The chi-square controversy: what if Pearson had R?

One of the most famous controversies in the history of Statistics regards the number of the degrees of freedom of a chi-square test. In 1900, Pearson introduced the chi-square test for goodness of fit without recognizing that the degrees of freedom depend on the number of estimated parameters under the null hypothesis. Yule tried an ‘experimental’ approach to check the results by a short series of ‘experiments’. Nowadays, an open-source language such as R gives the opportunity to empirically check the adequateness of Pearson’s arguments. Pearson paid crucial attention to the relative error, which he stated ‘will, as a rule, be small’. However, this point is fallacious, as is made evident by the simulations carried out with R. The simulations concentrate on 2 × 2 tables where the fallacy of the argument is most evident. Moreover, this is one of the most employed cases in the research field.