The Permutation Test as an Ancillary Procedure for Comparing Zero-Inflated Continuous Distributions

Jixiang Jixiang, Lei Zhang, William D Johnson

CSIRO Departments of Plant Science and Mathematics & Statistics, South Dakota State University, Brookings, SD 57007.
Office of Health Data and Research, Mississippi State Department of Health,570 East Woodrow Wilson, Jackson, MS 39215-1700.
Pennington Biomedical Research Center, Louisiana State University System, 6400 Perkins Road,Baton Rouge, LA 70808.

DOI: 10.4236/ojs.2012.23033                

Abstract

Empirical estimates of power and Type I error can be misleading if a statistical test does not perform at the stated rejection level under the null hypothesis. We employed the permutation test to control the empirical type I errors for zero-inflated exponential distributions. The simulation results indicated that the permutation test can be used effectively to control the type I errors near the nominal level even the sample sizes are small based on four statistical tests. Our results attest to the permutation test being a valuable adjunct to the current statistical methods for comparing distributions with underlying zero-inflated data structures.

Keywords

Central Limit Theorem; Likelihood Ratio Test; Modified Central Limit Theorem; Permutation Test; Wald Test; Zero-Inflated Distribution

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. M. Titterington, A. F. Smith and U. E. Makov, “Statistical Analysis of Finite Mixture Distributions,” John Wiley and Sons, New York, 1985.
[2]	J. Lawless, “Negative Binomial and Mixed Poisson Regression,” Canadian Journal of Statistics, Vol. 15, No. 3, 1987, pp. 209-225. doi:10.2307/3314912
[3]	D. C. Heibron, “Generalized Linear Models for Altered Zero Probability and Overdispersion in Count Data,” SIMS Technical Report No. 9, University of California, San Francisco, 1989.
[4]	R. Schall, “Estimation in Generalized Linear Models with Random Effects,” Biometrika, Vol. 78, No. 4, 1991, pp. 719-727. doi:10.1093/biomet/78.4.719
[5]	C. E. McCulloch, “Maximum Likelihood Algorithms for Generalized Linear Mixed Models,” Journal of American Statistical Association, Vol. 92, No. 437, 1997, pp. 162- 170.
[6]	D. B. Hall, “Zero-Inflated Poisson and Binomial Regression with Random Effects: A Case Study,” Biometrics, Vol. 56, No. 4, 2000, pp. 1030-1039. doi:10.1111/j.0006-341X.2000.01030.x
[7]	L. Zhang, J. Wu and W. D. Johnson, “Empirical Study of Six Tests for Equality of Populations with Zero-Inflated Continuous Distributions,” Communications in Statistics —Simulation and Computation, Vol. 39, No. 6, 2010, pp. 1196-1211. doi:10.1080/03610918.2010.489169
[8]	G. Casella and R. L. Berger, “Statistical Inference,” Duxbury Inc., San Francisco, 2002.
[9]	A. Wald, “Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations Is Large,” Transactions in American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482.
[10]	E. S. Edgington, “Statistical Inference and Nonrandom Samples,” Psychological Series A, Vol. 66, No. 6, 1966, pp. 485-487. doi:10.1037/h0023916
[11]	B. E. Wampold and N. L. Worsham, “Randomization Tests for Multiple Baseline Designs,” Behavioral Assessment, Vol. 8, 1986, pp. 135-143.
[12]	R. C. Blair and W. Karniski, “An Alternative Method for Significance Testing of Waveform Difference Potentials,” Psychophysiology, Vol. 30, No. 5, 1993, pp. 518-524. doi:10.1111/j.1469-8986.1993.tb02075.x
[13]	D. C. Adams and C. D. Anthony, “Using Randomization Techniques to Analyze Behavioural Data,” Animal Behaviour, Vol. 61, No. 1, 1996, pp. 733-738. doi:10.1006/anbe.2000.1576
[14]	J. Ludbrook and H. Dudley, “Why Permutation Tests Are Superior to t and F Tests in Biomedical Research,” American Statistician Association, Vol. 52, No. 2, 1998, pp. 127-132.
[15]	A. F. Hayes, “Randomization Tests and Equality of Variance Assumption When Comparing Group Means,” Animal Behaviour, Vol. 59, No. 3, 2000, pp. 653-656. doi:10.1006/anbe.1999.1366
[16]	L. H. Koopman, “Introduction of Contemporary Statistical Methods,” 2nd Edition, Duxbury Press, Boston, 1981.
[17]	J. Aitchison, “On the Distribution of a Positive Random Variable Having a Discrete Probability Mass at the Origin,” Journal of American Statistical Association, Vol. 50, No. 271, 1995, pp. 901-908.
[18]	S. C. Wang, “Analysis of Zero-Heavy Data Using a Mixture Model Approach,” Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, 1998.