Pearson's $\chi^2$-test for cross tables

$O_{ab}$ is the number of observations in row $a$ and column $b$ in a $A\times B$ cross table. $E_{ab}$ is the expected number of observations - given the marginals (row and column totals) - if $A$ and $B$ are independent. Pearson's $\chi^2$ is

$$\begin{equation*} \chi^2 = \sum_{a=1}^A\limits \sum_{b=1}^B\limits \frac{(O_{ab}-E_{ab})^2}{E_{ab}}, \end{equation*}$$ where $E_{ab}$ is the product of the appropriate row and column total divided by $n$ (row total times column total divided by the grand total). $\chi^2$ has $df=(A-1)(B-1)$ degrees of freedom.

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Effect sizes $\chi^2$

$$\begin{equation*} w = \sqrt{\frac{\chi^2}{n}} \end{equation*}$$

• $w=0.1$ small effect
• $w=0.3$ medium effect
• $w=0.5$ large effect

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Cohen's $\kappa$ (kappa)

For a $A$ by $A$ table

$$\begin{equation*} \kappa=\frac{F_{a}-E_{a}}{1-E_{a}}, \end{equation*}$$ where

• $F_a$ is the sum of the observations on the diagonal and
• $E_a$ is the sum of the observations on the diagonal if the rows and columns are independent (the product of the appropriate row and column total divided by $n$).