Abstract
Combining p values methods and uniformity tests are closely related subjects in meta analysis. In this context it is also known that publication bias can seriously impair an overall decision. The recent concepts of generalized p values and of random p values emphasize that, when faced with a significant number of results that casts some doubt on the null hypothesis, the correct approach to the problem should be to combine evidence under the alternative hypothesis. Following previous research, we investigate generalized p values and random p values for testing uniformity when the alternative is a mixture of a Beta(1,2) and standard uniform random variables.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The family (1) results from tilting the standard uniform density using the point (0.5,1) as a rotation center.
- 2.
If T is stochastically decreasing with θ, then \(p =\mathrm{ Pr}[T(X; \theta ,\eta ) \leq T(x; \theta ,\eta )\vert \theta = {\theta }_{0}]\).
- 3.
For m ∈ [0, 2], note that (1) is a mixture of a Beta(2,1) and Uniform(0,1) with mixing weights \(\frac{m} {2}\) and \(1 -\frac{m} {2}\), respectively.
References
Brilhante, M.F., Mendonça, S., Pestana, D., Sequeira, F.: Using products and powers of products to test uniformity. In: Luzar-Stiffler, V., et al. (eds.) Proceedings of the 32nd International Conference on Information Technology Interfaces, SRCE, Zagreb pp. 509–514 (2010)
Brilhante, M.F., Pestana, D., Sequeira, F.: Combining p-values and random p-values. In: Luzar-Stiffler, V., et al. (eds.) Proceedings of the 32nd International Conference on Information Technology Interfaces, pp. 515–520 (2010)
Dempster, A.P., Schatzoff, M.: Expected significance level as a sensibility index for test statistics. J. Am. Stat. Assoc. 60, 420–436 (1965)
Gomes, M.I., Pestana, D.D., Sequeira, F., Mendonça, S., Velosa, S.: Uniformity of offsprings from uniform and non-uniform parents. In: Luzar-Stiffler, V., et al. (eds.) Proceedings of the 31st International Conference on Information Technology Interfaces, pp. 243–248 (2009)
Iyer, H.K., Patterson, P.D.: In a recipe for constructing generalized pivotal quantities and generalized confidence intervals, Colorado State University Department of Statistics Technical Report 2002/10, http://www.state.colostate.edu/research/2002_2010.pdf (2002/2010)
Kulinskaya, E., Morgenthaler, S., Staudte, R.G.: Meta Analysis. A Guide to Calibrating and Combining Statistical Evidence. Wiley, Chichester (2008)
Pestana, D.: Combining p-values. In: Lovric, M. (ed.) International Encyclopedia of Statistical Science, pp. 1145–1147. Springer, Berlin (2011)
Royle, J.A., Dorazio, R.M., Link, W.A.: Analysis of multinomial models with unknown index using data augmentation. J. Comput. Grap. Stat. 16, 67–85 (2007)
Sackrowitz, H., Samuel-Cahn, E.: p-Values as Random Variables – Expected P Values. J. Am. Stat. Assoc. 53, 326–331 (1999)
Tsui, K., Weerahandi, S.: Generalized p-values in significance testing of hypotheses. J. Am. Stat. Assoc. 84, 602–607 (1989)
Acknowledgements
This research has been supported by National Funds through FCT—Fundação para a Ciência e a Tecnologia, project PEst-OE/MAT/UI0006/2011.
The author would also like to thank Professor Dinis Pestana for the stimulating conversations held about the topics discussed here, and the reviewers for their comments which led to a substantial improvement of the chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Brilhante, M.F. (2013). Generalized p Values and Random p Values When the Alternative to Uniformity Is a Mixture of a Beta(1,2) and Uniform. In: Oliveira, P., da Graça Temido, M., Henriques, C., Vichi, M. (eds) Recent Developments in Modeling and Applications in Statistics. Studies in Theoretical and Applied Statistics(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32419-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-32419-2_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32418-5
Online ISBN: 978-3-642-32419-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)