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.

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

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#).