Engineering statistics handbook

Table of contents :
nist.gov……Page 0
1. Exploratory Data Analysis……Page 1
1. Exploratory Data Analysis……Page 2
1.1. EDA Introduction……Page 10
1.1.1. What is EDA?……Page 11
1.1.2. How Does Exploratory Data Analysis differ from Classical Data Analysis?……Page 13
1.1.2.1. Model……Page 15
1.1.2.2. Focus……Page 16
1.1.2.3. Techniques……Page 17
1.1.2.4. Rigor……Page 18
1.1.2.5. Data Treatment……Page 19
1.1.2.6. Assumptions……Page 20
1.1.3. How Does Exploratory Data Analysis Differ from Summary Analysis?……Page 21
1.1.4. What are the EDA Goals?……Page 22
1.1.5. The Role of Graphics……Page 23
1.1.6. An EDA/Graphics Example……Page 25
1.1.7. General Problem Categories……Page 30
1.2. EDA Assumptions……Page 34
1.2.1. Underlying Assumptions……Page 35
1.2.2. Importance……Page 37
1.2.3. Techniques for Testing Assumptions……Page 38
1.2.4. Interpretation of 4-Plot……Page 41
1.2.5. Consequences……Page 43
1.2.5.1. Consequences of Non-Randomness……Page 44
1.2.5.2. Consequences of Non-Fixed Location Parameter……Page 46
1.2.5.3. Consequences of Non-Fixed Variation Parameter……Page 47
1.2.5.4. Consequences Related to Distributional Assumptions……Page 48
1.3. EDA Techniques……Page 50
1.3.1. Introduction……Page 51
1.3.2. Analysis Questions……Page 52
1.3.3. Graphical Techniques: Alphabetic……Page 54
1.3.3.1. Autocorrelation Plot……Page 57
1.3.3.1.1. Autocorrelation Plot: Random Data……Page 62
1.3.3.1.2. Autocorrelation Plot: Moderate Autocorrelation……Page 64
1.3.3.1.3. Autocorrelation Plot: Strong Autocorrelation and Autoregressive Model……Page 66
1.3.3.1.4. Autocorrelation Plot: Sinusoidal Model……Page 68
1.3.3.2. Bihistogram……Page 70
1.3.3.3. Block Plot……Page 73
1.3.3.4. Bootstrap Plot……Page 77
1.3.3.5. Box-Cox Linearity Plot……Page 80
1.3.3.6. Box-Cox Normality Plot……Page 83
1.3.3.7. Box Plot……Page 86
1.3.3.8. Complex Demodulation Amplitude Plot……Page 89
1.3.3.9. Complex Demodulation Phase Plot……Page 92
1.3.3.10. Contour Plot……Page 95
1.3.3.10.1. DEX Contour Plot……Page 98
1.3.3.11. DEX Scatter Plot……Page 102
1.3.3.12. DEX Mean Plot……Page 107
1.3.3.13. DEX Standard Deviation Plot……Page 110
1.3.3.14. Histogram……Page 113
1.3.3.14.1. Histogram Interpretation: Normal……Page 117
1.3.3.14.2. Histogram Interpretation: Symmetric, Non-Normal, Short-Tailed……Page 119
1.3.3.14.3. Histogram Interpretation: Symmetric, Non-Normal, Long-Tailed……Page 122
1.3.3.14.4. Histogram Interpretation: Symmetric and Bimodal……Page 124
1.3.3.14.5. Histogram Interpretation: Bimodal Mixture of 2 Normals……Page 126
1.3.3.14.6. Histogram Interpretation: Skewed (Non-Normal) Right……Page 128
1.3.3.14.7. Histogram Interpretation: Skewed (Non-Symmetric) Left……Page 131
1.3.3.14.8. Histogram Interpretation: Symmetric with Outlier……Page 132
1.3.3.15. Lag Plot……Page 134
1.3.3.15.1. Lag Plot: Random Data……Page 136
1.3.3.15.2. Lag Plot: Moderate Autocorrelation……Page 138
1.3.3.15.3. Lag Plot: Strong Autocorrelation and Autoregressive Model……Page 140
1.3.3.15.4. Lag Plot: Sinusoidal Models and Outliers……Page 142
1.3.3.16. Linear Correlation Plot……Page 145
1.3.3.17. Linear Intercept Plot……Page 148
1.3.3.18. Linear Slope Plot……Page 150
1.3.3.19. Linear Residual Standard Deviation Plot……Page 152
1.3.3.20. Mean Plot……Page 155
1.3.3.21. Normal Probability Plot……Page 158
1.3.3.21.1. Normal Probability Plot: Normally Distributed Data……Page 161
1.3.3.21.2. Normal Probability Plot: Data Have Short Tails……Page 163
1.3.3.21.3. Normal Probability Plot: Data Have Long Tails……Page 165
1.3.3.21.4. Normal Probability Plot: Data are Skewed Right……Page 167
1.3.3.22. Probability Plot……Page 169
1.3.3.23. Probability Plot Correlation Coefficient Plot……Page 173
1.3.3.24. Quantile-Quantile Plot……Page 177
1.3.3.25. Run-Sequence Plot……Page 180
1.3.3.26. Scatter Plot……Page 182
1.3.3.26.1. Scatter Plot: No Relationship……Page 185
1.3.3.26.2. Scatter Plot: Strong Linear (positive correlation) Relationship……Page 186
1.3.3.26.3. Scatter Plot: Strong Linear (negative correlation) Relationship……Page 187
1.3.3.26.4. Scatter Plot: Exact Linear (positive correlation) Relationship……Page 188
1.3.3.26.5. Scatter Plot: Quadratic Relationship……Page 190
1.3.3.26.6. Scatter Plot: Exponential Relationship……Page 192
1.3.3.26.7. Scatter Plot: Sinusoidal Relationship (damped)……Page 194
1.3.3.26.8. Scatter Plot: Variation of Y Does Not Depend on X (homoscedastic)……Page 196
1.3.3.26.9. Scatter Plot: Variation of Y Does Depend on X (heteroscedastic)……Page 198
1.3.3.26.10. Scatter Plot: Outlier……Page 200
1.3.3.26.11. Scatterplot Matrix……Page 202
1.3.3.26.12. Conditioning Plot……Page 205
1.3.3.27. Spectral Plot……Page 208
1.3.3.27.1. Spectral Plot: Random Data……Page 211
1.3.3.27.2. Spectral Plot: Strong Autocorrelation and Autoregressive Model……Page 213
1.3.3.27.3. Spectral Plot: Sinusoidal Model……Page 215
1.3.3.28. Standard Deviation Plot……Page 217
1.3.3.29. Star Plot……Page 220
1.3.3.30. Weibull Plot……Page 223
1.3.3.31. Youden Plot……Page 226
1.3.3.31.1. DEX Youden Plot……Page 228
1.3.3.32. 4-Plot……Page 231
1.3.3.33. 6-Plot……Page 236
1.3.4. Graphical Techniques: By Problem Category……Page 240
1.3.5. Quantitative Techniques……Page 244
1.3.5.1. Measures of Location……Page 248
1.3.5.2. Confidence Limits for the Mean……Page 253
1.3.5.3. Two-Sample t-Test for Equal Means……Page 257
1.3.5.3.1. Data Used for Two-Sample t-Test……Page 261
1.3.5.4. One-Factor ANOVA……Page 267
1.3.5.5. Multi-factor Analysis of Variance……Page 271
1.3.5.6. Measures of Scale……Page 276
1.3.5.7. Bartlett’s Test……Page 282
1.3.5.8. Chi-Square Test for the Standard Deviation……Page 285
1.3.5.8.1. Data Used for Chi-Square Test for the Standard Deviation……Page 289
1.3.5.9. F-Test for Equality of Two Standard Deviations……Page 292
1.3.5.10. Levene Test for Equality of Variances……Page 295
1.3.5.11. Measures of Skewness and Kurtosis……Page 299
1.3.5.12. Autocorrelation……Page 303
1.3.5.13. Runs Test for Detecting Non-randomness……Page 307
1.3.5.14. Anderson-Darling Test……Page 312
1.3.5.15. Chi-Square Goodness-of-Fit Test……Page 317
1.3.5.16. Kolmogorov-Smirnov Goodness-of-Fit Test……Page 323
1.3.5.17. Grubbs’ Test for Outliers……Page 329
1.3.5.18. Yates Analysis……Page 333
1.3.5.18.1. Defining Models and Prediction Equations……Page 338
1.3.5.18.2. Important Factors……Page 341
1.3.6. Probability Distributions……Page 348
1.3.6.1. What is a Probability Distribution……Page 349
1.3.6.2. Related Distributions……Page 351
1.3.6.3. Families of Distributions……Page 359
1.3.6.4. Location and Scale Parameters……Page 361
1.3.6.5. Estimating the Parameters of a Distribution……Page 366
1.3.6.5.1. Method of Moments……Page 367
1.3.6.5.2. Maximum Likelihood……Page 368
1.3.6.5.3. Least Squares……Page 371
1.3.6.5.4. PPCC and Probability Plots……Page 372
1.3.6.6. Gallery of Distributions……Page 374
1.3.6.6.1. Normal Distribution……Page 377
1.3.6.6.2. Uniform Distribution……Page 384
1.3.6.6.3. Cauchy Distribution……Page 391
1.3.6.6.4. t Distribution……Page 398
1.3.6.6.5. F Distribution……Page 402
1.3.6.6.6. Chi-Square Distribution……Page 406
1.3.6.6.7. Exponential Distribution……Page 410
1.3.6.6.8. Weibull Distribution……Page 417
1.3.6.6.9. Lognormal Distribution……Page 424
1.3.6.6.10. Fatigue Life Distribution……Page 432
1.3.6.6.11. Gamma Distribution……Page 439
1.3.6.6.12. Double Exponential Distribution……Page 446
1.3.6.6.13. Power Normal Distribution……Page 453
1.3.6.6.14. Power Lognormal Distribution……Page 460
1.3.6.6.15. Tukey-Lambda Distribution……Page 466
1.3.6.6.16. Extreme Value Type I Distribution……Page 470
1.3.6.6.17. Beta Distribution……Page 482
1.3.6.6.18. Binomial Distribution……Page 486
1.3.6.6.19. Poisson Distribution……Page 490
1.3.6.7. Tables for Probability Distributions……Page 494
1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution……Page 495
1.3.6.7.2. Upper Critical Values of the Student’s-t Distribution……Page 499
1.3.6.7.3. Upper Critical Values of the F Distribution……Page 507
1.3.6.7.4. Critical Values of the Chi-Square Distribution……Page 545
1.3.6.7.5. Critical Values of the t* Distribution……Page 560
1.3.6.7.6. Critical Values of the Normal PPCC Distribution……Page 563
1.4. EDA Case Studies……Page 567
1.4.1. Case Studies Introduction……Page 568
1.4.2. Case Studies……Page 572
1.4.2.1. Normal Random Numbers……Page 574
1.4.2.1.1. Background and Data……Page 575
1.4.2.1.2. Graphical Output and Interpretation……Page 578
1.4.2.1.3. Quantitative Output and Interpretation……Page 582
1.4.2.1.4. Work This Example Yourself……Page 589
1.4.2.2. Uniform Random Numbers……Page 591
1.4.2.2.1. Background and Data……Page 592
1.4.2.2.2. Graphical Output and Interpretation……Page 595
1.4.2.2.3. Quantitative Output and Interpretation……Page 602
1.4.2.2.4. Work This Example Yourself……Page 609
1.4.2.3. Random Walk……Page 612
1.4.2.3.1. Background and Data……Page 613
1.4.2.3.2. Test Underlying Assumptions……Page 625
1.4.2.3.3. Develop A Better Model……Page 632
1.4.2.3.4. Validate New Model……Page 634
1.4.2.3.5. Work This Example Yourself……Page 638
1.4.2.4. Josephson Junction Cryothermometry……Page 640
1.4.2.4.1. Background and Data……Page 641
1.4.2.4.2. Graphical Output and Interpretation……Page 645
1.4.2.4.3. Quantitative Output and Interpretation……Page 649
1.4.2.4.4. Work This Example Yourself……Page 657
1.4.2.5. Beam Deflections……Page 659
1.4.2.5.1. Background and Data……Page 660
1.4.2.5.2. Test Underlying Assumptions……Page 666
1.4.2.5.3. Develop a Better Model……Page 675
1.4.2.5.4. Validate New Model……Page 679
1.4.2.5.5. Work This Example Yourself……Page 682
1.4.2.6. Filter Transmittance……Page 685
1.4.2.6.1. Background and Data……Page 686
1.4.2.6.2. Graphical Output and Interpretation……Page 688
1.4.2.6.3. Quantitative Output and Interpretation……Page 692
1.4.2.6.4. Work This Example Yourself……Page 698
1.4.2.7. Standard Resistor……Page 700
1.4.2.7.1. Background and Data……Page 701
1.4.2.7.2. Graphical Output and Interpretation……Page 724
1.4.2.7.3. Quantitative Output and Interpretation……Page 728
1.4.2.7.4. Work This Example Yourself……Page 735
1.4.2.8. Heat Flow Meter 1……Page 738
1.4.2.8.1. Background and Data……Page 739
1.4.2.8.2. Graphical Output and Interpretation……Page 744
1.4.2.8.3. Quantitative Output and Interpretation……Page 748
1.4.2.8.4. Work This Example Yourself……Page 756
1.4.2.9. Airplane Polished Window Strength……Page 758
1.4.2.9.1. Background and Data……Page 759
1.4.2.9.2. Graphical Output and Interpretation……Page 761
1.4.2.9.3. Weibull Analysis……Page 768
1.4.2.9.4. Lognormal Analysis……Page 771
1.4.2.9.5. Gamma Analysis……Page 773
1.4.2.9.6. Power Normal Analysis……Page 775
1.4.2.9.7. Fatigue Life Analysis……Page 777
1.4.2.9.8. Work This Example Yourself……Page 779
1.4.2.10. Ceramic Strength……Page 783
1.4.2.10.1. Background and Data……Page 784
1.4.2.10.2. Analysis of the Response Variable……Page 797
1.4.2.10.3. Analysis of the Batch Effect……Page 800
1.4.2.10.4. Analysis of the Lab Effect……Page 805
1.4.2.10.5. Analysis of Primary Factors……Page 808
1.4.2.10.6. Work This Example Yourself……Page 815
1.4.3. References For Chapter 1: Exploratory Data Analysis……Page 818
2. Measurement Process Characterization……Page 823
2. Measurement Process Characterization……Page 824
2.1. Characterization……Page 832
2.1.1. What are the issues for characterization?……Page 833
2.1.1.1. Purpose……Page 835
2.1.1.2. Reference base……Page 836
2.1.1.3. Bias and Accuracy……Page 837
2.1.1.4. Variability……Page 839
2.1.2. What is a check standard?……Page 842
2.1.2.1. Assumptions……Page 844
2.1.2.2. Data collection……Page 846
2.1.2.3. Analysis……Page 848
2.2. Statistical control of a measurement process……Page 851
2.2.1. What are the issues in controlling the measurement process?……Page 852
2.2.2. How are bias and variability controlled?……Page 854
2.2.2.1. Shewhart control chart……Page 857
2.2.2.1.1. EWMA control chart……Page 859
2.2.2.2. Data collection……Page 861
2.2.2.3. Monitoring bias and long-term variability……Page 864
2.2.2.4. Remedial actions……Page 867
2.2.3. How is short-term variability controlled?……Page 869
2.2.3.1. Control chart for standard deviations……Page 870
2.2.3.2. Data collection……Page 872
2.2.3.3. Monitoring short-term precision……Page 874
2.2.3.4. Remedial actions……Page 876
2.3. Calibration……Page 877
2.3.1. Issues in calibration……Page 879
2.3.1.1. Reference base……Page 880
2.3.1.2. Reference standards……Page 882
2.3.2. What is artifact (single-point) calibration?……Page 883
2.3.3. What are calibration designs?……Page 885
2.3.3.1. Elimination of special types of bias……Page 888
2.3.3.1.1. Left-right (constant instrument) bias……Page 889
2.3.3.1.2. Bias caused by instrument drift……Page 891
2.3.3.2. Solutions to calibration designs……Page 893
2.3.3.2.1. General matrix solutions to calibration designs……Page 898
2.3.3.3. Uncertainties of calibrated values……Page 903
2.3.3.3.1. Type A evaluations for calibration designs……Page 904
2.3.3.3.2. Repeatability and level-2 standard deviations……Page 907
2.3.3.3.3. Combination of repeatability and level-2 standard deviations……Page 909
2.3.3.3.4. Calculation of standard deviations for 1,1,1,1 design……Page 910
2.3.3.3.5. Type B uncertainty……Page 913
2.3.3.3.6. Expanded uncertainties……Page 915
2.3.4. Catalog of calibration designs……Page 917
2.3.4.1. Mass weights……Page 920
2.3.4.1.1. Design for 1,1,1……Page 924
2.3.4.1.2. Design for 1,1,1,1……Page 926
2.3.4.1.3. Design for 1,1,1,1,1……Page 928
2.3.4.1.4. Design for 1,1,1,1,1,1……Page 931
2.3.4.1.5. Design for 2,1,1,1……Page 934
2.3.4.1.6. Design for 2,2,1,1,1……Page 936
2.3.4.1.7. Design for 2,2,2,1,1……Page 939
2.3.4.1.8. Design for 5,2,2,1,1,1……Page 941
2.3.4.1.9. Design for 5,2,2,1,1,1,1……Page 943
2.3.4.1.10. Design for 5,3,2,1,1,1……Page 946
2.3.4.1.11. Design for 5,3,2,1,1,1,1……Page 949
2.3.4.1.12. Design for 5,3,2,2,1,1,1……Page 952
2.3.4.1.13. Design for 5,4,4,3,2,2,1,1……Page 955
2.3.4.1.14. Design for 5,5,2,2,1,1,1,1……Page 958
2.3.4.1.15. Design for 5,5,3,2,1,1,1……Page 961
2.3.4.1.16. Design for 1,1,1,1,1,1,1,1 weights……Page 964
2.3.4.1.17. Design for 3,2,1,1,1 weights……Page 967
2.3.4.1.18. Design for 10-and 20-pound weights……Page 970
2.3.4.2. Drift-elimination designs for gauge blocks……Page 972
2.3.4.2.1. Doiron 3-6 Design……Page 976
2.3.4.2.2. Doiron 3-9 Design……Page 978
2.3.4.2.3. Doiron 4-8 Design……Page 980
2.3.4.2.4. Doiron 4-12 Design……Page 982
2.3.4.2.5. Doiron 5-10 Design……Page 984
2.3.4.2.6. Doiron 6-12 Design……Page 986
2.3.4.2.7. Doiron 7-14 Design……Page 988
2.3.4.2.8. Doiron 8-16 Design……Page 991
2.3.4.2.9. Doiron 9-18 Design……Page 994
2.3.4.2.10. Doiron 10-20 Design……Page 997
2.3.4.2.11. Doiron 11-22 Design……Page 1000
2.3.4.3. Designs for electrical quantities……Page 1003
2.3.4.3.1. Left-right balanced design for 3 standard cells……Page 1005
2.3.4.3.2. Left-right balanced design for 4 standard cells……Page 1007
2.3.4.3.3. Left-right balanced design for 5 standard cells……Page 1009
2.3.4.3.4. Left-right balanced design for 6 standard cells……Page 1011
2.3.4.3.5. Left-right balanced design for 4 references and 4 test items……Page 1014
2.3.4.3.6. Design for 8 references and 8 test items……Page 1017
2.3.4.3.7. Design for 4 reference zeners and 2 test zeners……Page 1021
2.3.4.3.8. Design for 4 reference zeners and 3 test zeners……Page 1024
2.3.4.3.9. Design for 3 references and 1 test resistor……Page 1027
2.3.4.3.10. Design for 4 references and 1 test resistor……Page 1029
2.3.4.4. Roundness measurements……Page 1031
2.3.4.4.1. Single-trace roundness design……Page 1033
2.3.4.4.2. Multiple-trace roundness designs……Page 1035
2.3.4.5. Designs for angle blocks……Page 1039
2.3.4.5.1. Design for 4 angle blocks……Page 1045
2.3.4.5.2. Design for 5 angle blocks……Page 1047
2.3.4.5.3. Design for 6 angle blocks……Page 1049
2.3.4.6. Thermometers in a bath……Page 1052
2.3.4.7. Humidity standards……Page 1054
2.3.4.7.1. Drift-elimination design for 2 reference weights and 3 cylinders……Page 1055
2.3.5. Control of artifact calibration……Page 1057
2.3.5.1. Control of precision……Page 1059
2.3.5.1.1. Example of control chart for precision……Page 1061
2.3.5.2. Control of bias and long-term variability……Page 1063
2.3.5.2.1. Example of Shewhart control chart for mass calibrations……Page 1066
2.3.5.2.2. Example of EWMA control chart for mass calibrations……Page 1069
2.3.6. Instrument calibration over a regime……Page 1072
2.3.6.1. Models for instrument calibration……Page 1075
2.3.6.2. Data collection……Page 1079
2.3.6.3. Assumptions for instrument calibration……Page 1081
2.3.6.4. What can go wrong with the calibration procedure……Page 1082
2.3.6.4.1. Example of day-to-day changes in calibration……Page 1084
2.3.6.5. Data analysis and model validation……Page 1087
2.3.6.5.1. Data on load cell #32066……Page 1090
2.3.6.6. Calibration of future measurements……Page 1092
2.3.6.7. Uncertainties of calibrated values……Page 1095
2.3.6.7.1. Uncertainty for quadratic calibration using propagation of error……Page 1097
2.3.6.7.2. Uncertainty for linear calibration using check standards……Page 1104
2.3.6.7.3. Comparison of check standard analysis and propagation of error……Page 1106
2.3.7. Instrument control for linear calibration……Page 1109
2.3.7.1. Control chart for a linear calibration line……Page 1112
2.4. Gauge R & R studies……Page 1115
2.4.1. What are the important issues?……Page 1117
2.4.2. Design considerations……Page 1118
2.4.3. Data collection for time-related sources of variability……Page 1120
2.4.3.1. Simple design……Page 1121
2.4.3.2. 2-level nested design……Page 1123
2.4.3.3. 3-level nested design……Page 1126
2.4.4. Analysis of variability……Page 1130
2.4.4.1. Analysis of repeatability……Page 1135
2.4.4.2. Analysis of reproducibility……Page 1139
2.4.4.3. Analysis of stability……Page 1142
2.4.4.4.4. Example of calculations……Page 1145
2.4.5. Analysis of bias……Page 1147
2.4.5.1. Resolution……Page 1148
2.4.5.2. Linearity of the gauge……Page 1150
2.4.5.3. Drift……Page 1152
2.4.5.4. Differences among gauges……Page 1153
2.4.5.5. Geometry/configuration differences……Page 1155
2.4.5.6. Remedial actions and strategies……Page 1158
2.4.6. Quantifying uncertainties from a gauge study……Page 1160
2.5. Uncertainty analysis……Page 1163
2.5.1. Issues……Page 1165
2.5.2. Approach……Page 1167
2.5.2.1. Steps……Page 1171
2.5.3. Type A evaluations……Page 1173
2.5.3.1. Type A evaluations of random components……Page 1175
2.5.3.1.1. Type A evaluations of time-dependent effects……Page 1178
2.5.3.1.2. Measurement configuration within the laboratory……Page 1181
2.5.3.2. Material inhomogeneity……Page 1184
2.5.3.2.1. Data collection and analysis……Page 1187
2.5.3.3. Type A evaluations of bias……Page 1190
2.5.3.3.1. Inconsistent bias……Page 1193
2.5.3.3.2. Consistent bias……Page 1195
2.5.3.3.3. Bias with sparse data……Page 1198
2.5.4. Type B evaluations……Page 1203
2.5.4.1. Standard deviations from assumed distributions……Page 1205
2.5.5. Propagation of error considerations……Page 1207
2.5.5.1. Formulas for functions of one variable……Page 1210
2.5.5.2. Formulas for functions of two variables……Page 1212
2.5.5.3. Propagation of error for many variables……Page 1214
2.5.6. Uncertainty budgets and sensitivity coefficients……Page 1218
2.5.6.1. Sensitivity coefficients for measurements on the test item……Page 1221
2.5.6.2. Sensitivity coefficients for measurements on a check standard……Page 1223
2.5.6.3. Sensitivity coefficients for measurements from a 2-level design……Page 1224
2.5.6.4. Sensitivity coefficients for measurements from a 3-level design……Page 1226
2.5.6.5. Example of uncertainty budget……Page 1228
2.5.7. Standard and expanded uncertainties……Page 1230
2.5.7.1. Degrees of freedom……Page 1231
2.5.8. Treatment of uncorrected bias……Page 1232
2.5.8.1. Computation of revised uncertainty……Page 1234
2.6. Case studies……Page 1236
2.6.1. Gauge study of resistivity probes……Page 1237
2.6.1.1. Background and data……Page 1238
2.6.1.1.1. Database of resistivity measurements……Page 1240
2.6.1.2. Analysis and interpretation……Page 1255
2.6.1.3. Repeatability standard deviations……Page 1261
2.6.1.4. Effects of days and long-term stability……Page 1265
2.6.1.5. Differences among 5 probes……Page 1270
2.6.1.6. Run gauge study example using Dataplot™……Page 1272
2.6.1.7. Dataplot macros……Page 1274
2.6.2. Check standard for resistivity measurements……Page 1278
2.6.2.1. Background and data……Page 1279
2.6.2.1.1. Database for resistivity check standard……Page 1280
2.6.2.2. Analysis and interpretation……Page 1283
2.6.2.2.1. Repeatability and level-2 standard deviations……Page 1285
2.6.2.3. Control chart for probe precision……Page 1287
2.6.2.4. Control chart for bias and long-term variability……Page 1288
2.6.2.5. Run check standard example yourself……Page 1289
2.6.2.6. Dataplot macros……Page 1291
2.6.3. Evaluation of type A uncertainty……Page 1293
2.6.3.1. Background and data……Page 1294
2.6.3.1.1. Database of resistivity measurements……Page 1296
2.6.3.1.2. Measurements on wiring configurations……Page 1311
2.6.3.2. Analysis and interpretation……Page 1314
2.6.3.2.1. Difference between 2 wiring configurations……Page 1319
2.6.3.3. Run the type A uncertainty analysis using Dataplot……Page 1322
2.6.3.4. Dataplot macros……Page 1324
2.6.4. Evaluation of type B uncertainty and propagation of error……Page 1332
2.7. References……Page 1337
3. Production Process Characterization……Page 1341
3. Production Process Characterization……Page 1343
3.1. Introduction to Production Process Characterization……Page 1346
3.1.1. What is PPC?……Page 1347
3.1.2. What are PPC Studies Used For?……Page 1349
3.1.3. Terminology/Concepts……Page 1350
3.1.3.1. Distribution (Location, Spread and Shape)……Page 1352
3.1.3.2. Process Variability……Page 1354
3.1.3.2.1. Controlled/Uncontrolled Variation……Page 1357
3.1.3.3. Propagating Error……Page 1359
3.1.3.4. Populations and Sampling……Page 1361
3.1.3.5. Process Models……Page 1363
3.1.3.6. Experiments and Experimental Design……Page 1367
3.1.4. PPC Steps……Page 1369
3.2. Assumptions / Prerequisites……Page 1371
3.2.1. General Assumptions……Page 1372
3.2.2. Continuous Linear Model……Page 1373
3.2.3. Analysis of Variance Models (ANOVA)……Page 1375
3.2.3.1. One-Way ANOVA……Page 1377
3.2.3.1.1. One-Way Value-Splitting……Page 1381
3.2.3.2. Two-Way Crossed ANOVA……Page 1384
3.2.3.2.1. Two-way Crossed Value-Splitting Example……Page 1388
3.2.3.3. Two-Way Nested ANOVA……Page 1391
3.2.3.3.1. Two-Way Nested Value-Splitting Example……Page 1395
3.2.4. Discrete Models……Page 1397
3.3. Data Collection for PPC……Page 1400
3.3.1. Define Goals……Page 1401
3.3.2. Process Modeling……Page 1402
3.3.3. Define Sampling Plan……Page 1405
3.3.3.1. Identifying Parameters, Ranges and Resolution……Page 1406
3.3.3.2. Choosing a Sampling Scheme……Page 1408
3.3.3.3. Selecting Sample Sizes……Page 1411
3.3.3.4. Data Storage and Retrieval……Page 1415
3.3.3.5. Assign Roles and Responsibilities……Page 1417
3.4. Data Analysis for PPC……Page 1419
3.4.1. First Steps……Page 1420
3.4.2. Exploring Relationships……Page 1421
3.4.2.1. Response Correlations……Page 1422
3.4.2.2. Exploring Main Effects……Page 1424
3.4.2.3. Exploring First Order Interactions……Page 1433
3.4.3. Building Models……Page 1436
3.4.3.1. Fitting Polynomial Models……Page 1438
3.4.3.2. Fitting Physical Models……Page 1440
3.4.4. Analyzing Variance Structure……Page 1442
3.4.5. Assessing Process Stability……Page 1444
3.4.6. Assessing Process Capability……Page 1446
3.4.7. Checking Assumptions……Page 1448
3.5. Case Studies……Page 1451
3.5.1. Furnace Case Study……Page 1452
3.5.1.1. Background and Data……Page 1453
3.5.1.2. Initial Analysis of Response Variable……Page 1460
3.5.1.3. Identify Sources of Variation……Page 1464
3.5.1.4. Analysis of Variance……Page 1468
3.5.1.5. Final Conclusions……Page 1470
3.5.1.6. Work This Example Yourself……Page 1471
3.5.2. Machine Screw Case Study……Page 1474
3.5.2.1. Background and Data……Page 1475
3.5.2.2. Box Plots by Factors……Page 1482
3.5.2.3. Analysis of Variance……Page 1486
3.5.2.4. Throughput……Page 1492
3.5.2.5. Final Conclusions……Page 1495
3.5.2.6. Work This Example Yourself……Page 1496
3.6. References……Page 1499
4. Process Modeling……Page 1501
4. Process Modeling……Page 1503
4.1. Introduction to Process Modeling……Page 1508
4.1.1. What is process modeling?……Page 1509
4.1.2. What terminology do statisticians use to describe process models?……Page 1513
4.1.3. What are process models used for?……Page 1516
4.1.3.1. Estimation……Page 1518
4.1.3.2. Prediction……Page 1522
4.1.3.3. Calibration……Page 1527
4.1.3.4. Optimization……Page 1531
4.1.4. What are some of the different statistical methods for model building?……Page 1535
4.1.4.1. Linear Least Squares Regression……Page 1537
4.1.4.2. Nonlinear Least Squares Regression……Page 1541
4.1.4.3. Weighted Least Squares Regression……Page 1545
4.1.4.4. LOESS (aka LOWESS)……Page 1547
4.2. Underlying Assumptions for Process Modeling……Page 1552
4.2.1. What are the typical underlying assumptions in process modeling?……Page 1554
4.2.1.1. The process is a statistical process…….Page 1555
4.2.1.2. The means of the random errors are zero…….Page 1557
4.2.1.3. The random errors have a constant standard deviation…….Page 1559
4.2.1.4. The random errors follow a normal distribution…….Page 1561
4.2.1.5. The data are randomly sampled from the process…….Page 1563
4.2.1.6. The explanatory variables are observed without error…….Page 1565
4.3. Data Collection for Process Modeling……Page 1570
4.3.1. What is design of experiments (aka DEX or DOE)?……Page 1571
4.3.2. Why is experimental design important for process modeling?……Page 1573
4.3.3. What are some general design principles for process modeling?……Page 1577
4.3.4. I’ve heard some people refer to “optimal” designs, shouldn’t I use those?……Page 1580
4.3.5. How can I tell if a particular experimental design is good for my application?……Page 1583
4.4. Data Analysis for Process Modeling……Page 1585
4.4.1. What are the basic steps for developing an effective process model?……Page 1587
4.4.2. How do I select a function to describe my process?……Page 1590
4.4.2.1. Incorporating Scientific Knowledge into Function Selection……Page 1592
4.4.2.2. Using the Data to Select an Appropriate Function……Page 1595
4.4.2.3. Using Methods that Do Not Require Function Specification……Page 1602
4.4.3. How are estimates of the unknown parameters obtained?……Page 1604
4.4.3.1. Least Squares……Page 1606
4.4.3.2. Weighted Least Squares……Page 1610
4.4.4. How can I tell if a model fits my data?……Page 1612
4.4.4.1. How can I assess the sufficiency of the functional part of the model?……Page 1616
4.4.4.2. How can I detect non-constant variation across the data?……Page 1622
4.4.4.3. How can I tell if there was drift in the measurement process?……Page 1628
4.4.4.4. How can I assess whether the random errors are independent from one to the next?……Page 1632
4.4.4.5. How can I test whether or not the random errors are distributed normally?……Page 1636
4.4.4.6. How can I test whether any significant terms are missing or misspecified in the functional part of the model?……Page 1643
4.4.4.7. How can I test whether all of the terms in the functional part of the model are necessary?……Page 1647
4.4.5. If my current model does not fit the data well, how can I improve it?……Page 1650
4.4.5.1. Updating the Function Based on Residual Plots……Page 1651
4.4.5.2. Accounting for Non-Constant Variation Across the Data……Page 1653
4.4.5.3. Accounting for Errors with a Non-Normal Distribution……Page 1667
4.5. Use and Interpretation of Process Models……Page 1674
4.5.1. What types of predictions can I make using the model?……Page 1675
4.5.1.1. How do I estimate the average response for a particular set of predictor variable values?……Page 1676
4.5.1.2. How can I predict the value and and estimate the uncertainty of a single response?……Page 1682
4.5.2. How can I use my process model for calibration?……Page 1687
4.5.2.1. Single-Use Calibration Intervals……Page 1688
4.5.3. How can I optimize my process using the process model?……Page 1693
4.6. Case Studies in Process Modeling……Page 1694
4.6.1. Load Cell Calibration……Page 1696
4.6.1.1. Background & Data……Page 1697
4.6.1.2. Selection of Initial Model……Page 1699
4.6.1.3. Model Fitting – Initial Model……Page 1701
4.6.1.4. Graphical Residual Analysis – Initial Model……Page 1703
4.6.1.5. Interpretation of Numerical Output – Initial Model……Page 1707
4.6.1.6. Model Refinement……Page 1709
4.6.1.7. Model Fitting – Model #2……Page 1711
4.6.1.8. Graphical Residual Analysis – Model #2……Page 1713
4.6.1.9. Interpretation of Numerical Output – Model #2……Page 1717
4.6.1.10. Use of the Model for Calibration……Page 1719
4.6.1.11. Work This Example Yourself……Page 1722
4.6.2. Alaska Pipeline……Page 1725
4.6.2.1. Background and Data……Page 1726
4.6.2.2. Check for Batch Effect……Page 1730
4.6.2.3. Initial Linear Fit……Page 1733
4.6.2.4. Transformations to Improve Fit and Equalize Variances……Page 1737
4.6.2.5. Weighting to Improve Fit……Page 1743
4.6.2.6. Compare the Fits……Page 1749
4.6.2.7. Work This Example Yourself……Page 1751
4.6.3. Ultrasonic Reference Block Study……Page 1754
4.6.3.1. Background and Data……Page 1755
4.6.3.2. Initial Non-Linear Fit……Page 1761
4.6.3.3. Transformations to Improve Fit……Page 1767
4.6.3.4. Weighting to Improve Fit……Page 1772
4.6.3.5. Compare the Fits……Page 1778
4.6.3.6. Work This Example Yourself……Page 1780
4.6.4. Thermal Expansion of Copper Case Study……Page 1783
4.6.4.1. Background and Data……Page 1784
4.6.4.2. Rational Function Models……Page 1790
4.6.4.3. Initial Plot of Data……Page 1794
4.6.4.4. Quadratic/Quadratic Rational Function Model……Page 1795
4.6.4.5. Cubic/Cubic Rational Function Model……Page 1800
4.6.4.6. Work This Example Yourself……Page 1805
4.7. References For Chapter 4: Process Modeling……Page 1807
4.8. Some Useful Functions for Process Modeling……Page 1809
4.8.1. Univariate Functions……Page 1810
4.8.1.1. Polynomial Functions……Page 1811
4.8.1.1.1. Straight Line……Page 1813
4.8.1.1.2. Quadratic Polynomial……Page 1815
4.8.1.1.3. Cubic Polynomial……Page 1820
4.8.1.2. Rational Functions……Page 1828
4.8.1.2.1. Constant / Linear Rational Function……Page 1832
4.8.1.2.2. Linear / Linear Rational Function……Page 1838
4.8.1.2.3. Linear / Quadratic Rational Function……Page 1844
4.8.1.2.4. Quadratic / Linear Rational Function……Page 1850
4.8.1.2.5. Quadratic / Quadratic Rational Function……Page 1856
4.8.1.2.6. Cubic / Linear Rational Function……Page 1863
4.8.1.2.7. Cubic / Quadratic Rational Function……Page 1869
4.8.1.2.8. Linear / Cubic Rational Function……Page 1875
4.8.1.2.9. Quadratic / Cubic Rational Function……Page 1880
4.8.1.2.10. Cubic / Cubic Rational Function……Page 1884
4.8.1.2.11. Determining m and n for Rational Function Models……Page 1888
5. Process Improvement……Page 1901
5. Process Improvement……Page 1903
5.1. Introduction……Page 1909
5.1.1. What is experimental design?……Page 1910
5.1.2. What are the uses of DOE?……Page 1913
5.1.3. What are the steps of DOE?……Page 1919
5.2. Assumptions……Page 1921
5.2.1. Is the measurement system capable?……Page 1922
5.2.2. Is the process stable?……Page 1923
5.2.3. Is there a simple model?……Page 1924
5.2.4. Are the model residuals well-behaved?……Page 1925
5.3. Choosing an experimental design……Page 1935
5.3.1. What are the objectives?……Page 1937
5.3.2. How do you select and scale the process variables?……Page 1939
5.3.3. How do you select an experimental design?……Page 1942
5.3.3.1. Completely randomized designs……Page 1945
5.3.3.2. Randomized block designs……Page 1948
5.3.3.2.1. Latin square and related designs……Page 1952
5.3.3.2.2. Graeco-Latin square designs……Page 1958
5.3.3.2.3. Hyper-Graeco-Latin square designs……Page 1962
5.3.3.3. Full factorial designs……Page 1965
5.3.3.3.1. Two-level full factorial designs……Page 1966
5.3.3.3.2. Full factorial example……Page 1969
5.3.3.3.3. Blocking of full factorial designs……Page 1975
5.3.3.4. Fractional factorial designs……Page 1979
5.3.3.4.1. A 23-1 design (half of a 23)……Page 1981
5.3.3.4.2. Constructing the 23-1 half-fraction design……Page 1985
5.3.3.4.3. Confounding (also called aliasing)……Page 1988
5.3.3.4.4. Fractional factorial design specifications and design resolution……Page 1991
5.3.3.4.5. Use of fractional factorial designs……Page 1998
5.3.3.4.6. Screening designs……Page 2000
5.3.3.4.7. Summary tables of useful fractional factorial designs……Page 2002
5.3.3.5. Plackett-Burman designs……Page 2005
5.3.3.6. Response surface designs……Page 2008
5.3.3.6.1. Central Composite Designs (CCD)……Page 2014
5.3.3.6.2. Box-Behnken designs……Page 2019
5.3.3.6.3. Comparisons of response surface designs……Page 2021
5.3.3.6.4. Blocking a response surface design……Page 2026
5.3.3.7. Adding centerpoints……Page 2031
5.3.3.8. Improving fractional factorial design resolution……Page 2035
5.3.3.8.1. Mirror-Image foldover designs……Page 2036
5.3.3.8.2. Alternative foldover designs……Page 2041
5.3.3.9. Three-level full factorial designs……Page 2044
5.3.3.10. Three-level, mixed-level and fractional factorial designs……Page 2048
5.4. Analysis of DOE data……Page 2053
5.4.1. What are the steps in a DOE analysis?……Page 2054
5.4.2. How to “look” at DOE data……Page 2056
5.4.3. How to model DOE data……Page 2059
5.4.4. How to test and revise DOE models……Page 2061
5.4.5. How to interpret DOE results……Page 2063
5.4.6. How to confirm DOE results (confirmatory runs)……Page 2064
5.4.7. Examples of DOE’s……Page 2066
5.4.7.1. Full factorial example……Page 2067
5.4.7.2. Fractional factorial example……Page 2082
5.4.7.3. Response surface model example……Page 2100
5.5. Advanced topics……Page 2116
5.5.1. What if classical designs don’t work?……Page 2118
5.5.2. What is a computer-aided design?……Page 2120
5.5.2.1. D-Optimal designs……Page 2122
5.5.2.2. Repairing a design……Page 2127
5.5.3. How do you optimize a process?……Page 2129
5.5.3.1. Single response case……Page 2131
5.5.3.1.1. Single response: Path of steepest ascent……Page 2132
5.5.3.1.2. Single response: Confidence region for search path……Page 2138
5.5.3.1.3. Single response: Choosing the step length……Page 2141
5.5.3.1.4. Single response: Optimization when there is adequate quadratic fit……Page 2145
5.5.3.1.5. Single response: Effect of sampling error on optimal solution……Page 2153
5.5.3.1.6. Single response: Optimization subject to experimental region constraints……Page 2154
5.5.3.2. Multiple response case……Page 2155
5.5.3.2.1. Multiple responses: Path of steepest ascent……Page 2156
5.5.3.2.2. Multiple responses: The desirability approach……Page 2159
5.5.3.2.3. Multiple responses: The mathematical programming approach……Page 2164
5.5.4. What is a mixture design?……Page 2167
5.5.4.1. Mixture screening designs……Page 2169
5.5.4.2. Simplex-lattice designs……Page 2170
5.5.4.3. Simplex-centroid designs……Page 2177
5.5.4.4. Constrained mixture designs……Page 2179
5.5.4.5. Treating mixture and process variables together……Page 2183
5.5.5. How can I account for nested variation (restricted randomization)?……Page 2186
5.5.6. What are Taguchi designs?……Page 2198
5.5.7. What are John’s 3/4 fractional factorial designs?……Page 2204
5.5.8. What are small composite designs?……Page 2210
5.5.9. An EDA approach to experimental design……Page 2213
5.5.9.1. Ordered data plot……Page 2216
5.5.9.2. Dex scatter plot……Page 2219
5.5.9.3. Dex mean plot……Page 2223
5.5.9.4. Interaction effects matrix plot……Page 2227
5.5.9.5. Block plot……Page 2236
5.5.9.6. Dex Youden plot……Page 2241
5.5.9.7. |Effects| plot……Page 2244
5.5.9.7.1. Statistical significance……Page 2248
5.5.9.7.2. Engineering significance……Page 2251
5.5.9.7.3. Numerical significance……Page 2253
5.5.9.7.4. Pattern significance……Page 2254
5.5.9.8. Half-normal probability plot……Page 2255
5.5.9.9. Cumulative residual standard deviation plot……Page 2260
5.5.9.9.1. Motivation: What is a Model?……Page 2266
5.5.9.9.2. Motivation: How do we Construct a Goodness-of-fit Metric for a Model?……Page 2267
5.5.9.9.3. Motivation: How do we Construct a Good Model?……Page 2269
5.5.9.9.4. Motivation: How do we Know When to Stop Adding Terms?……Page 2272
5.5.9.9.5. Motivation: What is the Form of the Model?……Page 2274
5.5.9.9.6. Motivation: Why is the 1/2 in the Model?……Page 2276
5.5.9.9.7. Motivation: What are the Advantages of the LinearCombinatoric Model?……Page 2280
5.5.9.9.8. Motivation: How do we use the Model to Generate Predicted Values?……Page 2282
5.5.9.9.9. Motivation: How do we Use the Model Beyond the Data Domain?……Page 2285
5.5.9.9.10. Motivation: What is the Best Confirmation Point for Interpolation?……Page 2287
5.5.9.9.11. Motivation: How do we Use the Model for Interpolation?……Page 2289
5.5.9.9.12. Motivation: How do we Use the Model for Extrapolation?……Page 2292
5.5.9.10. DEX contour plot……Page 2294
5.5.9.10.1. How to Interpret: Axes……Page 2298
5.5.9.10.2. How to Interpret: Contour Curves……Page 2301
5.5.9.10.3. How to Interpret: Optimal Response Value……Page 2303
5.5.9.10.4. How to Interpret: Best Corner……Page 2304
5.5.9.10.5. How to Interpret: Steepest Ascent/Descent……Page 2306
5.5.9.10.6. How to Interpret: Optimal Curve……Page 2307
5.5.9.10.7. How to Interpret: Optimal Setting……Page 2308
5.6. Case Studies……Page 2313
5.6.1. Eddy Current Probe Sensitivity Case Study……Page 2314
5.6.1.1. Background and Data……Page 2315
5.6.1.2. Initial Plots/Main Effects……Page 2317
5.6.1.3. Interaction Effects……Page 2321
5.6.1.4. Main and Interaction Effects: Block Plots……Page 2323
5.6.1.5. Estimate Main and Interaction Effects……Page 2325
5.6.1.6. Modeling and Prediction Equations……Page 2328
5.6.1.7. Intermediate Conclusions……Page 2330
5.6.1.8. Important Factors and Parsimonious Prediction……Page 2332
5.6.1.9. Validate the Fitted Model……Page 2336
5.6.1.10. Using the Fitted Model……Page 2339
5.6.1.11. Conclusions and Next Step……Page 2341
5.6.1.12. Work This Example Yourself……Page 2343
5.6.2. Sonoluminescent Light Intensity Case Study……Page 2346
5.6.2.1. Background and Data……Page 2347
5.6.2.2. Initial Plots/Main Effects……Page 2350
5.6.2.3. Interaction Effects……Page 2354
5.6.2.4. Main and Interaction Effects: Block Plots……Page 2356
5.6.2.5. Important Factors: Youden Plot……Page 2358
5.6.2.6. Important Factors: |Effects| Plot……Page 2360
5.6.2.7. Important Factors: Half-Normal Probability Plot……Page 2362
5.6.2.8. Cumulative Residual Standard Deviation Plot……Page 2364
5.6.2.9. Next Step: Dex Contour Plot……Page 2366
5.6.2.10. Summary of Conclusions……Page 2368
5.6.2.11. Work This Example Yourself……Page 2370
5.7. A Glossary of DOE Terminology……Page 2373
5.8. References……Page 2378
6. Process or Product Monitoring and Control……Page 2381
6. Process or Product Monitoring and Control……Page 2383
6.1. Introduction……Page 2387
6.1.1. How did Statistical Quality Control Begin?……Page 2388
6.1.2. What are Process Control Techniques?……Page 2390
6.1.3. What is Process Control?……Page 2392
6.1.4. What to do if the process is “Out of Control”?……Page 2393
6.1.5. What to do if “In Control” but Unacceptable?……Page 2394
6.1.6. What is Process Capability?……Page 2395
6.2. Test Product for Acceptability: Lot Acceptance Sampling……Page 2403
6.2.1. What is Acceptance Sampling?……Page 2404
6.2.2. What kinds of Lot Acceptance Sampling Plans (LASPs) are there?……Page 2406
6.2.3. How do you Choose a Single Sampling Plan?……Page 2409
6.2.3.1. Choosing a Sampling Plan: MIL Standard 105D……Page 2410
6.2.3.2. Choosing a Sampling Plan with a given OC Curve……Page 2413
6.2.4. What is Double Sampling?……Page 2419
6.2.5. What is Multiple Sampling?……Page 2424
6.2.6. What is a Sequential Sampling Plan?……Page 2426
6.2.7. What is Skip Lot Sampling?……Page 2429
6.3. Univariate and Multivariate Control Charts……Page 2431
6.3.1. What are Control Charts?……Page 2432
6.3.2. What are Variables Control Charts?……Page 2436
6.3.2.1. Shewhart X-bar and R and S Control Charts……Page 2441
6.3.2.2. Individuals Control Charts……Page 2446
6.3.2.3. Cusum Control Charts……Page 2449
6.3.2.3.1. Cusum Average Run Length……Page 2456
6.3.2.4. EWMA Control Charts……Page 2460
6.3.3. What are Attributes Control Charts?……Page 2464
6.3.3.1. Counts Control Charts……Page 2466
6.3.3.2. Proportions Control Charts……Page 2472
6.3.4. What are Multivariate Control Charts?……Page 2475
6.3.4.1. Hotelling Control Charts……Page 2477
6.3.4.2. Principal Components Control Charts……Page 2479
6.3.4.3. Multivariate EWMA Charts……Page 2481
6.4. Introduction to Time Series Analysis……Page 2485
6.4.1. Definitions, Applications and Techniques……Page 2487
6.4.2. What are Moving Average or Smoothing Techniques?……Page 2489
6.4.2.1. Single Moving Average……Page 2493
6.4.2.2. Centered Moving Average……Page 2495
6.4.3. What is Exponential Smoothing?……Page 2497
6.4.3.1. Single Exponential Smoothing……Page 2498
6.4.3.2. Forecasting with Single Exponential Smoothing……Page 2503
6.4.3.3. Double Exponential Smoothing……Page 2506
6.4.3.4. Forecasting with Double Exponential Smoothing(LASP)……Page 2508
6.4.3.5. Triple Exponential Smoothing……Page 2512
6.4.3.6. Example of Triple Exponential Smoothing……Page 2515
6.4.3.7. Exponential Smoothing Summary……Page 2518
6.4.4. Univariate Time Series Models……Page 2519
6.4.4.1. Sample Data Sets……Page 2520
6.4.4.1.1. Data Set of Monthly CO2 Concentrations……Page 2521
6.4.4.1.2. Data Set of Southern Oscillations……Page 2526
6.4.4.2. Stationarity……Page 2538
6.4.4.3. Seasonality……Page 2541
6.4.4.3.1. Seasonal Subseries Plot……Page 2546
6.4.4.4. Common Approaches to Univariate Time Series……Page 2548
6.4.4.5. Box-Jenkins Models……Page 2551
6.4.4.6. Box-Jenkins Model Identification……Page 2553
6.4.4.6.1. Model Identification for Southern Oscillations Data……Page 2557
6.4.4.6.2. Model Identification for the CO2 Concentrations Data……Page 2560
6.4.4.6.3. Partial Autocorrelation Plot……Page 2565
6.4.4.7. Box-Jenkins Model Estimation……Page 2568
6.4.4.8. Box-Jenkins Model Diagnostics……Page 2569
6.4.4.9. Example of Univariate Box-Jenkins Analysis……Page 2570
6.4.4.10. Box-Jenkins Analysis on Seasonal Data……Page 2574
6.4.5. Multivariate Time Series Models……Page 2580
6.4.5.1. Example of Multivariate Time Series Analysis……Page 2583
6.5. Tutorials……Page 2588
6.5.1. What do we mean by “Normal” data?……Page 2589
6.5.2. What do we do when data are non-normal……Page 2592
6.5.3. Elements of Matrix Algebra……Page 2595
6.5.3.1. Numerical Examples……Page 2599
6.5.3.2. Determinant and Eigenstructure……Page 2602
6.5.4. Elements of Multivariate Analysis……Page 2604
6.5.4.1. Mean Vector and Covariance Matrix……Page 2606
6.5.4.2. The Multivariate Normal Distribution……Page 2608
6.5.4.3. Hotelling’s T squared……Page 2610
6.5.4.3.1. T2 Chart for Subgroup Averages — Phase I……Page 2612
6.5.4.3.2. T2 Chart for Subgroup Averages — Phase II……Page 2615
6.5.4.3.3. Chart for Individual Observations — Phase I……Page 2617
6.5.4.3.4. Chart for Individual Observations — Phase II……Page 2619
6.5.4.3.5. Charts for Controlling Multivariate Variability……Page 2620
6.5.4.3.6. Constructing Multivariate Charts……Page 2621
6.5.5. Principal Components……Page 2622
6.5.5.1. Properties of Principal Components……Page 2625
6.5.5.2. Numerical Example……Page 2632
6.6. Case Studies in Process Monitoring……Page 2636
6.6.1. Lithography Process……Page 2637
6.6.1.1. Background and Data……Page 2638
6.6.1.2. Graphical Representation of the Data……Page 2650
6.6.1.3. Subgroup Analysis……Page 2658
6.6.1.4. Shewhart Control Chart……Page 2663
6.6.1.5. Work This Example Yourself……Page 2665
6.6.2. Aerosol Particle Size……Page 2668
6.6.2.1. Background and Data……Page 2669
6.6.2.2. Model Identification……Page 2683
6.6.2.3. Model Estimation……Page 2688
6.6.2.4. Model Validation……Page 2693
6.6.2.5. Work This Example Yourself……Page 2699
6.7. References……Page 2702
7. Product and Process Comparisons……Page 2706
7. Product and Process Comparisons……Page 2708
7.1. Introduction……Page 2711
7.1.1. What is the scope?……Page 2712
7.1.2. What assumptions are typically made?……Page 2713
7.1.3. What are statistical tests?……Page 2715
7.1.3.1. Critical values and p values……Page 2718
7.1.4. What are confidence intervals?……Page 2720
7.1.5. What is the relationship between a test and a confidence interval?……Page 2722
7.1.6. What are outliers in the data?……Page 2724
7.1.7. What are trends in sequential process or product data?……Page 2728
7.2. Comparisons based on data from one process……Page 2729
7.2.1. Do the observations come from a particular distribution?……Page 2732
7.2.1.1. Chi-square goodness-of-fit test……Page 2734
7.2.1.2. Kolmogorov- Smirnov test……Page 2736
7.2.1.3. Anderson-Darling and Shapiro-Wilk tests……Page 2737
7.2.2. Are the data consistent with the assumed process mean?……Page 2739
7.2.2.1. Confidence interval approach……Page 2742
7.2.2.2. Sample sizes required……Page 2744
7.2.3. Are the data consistent with a nominal standard deviation?……Page 2748
7.2.3.1. Confidence interval approach……Page 2750
7.2.3.2. Sample sizes required……Page 2752
7.2.4. Does the proportion of defectives meet requirements?……Page 2757
7.2.4.1. Confidence intervals……Page 2760
7.2.4.2. Sample sizes required……Page 2769
7.2.5. Does the defect density meet requirements?……Page 2771
7.2.6. What intervals contain a fixed percentage of the population values?……Page 2774
7.2.6.1. Approximate intervals that contain most of the population values……Page 2775
7.2.6.2. Percentiles……Page 2777
7.2.6.3. Tolerance intervals for a normal distribution……Page 2779
7.2.6.4. Two-sided tolerance intervals using EXCEL……Page 2784
7.2.6.5. Tolerance intervals based on the largest and smallest observations……Page 2787
7.3. Comparisons based on data from two processes……Page 2790
7.3.1. Do two processes have the same mean?……Page 2792
7.3.1.1. Analysis of paired observations……Page 2797
7.3.1.2. Confidence intervals for differences between means……Page 2799
7.3.2. Do two processes have the same standard deviation?……Page 2800
7.3.3. How can we determine whether two processes produce the same proportion of defectives?……Page 2804
7.3.4. Assuming the observations are failure times, are the failure rates (or Mean Times To Failure) for two distributions the same?……Page 2809
7.3.5. Do two arbitrary processes have the same mean?……Page 2812
7.4. Comparisons based on data from more than two processes……Page 2815
7.4.1. How can we compare several populations with unknown distributions (the Kruskal-Wallis test)?……Page 2816
7.4.2. Assuming the observations are normal, do the processes have the same variance?……Page 2819
7.4.3. Are the means equal?……Page 2822
7.4.3.1. 1-Way ANOVA overview……Page 2825
7.4.3.2. The 1-way ANOVA model and assumptions……Page 2827
7.4.3.3. The ANOVA table and tests of hypotheses about means……Page 2829
7.4.3.4. 1-Way ANOVA calculations……Page 2832
7.4.3.5. Confidence intervals for the difference of treatment means……Page 2834
7.4.3.6. Assessing the response from any factor combination……Page 2836
7.4.3.7. The two-way ANOVA……Page 2843
7.4.3.8. Models and calculations for the two-way ANOVA……Page 2845
7.4.4. What are variance components?……Page 2848
7.4.5. How can we compare the results of classifying according to several categories?……Page 2851
7.4.6. Do all the processes have the same proportion of defects?……Page 2855
7.4.7. How can we make multiple comparisons?……Page 2858
7.4.7.1. Tukey’s method……Page 2861
7.4.7.2. Scheffe’s method……Page 2864
7.4.7.3. Bonferroni’s method……Page 2868
7.4.7.4. Comparing multiple proportions: The Marascuillo procedure……Page 2872
7.5. References……Page 2875
8. Assessing Product Reliability……Page 2880
8. Assessing Product Reliability……Page 2882
8.1. Introduction……Page 2886
8.1.1. Why is the assessment and control of product reliability important?……Page 2888
8.1.1.1. Quality versus reliability……Page 2889
8.1.1.2. Competitive driving factors……Page 2890
8.1.1.3. Safety and health considerations……Page 2891
8.1.2. What are the basic terms and models used for reliability evaluation?……Page 2892
8.1.2.1. Repairable systems, non-repairable populations and lifetime distribution models……Page 2893
8.1.2.2. Reliability or survival function……Page 2896
8.1.2.3. Failure (or hazard) rate……Page 2897
8.1.2.4. “Bathtub” curve……Page 2899
8.1.2.5. Repair rate or ROCOF……Page 2901
8.1.3. What are some common difficulties with reliability data and how are they overcome?……Page 2902
8.1.3.1. Censoring……Page 2903
8.1.3.2. Lack of failures……Page 2905
8.1.4. What is “physical acceleration” and how do we model it?……Page 2907
8.1.5. What are some common acceleration models?……Page 2909
8.1.5.1. Arrhenius……Page 2910
8.1.5.2. Eyring……Page 2912
8.1.5.3. Other models……Page 2915
8.1.6. What are the basic lifetime distribution models used for non-repairable populations?……Page 2918
8.1.6.1. Exponential……Page 2919
8.1.6.2. Weibull……Page 2924
8.1.6.3. Extreme value distributions……Page 2930
8.1.6.4. Lognormal……Page 2934
8.1.6.5. Gamma……Page 2939
8.1.6.6. Fatigue life (Birnbaum-Saunders)……Page 2945
8.1.6.7. Proportional hazards model……Page 2951
8.1.7. What are some basic repair rate models used for repairable systems?……Page 2953
8.1.7.1. Homogeneous Poisson Process (HPP)……Page 2954
8.1.7.2. Non-Homogeneous Poisson Process (NHPP) – power law……Page 2956
8.1.7.3. Exponential law……Page 2959
8.1.8. How can you evaluate reliability from the “bottom-up” (component failure mode to system failure rate)?……Page 2960
8.1.8.1. Competing risk model……Page 2961
8.1.8.2. Series model……Page 2963
8.1.8.3. Parallel or redundant model……Page 2965
8.1.8.4. R out of N model……Page 2967
8.1.8.5. Standby model……Page 2969
8.1.8.6. Complex systems……Page 2971
8.1.9. How can you model reliability growth?……Page 2973
8.1.9.1. NHPP power law……Page 2975
8.1.9.2. Duane plots……Page 2978
8.1.9.3. NHPP exponential law……Page 2981
8.1.10. How can Bayesian methodology be used for reliability evaluation?……Page 2982
8.2. Assumptions/Prerequisites……Page 2986
8.2.1. How do you choose an appropriate life distribution model?……Page 2988
8.2.1.1. Based on failure mode……Page 2990
8.2.1.2. Extreme value argument……Page 2991
8.2.1.3. Multiplicative degradation argument……Page 2993
8.2.1.4. Fatigue life (Birnbaum-Saunders) model……Page 2995
8.2.1.5. Empirical model fitting – distribution free (Kaplan-Meier) approach……Page 2996
8.2.2. How do you plot reliability data?……Page 2999
8.2.2.1. Probability plotting……Page 3001
8.2.2.2. Hazard and cum hazard plotting……Page 3007
8.2.2.3. Trend and growth plotting (Duane plots)……Page 3011
8.2.3. How can you test reliability model assumptions?……Page 3015
8.2.3.1. Visual tests……Page 3016
8.2.3.2. Goodness of fit tests……Page 3018
8.2.3.3. Likelihood ratio tests……Page 3019
8.2.3.4. Trend tests……Page 3022
8.2.4. How do you choose an appropriate physical acceleration model?……Page 3027
8.2.5. What models and assumptions are typically made when Bayesian methods are used for reliability evaluation?……Page 3029
8.3. Reliability Data Collection……Page 3035
8.3.1. How do you plan a reliability assessment test?……Page 3036
8.3.1.1. Exponential life distribution (or HPP model) tests……Page 3037
8.3.1.2. Lognormal or Weibull tests……Page 3040
8.3.1.3. Reliability growth (Duane model)……Page 3044
8.3.1.4. Accelerated life tests……Page 3046
8.3.1.5. Bayesian gamma prior model……Page 3050
8.4. Reliability Data Analysis……Page 3053
8.4.1. How do you estimate life distribution parameters from censored data?……Page 3055
8.4.1.1. Graphical estimation……Page 3056
8.4.1.2. Maximum likelihood estimation……Page 3059
8.4.1.3. A Weibull maximum likelihood estimation example……Page 3062
8.4.2. How do you fit an acceleration model?……Page 3068
8.4.2.1. Graphical estimation……Page 3069
8.4.2.2. Maximum likelihood……Page 3075
8.4.2.3. Fitting models using degradation data instead of failures……Page 3087
8.4.3. How do you project reliability at use conditions?……Page 3094
8.4.4. How do you compare reliability between two or more populations?……Page 3097
8.4.5. How do you fit system repair rate models?……Page 3099
8.4.5.1. Constant repair rate (HPP/exponential) model……Page 3100
8.4.5.2. Power law (Duane) model……Page 3106
8.4.5.3. Exponential law model……Page 3109
8.4.6. How do you estimate reliability using the Bayesian gamma prior model?……Page 3111
8.4.7. References For Chapter 8: Assessing Product Reliability……Page 3114