Generalized p-value: Difference between revisions

Revision as of 09:04, 24 August 2010

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In [[statistics]], a '''generalized p-value''' is an extended version 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, 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.{{factCitation needed|date=August 2010}} 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 can be formulated based on generalized p-values.<ref name=TW>Tsui & Weerahandi (1989)</ref><ref>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-values so that one can obtain exact solutions for problems such as the [[Behrens&ndash;Fisher problem]].

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==A simple case==

To describe the idea of generalized p-values in a simple example, consider a situation of sampling from a normal population with mean <math>\mu</math>, and variance <math>\sigma ^2</math>, suppose <math>\overline{X}</math> and <math>S ^2</math> are the sample mean and the sample variance. Inferences on all unknown parameters can be based on the distributional results

:<math> Z = \sqrt{n}(\overline{X} - \mu)/ \sigma \sim N(0,1)</math>

and

:<math>U = n S^2 / \sigma^2 \sim \chi^2 _ {n-1} .</math>

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==Notes==

{{reflistReflist}}

==References==

*Tsui, K. and Weerahandi, S. (1989): [http://www.jstor.org/stable/2289949 "Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters"]. ''[[Journal of the American Statistical Association]]'', 84, 602&ndash;607

==References==

*Tsui, K. and Weerahandi, S. (1989): [http://www.jstor.org/stable/2289949 "Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters"]. ''[[Journal of the American Statistical Association]]'', 84, 602&ndash;607

*Weerahandi, S. (1995) [http://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-40621-3 ''Exact Statistical Method for Data Analysis'' ] Springer-Verlag, New York. ISBN 978-0-387-40621-3

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*[http://www.x-techniques.com/ XPro, Free software package for exact parametric statistics]

{{DEFAULTSORT:Generalized P-Value}}

[[Category:Hypothesis testing]]