A geometric test for the analysis of contingency tables

The Chi-squared test for contingency tables has good performance when sample sizes are sufficiently large but is not appropriate with small samples. When this condition is violated, Fisher’s exact test is often used as it is valid for all sample sizes. However, Fisher’s test has been criticised for being overly conservative. Alternatives have been proposed but typically involve more complex computations. In this paper we propose an alternative test based on a geometric projection of the multinomial distribution onto the n-sphere. Each multinomial observation is represented by a single point on this sphere. The angle between two points representing two different multinomial observations can then be compared to the distribution of the angle under the assumption that the realisations come from the same underlying distribution. The null hypothesis distribution is simulated and easy to compute. This new test is compared to the Chi-Squared test and Fisher’s Test in terms of both Type I and Type II errors and its potential use in reliability modelling is indicated using a real case study.