The open-ended categories in contingency tables are often expressed as 'Less than' or 'More than'. Generally, arbitrary scores are assigned to the open-ended categories before an application of a statistical method, for example, tests of linear trend. The methods developed here will address issues arising from the assignment of arbitrary scores to open- ended extreme categories when testing for linear trend. The issues of open-ended extreme categories are addressed separately for single 2xK ordered tables, multiple 2xK ordered tables and RxK ordered tables. Since the existing methods utilize arbitrary scores for the open-ended extreme categories to investigate the linear trend, the proposed methods are also intended to identify the situations where a specific set of scores for such categories may or may not influence the substantive data analytic conclusions. The appropriateness of the methods developed in the study will be exemplified using real life cancer data sets. The proposed method utilize volumes of scores that yield significant or (nonsignificant) values of the test statistic. Such volumes are closely related to the traditional confidence intervals, and will help an investigator what percent of possible scores for the open-ended category will produce significant values. Similar volumes are also proposed to determine what percentage of the score will yield a p-value less than the p-value yielded by a given score for the open-ended category. The volume approach proposed incorporate the results of optimization procedures also. If scores for the open-ended categories are needed, then such scores are extracted from the maximization of the criterion statistic, and the distribution of the statistic will also be derived. Simulation studies will be conducted to help understand the effect of the open-ended categories.
