Nonparametric Tests for Ordered Alternatives in a Two-Stage Nested Design

Date of Award


Degree Name

Doctor of Philosophy



First Advisor

Dr. Jeffrey T. Terpstra

Second Advisor

Dr. Joshua D. Naranjo

Third Advisor

Dr. Kevin Lee

Fourth Advisor

Dr. Jessaca Spybrook


order restricted inference, nonparametric, Nested design


Nested designs and order-restricted hypotheses are commonly encountered in statistics. In fact, order-restricted hypotheses are usually a primary research interest when factor levels are naturally ordered. However, tests for order-restricted hypotheses in a nested design have yet to be developed. To this end, we introduce some new nonparametric/robust tests for order-restricted hypotheses in a two-stage fixed effects nested design, for both the nested factor and the non-nested factor. For nested effects, Oron and Hoff in 2007 proposed the Nested Kruskal-Wallis test to test for the general alternative hypothesis. This test is essentially the sum of individual Kruskal-Wallis tests for the nested effects under each level of the non-nested factor. This dissertation proposes the Nested Jonckheere-Terpstra test, which is based on Oron and Hoff’s method, but uses Jonckheere-Terpstra tests instead to test for ordered alternatives among the nested effects. The simulation results showed that all of the versions of the proposed test (which incorporates the a-priori ordering) outperform the Nested Kruskal-Wallis test for ordered alternatives. With regard to ordered alterna- tives among the non-nested effects two tests are proposed. The so-called Aligned Rank Jonckheere-Terpstra test is the Jonckheere-Terpstra test applied to observations that are aligned using estimated nested e_ects. The so-called Averages-Based Jonckheere-Terpstra test is based on aggregating the data within each level of the non-nested factor and is essentially a multisample U-statistic. The simulation results indicate that the proposed tests (which incorporate the a-priori ordering) outperform the general alternative hypothesis test which is based on the reduction in dispersion F-test. Distributional results for all of the proposed tests were derived and shown to be asymptotically normal.

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