Examples of Designed Experiments With Nonnormal Responses SHARON L. LEWIS, DOUGLAS C. MONTGOMERY and...
-
Upload
elfreda-simmons -
Category
Documents
-
view
220 -
download
2
Transcript of Examples of Designed Experiments With Nonnormal Responses SHARON L. LEWIS, DOUGLAS C. MONTGOMERY and...
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