Classroom Simulation: Are Variance-Stabilizing Transformations Really Useful?

1. Classroom Simulation: AreVariance-StabilizingTransformations Really Useful?

2. Bruce E. Trumbo*Eric A. SuessRebecca E. Brafman† Department of Statistics California State University, Hayward † Presentation, JSM 2004, Toronto *btrumbo@csuhayward.edu

3. Introduction to One-way ANOVA In a one-way ANOVA, we test the null hypothesis that all group means i are equal against the alternative hypotheses that all group means are not equal. ANOVA Table Source DF SS MS F-Ratio . Factor I – 1 SS(Fact) MS(Fact) MS(Fact)/MS(Err)Error IJ – I SS(Err) MS(Err) .Total IJ – 1

4. Model and Assumptions We use the model: Xij i.i.d.NORM(i,2), for i = 1, …, I and j = 1, …, J. Assumptions: • normal data • independent groups • independent observations within groups • equal variances

5. When Data Are Not Normal… • If H0 True: Distributional difficulties arise • MS(Factor) and MS(Error) not chi-squared • MS(Factor) and MS(Error) not independent • F-ratio not distributed as F • If H0 False: • Different means may imply • Different variances

6. Commonly Recommended Method For Transformating Data to Stabilize Variances Based on two-term Taylor-series approximations. Given relationship between mean and variance: s2 =j(m). The following transformation makes variances approximately equal — even if means differ: Y = f(X), where f’(m)=[j(m)]–1/2

7. Some Types of Nonnormal Data and Their Variance-Stabilizing Transformations

8. Square Root Transformations (Right) of Three Poisson Samples Have Similar Variances

9. Arcsine of Square Root Transformations (Right) of Three Binomial Samples Have Similar Variances

10. Log Transformations (Right) of Three Exponential Samples Have Similar Variances

11. Additional Transformations We also consider rank transformations for exponential data. Possible future work (no results given here): Box-Cox Transformation of the type Y = Xa,where a is based on the data. Examples: • Square root if a = 1/2 • Reciprocal if a = –1 • Interpreted as log transformation if a = 0

12. Simulation Study 1. Simulations are based on data with known distributions: Poisson, binomial, or exponential. 2. Use R, S-Plus, and Minitab. (SAS can also be used but is very time consuming.) 3. In each simulation we generate 20,000 datasets from the nonnormal distribution under study. 4. Each dataset consists of I = 3 groups, usually with J = 5 or 10 observations per group. 5. For each distribution: Datasets under H0, and for a variety of cases with Ha.

13. Comparisons to JudgeUsefulness of Transformations All tests have nominal size  = 5%. P{Rej} is estimated as the proportion of 20,000 simulated datasets in which H0 is rejected. With and without transformation: When is H0 is true, does P{Rej} = 5% ? For various alternatives: When is P{Rej} larger, with or withouttransformation?

14. R / S-Plus Code for Exponential Simulation

15. Summary of Findings Within the limited scope of our study… For Poisson data, the square root transformation seems ineffective. For binomial data, the“arcsine” transformation seems ineffective. For exponential data, both the log and the rank transformations seem to be useful in some cases—particularly for small samples.

16. Some Specific Results: P{Rej} for Poisson DataThree groups, each with 5 observations

17. Some Specific Results: P{Rej} for Binomial ProportionsThree groups, each with 5 observations

18. For Exponential Data Log and Rank Transformations Sometimes UsefulPower = P{Rej|Ha} “often” larger for transformed data (one borderline exceptional case shown)

19. Exponential: Power Against Ha: 1, 10, 100For Various Numbers of ReplicationsLog and rank transformations work well when r is small and population means are widely separated. O = Original * = Log Transf + = Rank Transf.

20. Exponential: Power Against Ha: 1, 2, 4For Various Numbers r of ReplicationsWhen means are not so widely separated, log and rank transformations do some harm unless r is small . O = Original * = Log Transf + = Rank Transf.

21. Exponential: Power for Various AlternativesWhen M = 1, H0 is true; when M = 2, the group means are 1, 2, 4; when M = 4, the group means are 1, 4 , 16; etc. For r = 5 and M > 2 transformations are useful. Solid = Original Dotted = Log Transf Dashed = Rank Transf.

22. Exponential: Power for Various AlternativesWhen M = 1, H0 is true; when M = 2, the group means are 1, 2, 4; when M = 4, the group means a are 1, 4 , 16; etc. For r = 20, transformations may be harmful. Solid = OriginalDotted = Log TransfDashed = Rank Transf.

23. References / Acknowledgments REFERENCES ON VARIANCE STABILIZING TRANSFORMATIONS G. Oehlert: A First Course in Design and Analysis of Experiments, Freeman (2000), Chapter 6. D. Montgomery: Design and Analysis of Experiments,5th ed., Wiley (2001), Chapter 3. K. Brownlee: Statistical Theory and Methodology in Science and Engineering, 2nd ed., Wiley (1965). Chapter 3. H. Scheffé: The Analysis of Variance, Wiley 1959, Chapter 10. G. Snedecor and W. Cochran: Statistical Methods, 7th ed. Iowa State Univ. Press (1980), Chapter 15. WEB PAGES including computer code and results for this paper: www.sci.csuhayward.edu/~btrumbo/JSM2004/simtrans/. THANKS TO Jaimyoung Kwan (UC Berkeley/CSU Hayward) for suggestions, especially concerning the inclusion of power curves.Rebecca Brafman’s graduate study supported by NSF Graduate Research Fellowship.

24. About the Authors • Rebecca E. Brafman, presenting this poster at JSM 2004 in Toronto, has recently completed her M.S. in Statistics from CSU Hayward. • Eric A. Suess received his Ph.D. in Statistics from U.C. Davis and is Associate Professor of Statistics at CSU Hayward. His interests include statistical computation, time series and Bayesian statistics.esuess@csuhayward.edu • Bruce E. Trumbo is a fellow of ASA and IMS and has been a professor in the Statistics Department at CSU State University, Hayward for over 30 years.btrumbo@csuhayward.edu