On Formulating Pearson’s Chi-Squared Statistic in Two-Way Frequency Tables - Lynn Roy LaMotte (Austrian Journal of Statistics, 2016)
The standard form of Pearson’s chi-squared statistic ignores variation due to estimating the mean vector in settings where the mean vector is not completely specified by the null hypothesis, as is the case when testing for homogeneity or independence in two-way tables. The root form of the statistic is formulated here with and without that additional variance included, resulting in somewhat different expressions.