Sequential parameter structure,conditional inference,and likelihood drop

Successive tests of hypotheses, as exemplified with an analysis of variance table, impose a set theoretic structure on the parameter space and yet allow much arbitrariness in the definition of nuisance parameters. Two major types of statistical model, the exponential and transformation, are shown to have by basic theory well defined conditional testing procedures. The two types of testing procedure are then shown to have opposite forms of set theoretic structure on the sample space, and to differ sharply from the commonly used deviance or likelihood drop methods. The two types of model have the normal linear model as the intersection model and the two opposite forms of testing procedure manage to coincide by product space structure and independence. Details of the two types of testing procedure are discussed, related to the arbitrariness in nuisance parameter definition, and organized to provide a general-case pattern for the development of conditional procedures as an alternative to the default likelihood-drop methods.