Generalized p-value: Difference between revisions

Revision as of 19:49, 22 August 2010

Revision as of 20:08, 22 August 2010

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

Conventional statistical methods do not provide exact solutions to many statistical problems, especially when the problem involves many [[nuisance parameter]]s. As a result, practitioners often resort to approximate statistical methods or asymptotic statistical methods that are valid only with large samples. With small samples, such methods often have poor performance. Use of approximate and asymptotic methods may lead to misleading conclusions or may fail to detect truly significant results from experiments.

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In order to overcome the shortcomings of the classical p-values, Tsui and Weerahandi (1989) extended the definition of the classical p-values so that one can obtain exact solutions for problems such as the [[Behrens–Fisher problem]].

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 unknwonunknown paramtersparameters can be based on the distributional results

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

and

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>. ThisWhile this is not quite staright forward to do this 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>,