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Overview of life testing in Minitab

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Rob Schubert

This document provides an overview of reliability functions for life testing in Minitab. It discusses selecting a probability distribution, testing units to failure with right or arbitrary censoring, accelerated testing with single or multiple factors, and identifying the best-fitting distribution. The agenda includes introductions to reliability, probability distribution functions, parameters, censoring, distribution identification, and a case study of wire scrape testing to identify the best-fitting distribution for cable life data.

Engineering
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Overview of Reliability Functions for Life Testing in Minitab
Introduction Shure Inc. “The Most Trusted Audio Brand Worldwide” Industry: Consumer and professional audio electronics Founded: 1925 Products: Microphones, wireless microphone systems, headphones and earphones, mixers, conferencing systems Rob Schubert - Corporate Quality/Reliability Engineer at Shure Inc.
Agenda • Intro to Reliability • Selecting a distribution • Testing to failure – Right censoring • Testing to failure – Arbitrary censoring • Accelerated testing – Single Factor • Accelerated testing – Multiple factors • Summary • Questions
Quick Intro to Reliability • Reliability = quality over time • Field failures • Failure modes • Testing to replicate failure modes • Measuring time to failure • Attempt to predict failure rate
Probability Distribution Function (PDF) • A function that describes the relative likelihood for this random variable to take on a given value • Integral over a range = probability of the variable falling within that range Quick Intro to Reliability
Many Distributions Quick Intro to Reliability
Different ways to visualize the data Instantaneous probability Cumulative probability Straight line cumulative probability transformed so that the fitted distribution line forms a straight line Quick Intro to Reliability
Parameters Shape = Slope: slope of the line in a cumulative straight line probability plot Quick Intro to Reliability
Parameters 100000100001000100 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Data Percent Effect of Scale Parameter on Weibull plot Scale = Shifts the data “out” In a failure plot that means more time passes before each failure Quick Intro to Reliability
Parameters Threshold / Location: The 3rd parameter – shifts the zero point Forced to zero in a: 2-Parameter Lognormal Normally called just “Lognormal“ 1-Parameter Exponential Normally called just “Exponential“ 2-Parameter Weibull Normally called just “Weibull“ 2-Parameter Gamma Normally called just “gamma“ 2-Parameter Loglogistic Normally called just “Loglogistic“ (not represented on straight line cumulative probability plot) Quick Intro to Reliability
Testing to Failure: What is censoring? • Some tests will not be able to run to the end due to time constraints = Right censored data or Suspended data • Some tests cannot be continuously monitored = Arbitrary censoring Quick Intro to Reliability
Distribution Identification
Many Distributions – and more – which one? Distribution ID
• What do you know about your data? Reliability data starts at time=0 • Is it bound? Bound by zero (or something greater) • Any historical references? Weibull, Lognormal, Gumbel (extreme value) • Finally Does it look like it fits? Distribution ID
Statistics to look at: • Anderson-Darling test: How well your data fits the PDF • Smaller fits better • p value: Student’s t-test – does your data fit the PDF? • H0 = data fits the distribution • p is low, the null must go! i.e. if p<0.05, it doesn’t fit the distribution • Likelihood ratio test (p value) – does 3 parameter fit better? • H0 = 2 parameter fits better • i.e. if <0.05, the 3 parameter fits better Distribution ID
Example: Wire Scrape Test • Cable gets frayed in the field • Need to replicate • Scrape at several points • Measure cycles until breakthrough Distribution ID
3 steps to identify the distribution 1. Look at graphs 2. Look at warnings 3. Look at Anderson darling numbers Distribution ID
Wire Scrape Cable scrape 1kg 1046 1199 1249 874 842 957 1181 1134 933 1145 802 988 949 874 728 651 818 414 596 610 968 684 Distribution ID
Wire Scrape Cable scrape 1kg 1046 1199 1249 874 842 957 1181 1134 933 1145 802 988 949 874 728 651 818 414 596 610 968 684 Distribution ID
150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 221000220500220000219500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Lognormal A D = 0.489 P-V alue = 0.198 3-Parameter Lognormal A D = 0.224 P-V alue = * Goodness of Fit Test Normal A D = 0.212 P-V alue = 0.833 Box-C ox Transformation A D = 0.212 P-V alue = 0.833 A fter Box-C ox transformation (lambda = 1) Probability Plot for Cable scrape 1.0Kg Normal - 95% C I Normal - 95% C I Lognormal - 95% C I 3-Parameter Lognormal - 95% C I 10000100010010 90 50 10 1 Cable scrape 1.0Kg Percent 100001000100101 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 1000500 90 50 10 1 Cable scrape 1.0Kg Percent 1000500200 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Weibull A D = 0.205 P-V alue > 0.250 3-Parameter Weibull A D = 0.202 P-V alue > 0.500 Goodness of Fit Test Exponential A D = 5.590 P-V alue < 0.003 2-Parameter Exponential A D = 3.064 P-V alue < 0.010 Probability Plot for Cable scrape 1.0Kg Exponential - 95% C I 2-Parameter Exponential - 95% C I Weibull - 95% C I 3-Parameter Weibull - 95% C I 150010005000 90 50 10 1 Cable scrape 1.0Kg Percent 200015001000500 99 90 50 10 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 3300300027002400 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Gamma A D = 0.369 P-V alue > 0.250 3-Parameter Gamma A D = 0.271 P-V alue = * Goodness of Fit Test Smallest Extreme V alue A D = 0.270 P-V alue > 0.250 Largest Extreme V alue A D = 0.556 P-V alue = 0.152 Probability Plot for Cable scrape 1.0Kg Smallest Extreme V alue - 95% C I Largest Extreme V alue - 95% C I Gamma - 95% C I 3-Parameter Gamma - 95% C I 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 219000218500218000217500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 3-Parameter Loglogistic A D = 0.213 P-V alue = * Goodness of Fit Test Logistic A D = 0.213 P-V alue > 0.250 Loglogistic A D = 0.355 P-V alue > 0.250 Probability Plot for Cable scrape 1.0Kg Logistic - 95% C I Loglogistic - 95% C I 3-Parameter Loglogistic - 95% C I x x x x x x x Distribution ID
Note the Warnings! These are probably poor choices 3-Parameter Lognormal * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. 2-Parameter Exponential * WARNING * Variance/Covariance matrix of estimated parameters does not exist. The threshold parameter is assumed fixed when calculating confidence intervals. 3-Parameter Gamma * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. 3-Parameter Loglogistic * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. Distribution ID
150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 221000220500220000219500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Lognormal A D = 0.489 P-V alue = 0.198 3-Parameter Lognormal A D = 0.224 P-V alue = * Goodness of Fit Test Normal A D = 0.212 P-V alue = 0.833 Box-C ox Transformation A D = 0.212 P-V alue = 0.833 A fter Box-C ox transformation (lambda = 1) Probability Plot for Cable scrape 1.0Kg Normal - 95% C I Normal - 95% C I Lognormal - 95% C I 3-Parameter Lognormal - 95% C I 10000100010010 90 50 10 1 Cable scrape 1.0Kg Percent 100001000100101 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 1000500 90 50 10 1 Cable scrape 1.0Kg Percent 1000500200 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Weibull A D = 0.205 P-V alue > 0.250 3-Parameter Weibull A D = 0.202 P-V alue > 0.500 Goodness of Fit Test Exponential A D = 5.590 P-V alue < 0.003 2-Parameter Exponential A D = 3.064 P-V alue < 0.010 Probability Plot for Cable scrape 1.0Kg Exponential - 95% C I 2-Parameter Exponential - 95% C I Weibull - 95% C I 3-Parameter Weibull - 95% C I 150010005000 90 50 10 1 Cable scrape 1.0Kg Percent 200015001000500 99 90 50 10 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 3300300027002400 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Gamma A D = 0.369 P-V alue > 0.250 3-Parameter Gamma A D = 0.271 P-V alue = * Goodness of Fit Test Smallest Extreme V alue A D = 0.270 P-V alue > 0.250 Largest Extreme V alue A D = 0.556 P-V alue = 0.152 Probability Plot for Cable scrape 1.0Kg Smallest Extreme V alue - 95% C I Largest Extreme V alue - 95% C I Gamma - 95% C I 3-Parameter Gamma - 95% C I 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 219000218500218000217500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 3-Parameter Loglogistic A D = 0.213 P-V alue = * Goodness of Fit Test Logistic A D = 0.213 P-V alue > 0.250 Loglogistic A D = 0.355 P-V alue > 0.250 Probability Plot for Cable scrape 1.0Kg Logistic - 95% C I Loglogistic - 95% C I 3-Parameter Loglogistic - 95% C I x x x x x x x x x Last visual inspection kept: Normal 3 param. lognormal 2/3 param. Weibull Smallest Extreme Val 3 param. gamma Logistic x x Distribution ID
Goodness of Fit Test Distribution AD P LRT P Normal 0.212 0.833 Box-Cox Transformation 0.212 0.833 Lognormal 0.489 0.198 3-Parameter Lognormal 0.224 * 0.061 Exponential 5.590 <0.003 2-Parameter Exponential 3.064 <0.010 0.000 Weibull 0.205 >0.250 3-Parameter Weibull 0.202 >0.500 0.808 Smallest Extreme Value 0.270 >0.250 Largest Extreme Value 0.556 0.152 Gamma 0.369 >0.250 3-Parameter Gamma 0.271 * 0.229 Logistic 0.213 >0.250 Loglogistic 0.355 >0.250 3-Parameter Loglogistic 0.213 * 0.147 Likelihood ratio test = if small (<.05), 3 parameter fits better Distribution ID
Cable scrape 1kg 1046 1199 1249 874 842 957 1181 1134 933 1145 802 988 949 874 728 651 818 414 596 610 968 684 Wire Scrape: Alternatively… Distribution ID via Reliability/Survival
Wire Scrape Distribution ID via Reliability/Survival
Least Squares vs Maximum Likelihood Least squares (LSXY) • Better graphical display • Better for small samples without censoring Maximum likelihood (MLE) • More precise than least squares (XY) • Better for heavy censoring • MLE allows you to perform an analysis when there are no failures Distribution ID via Reliability/Survival
12008004000 90 50 10 1 Cable scrape 1kg Percent 12501000750500 99 90 50 10 1 Cable scrape 1kg Percent 15001000500 99 90 50 10 1 Cable scrape 1kg Percent Smallest Extreme V alue 0.874 Normal 0.771 Logistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Smallest Extreme Value Normal Logistic 1000500 90 50 10 1 Cable scrape 1kg - Threshold Percent 219900219600219300219000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 100010010 90 50 10 1 Cable scrape 1kg - Threshold Percent 218000217500217000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 3-Parameter Weibull 0.771 3-Parameter Lognormal 0.771 2-Parameter Exponential 3.745 3-Parameter Loglogistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data 3-Parameter Weibull 3-Parameter Lognormal 2-Parameter Exponential 3-Parameter Loglogistic 1000500 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent 10000100010010 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent Weibull 0.771 Lognormal 0.932 Exponential 6.208 Loglogistic 0.750 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Weibull Lognormal Exponential Loglogistic x x x x Distribution ID via Reliability/Survival
Review the Warnings Again 3-Parameter Lognormal * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. 2-Parameter Exponential * WARNING * Variance/Covariance matrix of estimated parameters does not exist. The threshold parameter is assumed fixed when calculating confidence intervals. 3-Parameter Loglogistic * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. Still poor choices Distribution ID via Reliability/Survival
12008004000 90 50 10 1 Cable scrape 1kg Percent 12501000750500 99 90 50 10 1 Cable scrape 1kg Percent 15001000500 99 90 50 10 1 Cable scrape 1kg Percent Smallest Extreme V alue 0.874 Normal 0.771 Logistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Smallest Extreme Value Normal Logistic 1000500 90 50 10 1 Cable scrape 1kg - Threshold Percent 219900219600219300219000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 100010010 90 50 10 1 Cable scrape 1kg - Threshold Percent 218000217500217000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 3-Parameter Weibull 0.771 3-Parameter Lognormal 0.771 2-Parameter Exponential 3.745 3-Parameter Loglogistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data 3-Parameter Weibull 3-Parameter Lognormal 2-Parameter Exponential 3-Parameter Loglogistic 1000500 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent 10000100010010 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent Weibull 0.771 Lognormal 0.932 Exponential 6.208 Loglogistic 0.750 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Weibull Lognormal Exponential Loglogistic x x x x x x Last visual inspection kept: Weibull 3 parameter Weibull Smallest extreme value Logistic Normal x Distribution ID via Reliability/Survival
Anderson-Darling Distribution (adj) Weibull 0.771 Lognormal 0.932 Exponential 6.208 Loglogistic 0.750 3-Parameter Weibull 0.771 3-Parameter Lognormal 0.771 2-Parameter Exponential 3.745 3-Parameter Loglogistic 0.738 Smallest Extreme Value 0.874 Normal 0.771 Logistic 0.738 Note: • Using Adjusted Anderson-Darling statistic* • No P value provided • no Likelihood ratio test BUT! Can use censored data *Even when the data are uncensored, the adjusted Anderson-Darling statistic will not necessarily yield the same result as the non-adjusted Anderson-Darling statistic for small samples. However, for large sample sizes, the disparity between the two approaches vanishes. Distribution ID via Reliability/Survival
Table of Percentiles Standard 95% Normal CI Distribution Percent Percentile Error Lower Upper Weibull 1 376.505 68.5763 263.471 538.032 Lognormal 1 464.956 50.8249 375.289 576.047 Exponential 1 8.97312 1.91308 5.90836 13.6276 Loglogistic 1 445.882 60.9477 341.090 582.869 3-Parameter Weibull 1 373.325* 155.365 68.8161 677.835 3-Parameter Lognormal 1 393.301* 88.0037 220.817 565.785 2-Parameter Exponential 1 396.241 1.07483 394.140 398.353 3-Parameter Loglogistic 1 320.697 112.297 100.599 540.795 Smallest Extreme Value 1 123.762 162.753 -195.228 442.752 Normal 1 392.831 88.2110 219.940 565.721 Logistic 1 320.018 112.575 99.3745 540.661 Table of MTTF Standard 95% Normal CI Distribution Mean Error Lower Upper Weibull 894.250 45.098 810.087 987.16 Lognormal 895.154 51.723 799.309 1002.49 Exponential 892.818 190.349 587.877 1355.94 Loglogistic 917.307 51.467 821.783 1023.94 3-Parameter Weibull 894.247* 45.156 805.742 982.75 3-Parameter Lognormal 892.885* 45.828 803.065 982.71 2-Parameter Exponential 892.816 106.945 705.995 1129.07 3-Parameter Loglogistic 898.124 47.193 805.628 990.62 Smallest Extreme Value 887.984 51.629 786.794 989.18 Normal 892.818 45.822 803.009 982.63 Logistic 898.053 47.191 805.560 990.55 Other Output: *note: the 3rd parameter (threshold) is not commonly accurate for sample sizes below 10, 20 or greater is recommended Distribution ID via Reliability/Survival
Many Distributions – notice the similarity Distribution ID via Reliability/Survival
Which distribution ID to use? • Recommend using Stat > Quality Tools > Individual Distribution Identification, Since it includes P values and Maximum Likelihood Ratio • Unless you have censored data Distribution ID
Testing to failure (Right censoring)
Testing to Failure - Remember censoring? • Some tests will not be able to run to the end due to time constraints = Right censored data or suspended data • Some tests cannot be continuously monitored = Arbitrary censoring
Testing to Failure • Based on MIL DTL -915G • Checking for continuity on 10-20 stations • Failure mode – open on any conductor • Each unit is independently measured every millisecond and cycles are recorded Cable Flex -Right Censored
cycles susp qty 218457 s 3 78014 n 1 124859 n 1 36657 n 1 109032 n 1 58111 n 1 169316 n 1 204050 n 1 Testing to Failure Cable Flex -Right Censored
Testing to Failure Cable Flex -Right Censored cycles susp qty 218457 s 3 78014 n 1 124859 n 1 36657 n 1 109032 n 1 58111 n 1 169316 n 1 204050 n 1
Testing to Failure Cable Flex -Right Censored
Least Squares vs Maximum Likelihood Least squares (LSXY) • Better graphical display • Better for small samples without censoring Maximum likelihood (MLE) • More precise than least squares (XY) • Better for heavy censoring • MLE allows you to perform an analysis when there are no failures Testing to Failure
5000004000003000002000001000000 100 80 60 40 20 0 Shape 1 .621 35 Scale 1 91 31 4 Mean 1 71 324 StDev 1 0831 1 Median 1 52607 IQR 1 45293 Failure 7 Censor 3 AD* 22.41 6 Table of Statistics cycles Percent Cumulative Failure Plot for cycles Censoring Column in susp - ML Estimates Weibull - 95% CI Testing to Failure Cable Flex -Right Censored
Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 1.62135 0.533550 0.850680 3.09021 Scale 191314 44798.7 120900 302738 Log-Likelihood = -91.724 Goodness-of-Fit Anderson-Darling (adjusted) = 22.416 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 171324 40868.3 107342 273444 Standard Deviation 108311 45660.1 47406.2 247462 Median 152607 36459.0 95546.4 243743 First Quartile(Q1) 88719.9 29101.9 46645.5 168746 Third Quartile(Q3) 234013 58343.0 143556 381467 Interquartile Range(IQR) 145293 53304.7 70787.5 298216 Testing to Failure Cable Flex -Right Censored
Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 1 11208.6 10546.2 1772.79 70867.5 2 17241.5 13868.6 3563.63 83417.6 3 22210.1 16102.7 5363.08 91978.8 4 26606.6 17800.1 7170.08 98731.0 5 30630.3 19167.9 8984.34 104428 6 34386.8 20309.6 10805.7 109429 7 37940.2 21285.8 12634.1 113934 8 41332.9 22134.9 14469.6 118069 9 44595.1 22883.4 16311.9 121918 10 47749.0 23550.3 18161.2 125541 20 75853.2 27772.4 37010.2 155463 30 101298 30281.9 56382.8 181995 40 126421 32818.9 76005.8 210277 50 152607 36459.0 95546.4 243743 60 181272 42265.0 114780 286283 70 214521 51626.6 133850 343811 80 256577 67119.2 153655 428437 90 320001 96517.1 177180 577945 91 328954 101139 180064 600957 92 338787 106333 183133 626741 93 349725 112247 186434 656036 94 362093 119102 190037 689926 95 376392 127234 194045 730092 96 393445 137211 198625 779351 97 414776 150087 204083 842987 98 443734 168222 211070 932863 99 490701 199090 221545 1086856 Testing to Failure Cable Flex -Right Censored
Testing to failure (Arbitrary censoring)
Drop on the connector 10 drops at 40” then 10 drops at 50” then 10 drops at 60” etc. until failure Testing to Failure Stepped Drop Test - Arbitrary Censoring
Tabulated Data Number of drops before failure @ 40 inch(if <10) Number of drops before failure @ 50 inch(if <10) Number of drops before failure @ 60 inch(if <10) Total inches Passed Total inches Failed 1 10 10 6 1260 1320 2 10 4 600 650 3 10 10 2 1020 1080 4 10 8 800 850 5 10 7 750 800 6 10 10 2 1020 1080 Testing to Failure Stepped Drop Test - Arbitrary Censoring
Testing to Failure Stepped Drop Test - Arbitrary Censoring Total inches Passed Total inches Failed 1260 1320 600 650 1020 1080 800 850 750 800 1020 1080
Testing to Failure Stepped Drop Test - Arbitrary Censoring Total inches Passed Total inches Failed 1260 1320 600 650 1020 1080 800 850 750 800 1020 1080
2000 1500 1000 900 800 700 600 500 400 300 200 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Total inches Passed Percent Shape 4.76974 Scale 1022.65 Mean 936.407 StDev 223.914 Median 947.015 IQ R 307.573 A D* 2.852 Table of Statistics Probability Plot for Total inches Passed Arbitrary Censoring - ML Estimates Weibull - 95% CI 150012501000750500 100 80 60 40 20 0 Total inches Passed Percent Shape 4.76974 Scale 1022.65 Mean 936.407 StDev 223.914 Median 947.015 IQ R 307.573 A D* 2.852 Table of Statistics Cumulative Failure Plot for Total inches Passed Arbitrary Censoring - ML Estimates Weibull - 95% CI Testing to Failure Stepped Drop Test - Arbitrary Censoring
How Is It Different? 2000 1500 1000 900 800 700 600 500 400 300 200 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Total inches Passed Percent Shape 4.76974 Scale 1022.65 Mean 936.407 StDev 223.914 Median 947.015 IQ R 307.573 A D* 2.852 Table of Statistics Probability Plot for Total inches Passed Arbitrary Censoring - ML Estimates Weibull - 95% CI 2000 1500 1000 900 800 700 600 500 400 300 200 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Data Percent 4.66343 994.17 2.174 6 0 4.85171 1051.86 2.178 6 0 4.75849 1023.02 2.176 6 0 Shape Scale A D* F C Table of Statistics Total inches Passed Total inches Failed average Variable Probability Plot for Total inches Passed, Total inches Failed, average Complete Data - ML Estimates Weibull - 95% CI In this case, fairly close to the average. But with wider intervals it becomes very different. Testing to Failure Arbitrary vs Right censoring
Acceleration Factors
Acceleration Factors • Reliability engineers attempt to speed up testing by “accelerating” the test • Environmental: • Increased temperature • Increased temperature swing • Adding environmental factors together • Sequentially testing • Mechanical • Increased speed • Increased weight • Others? • Multiple factors
Wire Scrape Test Back to: • This test can use different weights, what effect does that have? • In this case, we used 4 weights: 0.85 kg, 1 kg, 1.1 kg, 1.25 kg Acceleration Factors
Acceleration Factors Wire Scrape cycles weight 3055 0.85 3047 0.85 1728 0.85 3155 0.85 2498 0.85 2341 0.85 4140 0.85 1583 0.85 2908 0.85 4581 0.85 2286 0.85 1630 0.85 2521 0.85 2849 0.85 2283 0.85 6018 0.85 1394 0.85 2345 0.85 2984 0.85 1046 1 1199 1 1249 1 874 1 842 1 957 1 1181 1 1134 1 933 1 1145 1 802 1 988 1 949 1 874 1 728 1 651 1 818 1 414 1 596 1 610 1 968 1 684 1 845 1.1 591 1.1 852 1.1 655 1.1 382 1.1 397 1.1 315 1.1 299 1.1 471 1.1 806 1.1 699 1.1 440 1.1 547 1.1 317 1.1 270 1.1 1304 1.1 691 1.1 543 1.1 360 1.1 415 1.1 226 1.25 134 1.25 270 1.25 200 1.25 155 1.25 139 1.25 446 1.25 474 1.25 199 1.25 164 1.25 146 1.25 247 1.25 223 1.25 205 1.25 412 1.25 231 1.25 225 1.25 310 1.25 173 1.25 132 1.25
Acceleration Factors Wire Scrape
Acceleration Factors Wire Scrape 100001000100 99 90 80 70 60 50 40 30 20 10 5 3 2 1 cycles Percent 2.58992 2854.89 2.101 19 0 2.58992 1154.19 2.433 22 0 2.58992 631.06 1.179 20 0 2.58992 255.13 1.735 20 0 Shape Scale A D* F C Table of Statistics 0.85 1.00 1.10 1.25 weight Probability Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Relationship with accelerating variable(s): Linear Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 13.0887 0.292537 44.74 0.000 12.5154 13.6621 weight -6.03758 0.275137 -21.94 0.000 -6.57684 -5.49832 Shape 2.58992 0.202125 2.22257 3.01798 Log-Likelihood = -574.192 Confidence intervals removed for clarity
(The error term depends on the percentile of interest. For the 50 percentile, -0.3665 is the error term) 13.0887 -6.03758 2.58992 Relationship with accelerating variable(s): Linear Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 13.0887 0.292537 44.74 0.000 12.5154 13.6621 weight -6.03758 0.275137 -21.94 0.000 -6.57684 -5.49832 Shape 2.58992 0.202125 2.22257 3.01798 Log-Likelihood = -574.192 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*error term] Weight (kg) Prediction (cycles) 0.50 20504.5 0.75 4532.4 1.00 1001.9 1.25 221.5 1.50 49.0 1.75 10.8 point estimates Acceleration Factors Wire Scrape
Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 0.75 4532.4 1.00 1001.9 1.25 221.5 1.50 49.0 1.75 10.8 Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 20504.5 4532.4 = 4.52 point estimates Acceleration Factors
Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 0.75 4532.4 1.00 1001.9 1.25 221.5 1.50 49.0 1.75 10.8 Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 4.52 Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 0.75 4532.4 4.52 1.00 1001.9 4.52 1.25 221.5 4.52 1.50 49.0 4.52 1.75 10.8 … point estimates Acceleration Factors
Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 0.75 4532.4 4.52 1.00 1001.9 4.52 1.25 221.5 4.52 1.50 49.0 4.52 1.75 10.8 … Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 20504.5 49.0 = 418.9 point estimates Acceleration Factors
Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 0.75 4532.4 4.52 1.00 1001.9 4.52 1.25 221.5 4.52 1.50 49.0 4.52 1.75 10.8 … Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 418.9 Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 418.9 0.75 4532.4 4.52 418.9 1.00 1001.9 4.52 … 1.25 221.5 4.52 … 1.50 49.0 4.52 … 1.75 10.8 … … point estimates Acceleration Factors
Acceleration Factors Multiple factors
Acceleration Factors • Reliability engineers attempt to speed up testing by “accelerating” the test • Environmental: • Increased temperature • Increased temperature swing • Adding environmental factors together • Sequentially testing • Mechanical • Increased speed • Increased weight • Others? • Multiple factors
Cable Flex • Checking for continuity on 10-20 stations • Failure mode – open on any conductor • Varying Mandrel • Varying Weight Multiple Acceleration Factors Back to: Wt
cycles susp weight (kg) mandrel (mm) qty 942 n 0.5 4 1 735 n 0.5 4 1 1289 n 0.5 4 1 1487 n 0.5 4 1 1222 n 0.5 4 1 2750 n 0.25 4 1 3074 n 0.25 4 1 2547 n 0.25 4 1 2992 n 0.25 4 1 4338 n 0.25 4 1 1954 n 0.25 4 1 1845 n 0.4 4 1 867 n 0.4 4 1 1004 n 0.4 4 1 1182 n 0.4 4 1 3451 n 0.4 4 1 2715 n 0.4 4 1 1071 n 0.5 4 1 831 n 0.5 4 1 1114 n 0.5 4 1 1159 n 0.5 4 1 934 n 0.5 4 1 1130 n 0.5 4 1 1150 n 0.5 4 1 1115 n 0.5 4 1 1560 n 0.5 4 1 1465 n 0.5 4 1 939 n 0.5 4 1 1343 n 0.5 4 1 1331 n 0.5 4 1 1329 n 0.5 4 1 2264 n 0.5 4 1 17701 n 1 25.4 1 12741 n 1 25.4 1 12805 n 1 25.4 1 14467 n 1 25.4 1 17301 n 1 25.4 1 18815 n 1 25.4 1 21800 n 0.5 25.4 1 19710 n 0.5 25.4 1 29224 n 0.5 25.4 1 35348 n 0.5 25.4 1 24332 n 0.5 25.4 1 24865 n 0.5 25.4 1 33948 n 0.5 25.4 1 33944 n 0.25 25.4 1 29157 n 0.25 25.4 1 32391 n 0.25 25.4 1 34150 n 0.25 25.4 1 31960 n 0.25 25.4 1 29278 n 0.25 25.4 1 70000 s 0.25 25.4 1 Cable Flex Multiple Acceleration Factors
Cable Flex Multiple Acceleration Factors
Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 7.66490 0.106594 71.91 0.000 7.45598 7.87382 weight -1.65024 0.225415 -7.32 0.000 -2.09205 -1.20843 Mandrel-mm 0.138599 0.0056508 24.53 0.000 0.127524 0.149675 Shape 2.89069 0.296798 2.36377 3.53506 Log-Likelihood = -449.260 Confidence intervals removed for clarity Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 7.66490 0.106594 71.91 0.000 7.45598 7.87382 weight -1.65024 0.225415 -7.32 0.000 -2.09205 -1.20843 Mandrel-mm 0.138599 0.0056508 24.53 0.000 0.127524 0.149675 Shape 2.89069 0.296798 2.36377 3.53506 Log-Likelihood = -449.260 7.66490 -1.65024 2.89069 *Error term= -0.3665 for 50% 0.138599 Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*error term] Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
710.6 Acceleration factor in a traditional sense Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*(-0.3665)] Weight (kg) Mandrel (mm) Prediction (cycle) Acceleration factor per 1/2 Kg Acceleration factor per 5mm mandrel increase 0.5 5.0 1649.0 0.5 10.0 3304.0 0.5 15.0 6620.0 1.0 5.0 722.6 1.0 10.0 1447.8 1.0 15.0 2900.7 1621.8 = 2.28 point estimates Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
3304.0 Acceleration factor in a traditional sense Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*(-0.3665)] Weight (kg) Mandrel (mm) Prediction (cycle) Acceleration factor per 1/2 Kg Acceleration factor per 5mm mandrel increase 0.5 5.0 1649.0 2.28 0.5 10.0 3304.0 2.28 0.5 15.0 6620.0 2.28 1.0 5.0 722.6 … 1.0 10.0 1447.8 … 1.0 15.0 2900.7 … 1649.0 = 2.00 point estimates Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
Acceleration factor in a traditional sense Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*(-0.3665)] point estimates Confidence intervals removed for clarity Weight (kg) Mandrel (mm) Prediction (cycle) Acceleration factor per 1/2 Kg Acceleration factor per 5mm mandrel increase 0.5 5.0 1649.0 2.28 2.00 0.5 10.0 3304.0 2.28 2.00 0.5 15.0 6620.0 2.28 … 1.0 5.0 722.6 … 2.00 1.0 10.0 1447.8 … 2.00 1.0 15.0 2900.7 … … Cable Flex Multiple Acceleration Factors
Summary • Distribution selection can be assisted with statistics, but knowledge about your data is key • Use Right Censoring if an exact count is available, Arbitrary Censoring is for periodic testing • Accelerated testing can provide faster answers and testing at different levels can assist in understanding the relationship for the acceleration
Author Rob Schubert Corporate Quality/Reliability Engineer Shure Inc – 9 years Schubert_Rob@shure.com Certified Reliability Engineer (ASQ) Master’s in Acoustical Engineering, Penn State Thesis: Use of Multiple-Input Single-Output Methods to Increase Repeatability of Measurements of Road Noise in Automobiles Recent presentations: 2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test – ARS 2015 Previous work experience: Ford (13 years) - Quality/Reliability Engineer, 6 Sigma Black belt, Noise & Vibration Engineer
Thank you for your attention. Do you have any questions?

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Overview of life testing in Minitab

  • 1. Overview of Reliability Functions for Life Testing in Minitab
  • 2. Introduction Shure Inc. “The Most Trusted Audio Brand Worldwide” Industry: Consumer and professional audio electronics Founded: 1925 Products: Microphones, wireless microphone systems, headphones and earphones, mixers, conferencing systems Rob Schubert - Corporate Quality/Reliability Engineer at Shure Inc.
  • 3. Agenda • Intro to Reliability • Selecting a distribution • Testing to failure – Right censoring • Testing to failure – Arbitrary censoring • Accelerated testing – Single Factor • Accelerated testing – Multiple factors • Summary • Questions
  • 4. Quick Intro to Reliability • Reliability = quality over time • Field failures • Failure modes • Testing to replicate failure modes • Measuring time to failure • Attempt to predict failure rate
  • 5. Probability Distribution Function (PDF) • A function that describes the relative likelihood for this random variable to take on a given value • Integral over a range = probability of the variable falling within that range Quick Intro to Reliability
  • 7. Different ways to visualize the data Instantaneous probability Cumulative probability Straight line cumulative probability transformed so that the fitted distribution line forms a straight line Quick Intro to Reliability
  • 8. Parameters Shape = Slope: slope of the line in a cumulative straight line probability plot Quick Intro to Reliability
  • 9. Parameters 100000100001000100 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Data Percent Effect of Scale Parameter on Weibull plot Scale = Shifts the data “out” In a failure plot that means more time passes before each failure Quick Intro to Reliability
  • 10. Parameters Threshold / Location: The 3rd parameter – shifts the zero point Forced to zero in a: 2-Parameter Lognormal Normally called just “Lognormal“ 1-Parameter Exponential Normally called just “Exponential“ 2-Parameter Weibull Normally called just “Weibull“ 2-Parameter Gamma Normally called just “gamma“ 2-Parameter Loglogistic Normally called just “Loglogistic“ (not represented on straight line cumulative probability plot) Quick Intro to Reliability
  • 11. Testing to Failure: What is censoring? • Some tests will not be able to run to the end due to time constraints = Right censored data or Suspended data • Some tests cannot be continuously monitored = Arbitrary censoring Quick Intro to Reliability
  • 13. Many Distributions – and more – which one? Distribution ID
  • 14. • What do you know about your data? Reliability data starts at time=0 • Is it bound? Bound by zero (or something greater) • Any historical references? Weibull, Lognormal, Gumbel (extreme value) • Finally Does it look like it fits? Distribution ID
  • 15. Statistics to look at: • Anderson-Darling test: How well your data fits the PDF • Smaller fits better • p value: Student’s t-test – does your data fit the PDF? • H0 = data fits the distribution • p is low, the null must go! i.e. if p<0.05, it doesn’t fit the distribution • Likelihood ratio test (p value) – does 3 parameter fit better? • H0 = 2 parameter fits better • i.e. if <0.05, the 3 parameter fits better Distribution ID
  • 16. Example: Wire Scrape Test • Cable gets frayed in the field • Need to replicate • Scrape at several points • Measure cycles until breakthrough Distribution ID
  • 17. 3 steps to identify the distribution 1. Look at graphs 2. Look at warnings 3. Look at Anderson darling numbers Distribution ID
  • 18. Wire Scrape Cable scrape 1kg 1046 1199 1249 874 842 957 1181 1134 933 1145 802 988 949 874 728 651 818 414 596 610 968 684 Distribution ID
  • 19. Wire Scrape Cable scrape 1kg 1046 1199 1249 874 842 957 1181 1134 933 1145 802 988 949 874 728 651 818 414 596 610 968 684 Distribution ID
  • 20. 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 221000220500220000219500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Lognormal A D = 0.489 P-V alue = 0.198 3-Parameter Lognormal A D = 0.224 P-V alue = * Goodness of Fit Test Normal A D = 0.212 P-V alue = 0.833 Box-C ox Transformation A D = 0.212 P-V alue = 0.833 A fter Box-C ox transformation (lambda = 1) Probability Plot for Cable scrape 1.0Kg Normal - 95% C I Normal - 95% C I Lognormal - 95% C I 3-Parameter Lognormal - 95% C I 10000100010010 90 50 10 1 Cable scrape 1.0Kg Percent 100001000100101 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 1000500 90 50 10 1 Cable scrape 1.0Kg Percent 1000500200 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Weibull A D = 0.205 P-V alue > 0.250 3-Parameter Weibull A D = 0.202 P-V alue > 0.500 Goodness of Fit Test Exponential A D = 5.590 P-V alue < 0.003 2-Parameter Exponential A D = 3.064 P-V alue < 0.010 Probability Plot for Cable scrape 1.0Kg Exponential - 95% C I 2-Parameter Exponential - 95% C I Weibull - 95% C I 3-Parameter Weibull - 95% C I 150010005000 90 50 10 1 Cable scrape 1.0Kg Percent 200015001000500 99 90 50 10 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 3300300027002400 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Gamma A D = 0.369 P-V alue > 0.250 3-Parameter Gamma A D = 0.271 P-V alue = * Goodness of Fit Test Smallest Extreme V alue A D = 0.270 P-V alue > 0.250 Largest Extreme V alue A D = 0.556 P-V alue = 0.152 Probability Plot for Cable scrape 1.0Kg Smallest Extreme V alue - 95% C I Largest Extreme V alue - 95% C I Gamma - 95% C I 3-Parameter Gamma - 95% C I 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 219000218500218000217500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 3-Parameter Loglogistic A D = 0.213 P-V alue = * Goodness of Fit Test Logistic A D = 0.213 P-V alue > 0.250 Loglogistic A D = 0.355 P-V alue > 0.250 Probability Plot for Cable scrape 1.0Kg Logistic - 95% C I Loglogistic - 95% C I 3-Parameter Loglogistic - 95% C I x x x x x x x Distribution ID
  • 21. Note the Warnings! These are probably poor choices 3-Parameter Lognormal * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. 2-Parameter Exponential * WARNING * Variance/Covariance matrix of estimated parameters does not exist. The threshold parameter is assumed fixed when calculating confidence intervals. 3-Parameter Gamma * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. 3-Parameter Loglogistic * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. Distribution ID
  • 22. 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 221000220500220000219500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Lognormal A D = 0.489 P-V alue = 0.198 3-Parameter Lognormal A D = 0.224 P-V alue = * Goodness of Fit Test Normal A D = 0.212 P-V alue = 0.833 Box-C ox Transformation A D = 0.212 P-V alue = 0.833 A fter Box-C ox transformation (lambda = 1) Probability Plot for Cable scrape 1.0Kg Normal - 95% C I Normal - 95% C I Lognormal - 95% C I 3-Parameter Lognormal - 95% C I 10000100010010 90 50 10 1 Cable scrape 1.0Kg Percent 100001000100101 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 1000500 90 50 10 1 Cable scrape 1.0Kg Percent 1000500200 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Weibull A D = 0.205 P-V alue > 0.250 3-Parameter Weibull A D = 0.202 P-V alue > 0.500 Goodness of Fit Test Exponential A D = 5.590 P-V alue < 0.003 2-Parameter Exponential A D = 3.064 P-V alue < 0.010 Probability Plot for Cable scrape 1.0Kg Exponential - 95% C I 2-Parameter Exponential - 95% C I Weibull - 95% C I 3-Parameter Weibull - 95% C I 150010005000 90 50 10 1 Cable scrape 1.0Kg Percent 200015001000500 99 90 50 10 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 3300300027002400 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent Gamma A D = 0.369 P-V alue > 0.250 3-Parameter Gamma A D = 0.271 P-V alue = * Goodness of Fit Test Smallest Extreme V alue A D = 0.270 P-V alue > 0.250 Largest Extreme V alue A D = 0.556 P-V alue = 0.152 Probability Plot for Cable scrape 1.0Kg Smallest Extreme V alue - 95% C I Largest Extreme V alue - 95% C I Gamma - 95% C I 3-Parameter Gamma - 95% C I 150010005000 99 90 50 10 1 Cable scrape 1.0Kg Percent 20001000500 99 90 50 10 1 Cable scrape 1.0Kg Percent 219000218500218000217500 99 90 50 10 1 Cable scrape 1.0Kg - T hreshold Percent 3-Parameter Loglogistic A D = 0.213 P-V alue = * Goodness of Fit Test Logistic A D = 0.213 P-V alue > 0.250 Loglogistic A D = 0.355 P-V alue > 0.250 Probability Plot for Cable scrape 1.0Kg Logistic - 95% C I Loglogistic - 95% C I 3-Parameter Loglogistic - 95% C I x x x x x x x x x Last visual inspection kept: Normal 3 param. lognormal 2/3 param. Weibull Smallest Extreme Val 3 param. gamma Logistic x x Distribution ID
  • 23. Goodness of Fit Test Distribution AD P LRT P Normal 0.212 0.833 Box-Cox Transformation 0.212 0.833 Lognormal 0.489 0.198 3-Parameter Lognormal 0.224 * 0.061 Exponential 5.590 <0.003 2-Parameter Exponential 3.064 <0.010 0.000 Weibull 0.205 >0.250 3-Parameter Weibull 0.202 >0.500 0.808 Smallest Extreme Value 0.270 >0.250 Largest Extreme Value 0.556 0.152 Gamma 0.369 >0.250 3-Parameter Gamma 0.271 * 0.229 Logistic 0.213 >0.250 Loglogistic 0.355 >0.250 3-Parameter Loglogistic 0.213 * 0.147 Likelihood ratio test = if small (<.05), 3 parameter fits better Distribution ID
  • 25. Wire Scrape Distribution ID via Reliability/Survival
  • 26. Least Squares vs Maximum Likelihood Least squares (LSXY) • Better graphical display • Better for small samples without censoring Maximum likelihood (MLE) • More precise than least squares (XY) • Better for heavy censoring • MLE allows you to perform an analysis when there are no failures Distribution ID via Reliability/Survival
  • 27. 12008004000 90 50 10 1 Cable scrape 1kg Percent 12501000750500 99 90 50 10 1 Cable scrape 1kg Percent 15001000500 99 90 50 10 1 Cable scrape 1kg Percent Smallest Extreme V alue 0.874 Normal 0.771 Logistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Smallest Extreme Value Normal Logistic 1000500 90 50 10 1 Cable scrape 1kg - Threshold Percent 219900219600219300219000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 100010010 90 50 10 1 Cable scrape 1kg - Threshold Percent 218000217500217000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 3-Parameter Weibull 0.771 3-Parameter Lognormal 0.771 2-Parameter Exponential 3.745 3-Parameter Loglogistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data 3-Parameter Weibull 3-Parameter Lognormal 2-Parameter Exponential 3-Parameter Loglogistic 1000500 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent 10000100010010 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent Weibull 0.771 Lognormal 0.932 Exponential 6.208 Loglogistic 0.750 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Weibull Lognormal Exponential Loglogistic x x x x Distribution ID via Reliability/Survival
  • 28. Review the Warnings Again 3-Parameter Lognormal * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. 2-Parameter Exponential * WARNING * Variance/Covariance matrix of estimated parameters does not exist. The threshold parameter is assumed fixed when calculating confidence intervals. 3-Parameter Loglogistic * WARNING * Newton-Raphson algorithm has not converged after 50 iterations. * WARNING * Convergence has not been reached for the parameter estimates criterion. Still poor choices Distribution ID via Reliability/Survival
  • 29. 12008004000 90 50 10 1 Cable scrape 1kg Percent 12501000750500 99 90 50 10 1 Cable scrape 1kg Percent 15001000500 99 90 50 10 1 Cable scrape 1kg Percent Smallest Extreme V alue 0.874 Normal 0.771 Logistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Smallest Extreme Value Normal Logistic 1000500 90 50 10 1 Cable scrape 1kg - Threshold Percent 219900219600219300219000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 100010010 90 50 10 1 Cable scrape 1kg - Threshold Percent 218000217500217000 99 90 50 10 1 Cable scrape 1kg - Threshold Percent 3-Parameter Weibull 0.771 3-Parameter Lognormal 0.771 2-Parameter Exponential 3.745 3-Parameter Loglogistic 0.738 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data 3-Parameter Weibull 3-Parameter Lognormal 2-Parameter Exponential 3-Parameter Loglogistic 1000500 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent 10000100010010 90 50 10 1 Cable scrape 1kg Percent 1000500 99 90 50 10 1 Cable scrape 1kg Percent Weibull 0.771 Lognormal 0.932 Exponential 6.208 Loglogistic 0.750 A nderson-Darling (adj) Probability Plot for Cable scrape 1kg ML Estimates-Complete Data Weibull Lognormal Exponential Loglogistic x x x x x x Last visual inspection kept: Weibull 3 parameter Weibull Smallest extreme value Logistic Normal x Distribution ID via Reliability/Survival
  • 30. Anderson-Darling Distribution (adj) Weibull 0.771 Lognormal 0.932 Exponential 6.208 Loglogistic 0.750 3-Parameter Weibull 0.771 3-Parameter Lognormal 0.771 2-Parameter Exponential 3.745 3-Parameter Loglogistic 0.738 Smallest Extreme Value 0.874 Normal 0.771 Logistic 0.738 Note: • Using Adjusted Anderson-Darling statistic* • No P value provided • no Likelihood ratio test BUT! Can use censored data *Even when the data are uncensored, the adjusted Anderson-Darling statistic will not necessarily yield the same result as the non-adjusted Anderson-Darling statistic for small samples. However, for large sample sizes, the disparity between the two approaches vanishes. Distribution ID via Reliability/Survival
  • 31. Table of Percentiles Standard 95% Normal CI Distribution Percent Percentile Error Lower Upper Weibull 1 376.505 68.5763 263.471 538.032 Lognormal 1 464.956 50.8249 375.289 576.047 Exponential 1 8.97312 1.91308 5.90836 13.6276 Loglogistic 1 445.882 60.9477 341.090 582.869 3-Parameter Weibull 1 373.325* 155.365 68.8161 677.835 3-Parameter Lognormal 1 393.301* 88.0037 220.817 565.785 2-Parameter Exponential 1 396.241 1.07483 394.140 398.353 3-Parameter Loglogistic 1 320.697 112.297 100.599 540.795 Smallest Extreme Value 1 123.762 162.753 -195.228 442.752 Normal 1 392.831 88.2110 219.940 565.721 Logistic 1 320.018 112.575 99.3745 540.661 Table of MTTF Standard 95% Normal CI Distribution Mean Error Lower Upper Weibull 894.250 45.098 810.087 987.16 Lognormal 895.154 51.723 799.309 1002.49 Exponential 892.818 190.349 587.877 1355.94 Loglogistic 917.307 51.467 821.783 1023.94 3-Parameter Weibull 894.247* 45.156 805.742 982.75 3-Parameter Lognormal 892.885* 45.828 803.065 982.71 2-Parameter Exponential 892.816 106.945 705.995 1129.07 3-Parameter Loglogistic 898.124 47.193 805.628 990.62 Smallest Extreme Value 887.984 51.629 786.794 989.18 Normal 892.818 45.822 803.009 982.63 Logistic 898.053 47.191 805.560 990.55 Other Output: *note: the 3rd parameter (threshold) is not commonly accurate for sample sizes below 10, 20 or greater is recommended Distribution ID via Reliability/Survival
  • 32. Many Distributions – notice the similarity Distribution ID via Reliability/Survival
  • 33. Which distribution ID to use? • Recommend using Stat > Quality Tools > Individual Distribution Identification, Since it includes P values and Maximum Likelihood Ratio • Unless you have censored data Distribution ID
  • 35. Testing to Failure - Remember censoring? • Some tests will not be able to run to the end due to time constraints = Right censored data or suspended data • Some tests cannot be continuously monitored = Arbitrary censoring
  • 36. Testing to Failure • Based on MIL DTL -915G • Checking for continuity on 10-20 stations • Failure mode – open on any conductor • Each unit is independently measured every millisecond and cycles are recorded Cable Flex -Right Censored
  • 37. cycles susp qty 218457 s 3 78014 n 1 124859 n 1 36657 n 1 109032 n 1 58111 n 1 169316 n 1 204050 n 1 Testing to Failure Cable Flex -Right Censored
  • 38. Testing to Failure Cable Flex -Right Censored cycles susp qty 218457 s 3 78014 n 1 124859 n 1 36657 n 1 109032 n 1 58111 n 1 169316 n 1 204050 n 1
  • 39. Testing to Failure Cable Flex -Right Censored
  • 40. Least Squares vs Maximum Likelihood Least squares (LSXY) • Better graphical display • Better for small samples without censoring Maximum likelihood (MLE) • More precise than least squares (XY) • Better for heavy censoring • MLE allows you to perform an analysis when there are no failures Testing to Failure
  • 41. 5000004000003000002000001000000 100 80 60 40 20 0 Shape 1 .621 35 Scale 1 91 31 4 Mean 1 71 324 StDev 1 0831 1 Median 1 52607 IQR 1 45293 Failure 7 Censor 3 AD* 22.41 6 Table of Statistics cycles Percent Cumulative Failure Plot for cycles Censoring Column in susp - ML Estimates Weibull - 95% CI Testing to Failure Cable Flex -Right Censored
  • 42. Parameter Estimates Standard 95.0% Normal CI Parameter Estimate Error Lower Upper Shape 1.62135 0.533550 0.850680 3.09021 Scale 191314 44798.7 120900 302738 Log-Likelihood = -91.724 Goodness-of-Fit Anderson-Darling (adjusted) = 22.416 Characteristics of Distribution Standard 95.0% Normal CI Estimate Error Lower Upper Mean(MTTF) 171324 40868.3 107342 273444 Standard Deviation 108311 45660.1 47406.2 247462 Median 152607 36459.0 95546.4 243743 First Quartile(Q1) 88719.9 29101.9 46645.5 168746 Third Quartile(Q3) 234013 58343.0 143556 381467 Interquartile Range(IQR) 145293 53304.7 70787.5 298216 Testing to Failure Cable Flex -Right Censored
  • 43. Table of Percentiles Standard 95.0% Normal CI Percent Percentile Error Lower Upper 1 11208.6 10546.2 1772.79 70867.5 2 17241.5 13868.6 3563.63 83417.6 3 22210.1 16102.7 5363.08 91978.8 4 26606.6 17800.1 7170.08 98731.0 5 30630.3 19167.9 8984.34 104428 6 34386.8 20309.6 10805.7 109429 7 37940.2 21285.8 12634.1 113934 8 41332.9 22134.9 14469.6 118069 9 44595.1 22883.4 16311.9 121918 10 47749.0 23550.3 18161.2 125541 20 75853.2 27772.4 37010.2 155463 30 101298 30281.9 56382.8 181995 40 126421 32818.9 76005.8 210277 50 152607 36459.0 95546.4 243743 60 181272 42265.0 114780 286283 70 214521 51626.6 133850 343811 80 256577 67119.2 153655 428437 90 320001 96517.1 177180 577945 91 328954 101139 180064 600957 92 338787 106333 183133 626741 93 349725 112247 186434 656036 94 362093 119102 190037 689926 95 376392 127234 194045 730092 96 393445 137211 198625 779351 97 414776 150087 204083 842987 98 443734 168222 211070 932863 99 490701 199090 221545 1086856 Testing to Failure Cable Flex -Right Censored
  • 45. Drop on the connector 10 drops at 40” then 10 drops at 50” then 10 drops at 60” etc. until failure Testing to Failure Stepped Drop Test - Arbitrary Censoring
  • 46. Tabulated Data Number of drops before failure @ 40 inch(if <10) Number of drops before failure @ 50 inch(if <10) Number of drops before failure @ 60 inch(if <10) Total inches Passed Total inches Failed 1 10 10 6 1260 1320 2 10 4 600 650 3 10 10 2 1020 1080 4 10 8 800 850 5 10 7 750 800 6 10 10 2 1020 1080 Testing to Failure Stepped Drop Test - Arbitrary Censoring
  • 47. Testing to Failure Stepped Drop Test - Arbitrary Censoring Total inches Passed Total inches Failed 1260 1320 600 650 1020 1080 800 850 750 800 1020 1080
  • 48. Testing to Failure Stepped Drop Test - Arbitrary Censoring Total inches Passed Total inches Failed 1260 1320 600 650 1020 1080 800 850 750 800 1020 1080
  • 49. 2000 1500 1000 900 800 700 600 500 400 300 200 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Total inches Passed Percent Shape 4.76974 Scale 1022.65 Mean 936.407 StDev 223.914 Median 947.015 IQ R 307.573 A D* 2.852 Table of Statistics Probability Plot for Total inches Passed Arbitrary Censoring - ML Estimates Weibull - 95% CI 150012501000750500 100 80 60 40 20 0 Total inches Passed Percent Shape 4.76974 Scale 1022.65 Mean 936.407 StDev 223.914 Median 947.015 IQ R 307.573 A D* 2.852 Table of Statistics Cumulative Failure Plot for Total inches Passed Arbitrary Censoring - ML Estimates Weibull - 95% CI Testing to Failure Stepped Drop Test - Arbitrary Censoring
  • 50. How Is It Different? 2000 1500 1000 900 800 700 600 500 400 300 200 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Total inches Passed Percent Shape 4.76974 Scale 1022.65 Mean 936.407 StDev 223.914 Median 947.015 IQ R 307.573 A D* 2.852 Table of Statistics Probability Plot for Total inches Passed Arbitrary Censoring - ML Estimates Weibull - 95% CI 2000 1500 1000 900 800 700 600 500 400 300 200 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Data Percent 4.66343 994.17 2.174 6 0 4.85171 1051.86 2.178 6 0 4.75849 1023.02 2.176 6 0 Shape Scale A D* F C Table of Statistics Total inches Passed Total inches Failed average Variable Probability Plot for Total inches Passed, Total inches Failed, average Complete Data - ML Estimates Weibull - 95% CI In this case, fairly close to the average. But with wider intervals it becomes very different. Testing to Failure Arbitrary vs Right censoring
  • 52. Acceleration Factors • Reliability engineers attempt to speed up testing by “accelerating” the test • Environmental: • Increased temperature • Increased temperature swing • Adding environmental factors together • Sequentially testing • Mechanical • Increased speed • Increased weight • Others? • Multiple factors
  • 53. Wire Scrape Test Back to: • This test can use different weights, what effect does that have? • In this case, we used 4 weights: 0.85 kg, 1 kg, 1.1 kg, 1.25 kg Acceleration Factors
  • 54. Acceleration Factors Wire Scrape cycles weight 3055 0.85 3047 0.85 1728 0.85 3155 0.85 2498 0.85 2341 0.85 4140 0.85 1583 0.85 2908 0.85 4581 0.85 2286 0.85 1630 0.85 2521 0.85 2849 0.85 2283 0.85 6018 0.85 1394 0.85 2345 0.85 2984 0.85 1046 1 1199 1 1249 1 874 1 842 1 957 1 1181 1 1134 1 933 1 1145 1 802 1 988 1 949 1 874 1 728 1 651 1 818 1 414 1 596 1 610 1 968 1 684 1 845 1.1 591 1.1 852 1.1 655 1.1 382 1.1 397 1.1 315 1.1 299 1.1 471 1.1 806 1.1 699 1.1 440 1.1 547 1.1 317 1.1 270 1.1 1304 1.1 691 1.1 543 1.1 360 1.1 415 1.1 226 1.25 134 1.25 270 1.25 200 1.25 155 1.25 139 1.25 446 1.25 474 1.25 199 1.25 164 1.25 146 1.25 247 1.25 223 1.25 205 1.25 412 1.25 231 1.25 225 1.25 310 1.25 173 1.25 132 1.25
  • 56. Acceleration Factors Wire Scrape 100001000100 99 90 80 70 60 50 40 30 20 10 5 3 2 1 cycles Percent 2.58992 2854.89 2.101 19 0 2.58992 1154.19 2.433 22 0 2.58992 631.06 1.179 20 0 2.58992 255.13 1.735 20 0 Shape Scale A D* F C Table of Statistics 0.85 1.00 1.10 1.25 weight Probability Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Relationship with accelerating variable(s): Linear Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 13.0887 0.292537 44.74 0.000 12.5154 13.6621 weight -6.03758 0.275137 -21.94 0.000 -6.57684 -5.49832 Shape 2.58992 0.202125 2.22257 3.01798 Log-Likelihood = -574.192 Confidence intervals removed for clarity
  • 57. (The error term depends on the percentile of interest. For the 50 percentile, -0.3665 is the error term) 13.0887 -6.03758 2.58992 Relationship with accelerating variable(s): Linear Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 13.0887 0.292537 44.74 0.000 12.5154 13.6621 weight -6.03758 0.275137 -21.94 0.000 -6.57684 -5.49832 Shape 2.58992 0.202125 2.22257 3.01798 Log-Likelihood = -574.192 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*error term] Weight (kg) Prediction (cycles) 0.50 20504.5 0.75 4532.4 1.00 1001.9 1.25 221.5 1.50 49.0 1.75 10.8 point estimates Acceleration Factors Wire Scrape
  • 58. Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 0.75 4532.4 1.00 1001.9 1.25 221.5 1.50 49.0 1.75 10.8 Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 20504.5 4532.4 = 4.52 point estimates Acceleration Factors
  • 59. Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 0.75 4532.4 1.00 1001.9 1.25 221.5 1.50 49.0 1.75 10.8 Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 4.52 Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 0.75 4532.4 4.52 1.00 1001.9 4.52 1.25 221.5 4.52 1.50 49.0 4.52 1.75 10.8 … point estimates Acceleration Factors
  • 60. Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 0.75 4532.4 4.52 1.00 1001.9 4.52 1.25 221.5 4.52 1.50 49.0 4.52 1.75 10.8 … Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 20504.5 49.0 = 418.9 point estimates Acceleration Factors
  • 61. Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 0.75 4532.4 4.52 1.00 1001.9 4.52 1.25 221.5 4.52 1.50 49.0 4.52 1.75 10.8 … Wire Scrape Prediction = exp [13.0887 -6.03758 * (weight) + (1.0/2.58992)*(-0.3665)] 1.31.21.11.00.90.8 10000 1000 100 weight cycles 50 Percentiles Relation Plot (Fitted Linear) for cycles Complete Data - ML Estimates Weibull - 95% CI Acceleration factor in a traditional sense 418.9 Weight (kg) Prediction (cycles) Acceleration factor per 1/4 Kg Acceleration factor per 1Kg 0.50 20504.5 4.52 418.9 0.75 4532.4 4.52 418.9 1.00 1001.9 4.52 … 1.25 221.5 4.52 … 1.50 49.0 4.52 … 1.75 10.8 … … point estimates Acceleration Factors
  • 63. Acceleration Factors • Reliability engineers attempt to speed up testing by “accelerating” the test • Environmental: • Increased temperature • Increased temperature swing • Adding environmental factors together • Sequentially testing • Mechanical • Increased speed • Increased weight • Others? • Multiple factors
  • 64. Cable Flex • Checking for continuity on 10-20 stations • Failure mode – open on any conductor • Varying Mandrel • Varying Weight Multiple Acceleration Factors Back to: Wt
  • 65. cycles susp weight (kg) mandrel (mm) qty 942 n 0.5 4 1 735 n 0.5 4 1 1289 n 0.5 4 1 1487 n 0.5 4 1 1222 n 0.5 4 1 2750 n 0.25 4 1 3074 n 0.25 4 1 2547 n 0.25 4 1 2992 n 0.25 4 1 4338 n 0.25 4 1 1954 n 0.25 4 1 1845 n 0.4 4 1 867 n 0.4 4 1 1004 n 0.4 4 1 1182 n 0.4 4 1 3451 n 0.4 4 1 2715 n 0.4 4 1 1071 n 0.5 4 1 831 n 0.5 4 1 1114 n 0.5 4 1 1159 n 0.5 4 1 934 n 0.5 4 1 1130 n 0.5 4 1 1150 n 0.5 4 1 1115 n 0.5 4 1 1560 n 0.5 4 1 1465 n 0.5 4 1 939 n 0.5 4 1 1343 n 0.5 4 1 1331 n 0.5 4 1 1329 n 0.5 4 1 2264 n 0.5 4 1 17701 n 1 25.4 1 12741 n 1 25.4 1 12805 n 1 25.4 1 14467 n 1 25.4 1 17301 n 1 25.4 1 18815 n 1 25.4 1 21800 n 0.5 25.4 1 19710 n 0.5 25.4 1 29224 n 0.5 25.4 1 35348 n 0.5 25.4 1 24332 n 0.5 25.4 1 24865 n 0.5 25.4 1 33948 n 0.5 25.4 1 33944 n 0.25 25.4 1 29157 n 0.25 25.4 1 32391 n 0.25 25.4 1 34150 n 0.25 25.4 1 31960 n 0.25 25.4 1 29278 n 0.25 25.4 1 70000 s 0.25 25.4 1 Cable Flex Multiple Acceleration Factors
  • 67. Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 7.66490 0.106594 71.91 0.000 7.45598 7.87382 weight -1.65024 0.225415 -7.32 0.000 -2.09205 -1.20843 Mandrel-mm 0.138599 0.0056508 24.53 0.000 0.127524 0.149675 Shape 2.89069 0.296798 2.36377 3.53506 Log-Likelihood = -449.260 Confidence intervals removed for clarity Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
  • 68. Regression Table Standard 95.0% Normal CI Predictor Coef Error Z P Lower Upper Intercept 7.66490 0.106594 71.91 0.000 7.45598 7.87382 weight -1.65024 0.225415 -7.32 0.000 -2.09205 -1.20843 Mandrel-mm 0.138599 0.0056508 24.53 0.000 0.127524 0.149675 Shape 2.89069 0.296798 2.36377 3.53506 Log-Likelihood = -449.260 7.66490 -1.65024 2.89069 *Error term= -0.3665 for 50% 0.138599 Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*error term] Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
  • 69. 710.6 Acceleration factor in a traditional sense Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*(-0.3665)] Weight (kg) Mandrel (mm) Prediction (cycle) Acceleration factor per 1/2 Kg Acceleration factor per 5mm mandrel increase 0.5 5.0 1649.0 0.5 10.0 3304.0 0.5 15.0 6620.0 1.0 5.0 722.6 1.0 10.0 1447.8 1.0 15.0 2900.7 1621.8 = 2.28 point estimates Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
  • 70. 3304.0 Acceleration factor in a traditional sense Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*(-0.3665)] Weight (kg) Mandrel (mm) Prediction (cycle) Acceleration factor per 1/2 Kg Acceleration factor per 5mm mandrel increase 0.5 5.0 1649.0 2.28 0.5 10.0 3304.0 2.28 0.5 15.0 6620.0 2.28 1.0 5.0 722.6 … 1.0 10.0 1447.8 … 1.0 15.0 2900.7 … 1649.0 = 2.00 point estimates Confidence intervals removed for clarity Cable Flex Multiple Acceleration Factors
  • 71. Acceleration factor in a traditional sense Prediction = exp [7.66490 -1.65024 * (weight) +0.138599 (Mandrel-mm)+ (1.0/2.89069)*(-0.3665)] point estimates Confidence intervals removed for clarity Weight (kg) Mandrel (mm) Prediction (cycle) Acceleration factor per 1/2 Kg Acceleration factor per 5mm mandrel increase 0.5 5.0 1649.0 2.28 2.00 0.5 10.0 3304.0 2.28 2.00 0.5 15.0 6620.0 2.28 … 1.0 5.0 722.6 … 2.00 1.0 10.0 1447.8 … 2.00 1.0 15.0 2900.7 … … Cable Flex Multiple Acceleration Factors
  • 72. Summary • Distribution selection can be assisted with statistics, but knowledge about your data is key • Use Right Censoring if an exact count is available, Arbitrary Censoring is for periodic testing • Accelerated testing can provide faster answers and testing at different levels can assist in understanding the relationship for the acceleration
  • 73. Author Rob Schubert Corporate Quality/Reliability Engineer Shure Inc – 9 years Schubert_Rob@shure.com Certified Reliability Engineer (ASQ) Master’s in Acoustical Engineering, Penn State Thesis: Use of Multiple-Input Single-Output Methods to Increase Repeatability of Measurements of Road Noise in Automobiles Recent presentations: 2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test – ARS 2015 Previous work experience: Ford (13 years) - Quality/Reliability Engineer, 6 Sigma Black belt, Noise & Vibration Engineer
  • 74. Thank you for your attention. Do you have any questions?

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