Design and Analysis of Gauge R&R Studies: Making Decisions with Confidence Intervals in Random and Mixed Anova Models (ASA-SIAM Series on Statistics and Applied Probability)

220 Pages·2005·22.76 MB·English

by  Richard K. Burdick

Design and Analysis
of Gauge R&R Studies
ASA-SIAM Series on
Statistics and Applied Probability
ABA
SIAM
The ASA-SIAM Series on Statistics and Applied Probability is published
jointly by the American Statistical Association and the Society for Industrial and Applied
Mathematics. The series consists of a broad spectrum of books on topics in statistics and
applied probability. The purpose of the series is to provide inexpensive, quality publications
of interest to the intersecting membership of the two societies.
Editorial Board
Martin T. Wells Douglas M. Hawkins
Cornell University, Editor-in-Chief University of Minnesota
David Banks Susan Holmes
Duke University Stanford University
H. T. Banks Lisa LaVange
North Carolina State University Inspire Pharmaceuticals, Inc.
Richard K. Burdick Francoise Seillier-Moiseiwitsch
Arizona State University University of Maryland - Baltimore County
Joseph Gardiner Mark van der Laan
Michigan State University University of California, Berkeley
Burdick, R. K., Borror, C. M., and Montgomery, D. C, Design and Analysis of Gauge R&R
Studies: Making Decisions with Confidence Intervals in Random and Mixed ANOVA Models
Albert, J., Bennett, J., and Cochran, J. J., eds., Anthology of Statistics in Sports
Smith, W. F., Experimental Design for Formulation
Baglivo, J. A., Mathematica Laboratories for Mathematical Statistics: Emphasizing Simulation
and Computer Intensive Methods
Lee, H. K. H., Bayesian Nonparametrics via Neural Networks
O'Gorman, T. W., Applied Adaptive Statistical Methods: Tests of Significance and
Confidence Intervals
Ross, T. J., Booker, J. M., and Parkinson, W. J., eds., Fuzzy Logic and Probability Applications:
Bridging the Gap
Nelson, W. B., Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and
Other Applications
Mason, R. L. and Young, J. C., Multivariate Statistical Process Control with Industrial
Applications
Smith, P. L., A Primer for Sampling Solids, Liquids, and Gases: Based on the Seven Sampling
Errors of Pierre Gy
Meyer, M. A. and Booker, J. M., Eliciting and Analyzing Expert Judgment: A Practical Guide
Latouche, G. and Ramaswami, V., Introduction to Matrix Analytic Methods in Stochastic
Modeling
Peck, R., Haugh, L., and Goodman, A., Statistical Case Studies: A Collaboration Between
Academe and Industry, Student Edition
Peck, R., Haugh, L., and Goodman, A., Statistical Case Studies: A Collaboration Between
Academe and Industry
Barlow, R., Engineering Reliability
Czitrom, V. and Spagon, P. D., Statistical Case Studies for Industrial Process Improvement
Design and Analysis
of Gauge R&R Studies
Making Decisions with Confidence Intervals
in Random and Mixed ANOVA Models
Richard K. Burdick
Arizona State University
Tempe, Arizona
Connie M. Borror
University of Illinois at Urbana-Champaign
Urbana, Illinois
Douglas C. Montgomery
Arizona State University
Tempe, Arizona
Society for Industrial and Applied Mathematics American Statistical Association
Philadelphia, Pennsylvania Alexandria, Virginia
The correct bibliographic citation for this book is as follows: Burdick, Richard K.,
Connie M. Borror, and Douglas C. Montgomery, Design and Analysis of Gauge R&R
Studies: Making Decisions with Confidence Intervals in Random and Mixed ANOVA
Models, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia,
ASA, Alexandria, VA, 2005.
Copyright © 2005 by the American Statistical Association and the Society for
Industrial and Applied Mathematics.
109876 543 2 1
All rights reserved. Printed in the United States of America. No part of this book
may be reproduced, stored, or transmitted in any manner without the written
permission of the publisher. For information, write to the Society for Industrial and
Applied Mathematics, 3600 University City Science Center, Philadelphia, PA
19104-2688.
Library of Congress Cataloging-in-Publication Data
Burdick, Richard K.
Design and analysis of gauge R&R studies : making decisions with confidence
intervals in random and mixed ANOVA models / Richard K. Burdick, Connie M.
Borror, and Douglas c. Montgomery,
p. cm.
ISBN 0-89871-588-1 (pbk.)
1. Confidence intervals. 2. Analysis of variance. 3. Experimental design.
I. Borror, Connie M. II. Montgomery, Douglas C. III. Title.
QA276.74.B88 2005
519.5'4-dc22
2005044113
is a registered trademark.
Contents
List of Figures xi
List of Tables xiii
Preface xvii
1 Introduction 1
1.1 Objectives 1
1.2 Gauge R&R Studies 1
1.3 Capability Measures 3
1.4 Confidence Intervals 5
1.5 R&R Graphs 6
1.6 Misclassification Rates 7
1.7 Determination of F-Values 10
1.8 Computer Programs 10
1.9 Summary 11
2 Balanced One-Factor Random Models 13
2.1 Introduction 13
2.2 The Model 14
2.3 MLS Confidence Intervals 14
2.3.1 Interval for p 16
Y
2.3.2 Interval for y 16
p
2.3.3 Interval for y 17
u
2.3.4 Interval for y 17
R
2.4 Generalized Confidence Intervals 18
2.5 Numerical Example 19
2.5.1 Interval for/it 19
y
2.5.2 Intervals for y 20
p
2.5.3 Interval for X 21
M
2.5.4 Interval for y 21
R
2.5.5 Interval for PTR 22
2.5.6 Interval for SNR 22
2.5.7 Interval for C 23
p
v
vi Contents
2.5.8 Intervals for Misclassification Rates 23
2.5.9 Conclusions 24
2.6 Summary 25
3 Balanced Two-Factor Crossed Random Models with Interaction 27
3.1 Introduction 27
3.2 The Model 27
3.3 MLS Intervals, Gauge R&R Parameters 29
3.3.1 Interval for/A 31
y
3.3.2 Interval for y 31
p
3.3.3 Interval for y 32
u
3.3.4 Interval for y 32
R
3.4 Generalized Intervals, Gauge R&R Parameters 32
3.5 Numerical Example 34
3.5.1 Intervals for (JL 34
Y
3.5.2 Intervals for y 35
f
3.5.3 Intervals for y 36
u
3.5.4 Interval for y 37
R
3.5.5 Interval for PTR 38
3.5.6 Interval for SNR 38
3.5.7 Interval for C 38
p
3.5.8 Intervals for Misclassification Rates 39
3.5.9 Conclusions 39
3.6 Comparing Repeatability and Reproducibility 39
3.7 Comparing Two Measurement Systems 44
3.8 Intervals for Other Parameters 47
3.8.1 Intervals for OT0,<7| , and crj 48
O
3.8.2 Interval for y 49
Y
3.8.3 Intervalsforcr^/cr|, crj/o-|, anda| /or| 49
0
3.8.4 Intervals for ffQ/x,orp /y , and cr|/x 50
r O x y
3.9 Summary 51
4 Design of Gauge R&R Experiments S3
4.1 Introduction 53
4.2 Guidelines for Designing Experiments 54
4.3 Factor Structures in Gauge R&R Experiments 57
4.4 Model Adequacy Checking 58
4.5 Summary 62
5 Balanced Two-Factor Crossed Random Models with No Interaction 65
5.1 Introduction 65
5.2 The Model 65
5.3 MLS Intervals, Gauge R&R Parameters 65
5.3.1 Interval for ^ 67
5.3.2 Interval for y 68
p
Contents vii
5.3.3 Interval for y 68
M
5.3.4 Interval for y 69
R
5.4 Generalized Intervals, Gauge R&R Parameters 69
5.5 Numerical Example 70
5.5.1 Interval for Ai 70
y
5.5.2 Interval for y 72
p
5.5.3 Interval for y 73
u
5.5.4 Interval for y 73
s
5.5.5 Interval for PTR 74
5.5.6 Interval for SNR 74
5.5.7 Interval for C 75
p
5.5.8 Intervals for Misclassification Rates 75
5.5.9 Conclusions 75
5.6 Intervals for Other Parameters 76
5.6.1 Intervals for 00 and<r| 77
5.6.2 Interval for y 78
r
5.6.3 Intervals for cr|/or| and <r^/a| 78
5.6.4 Intervals for 0-0/x,, andcrj/x 79
y
5.7 Summary 80
6 Balanced Two-Factor Crossed Mixed Models 81
6.1 Introduction 81
6.2 The Mixed Model Formulation 81
6.3 The Two-Factor Mixed Model with Interaction 84
6.4 MLS Intervals, Gauge R&R Parameters 84
6.4.1 Interval for/A 88
r
6.4.2 Interval for £t 88
;
6.4.3 Interval for y 88
p
6.4.4 Interval for y . 89
u
6.4.5 Interval for y 89
K
6.5 Generalized Intervals, Gauge R&R Parameters 90
6.6 Misclassification Rates 91
6.7 Numerical Example . 92
6.7.1 Interval for ^ 93
6.7.2 Interval for p. 93
6.7.3 Interval for y 94
f
6.7.4 Interval for y 95
u
6.7.5 Interval for y 96
K
6.7.6 Interval for PTR 96
6.7.7 Interval for SNR 97
6.7.8 Interval for C 97
p
6.7.9 Intervals for Misclassification Rates 97
6.7.10 Conclusions 98
6.8 Confidence Intervals for Contrasts of/>t 101
;
viii Contents
6.9 The Two-Factor Mixed Model with No Interaction 102
6.9.1 Intervals for R&R Parameters 104
6.9.2 Intervals for Misclassification Rates 105
6.9.3 Intervals for n, fij, and a) = E/cy/Li/ 106
r
6.9.4 Numerical Example 106
6.10 Summary 109
7 Unbalanced One- and Two-Factor Models 111
7.1 Introduction Ill
7.2 Unbalanced One-Factor Random Models Ill
7.2.1 Interval for ^ 114
y
7.2.2 Interval for y 115
p
7.2.3 Interval for y 116
M
7.2.4 Interval for y 116
R
7.2.5 GCIs 117
7.2.6 Intervals for Misclassification Rates 117
7.2.7 Numerical Example 118
7.3 Unbalanced Two-Factor Random Model with Interaction 124
7.3.1 Confidence Intervals for Gauge R&R Parameters 125
7.3.2 Intervals for Misclassification Rates 128
7.3.3 Numerical Example 128
7.3.4 Modification for Fixed Operators 134
7.4 Unbalanced Two-Factor Random Model with No Interaction 137
7.4.1 Confidence Intervals for Gauge R&R Parameters 137
7.4.2 Intervals for Misclassification Rates 139
7.4.3 Numerical Example 139
7.4.4 Modification for Fixed Operators 140
7.5 Summary 141
7.6 Appendix: Generalized Confidence Interval for y in Model (7.1) . . . . 141
p
8 Strategies for Constructing Intervals with ANOVA Models 143
8.1 Introduction 143
8.2 General Strategies 143
8.2.1 MLS Intervals for y 146
8.2.2 MLS Intervals for y 147
2
8.2.3 MLS Intervals for y 150
3
8.2.4 MLS Intervals for y 152
4
8.2.5 Modification for Mixed Models 153
8.2.6 Modification for Unbalanced Designs 153
8.2.7 Summary 153
8.3 A Two-Fold Nested Design 153
8.3.1 Interval for /u 155
y
8.3.2 Interval for y 157
p
8.3.3 Interval for y 158
M
8.3.4 Interval for y 158
R
Contents ix
8.3.5 Generalized Confidence Intervals 159
8.3.6 Intervals for Misclassification Rates 159
8.3.7 Conclusions 161
8.3.8 Other MLS Intervals 161
8.3.9 Unbalanced Modifications 162
8.4 Three-Factor Crossed Random Designs 163
8.4.1 Interval for y 165
p
8.4.2 Interval for y 166
M
8.4.3 Interval for y 166
s
8.4.4 Intervals for Misclassification Rates 167
8.4.5 Conclusions 167
8.5 Designs with Both Nested and Crossed Factors 169
8.5.1 Interval for y 170
p
8.5.2 Interval for y 172
u
8.5.3 Interval for y 172
R
8.5.4 Intervals for Misclassification Rates 173
8.5.5 Conclusions 173
8.6 Summary 174
A The Analysis of Variance 175
B MLS and GCI Methods 179
B.I MLS Method 179
B.2 GCIs 181
C Tables of F- Values 183
Bibliography 193
Index 199