2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test

2015 ARS, North America, Tucson2015 ARS, North America, Tucson
Red Room, Session #15Red Room, Session #15 Current Time:
09:40 PM
2 Parameter vs. 3 Parameter
Weibull with a Cable Flex Test
Rob Schubert, Shure Inc.
Jeff Whalen, Shure Inc.
Alexander Ho, Shure Inc.
Begins at 3:30 PM, Thursday, June 4thBegins at 3:30 PM, Thursday, June 4th

Rob Schubert, Shure Inc. Slide Number: 2Session #15Red Room
AppliedReliabilitySymposium,NorthAmerica2015
IntroductionIntroduction
Rob Schubert - Corporate Quality/Reliability Engineer at 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
All which use cables!

AgendaAgenda
 Basic definitions 5 min
 Shure’s cable flex test 5 min
 Weibull comparison with generated data 10 min
 Weibull comparison with real data which fits 3 parameter 10 min
 Weibull comparison with real data which doesn’t quite fit 10 min
 Advantages of each 5 min
 Summary 5 min
 Questions 10 min

DefinitionsDefinitions
• Shape = Slope = β : slope of the line in a probability plot
• Scale = η = shifts the data “out”
• Threshold = Location = γ : forced to zero in 2 parameter Weibull
• Anderson Darling test = How well your data fits a PDF - Smaller fits
better
• Likelihood ratio test = if small (<.05), 3 parameter fits better

 Shape = Slope = β : slope of the line in a probability plot

100000100001000100
99
90
80
70
60
50
40
30
20
10
5
3
2
1
Dat a
Percent
Effect of Scale Parameter on Weibull plot
 Scale = η = Shifts the data “out”

 Threshold = Location = γ : forced
to zero in 2 parameter Weibull
(not represented on probability plot)

Shure’s cable flex testShure’s cable flex test
Checking for continuity on 20 stations
Failure mode – open on any conductor
Each unit is independently measured
every millisecond and cycles are recorded

Based on MIL-DTL-915G

Shure’s cable flex requirementsShure’s cable flex requirements
Purpose: to ensure cables last in the field
how can we measure that?
how many samples should we use?
Originally based on 3 parameter Weibull
Later changed to 2 parameter Weibull
Now re-investigating 3 parameter Weibull
Which is correct?

100001000100
99.9
95
80
50
20
5
2
1
0.1
cable 1
Percent
100001000100
99.9
95
80
50
20
5
2
1
0.1
cable 1 - Threshold
Percent
Goodness of Fit Test
Weibull
AD = 0.999
P-Value = 0.012
3-Parameter Weibull
AD = 0.232
P-Value > 0.500
Probability Plot for cable 1
Weibull - 95% CI 3-Parameter Weibull - 95% CI
150 data points for 1 single cable (Cable #1)150 data points for 1 single cable (Cable #1)
Distribution Shape Scale Threshold L10
Weibull 2.306 9313 4050
3-Parameter 1.710 6996 1994 3946
3 parameter Weibull fits better

3 parameter Weibull – Cable #13 parameter Weibull – Cable #1
Even with 150 points, the parameters have fairly wide confidence bandsEven with 150 points, the parameters have fairly wide confidence bands
shape scale threshold
estimate 1.710 6996 1994.3
lower bound 1.484 6245 1695.1
upper bound 1.972 7836 2293.6
100001000100
99.9
99
90
80
70
60
50
40
30
20
10
5
3
2
1
0.1
cable 1 - Threshold
Percent
AD* 0.331
Shape 1.71046
Scale 6995.97
Thres 1994.34
Mean 8233.89
StDev 3756.91
Median 7640.96
I QR 5091.21
Failure 150
Censor 0
Table of Statistics
Complete Data - ML Estimates
3-Parameter Weibull - 95% CI

Let’s take out the real world variability andLet’s take out the real world variability and
look at some ideal datalook at some ideal data

200 Generated points with same shape, scale and200 Generated points with same shape, scale and thresholdthreshold
100001000
99.9
95
80
50
20
5
2
1
0.1
generat ed dat a
Percent
100001000100
99.9
95
80
50
20
5
2
1
0.1
generat ed dat a - Threshold
Percent
Weibull
AD = 0.510
P-Value = 0.210
3-Parameter Weibull
AD = 0.118
P-Value > 0.500
Probability Plot for generated data
Distribution Shape Scale Threshold L10
3-Parameter generated 1.710 6995 1994 -
Resulted in:
Weibull 2.436 9698 3850
3-Parameter 1.802 7428 1967 4099

3 parameter Weibull on generated data3 parameter Weibull on generated data
100001000100
99.9
99
90
80
70
60
50
40
30
20
10
5
3
2
1
0.1
generat ed dat a - Threshold
Percent
AD* 0.179
Shape 1.80269
Scale 7428.49
Thres 1967.72
Mean 8573.30
StDev 3792.23
Median 8029.53
I QR 5182.39
Failure 200
Censor 0
Table of Statistics
Probability Plot for generated data
estimate 1.80 7428 1967
lower bound 1.55 6617 1483
upper bound 2.10 8339 2452

Let’s take sets of this data:Let’s take sets of this data:
ETC.
ETC.
Then randomize…
Continue until 1000 sets are made:
3 samples, 5 samples, 10 samples, 20 samples and 40 samples
All in all, 78000 data points analyzed per cable
Then randomize…

Let’s first look at Anderson DarlingLet’s first look at Anderson Darling

Anderson Darling by sample sizeAnderson Darling by sample size
(Statistically this is ±3-5%)

Distribution of differences in theDistribution of differences in the
Anderson Darling statisticAnderson Darling statistic
3 parameter
lower
2 parameter
lower
Setsofdata

Let’s review the effect of sampleLet’s review the effect of sample
size on thresholdsize on threshold

Threshold by sample sizeThreshold by sample size
(Statistically this is ±3-5%)

Distribution of differences in ThresholdDistribution of differences in ThresholdSetsofdata

What about confidence intervals ofWhat about confidence intervals of
those thresholds?those thresholds?

40 samples – Threshold confidence intervals40 samples – Threshold confidence intervals
As threshold increases, confidence interval goes down
40003000200010000
16000
14000
12000
10000
8000
6000
4000
2000
0
40 samples t hreshold
40sampleconfrange

What about slope? Let’s reviewWhat about slope? Let’s review
slopeslope

Difference in slope (2 vs 3 parameter)Difference in slope (2 vs 3 parameter)
2 param.
Slope
steeper
3 param.
Slope
steeper
Setsofdata

1000 different generated Weibull sets of 40 samples1000 different generated Weibull sets of 40 samples
with different slope, scale and thresholdwith different slope, scale and threshold
50.037.525.012.50.0-12.5-25.0
Median
Mean
5.55.04.54.03.53.0
1st Quartile 1.7403
Median 3.4172
3rd Quartile 6.3986
Maximum 48.5880
4.6248 5.3782
3.1649 3.6993
5.7641 6.2974
A-Squared 68.33
P-Value < 0.005
Mean 5.0015
StDev 6.0189
Variance 36.2270
Skew ness 1.6762
Kurtosis 10.9488
N 983
Minimum -32.8531
Anderson-Darling Normality Test
95% Confidence I nterv al for Mean
95% Confidence I nterv al for Median
95% Confidence I nterv al for StDev
9 5 % Conf idence I nt er v als
Summary for difference
2 param.
Slope
steeper
3 param.
Slope
steeper
2 parameter slope is generally steeper

Do not ignore the WarningsDo not ignore the Warnings
* WARNING * Variance/Covariance matrix of estimated parameters
does not exist. The threshold parameter is assumed fixed when
calculating confidence intervals.
* WARNING * Convergence has not been reached for the log-
likelihood criterion.
* WARNING * Newton-Raphson algorithm has not converged after
20 iterations.
* WARNING * Convergence has not been reached for the parameter
estimates criterion.
40 samples 20 samples 10 samples 5 samples 3 samples
Warnings 3 131 1193 countless countless

Back to real data! Cable #1Back to real data! Cable #1

3 parameter Weibull – Cable #13 parameter Weibull – Cable #1
Estimate 1.710 6996 1994.3
lower bound 1.484 6245 1695.1
upper bound 1.972 7836 2293.6
100001000100
99.9
99
90
80
70
60
50
40
30
20
10
5
3
2
1
0.1
cable 1 - Threshold
Percent
AD* 0.331
Shape 1.71046
Scale 6995.97
Thres 1994.34
Mean 8233.89
StDev 3756.91
Median 7640.96
I QR 5091.21
Failure 150
Censor 0
Table of Statistics

Anderson Darling by sample size (Cable #1)Anderson Darling by sample size (Cable #1)

Anderson Darling – compared to generatedAnderson Darling – compared to generated
Does not fit 2 parameterDoes not fit 2 parameter
Cable 1 = real data

Cable 1 = real data
O
rder reversed
for clarity

Likelihood ratio testLikelihood ratio test

3 parameter scores lower than 2 parameter3 parameter scores lower than 2 parameter

Threshold by sample size (Cable #1)Threshold by sample size (Cable #1)

Threshold – compared to generated:Threshold – compared to generated:
Above upper confidence limitAbove upper confidence limit
Cable 1 = real data

Below lower confidence limit (and positive)Below lower confidence limit (and positive)
Cable 1 = real data

NegativeNegative
Cable 1 = real data
O
rder reversed
for clarity

Within confidence limitsWithin confidence limits
Cable 1 = real data

Median of positive results - Delta toMedian of positive results - Delta to
“true”“true” (150 – 200 data points)(150 – 200 data points)
O
rder reversed
for clarity

Let’s try a cable that doesn’t quite fitLet’s try a cable that doesn’t quite fit
either Weibulleither Weibull

100001000
99.9
95
80
50
20
5
2
1
0.1
cable 2
Percent
1000100
99.9
95
80
50
20
5
2
1
0.1
cable 2 - Threshold
Percent
Weibull
AD = 3.961
P-Value < 0.010
3-Parameter Weibull
AD = 0.940
P-Value = 0.019
170 data points for Cable #2170 data points for Cable #2

3 parameter Weibull – cable #23 parameter Weibull – cable #2
estimate 2.00 965 968.9
lower bound 1.73 864 906.9
upper bound 2.31 1078 1030.8
1000100
99.9
99
90
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40
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5
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2
1
0.1
Cable 2 - Threshold
Percent
AD* 1.034
Shape 1.99766
Scale 965.093
Thres 968.853
Mean 1824.16
StDev 447.561
Median 1772.17
I QR 619.268
Failure 170
Censor 0
Table of Statistics
Probability Plot for Cable 2

Anderson Darling by sample sizeAnderson Darling by sample size

Anderson Darling – compare to previous:Anderson Darling – compare to previous:
Cable 1 & 2 = real data

O
rder reversed
for clarity

Likelihood ratio testLikelihood ratio test

3 parameter scores lower than 2 parameter3 parameter scores lower than 2 parameter

Threshold by sample size (Cable #2)Threshold by sample size (Cable #2)

Threshold – compare to generated:Threshold – compare to generated:
Above upper confidence limitAbove upper confidence limit

Below lower confidence limitBelow lower confidence limit
O
rder reversed
for clarity

NegativeNegative
O
rder reversed
for clarity

Within Confidence limitsWithin Confidence limits

Median of positive results - Delta to “true”Median of positive results - Delta to “true”
(150 – 200 data points)(150 – 200 data points)
negative

Advantages of the 2 parameterAdvantages of the 2 parameter
WeibullWeibull
Works with a very small set of data
Negative threshold doesn’t occur (As
seen, negative threshold makes little
sense)
Simpler to understand for reliability
engineers, most common distribution used
(little need to discuss)

Advantages of the 3 parameterAdvantages of the 3 parameter
WeibullWeibull
Simple specification setting (can set
threshold as a minimum life)
May be more accurate, but needs
significant data to actualize

““Schubert’s Six” rulesSchubert’s Six” rules
3 parameter vs 2 parameter3 parameter vs 2 parameter
 3 and 5 samples are unusable (10 is barely
acceptable) with 3 parameter.
 3 parameter slope is less steep.
 Negative threshold means you should take more
samples.
 Positive threshold generally measures high, but
asymptotically approaches the true answer with
more samples.
 Confidence interval of threshold is smaller as
threshold gets larger.
 Do not ignore the *Warnings*!

Shure’s directionShure’s direction
Specify minimum flex life and test via 3
parameter Weibull (all cases, regardless
of fit)
Set specifications by use case with a 25%
safety factor
Test with a minimum of 20 samples, retest
(add data) if threshold is negative or if
calculation shows warnings

Where to Get More InformationWhere to Get More Information
 MIL-DTL-915G
 www.shure.com (product line)
 Presentation, Data, Macros, etc. available here:
https://www.dropbox.com/sh/5qyncmbshmc4af1/AAD-EZcTF0jvcPxBQZUB-UCua?dl=0

Authors/ContributorsAuthors/Contributors
 Rob Schubert
Corporate Quality/Reliability Engineer
Shure Inc – 8 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 Reliability of
Measurements of Road Noise in Automobiles
Previous work experience: Ford (13 years) - Quality/Reliability Engineer, 6 Sigma
Black belt, Noise & Vibration Engineer
Contributors

QuestionsQuestions
Thank you for your attention.
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2 Parameter vs. 3 Parameter Weibull with a Cable Flex Test

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