Generalized p-value: Difference between revisions

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In [[statistics]], a '''generalized p-value''' is an extended version of the classical [[p-value]], which except in a limited number of applications, provide only approximate solutions.

Conventional statistical methods do not provide exact solutions to many statistical problems such as those arise in [[mixed model]]s and MANOVA, especially when the problem involves many [[nuisance parameter]]s. As a result, practitioners often resort to approximate statistical methods or [[Asymptotic theory (statistics)|asymptotic statistical methods]] that are valid only with large samples. With small samples, such methods often have poor performance.{{Citation needed|date<ref name=August 2010}}WE/>. Use of approximate and asymptotic methods may lead to misleading conclusions or may fail to detect truly [[Statistical significance|significant]] results from [[experiment]]s.

Tests based on generalized p-values are exact statistical methods in that they are based on exact probability statements. While conventional statistical methods do not provide exact solutions to such problems as testing [[variance components]] or [[ANOVA]] under unequal variances, exact tests for such problems can be formulatedobtained based on generalized p-values.<ref name=TW>Tsui & Weerahandi (1989)</ref><ref name=WE>Weerahandi (1995)</ref>

In order to overcome the shortcomings of the classical p-values, Tsui and Weerahandi<ref name=TW/> extended the definition of the classical p-valuesdefinition so that one can obtain exact solutions for problems such problems as the [[Behrens&ndash;Fisher problem]] and testing variance components. This is accomplished by allowing test variables to depend on observable random vectors as well as their observed values, as in the Bayesian treatment of the problem, but without having to treat constant parameters as random variables.

==A simple case==