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

Affiliation(s)

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.

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.

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.

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.

Cite this paper

J. Jixiang, L. Zhang and W. Johnson, "The Permutation Test as an Ancillary Procedure for Comparing Zero-Inflated Continuous Distributions,"*Open Journal of Statistics*, Vol. 2 No. 3, 2012, pp. 274-280. doi: 10.4236/ojs.2012.23033.

J. Jixiang, L. Zhang and W. Johnson, "The Permutation Test as an Ancillary Procedure for Comparing Zero-Inflated Continuous Distributions,"

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.

[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.