Test of Independence

Used when two categorical variables are measured on the same sample. Tests if there is a relationship between the variables.

Suppose a table of $r$ rows and $c$ columns, containing observed data, is provided. Observed values are dentoed by $O_{ij}$.

Hypotheses

$H_0 : \text{There is no association between the two variables.}$

$H_1 : \text{There is a significant association between the two variables.}$

Degrees of freedom

$$\nu = (r - 1)(c - 1)$$

Expected Frequency

$$E_{ij} = \frac{(\text{Row total}) \times (\text{Column total})}{\text{Grand total}}$$

Test Statistic

$$\chi_{\text{calc}}^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \quad \sim \chi^2_\nu$$

$H_0$ is rejected if $\chi_{\text{calc}}^2 \gt \chi_{\nu,\alpha}$.