Generalized p-value: Difference between revisions

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In Statistics[[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[[mixed Modelsmodel]]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.{{fact}} Use of approximate and asymptotic methods may lead to misleading conclusions or may fail to detect truly [[Statistical significance|significant]] results from experiments[[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, theexact referencestests belowcan providebe exact testsformulated 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 (1989)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]].

==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>.

Now suppose we need to test the coefficient of variation, <math>\rho = \mu /\sigma </math>. While the problem is not trivial with conventional p-values, the task can be easily accomplished based on the generalized test variable

:<math>R = \frac {\overline{x} S} {s \sigma} - \frac{\overline{X}- \mu} {\sigma}

= \frac {\overline{x}} {s} \frac {\sqrt{U}} {\sqrt{n}} ~-~ \frac {Z} {\sqrt{n}} ,</math>,

where <math>\overline{x}</math> is the observed value of <math>\overline{X}</math> and <math>S</math> is the observed value of <math>s</math>. Note that the distribution of <math>R</math> and its observed value are both free of nuisance parameters. Therefore, one-sided hypotheses such as <math> H_0 : \rho < \rho_0 </math> can be tested based on the generalized p-value <math> p = Pr( R \ge \rho_0 )</math>, a quantity that can be evaluated via Monte Carlo simulation or using the non-central t-distribution.

==Notes==

{{reflist}}

==References==

[1] *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 (1989). [http://www.jstor.org/stable/2289949]

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

==External links==

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

[[Category:Hypothesis testing]]