On Some Procedures Based on Fisher’s Inverse Chi-Square Statistic


We present approximations to the distribution of the weighted combination of independent and dependent p - values, . In case that independence of Pi 's is not assumed, it is argued that the quantity A is implic-itly dominated by positive definite quadratic forms that induce a chi-square distribution. This gives way to the approximation of the associated degrees of freedom using Satterthwaite (1946) or Patnaik (1949) method. An approximation by Brown (1975) is used to estimate the covariance between the log transformed P - values. The performance of the approximations is compared using simulations. For both the independent and dependent cases, the approximations are shown to yield probability values close to the nominal level, even for arbitrary weights, ωi ’s.

Share and Cite:

K. Makambi, "On Some Procedures Based on Fisher’s Inverse Chi-Square Statistic," Applied Mathematics, Vol. 4 No. 8, 2013, pp. 1109-1114. doi: 10.4236/am.2013.48150.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. Solomon and M. A. Stephens, “Distribution of a Weighted Sum of Chi-Squared Variables,” Journal of the American Statistical Association, Vol. 72, No. 360a, 1977, pp. 881-885.
[2] M. J. Buckley and G. G. Eagleson, “An Approximation to the Distribution of Quadratic Forms in Normal Random Variables,” Australian Journal of Statistics, Vol. 30A, No. 1, 1988, pp. 150-159. doi:10.1111/j.1467-842X.1988.tb00471.x
[3] J.-T. Zhang, “Approximate and Asymptotic Distributions of Chi-Squared-Type Mixtures with Applications,” Jour nal of the American Statistical Association, Vol. 100, No. 469, 2005, pp. 273-285. doi:10.1198/016214504000000575
[4] R. L. Eubank and C. H. Spiegelman, “Testing the Good ness of Fit of a Linear Model via Non-Parametric Regres sion techniques,” Journal of the American Statistical As sociation, Vol. 85, No. 410, 1990, pp. 387-392. doi:10.1080/01621459.1990.10476211
[5] A. Azzalini and A. W. Bowman, “On the Use of Non Parametric Regression for Checking Linear Relation ships,” Journal of the Royal Statistical Society Series B, Vol. 55, No. 2, 1993, pp. 549-557.
[6] J. C. Chen, “Testing the Goodness of Fit of Polynomial Models via Spline Smoothing Techniques,” Statistics and Probability Letters, Vol. 19, No. 1, 1994, pp. 65-76. doi:10.1016/0167-7152(94)90070-1
[7] W. Gonzalez-Manteiga and R. Cao, “Testing the Hypo thesis of a General Linear Model Using Non-Parametric Regression Estimation,” Test, Vol. 2, No. 1-2, 1993, pp. 161-188. doi:10.1007/BF02562674
[8] J. Fan, C. Zhang, J. Zhang, “Generalized Likelihood Ra tio Statistics and Wilks Phenomenon,” The Annals of Sta tistics, Vol. 29, No. 1, 2001, pp. 153-193. doi:10.1214/aos/996986505
[9] K. H. Makambi, “Weighted Inverse Chi-Square Method for Correlated Significance Tests,” Journal of Applied Statistics, Vol. 30, No. 2, 2003, pp. 225-234. doi:10.1080/0266476022000023767
[10] I. J. Good, “On the Weighted Combination of Signifi cance Tests,” Journal of the Royal Statistical Society Se ries B, Vol. 17, No. 1, 1995, pp. 264-265.
[11] D. S. Bhoj, “On the Distribution of the Weighted Combi nation of Independent Probabilities,” Statistics & Prob ability Letters, Vol. 15, No. 1, 1992, pp. 37-40. doi:10.1016/0167-7152(92)90282-A
[12] F. E. Satterthwaite, “An Approximate Distribution of the Estimates of Variance Components,” Biometrics Bulletin, Vol. 2, No. 6, 1946, pp. 110-114. doi:10.2307/3002019
[13] P. B. Patnaik, “The Non-Central -and -Distribu tions and Their Applications,” Biometrika, Vol. 36, No. 1-2, 1049, pp. 202-232.
[14] C. D. Hou, “A Simple Approximation for the Distribution of the Weighted Combination of Nonindependent or In dependent Probabilities,” Statistics and Probability Let ters, Vol. 73, No. 2, 2005, pp. 179-187. doi:10.1016/j.spl.2004.11.028
[15] P. L. Canner, “An Overview of Six Clinical Trials of Aspirin in the Coronary Heart Disease,” Statistics in Medicine, Vol. 6, No. 3, 1987, pp. 255-263.
[16] M. B. Brown, “A Method for combining Non-Indepen dent, One-Sided Tests of Significance,” Biometrics, Vol. 31, No. 4, 1975, pp. 987-992. doi:10.2307/2529826

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.