptual understanding and mathematical calculation of the Chi-Square test of independence as presented in PSYC20008.

How is categorical data defined in the context of these lecture notes?

Information that can be transformed and organised into a manner that can be manipulated, analysed, and used to infer relationships, specifically categorised into Binary, Nominal, or Ordinal types.

What is the recommended tool for cross-tabulating variables instead of using simple frequency tables?

Contingency tables.

What is the formula for calculating Expected Frequencies ($E_{ij}$)?

E_{ij} = rac{ ext{row total}_i imes ext{column total}_j}{n}

What question does the Chi-square test of independence aim to answer regarding variation?

How much of the variation of frequencies within a contingency table is due to the relationship between the variables (and how much is due to the distribution of the variables)?

What is the formula for the Chi-square statistic ($ext{c}^2$)?

ext{c}^2 = ext{S} rac{( ext{observed}{ij} - ext{expected}{ij})^2}{ ext{expected}_{ij}}

In Chi-Square calculations, what does the 'residual' represent?

The difference between the observed frequency and the expected frequency ($ext{observed} - ext{expected}$).

How are Degrees of Freedom ($df$) calculated for a contingency table?

$$df = (n_{ ext{rows}} - 1) imes (n_{ ext{columns}} - 1)$$

What is the definition of an Adjusted Standardised Residual?

The difference between the observed and expected count for any cell, considering expected counts and row and column totals (i.e., variation relative to the rest of the table).

In JASP software, what title is used for adjusted standardised residuals?

Standardized (adjusted Pearson).

What threshold values indicate that an Adjusted Standardised Residual is statistically significant?

Values that are greater than $1.96$ (> 1.96) or less than $-1.96$ (< -1.96).

What does a positive valence in an adjusted standardised residual indicate?

That there are more people in that category than expected by chance.

What does a negative valence in an adjusted standardised residual indicate?

That there are fewer people in that category than expected by chance.

According to the 'Best' reporting example, what should be revealed to describe specific significant findings in a contingency table?

Examination of the standardised residuals.

What six elements must be presented when reporting a chi-square test of independence?

1. Observed counts, 2. Expected counts, 3. The chi-square statistic, 4. Degrees of Freedom & sample size, 5. The p value, and 6. The adjusted standardised residuals.