Approximate unconditional test procedure for comparing two ordered multinomials

The asymptotic and exact conditional methods are widely used to compare two ordered multinomials. The asymptotic method is well known for its good performance when the sample size is sufficiently large. However, Brown et al. (2001) gave a contrary example in which this method performed liberally even when the sample size was large. In practice, when the sample size is moderate, the exact conditional method is a good alternative, but it is often criticised for its conservativeness. Exact unconditional methods are less conservative, but their computational burden usually renders them infeasible in practical applications. To address these issues, we develop an approximate unconditional method in this paper. Its computational burden is successfully alleviated by using an algorithm that is based on polynomial multiplication. Moreover, the proposed method not only corrects the conservativeness of the exact conditional method, but also produces a satisfactory type I error rate. We demonstrate the practicality and applicability of this proposed procedure with two real examples, and simulation studies are conducted to assess its performance. The results of these simulation studies suggest that the proposed procedure outperforms the existing procedures in terms of the type I error rate and power, and is a reliable and attractive method for comparing two ordered multinomials.