Generalized p-value
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Conventional statistical methods do not provide exact solutions to many statistical problems, especially when the problem involves many nuisance parameters. As a result, practitioners often resort to approximate statistical methods or asymptotic statistical methods that are primarily based on large samples. With small samples, in most cases, these approximate methods and asymptotic methods perform very poorly. Due to the utilization of these approximate and asymptotic methods, experimenters often fail to detect the significance of their experiments. Furthermore, there are well-documented cases where these methods not only fail to detect the statistical significance, but may also lead to misleading conclusions.
Generalized p-values method is an exact statistical method based on exact probability statements rather than asymptotic statistical methods to tackle difficult statistical problems where conventional statistical methods do not provide exact solutions. Moreover, the generalized p-value approach is an extension of the classical p-value approach.
In order to over come the shortcomings of the classical p-values, Tsui and Weerahandi (1989) extended the definition of the classical p-values so that one can obtain the exact solutions for problems such as the Behrens–Fisher problem.
Later, Weerahandi (1993) showed how one can utilize the generalized p-values to construct generalized confidence intervals.
A complete coverage of the definitions and applications of generalized p-values can be found in Weerahandi (1995) and Weerahandi (2004).
Many of the generalize p-values procedures are incorporated into software packages that are freely available in the Internet (eg. XPro).
References
[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). [[1]]
[2] Weerahandi, S. (1993): Generalized confidence intervals. Journal of the American Statistical Association, 88, 899-905 (1993). [[2]]
[3] Weerahandi, S. 1995. Exact Statistical Method for Data Analysis. Springer-Verlag, New York.
[5] XPro, Windows software package for exact parametric statistics