Generalized p-value: Difference between revisions

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{{Short description|Statistics concept}}

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

{{More footnotes|date=January 2017}}

{{DISPLAYTITLE:Generalized ''p''-value}}

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

Conventional statistical methods do not provide exact solutions to many statistical problems, such as those arising in [[mixed model]]s and [[MANOVA]], especially when the problem involves manya number of [[nuisance parameter]]s. As a result, practitioners often resort to approximate statistical methods or [[Asymptotic theory (statistics)|asymptotic statistical methods]] that are valid only withwhen largethe samplessample size is large. With small samples, such methods often have poor performance.<ref name=WE/> 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 belowfor providesuch exactproblems testscan be obtained based on generalized ''p''-values..<ref name=WE>Weerahandi (1995)</ref><ref name=TW>Tsui & Weerahandi (1989)</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-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.

==Example==

To describe the idea in 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 unknwon paramters can be based on the distributional results

To describe the idea of generalized ''p''-values in a simple example, consider a situation of sampling from a normal population with the mean <math>\mu</math>, and the variance <math>\sigma ^2</math>,. supposeLet <math>\overline{X}</math> and <math>S ^2</math> arebe the sample mean and the sample variance. Inferences on all unknwonunknown paramtersparameters 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>.

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

Now suppose we need to test the coefficient of variation, <math>\rho = \mu /\sigma </math>. This can be easily accomplished based on the generalized test variable

and

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

:<math>U = n S^2 =/ \fracsigma^2 {\overline{x}} {s}sim \fracchi^2 {\sqrt{U}}_ {\sqrt{n}} ~-~ \frac {Z} {\sqrt{n}1} .</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.

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

==References==

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

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

= \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>Ss</math> is the observed value of <math>sS</math>. Note that the distribution of <math>R</math> and its observed value are both free of nuisance parameters. Therefore, a test of a hypothesis with a one-sided hypothesesalternative such as <math> H_0H_A : \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 easily evaluated via Monte Carlo simulation or using the non-central t-distribution.

[2] [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. ]

==Notes==

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

{{Reflist}}

==References==

[[Category:Hypothesis testing]]

*Gamage J, Mathew T, and Weerahandi S. (2013). Generalized prediction intervals for BLUPs in mixed models, Journal of Multivariate Analysis}, 220, 226-233.

*Hamada, M., and Weerahandi, S. (2000). Measurement System Assessment via Generalized Inference. Journal of Quality Technology, 32, 241-253.

*Krishnamoorthy, K. and Tian, L. (2007), “Inferences on the ratio of means of two inverse Gaussian distributions: the generalized variable approach”, Journal of Statistical Planning and Inferences, Volume 138, Issue 7, 1, Pages 2082-2089.

*Li, X., Wang J., Liang H. (2011). Comparison of several means: a fiducial based approach. Computational Statistics and Data Analysis, 55, 1993-2002.

* Mathew, T. and Webb, D. W. (2005). Generalized p-values and confidence intervals for variance components: Applications to Army test and evaluation, Technometrics, 47, 312-322.

*Wu, J. and Hamada, M. S. (2009) Experiments: Planning, Analysis, and Optimization. Wiley, Hoboken, New Jersey.

*Zhou, L., and Mathew, T. (1994). Some Tests for Variance Components Using Generalized p-Values, Technometrics, 36, 394-421.

*Tian, L. and Wu, Jianrong (2006) “Inferences on the Common Mean of Several Log-normal Populations: The Generalized Variable Approach”, Biometrical Journal.

[1] *Tsui, K. and Weerahandi, S. (1989): [https://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) [httphttps://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-40621-3 Weerahandi, S. 1995. ''Exact Statistical MethodMethods 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]

{{DEFAULTSORT:Generalized P-Value}}

[[Category:HypothesisStatistical hypothesis testing]]