Examples of Designed Experiments With Nonnormal Responses

SHARON L. LEWIS, DOUGLAS C. MONTGOMERY and

RAYMOND H. MYERS

Journal of Quality Technology, 33, pp. 265-278, 2001

演講者 : 張秉鈞

Outline

Introduction

Example 1: The Drill Experiment

Example 2: The Windshield Molding Slugging Experiment

Conclusion

Introduction

In general, linear model :

Check model’s three basic assumption 1. Normal probability plot 2. Residuals plot

Nonnormal responses 1. data transformations 2. GLM (Generalized Linear Models)

),0(~ , 2110 Nxxy kk

Generalized Linear Models Three components: (1) Response distribution is exponential family (Binomial, Poisson, Gamma, Normal, etc)

(2) Linear predictor

(3) Link function (relationship between the and )

kk xxx 110'

'x

')( xs

More details: Introduction to Linear Regression Analysis

(Chapter 13) Software packages: SAS, S-PLUS

Objective: To compare two approaches by designed experiments with nonnormal responses

Criterion: Lengths of confidence intervals of mean response

The Drill Experiment

unreplicated factorial design

advance rate drill load flow rate rotational speed type of drilling mud used

GLM: Gamma distribution, log link function

42

:y

:1x

:2x

:3x

:4x

Effect Estimates

Half -Normal Probability Plot

3x 4x, and are significant effects

2x

432 1633.05772.02900.05977.1ˆ xxxey

432 1656.05789.02895.06032.1ˆ xxxey

data transformation model:

GLM model:

95% Confidence Interval On the Means

The Windshield Molding Slugging Experiment

During the stamping process, debris carried into the die appears as slugs in the product

fractional factorial design, and resolution III

number of good parts out of 1000 poly-film thickness (0.0025, 0.00175) oil mixture (1:20, 1:10) gloves (cotton, nylon) metal blanks (dry underside, oily underside)

142

:y

:1x

:2x

:3x

:4x

431 xxxI

Design Matrix and Response Data

data transformation: logistic GLM: Binomial distribution, logistic link function

Hamada and Nelder (1997)

Effect Estimates

Std. Err. t

Intercept

-0.513 0.28 -1.80

2.971 0.46 6.43

-0.270 0.32 -0.84

1.329 0.46 2.88

0.351 0.46 0.76

1x

2x

3x

4x

Refit the Model (GLM)

We fit the model with factors 2132431 and , , , xxxxxxx

95% Confidence Intervals On the Means

Conclusion Data transformations may be inappropriate for

some situations

With the GLM, normality and constant variance are not required

With the GLM, length of confidence interval is short

References

HAMADA, M. and NELDER, J. A. (1997). “Generalized Linear Models for Quality-Improvement Experiments”. Journal of Quality Technology 29, pp. 292-304

MONTGOMERY, D. C. (2001). Design and Analysis of Experiments, 5th ed. John Wiley & Sons, Inc., New York, NY

MONTGOMERY, D. C. and PECK, E. A. (1992). Introduction to Linear Regression Analysis, 2th ed. John Wiley & Sons, Inc., New York, NY

MYERS, R. H. and MONTGOMERY, D. C. (1997). “A Tutorial on Generalized Linear Models”. Journal of Quality Technology 29, pp. 274-291