COMPARING GROUPS WITH UMBRELLA ORDERINGS

One commonly observed response pattern in a one-factor design with ord ered treatment levels is the umbrella or unimodal ordering, in which t he response variable increases with an increase in the treatment level up to a point, then decreases with further increase in the treatment level. The turning point is called the peak of an umbrella ordering. I n many practical settings, there are two or more different populations with umbrella orderings, and it is of interest to compare these popul ations. In this article, likelihood ratio tests are developed for test ing whether two groups with umbrella orderings have the same peaks. Th ree cases are considered: variances known, variances unknown and diffe rent for each group, and variances unknown and the same for both group s. Extensions to randomized block designs and to a more general class of testing problems are discussed.