Examples of Designed Experiments With Nonnormal Responses

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Presentation transcript:

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)

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 )

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

Effect Estimates

Half -Normal Probability Plot , and are significant effects 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)

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 2.88 0.351 0.76

Refit the Model (GLM) We fit the model with factors

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