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# Classroom Simulation: Are Variance-Stabilizing Transformations Really Useful? - PowerPoint PPT Presentation

Classroom Simulation: Are Variance-Stabilizing Transformations Really Useful?. Bruce E. Trumbo * Eric A. Suess Rebecca E. Brafman †. Department of Statistics California State University, Hayward † Presentation, JSM 2004, Toronto * [email protected] Introduction to One-way ANOVA.

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### Bruce E. Trumbo*Eric A. SuessRebecca E. Brafman†

Department of Statistics

California State University, Hayward

† Presentation, JSM 2004, Toronto

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

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

• 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

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

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

Arcsine of Square Root Transformations (Right) of TransformationsThree Binomial Samples Have Similar Variances

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

Simulation Study Have Similar Variances

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.

Comparisons to Judge Have Similar VariancesUsefulness 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?

Summary of Findings Have Similar Variances

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.

Some Specific Results: Have Similar VariancesP{Rej} for Poisson DataThree groups, each with 5 observations

Some Specific Results: Have Similar VariancesP{Rej} for Binomial ProportionsThree groups, each with 5 observations

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

Exponential: Power Against H Usefula: 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.

Exponential: Power Against H Usefula: 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.

Exponential: Power for Various Alternatives UsefulWhen 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.

Exponential: Power for Various Alternatives UsefulWhen 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.

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.