Weibull Reliability Analysis

Home , Uncensored (1999), Weibull distribution

=⇒ http://www.rt.cs.boeing.com/MEA/stat/reliability.html

Fritz Scholz (425-865-3623, 7L-22) Boeing Phantom Works Mathematics & Computing Technology

Weibull Reliability Analysis—FWS-5/1999—1 Wallodi Weibull

Weibull Reliability Analysis—FWS-5/1999—2 Seminal Paper

Weibull Reliability Analysis—FWS-5/1999—3 The Weibull Distribution

• Weibull distribution, useful uncertainty model for – wearout failure time T when governed by wearout of weakest subpart – material strength T when governed by embedded ﬂaws or weaknesses, • It has often been found useful based on empirical data (e.g. Y2K) • It is also theoretically founded on the weakest link principle

T = min (X1,...,Xn) ,

with X1,...,Xn statistically independent random strengths or failure times of the n “links” comprising the whole.

The Xi must have a natural ﬁnite lower endpoint, e.g., link strength ≥ 0 or subpart time to failure ≥ 0. ,→

Weibull Reliability Analysis—FWS-5/1999—4 Theoretical Basis

• Under weak conditions Extreme Value Theory shows1 that for large n    β  −   t τ   P (T ≤ t) ≈ 1 − exp −    for t ≥ τ,α > 0,β >0 α • The above approximation has very much the same spirit as the Central Limit Theorem which under some weak conditions on the Xi asserts that the distribution of T = X1 + ...+ Xn is approximately bell-shaped normal or Gaussian • Assuming a Weibull model for T , material strength or cycle time to failure, amounts to treating the above approximation as an equality    β  −   t τ   F (t)=P (T ≤ t)=1− exp −    for t ≥ τ,α > 0,β >0 α

1see: E. Castillo, Extreme Value Theory in Engineering, Academic Press, 1988

Weibull Reliability Analysis—FWS-5/1999—5 Weibull Reproductive Property

If X1,...,Xn are statistically independent

with Xi ∼ Weibull(αi,β)then

P (min(X1,...,Xn) >t)=P (X1 >t) ×···×P (Xn >t)      β  β Yn    Xn    t     t   = exp −    =exp−    i=1 αi i=1 αi    β     t   =exp−    α

Hence T = min(X1,...,Xn) ∼ Weibull(α, β) with

 −1/β  Xn −β α =  αi  i=1

Weibull Reliability Analysis—FWS-5/1999—6 Weibull Parameters

The Weibull distribution may be controlled by 2 or 3 parameters: • the threshold parameter τ T ≥ τ with probability 1 τ =0 =⇒ 2-parameter Weibull model.

• the characteristic life or scale parameter α>0

   β  α  P (T ≤ τ + α)=1− exp −    =1− exp(−1) = .632 α regardless of the value β>0

• the shape parameter β>0, usually β ≥ 1

Weibull Reliability Analysis—FWS-5/1999—7 2-Parameter Weibull Model

• We focus on analysis using the 2-parameter Weibull model

• Methods and software tools much better developed

• Estimation of τ in the 3-parameter Weibull model leads to complications

• When a 3-parameter Weibull model is assumed, it will be stated explicitly

Weibull Reliability Analysis—FWS-5/1999—8 Relation of α & β to Statistical Parameters

• The expectation or mean value of T Z ∞ µ = E(T )= 0 tf(t) dt = αΓ(1 + 1/β)

Z ∞ − t−1 with Γ(t)= 0 exp( x) x dx

• The variance of T Z 2 − 2 ∞ − 2 2 − 2 σ = E(T µ) = 0 (t µ) f(t) dt = α Γ(1 + 2/β) Γ (1 + 1/β)

• p-quantile tp of T , i.e., by deﬁnition P (T ≤ tp)=p

1/β tp = α [− log(1 − p)] , for p =1− exp(−1) = .632 =⇒ tp = α

Weibull Reliability Analysis—FWS-5/1999—9 Weibull Density

• The cumulative distribution function F (t)=P (T ≤ t)isjust one way to describe the distribution of the random quantity T

• The density function f(t) is another representation (τ =0)

   β−1  β   0 dF (t) β  t    t   f(t)=F (t)= =   exp −    t ≥ 0 dt α α α

P (t ≤ T ≤ t + dt) ≈ f(t) dt

Z t F (t)= 0 f(x) dx

Weibull Reliability Analysis—FWS-5/1999—10 Weibull Density & Distribution Function

Weibull density α = 10000, β = 2.5 total area under density = 1

p cumulative distribution function 0

0 5000 10000 15000 20000 cycles

Weibull Reliability Analysis—FWS-5/1999—11 Weibull Densities: Eﬀect of τ

τ = 0 τ = 1000 τ = 2000

α = 1000, β = 2.5 probability density

0 2000 4000

cycles

Weibull Reliability Analysis—FWS-5/1999—12 Weibull Densities: Eﬀect of α

α = 1000

τ = 0, β = 2.5

α = 2000

α = 3000 probability density

0 2000 4000 6000 8000

cycles

Weibull Reliability Analysis—FWS-5/1999—13 Weibull Densities: Eﬀect of β

β = 7

β = .5

β = 4

β = 1 β = 2

probability density τ = 0, α = 1000

0 1000 2000 3000 4000

cycles

Weibull Reliability Analysis—FWS-5/1999—14 Failure Rate or Hazard Function

• A third representation of the Weibull distribution is through the hazard or failure rate function  β−1 f(t) β  t  λ(t)= =   1 − F (t) α α • λ(t) is increasing t for β>1 (wearout) • λ(t) is decreasing t for β<1 • λ(t) is constant for β = 1 (exponential distribution) P (t ≤ T ≤ t + dt) f(t) dt P (t ≤ T ≤ t + dt|T ≥ t)= ≈ = λ(t) dt P (T ≥ t) 1 − F (t)

Z ! Z ! − − t − t F (t)=1 exp 0 λ(x) dx and f(t)=λ(t)exp 0 λ(x) dx

Weibull Reliability Analysis—FWS-5/1999—15 Exponential Distribution

• The exponential distribution is a special case: β =1&τ =0

   t  F (t)=P (T ≤ t)=1− exp −  for t ≥ 0 α

• This distribution is useful when parts fail due to random external inﬂuences and not due to wear out • Characterized by the memoryless property, a part that has not failed by time t is as good as new, past stresses without failure are water under the bridge • Good for describing lifetimes of electronic components, failures due to external voltage spikes or overloads

Weibull Reliability Analysis—FWS-5/1999—16 Unknown Parameters

• Typically will not know the Weibull distribution: α, β unknown

c • Will only have sample data =⇒ estimates α,c β get estimated Weibull model for failure time distribution =⇒ double uncertainty uncertainty of failure time & uncertainty of estimated model • Samples of failure times are sometimes very small, only 7 fuse pins or 8 ball bearings tested until failure, long lifetimes make destructive testing difficult • Variability issues are often not sufficiently appreciated how do small sample sizes affect our confidence in estimates and predictions concerning future failure experiences?

Weibull Reliability Analysis—FWS-5/1999—17 Estimation Uncertainty

A Weibull Population: Histogram for N = 10,000 & Density

63.2% 36.8% characteristic life = 30,000 shape = 2.5

true model estimated model from 9 data points

····· ··· · 0.0 0.00002 0.00004

0 20000 40000 60000 80000

cycles

Weibull Reliability Analysis—FWS-5/1999—18 Weibull Parameters & Sample Estimates

shape parameter β = 4

63.2% 36.8%

characteristic lifeα = 30,000 p=P(T < t )

t =t p p-quantile ········· · ····· ··· · · ··········

25390 3.02 33860 4.27 29410 5.01

parameter estimates from three samples of size n = 10

Weibull Reliability Analysis—FWS-5/1999—19 Generation of Weibull Samples

1/β • Using the quantile relationship tp = α [− log(1 − p)] one can generate a Weibull random sample of size n by

– generating a random sample U1,...,Un from a uniform [0, 1] distribution

1/β – and computing Ti = α [− log(1 − Ui)] , i =1,...,n.

– Then T1,...,Tn canbeviewedasarandomsampleofsizen from a Weibull population or Weibull distribution with parameters α & β.

• Simulations are useful in gaining insight on estimation procedures

Weibull Reliability Analysis—FWS-5/1999—20 Graphical Methods

• Suppose we have a complete Weibull sample of size n: T1,...,Tn

• Sort these values from lowest to highest: T(1) ≤ T(2) ≤ ...≤ T(n) 1/β • Recall that the p-quantile is tp = α [− log(1 − p)]

• − Compute tp1 <...

• Plot the points (T(i),tpi), i =1,...,n and expect

these points to cluster around main diagonal

Weibull Reliability Analysis—FWS-5/1999—21 Weibull Quantile-Quantile Plot: Known Parameters

· · · · · ·· ··· ·· ··· ··· · ··· ·· ··· ···· ··· tp ···· ····· · ····· ···· · · ·· ···· ···· ··· ·· · · ·· · · 0 5000 10000 15000

0 5000 10000 15000

Weibull Reliability Analysis—FWS-5/1999—22 Weibull QQ-Plot: Unknown Parameters

• Previous plot requires knowledge of the unknown parameters α & β

• Note that

log (tp)=log(α)+wp/β , where wp =log[− log(1 − p)]

• Expect points log[T(i)],wpi , i =1,...,n, to cluster around line with slope 1/β and intercept log(α)

• This suggests estimating α & β from a ﬁtted least squares line

Weibull Reliability Analysis—FWS-5/1999—23 Maximum Likelihood Estimation

• If t1,...,tn are the observed sample values one can contemplate the probability of obtaining such a sample or of values nearby, i.e.,

P (T1 ∈ [t1 − dt/2,t1 + dt/2],...,Tn ∈ [tn − dt/2,tn + dt/2])

= P (T1 ∈ [t1 − dt/2,t1 + dt/2]) ×···×P (Tn ∈ [tn − dt/2,tn + dt/2])

≈ fα,β(t1)dt ×···×fα,β(tn)dt

where f(t)=fα,β(t) is the Weibull density with parameters (α, β)

• Maximum likelihood estimation maximizes this probability c over α & β =⇒ maximum likelihood estimates (m.l.e.s) αc and β

Weibull Reliability Analysis—FWS-5/1999—24 General Remarks on Estimation

• MLEs tend to be optimal in large samples (lots of theory)

• Method is very versatile in extending to may other data scenarios censoring and covariates

• Least squares method applied to QQ-plot is not entirely appropriate tends to be unduly aﬀected by stray observations not as versatile to extend to other situations

Weibull Reliability Analysis—FWS-5/1999—25 Weibull Plot: n =20

.990 · · .900 · ·· .632 · ·· .500 ·· ·· ·· .200 · · .100 · probability ·

.010 true model, Weibull( 100 , 3 ) m.l.e. model, Weibull( 108 , 3.338 ) least squares model, Weibull( 107.5 , 3.684 )

.001

10 20 50 100 200 500 1000

cycles/hours

Weibull Reliability Analysis—FWS-5/1999—26 Weibull Plot: n = 100

.990 · ·· · ··· .900 ··· ··· ····· .632 ··· ······· .500 ····· ···· ···· ···· .200 ·· ·· · .100 ·· · · ·· · probability · · .010 true model, Weibull( 100 , 3 ) · m.l.e. model, Weibull( 102.8 , 2.981 ) least squares model, Weibull( 102.2 , 3.144 )

.001

10 20 50 100 200 500 1000

cycles/hours

Weibull Reliability Analysis—FWS-5/1999—27 Tests of Fit (Graphical)

• The Weibull plots provide an informal diagnostic for checking the Weibull model assumption

• The anticipated linearity is based on the Weibull model properties

• Strong nonlinearity indicates that the model is not Weibull

• Sorting out nonlinearity from normal statistical point scatter takes a lot of practice and a good sense for the eﬀect of sample size on the variation in point scatter

• Formal tests of ﬁt are available for complete samples2 and also for some other censored data scenarios

2R.B. D’Agostino and M.A. Stephens, Goodness-of-Fit Techniques, Marcel Dekker 1986

Weibull Reliability Analysis—FWS-5/1999—28 Formal Goodness-of-Fit Tests

• Let Fα,b βb(t) be the ﬁtted Weibull distribution function

d { ≤ } • # Ti t; i=1,...,n Let Fn(t)= n be the empirical distribution function d • b Compute a discrepancy metric D between Fα,b β and Fn,

d d − DKS(Fα,b βb, Fn)=sup Fα,b βb(t) Fn(t) Kolmogorov-Smirnov t Z ! d ∞ d 2 b b − b DCvM(Fα,b β, Fn)= 0 Fα,b β(t) Fn(t) fα,b β(t) dt Cramer-von Mises ! d 2 Z b − d ∞ Fα,b β(t) Fn(t) DAD(Fα,b βb, Fn)= fα,b βb(t) dt Anderson-Darling 0 − b Fα,b βb(t)(1 Fα,b β(t)) • The distributions of D, when sampling from a Weibull population, are known and p-values of observed values d of D can be calculated p = P (D ≥ d)=⇒ BCSLIB: HSPFIT

Weibull Reliability Analysis—FWS-5/1999—29 Kolmogorov-Smirnov Distance

n = 10

K-S distance

··· ·· · ·· · · 0.0 0.2 0.4 0.6 0.8 1.0

0 50 100 150 200 250

Weibull Reliability Analysis—FWS-5/1999—30 Weibull Plots: n =10

p(KS) = 0.62 · p(KS) = 0.28 · p(KS) = 0.88 · p(KS) = 0.88 · p(CvM) = 0.54 · p(CvM) = 0.27 · p(CvM) = 0.62 · p(CvM) = 0.64 · p(AD) = 0.62 · p(AD) = 0.35 · p(AD) = 0.66 · p(AD) = 0.69 · · · · · · · · · · · · · · · · · · · · ·

· · · ·

· n = 10 · · · -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1 3.8 4.2 4.6 5.0 4.0 4.4 4.8 3.8 4.4 5.0 3.8 4.2 4.6 5.0

p(KS) = 0.72 · p(KS) = 0.16 · p(KS) = 0.34 · p(KS) = 0.86 · p(CvM) = 0.53 · p(CvM) = 0.21 · p(CvM) = 0.23 · p(CvM) = 0.56 · p(AD) = 0.6 · p(AD) = 0.32 · p(AD) = 0.22 · p(AD) = 0.57 · · · · · · · · · · · · · · · · · · · · ·

· · · ·

· · · · -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1 4.0 4.4 4.8 3.8 4.2 4.6 4.0 4.4 4.8 3.8 4.2 4.6

Weibull Reliability Analysis—FWS-5/1999—31 Weibull Plots: n =20

p(KS) = 0.55 · p(KS) = 0.13 · p(KS) = 0.12 · p(KS) = 0.46 · · · · · p(CvM) = 0.49 · p(CvM) = 0.027 · p(CvM) = 0.053 · p(CvM) = 0.39 · p(AD) = 0.49 ·· p(AD) = 0.028 · p(AD) = 0.048 ·· p(AD) = 0.38 ·· · · · · · · ·· ·· ·· · · · ·· ·· · · · ·· ·· ·· · · · · · · · · · · · · · · · · · · · · · · · · -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1

· n = 20 · · ·

3.5 4.0 4.5 5.0 2.5 3.5 4.5 4.2 4.6 3.8 4.2 4.6 5.0

p(KS) = 0.46 · p(KS) = 0.47 · p(KS) = 0.85 · p(KS) = 0.04 · · · · · p(CvM) = 0.41 · p(CvM) = 0.63 · p(CvM) = 0.6 · p(CvM) = 0.017 · p(AD) = 0.41 · p(AD) = 0.69 · p(AD) = 0.64 ·· p(AD) = 0.023 ·· ·· ·· ·· ·· ·· ·· · · · · ·· · ·· ·· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1 -3 -2 -1 0 1

· · · ·

3.6 4.0 4.4 4.8 3.0 4.0 5.0 4.0 4.4 4.8 3.6 4.0 4.4 4.8

Weibull Reliability Analysis—FWS-5/1999—32 Weibull Plots: n =50

p(KS) = 0.7 · p(KS) = 0.22 · p(KS) = 0.15 · p(KS) = 0.76 · · · · · p(CvM) = 0.41 · p(CvM) = 0.35 · p(CvM) = 0.23 · p(CvM) = 0.49 · ·· ·· ·· ·· p(AD) = 0.43 ·· p(AD) = 0.32 ·· p(AD) = 0.12 ··· p(AD) = 0.52 ··· ··· ··· ··· ··· · ·· ···· ··· ··· ··· ··· · ·· ·· ·· ·· ··· ··· ··· ··· ··· ·· ··· ··· ··· ·· ··· ·· ··· · · · · ·· ·· ·· ·· · ·· · · ·· ·· · · · ·· · · · · · · · · · · · · · · · · -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 · n = 50 · · ·

2.0 3.0 4.0 5.0 3.0 4.0 5.0 3.5 4.0 4.5 3.5 4.5

p(KS) = 0.88 · p(KS) = 0.27 · p(KS) = 0.85 · p(KS) = 0.18 · · · · · p(CvM) = 0.66 · p(CvM) = 0.22 · p(CvM) = 0.56 · p(CvM) = 0.035 · ·· ··· ··· ··· p(AD) = 0.72 ·· p(AD) = 0.23 ·· p(AD) = 0.62 ·· p(AD) = 0.039 ·· · ··· ·· · ·· ···· ·· ··· ··· ·· ··· ··· ··· ··· ··· ·· ·· ··· ··· ··· ··· ··· ·· ·· ··· ·· ··· ··· ··· ··· ·· ·· ·· ·· ·· ·· ·· ·· · · ·· · ·· ·· · · · · · · · · · · · · · · · · · -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 · · · ·

3.5 4.0 4.5 3.5 4.0 4.5 5.0 3.5 4.0 4.5 5.0 3.5 4.0 4.5 5.0

Weibull Reliability Analysis—FWS-5/1999—33 Weibull Plots: n = 100

p(KS) = 0.58 · p(KS) = 0.86 · p(KS) = 0.61 · p(KS) = 0.21 · ·· ·· ·· · ·· p(CvM) = 0.26 ·· p(CvM) = 0.68 ·· p(CvM) = 0.48 ·· p(CvM) = 0.26 ··· ···· ··· ··· ·· p(AD) = 0.3 ··· p(AD) = 0.72 ···· p(AD) = 0.55 ··· p(AD) = 0.43 ·· ··· ··· ···· ··· ··· ···· ·· ····· ·· ··· ··· ··· ···· ···· ···· ··· ··· ··· ·· ··· ·· ····· ··· ··· ·· ··· ··· ··· ··· ··· ··· ·· ·· ··· ···· ··· ·· ·· ·· ·· ·· ·· · ·· · ·· ·· ·· · · ·· ·· ·· · · · · · · · · -4 -2 0 -4 -2 0 -4· -2 0 -4· -2 0 · ·

· n = 100 · · ·

3.0 4.0 5.0 3.5 4.5 3.0 4.0 5.0 3.5 4.5

p(KS) = 0.47 · p(KS) = 0.053 · p(KS) = 0.87 · p(KS) = 0.72 · · ··· ·· ·· p(CvM) = 0.3 ··· p(CvM) = 0.025 ·· p(CvM) = 0.58 ·· p(CvM) = 0.45 ·· ··· ·· ·· ··· p(AD) = 0.37 ·· p(AD) = 0.032 ·· p(AD) = 0.6 ···· p(AD) = 0.43 ···· ··· ·· ···· ·· ···· ··· ····· ··· ··· ···· ··· ·· ·· ····· ···· ·· ·· ····· ···· ·· ·· ··· ··· ··· ···· ·· ····· ··· ··· ··· ·· ··· ·· ··· ·· ·· ··· ··· ··· ·· ··· ·· ·· ·· ·· ·· ·· · · ·· ·· ·· ·· · · · · · · · · -4 -2 0 -4 -2 0 -4 -2 0 -4· -2 0 · · ·

· · · ·

3.0 4.0 5.0 3.0 4.0 5.0 3.5 4.5 2.5 3.5 4.5

Weibull Reliability Analysis—FWS-5/1999—34 Estimates and Conﬁdence Bounds/Intervals

• For a target θ 1/β θ = α, θ = β, θ = tp = α[− log(1 − p)] , or θ = Pα,β(T ≤ t) c c one gets corresponding m.l.e θ by replacing (α, β)by(α,c β)

• Such estimates vary around the target due to sampling variation • Capture the estimation uncertainty via conﬁdence bounds

c c • For 0 <γ<1 get 100γ% lower/upper conﬁdence bounds θL,γ & θU,γ

c c P θL,γ ≤ θ = γ or P θ ≤ θU,γ = γ

c c • For γ>.5 get a 100(2γ − 1)% conﬁdence interval by [θL,γ, θU,γ]

c c ? P θL,γ ≤ θ ≤ θU,γ =2γ − 1=γ

Weibull Reliability Analysis—FWS-5/1999—35 Conﬁdence Bounds & Sampling Variation, n =10

L: Lawless Exact Method (RAP); B: Bain Exact Method (Tables); M: MLE Approximate Method

true α 25000 35000 90% confidence intervals 90% confidence LBM

samples 12345

true β

LBM 90% confidence intervals 90% confidence 02468

12345

Weibull Reliability Analysis—FWS-5/1999—36 Conﬁdence Bounds & Sampling Variation, n =10

true 10-percentile LBM

L: Lawless Exact Method (RAP); B: Bain Exact Method (Tables); M: MLE Approximate Method 0 5000 15000 25000 95% lower confidence bounds 95% lower confidence

samples 12345

LBM

true P(T < 10,000) 0.0 0.10 0.20 95% upper confidence bounds 95% upper confidence

12345

Weibull Reliability Analysis—FWS-5/1999—37 Conﬁdence Bounds & Eﬀect of n

· · · · · · · · · · · · · · · · · · · true α · · 95% confidence intervals 95% confidence 5 10 20 50 100 200 500 1000

15000 25000 35000 sample size

· · · · · · · · · · · · · · · · · · · true β · 95% confidence intervals 95% confidence · 1234567

Weibull Reliability Analysis—FWS-5/1999—38 Conﬁdence Bounds & Eﬀect of n

true P(T < 10000) · · · · · · · · · · · · 0.0 0.05 0.15

95% upper confidence bounds 95% upper confidence 5 10 20 50 100 200 500 1000 sample size

· · · · · · · · · · · · · true t .10 · ·

· 95% lower confidence bounds 95% lower confidence 5000 10000 20000

Weibull Reliability Analysis—FWS-5/1999—39 Incomplete Data or Censored Failure Times

• Type I censoring or time censoring: units are tested until failure or until a prespeciﬁed time has elapsed

• Type II censoring or failure censoring: only the r lowest values of the total sample of size n become known this shows up when we put n units on test simultaneously and terminate the test after the ﬁrst r units have failed

• Interval censoring, inspection data, grouped data: ith unit is only known to have failed between two known

inspection time points, i.e., failure time Ti falls in (si,ei] only bracketing intervals (si,ei), i =1,...,n, become known

Weibull Reliability Analysis—FWS-5/1999—40 Incomplete Data or Censored Failure Times (continued)

• Random right censoring: units are observed until failed or removed from observation due to other causes (diﬀerent failure modes, competing risks)

• Multiple right censoring: units are put into service at diﬀerent times and times to failure or censoring are observed

• Data can also combine several of the above censoring phenomena

• It is important that the censoring mechanism should not correlate with the (potential) failure times, i.e., no censoring of anticipated failures

• All data (censored & uncensored) should enter analysis

Weibull Reliability Analysis—FWS-5/1999—41 Type I or Time Censored Data: n =10

unit 10 X

9 ?

8 ?

7 X

6 X

5 ?

4 X

3 X

2 ?

1 ?

0 2000 4000 6000 8000 10000

cycles

Weibull Reliability Analysis—FWS-5/1999—42 Type II or Failure Censored Data: n =10

unit 10 ?

9 ?

8 ?

7 ?

6 X

5 X

4 X

3 X

2 X

1 X

0 2000 4000 6000 8000 10000

cycles

Weibull Reliability Analysis—FWS-5/1999—43 Interval Censored Data: n =10

unit 10 ?

9 (]·

8 ?

7 ( · ?

6 (]·

5 ?

4 (]·

3 ( · ?

2 ?

1 (]·

0 2000 4000 6000 8000 10000

cycles

Weibull Reliability Analysis—FWS-5/1999—44 Multiply Right Censored Data: n =10

unit 10 ?

9 ?

8 ?

7 ?

6 X

5 X

4 X

3 X

2 ?

1 X

0 2000 4000 6000 8000 10000

cycles

Weibull Reliability Analysis—FWS-5/1999—45 Nonparametric MLEs of CDF

• For complete, type I & II censored data the nonparametric MLE is

d #{T ≤ t; i =1,...,n} F (t)= i , n −∞

−∞ t ? wherep ˆi = di/ni, ni = # units known to be at risk just prior to ti ? and di = # units that failed at ti • The nonparametric MLE for interval censored data is complicated

Weibull Reliability Analysis—FWS-5/1999—46 Empirical CDF (complete samples)

empirical and true and Weibull mle distribution functions

n = 10 n = 30 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

n = 50 n = 100 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

Weibull Reliability Analysis—FWS-5/1999—47 Nonparametric MLE for Type I Censored Data

nonparametric mle of CDF Weibull mle of CDF and true CDF

n = 10 n = 30 ...... 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

n = 50 n = 100 ...... 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

Weibull Reliability Analysis—FWS-5/1999—48 Nonparametric MLE for Type II Censored Data

nonparametric mle of CDF Weibull mle of CDF and true CDF

n = 10 n = 30 ...... 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

n = 50 n = 100 ...... 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

Weibull Reliability Analysis—FWS-5/1999—49 Kaplan-Meier Estimates (multiply right censored samples)

Kaplan-Meier CDF Weibull mle for CDF true CDF

n = 10 n = 30 ...... 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

n = 50 n = 100 ...... 0.0 0.4 0.8 0.0 0.4 0.8

0 5000 15000 25000 0 5000 15000 25000

Weibull Reliability Analysis—FWS-5/1999—50 Nonparametric MLE for Interval Censored Data

superimposed is the true Weibull(10000,2.5) that generated the 1,000 interval censored data cases inspection points roughly 3000 cycles apart CDF 0.0 0.2 0.4 0.6 0.8 1.0

0 5000 10000 15000

cycles

Weibull Reliability Analysis—FWS-5/1999—51 Nonparametric MLE for Interval Censored Data

superimposed is the true Weibull(10000,2.5) that generated the 10,000 interval censored data cases inspection intervals randomly generated from same Weibull CDF 0.0 0.2 0.4 0.6 0.8 1.0

0 5000 10000 15000

cycles

Weibull Reliability Analysis—FWS-5/1999—52 Application to Y2K Questionnaire Return Data

8124 cases

estimated .99-quantile probability of questionnaire return of questionnaire probability 0.0 0.2 0.4 0.6 0.8 1.0

0 50 100 150 200 250 300

days

Weibull Reliability Analysis—FWS-5/1999—53 Plotting Positions for Weibull Plots (Censored Case)

• For complete samples we plotted (log[T(i)], log [− log(1 − pi)]) with   − − i .5 1 d d 1  i i 1 p = = F (T )+F (T − ) =  +  i n 2 (i) (i 1) 2 n n

• For type I & II & multiply right censored samples ? − − we plot in analogy (log[T(i)], log [ log(1 pi)]), where ? ? T(1) <...

" # 1 d d p = F (T ? )+F (T ? ) i 2 (i) (i−1)

and Fd(t) is the nonparametric MLE of F (t) for the censored sample

Weibull Reliability Analysis—FWS-5/1999—54 Weibull Plot, Multiply Right Censored Sample n =50

.990 .900 · · ·· .632 · · .500 ·· ··· ·· ·· · .200 ·· ·· .100 ·· · ·

probability ·

.010 · true model, Weibull( 100 , 3 ) m.l.e. model, Weibull( 96.27 , 2.641 ) n = 50 least squares model, Weibull( 96.3 , 2.541 ) n = 50

.001

10 20 50 100 200 500 1000

cycles/hours

Weibull Reliability Analysis—FWS-5/1999—55 Conﬁdence Bounds for Censored Data

• For complete & type II censored data RAP computes exact coverage

conﬁdence bounds for α, β, tp,andP (T ≤ t). • WEIBREG and commercial software (Weibull++, WeibullSMITH, common statistical packages) compute approximate conﬁdence bounds for above targets, based on large sample m.l.e. theory • RAP & WEIBREG are Boeing code, runs within DOS mode of Windows95 (interface clumsy, but job gets done), contact me • Boeing has a site license for Weibull++ Version 4 in the Puget Sound area contact David Twigg (425) 717-1221, [email protected] • There is a Weibull++ Version 5 out

Weibull Reliability Analysis—FWS-5/1999—56 Weibull Plot With Conﬁdence Bound

0.999 0.990 alpha = 102.4 [ 82.52 , 143.8 ] 95 %

0.900 beta = 2.524 [ 1.352 , 3.696 ] 95 % n = 30 , r = 13 · · 0.500 · · · · · 0.200 · · 0.100 · · ·

0.010

0.001

1 2 3 5 10 20 50 100 200 500 1000

Weibull Reliability Analysis—FWS-5/1999—57 Weibull Plot With Conﬁdence Bound (Variation)

0.999 0.990 alpha = 89.79 [ 77.4 , 109.4 ] 95 %

0.900 beta = 3.674 [ 2.15 , 5.198 ] 95 % n = 30 , r = 13 · · · 0.500 · · · · · 0.200 · · 0.100 · · ·

0.010

0.001

1 2 3 5 10 20 50 100 200 500 1000

Weibull Reliability Analysis—FWS-5/1999—58 Weibull Regression Model & Analysis

1/β • Recall tp = α[− log(1 − p)] or

log(tp)=log(α)+wp/β with wp =log[− log(1 − p)] • Often we deal with failure data collected under different conditions – different part types – different environmental conditions – different part users • Regression model on log(α) (multiplicative on α)

log(αi)=b1Zi1 + ...+ bpZip th where Zi1,...,Zip are known covariates for i unit

• Now we have p + 1 unknown parameters β,b1,...,bp • parameter estimates & conﬁdence bounds using MLEs

Weibull Reliability Analysis—FWS-5/1999—59 Accelerated Life Testing: Inverse Power Law

• Units last too long under normal usage conditions • Increase “stress” to accelerate failure time • Increase voltage and accelerate life via inverse power law  c  Volt  T (Volt) = T (VoltU )   , where usually c<0 VoltU

this means  c  Volt  α(Volt) = α (VoltU )   VoltU

or log [α(Volt)] = log [α (VoltU )] + c log (Volt) − c log (VoltU )

= b1 + b2Z with

b1 =log[α (VoltU )] − c log (VoltU ) ,b2 = c, and Z = log (Volt)

Weibull Reliability Analysis—FWS-5/1999—60 Accelerated Life Testing: Arrhenius Model

• Another way of accelerating failure in processes involving chemical reaction rates is to increase the temperature • Arrhenius proposed the following acceleration model with temperature measured in Kelvin (tempK =temp◦C + 273.15)    κ κ   −  T (temp) = T (tempU )exp  temp tempU or κ − κ log [α(temp)] = log [α(tempU )] + temp tempU

= b1 + b2 Z with − κ −1 b1 =log[α(tempU )] ,b2 = κ, and Z =temp tempU

Weibull Reliability Analysis—FWS-5/1999—61 Pooling Data With Diﬀerent α’s

• Suppose we have three groups of failure data ∼W ∼W T1,...,Tn1 (˜α1,β),Tn1+1,...,Tn1+n2 (˜α2,β), ∼W Tn1+n2+1,...,Tn1+n2+n3 (˜α3,β)

• We can analyze the whole data set of N = n1 + n2 + n3 values jointly

using the following model for αj and dummy covariates Z1,j, Z2,j & Z3,j

log(αj)=b1Z1,j + b2Z2,j + b3Z3,j ,j=1, 2,...,N

where b1 =log(˜α1),b2 =log(˜α2)−log(˜α1), and b3 =log(˜α3)−log(˜α1)

Z1,j =1for j =1,...,N, Z2,j =1for j = n1 +1,...,n1 + n2,&Z2,j =0else,

and Z3,j =1for j = n1 + n2 +1,...,N,&Z3,j =0else. • Advantage: Smaller estimation error

Weibull Reliability Analysis—FWS-5/1999—62 Pooling Data (continued)

log(α1)=b1 · 1+b2 · 0+b3 · 0=log(˜α1) . . · · · log(αn1)=b1 1+b2 0+b3 0=log(˜α1) . . · · · − log(αn1+1)=b1 1+b2 1+b3 0=log(˜α1) + [log(˜α2) log(˜α1)] = log(˜α2) . . · · · − log(αn1+n2)=b1 1+b2 1+b3 0=log(˜α1) + [log(˜α2) log(˜α1)] = log(˜α2) . . · · · − log(αn1+n2+1)=b1 1+b2 0+b3 1=log(˜α1) + [log(˜α3) log(˜α1)] = log(˜α3) . . log(αN )=b1 · 1+b2 · 0+b3 · 1=log(˜α1) + [log(˜α3) − log(˜α1)] = log(˜α3)

Weibull Reliability Analysis—FWS-5/1999—63 Accelerated Life Testing: Model & Data

200

100

20 · · 10 · · · · thousand cycles · · 5 · ·

50 100 150 200 250 300 350 400 450 500

Voltage

Weibull Reliability Analysis—FWS-5/1999—64 Accelerated Life Testing: Model, Data & MLEs

200

100

50 true line mle line

20 · · 10 · · · mle for .01-quantile · · thousand cycles · · 5 · · 95% lower bound for .01-quantile

50 100 150 200 250 300 350 400 450 500

Voltage

Weibull Reliability Analysis—FWS-5/1999—65 Analysis for Data from Exponential Distribution

• T ∼E(θ) Exponential with mean θ, i.e., E(θ)=W(α = θ, β =1)

• It suﬃces to get conﬁdence bounds for θ since all other quantities of interest are explicit and monotone functions of θ

Fθ(t0)=Pθ(T ≤ t0)=1− exp(−t0/θ)

Rθ(t0)=Pθ(T>t0)=exp(−t0/θ) and tp(θ)=θ [− log(1 − p)]

Weibull Reliability Analysis—FWS-5/1999—66 Complete & Type II Censored Exponential Samples

• T ∼E(θ) Exponential with mean θ, E(θ)=W(α = θ, β =1)

• For complete exponential samples T1,...,Tn ∼E(θ) or for a type II censored sample of this type,

i.e., with observed r lowest values T(1) ≤ ...≤ T(r),1≤ r ≤ n, one gets 100γ% lower conﬁdence bounds for θ by computing × ˆ 2 TTT θL,γ = 2 , χ2r,γ

where TTT = T(1) + ···+ T(r) +(n − r)T(r) = Total Time on Test, 2 2 and χ2r,γ = γ-quantile of the χ2r distribution get this in Excel via = GAMMAINV(γ,r,1) or = CHIINV(1 − γ,2r)/2 • These bounds have exact conﬁdence coverage properties

Weibull Reliability Analysis—FWS-5/1999—67 Multiply Right Censored Exponential Data

• Here we deﬁne TTT = T1 + ...+ Tn as the total time on test th but now Ti is either the observed failure time of the i part or the observed right censoring time of the ith part The number r of observed failures is random here, 0 ≤ r ≤ n

• Approximate 100γ% lower conﬁdence bounds for θ are computed as × ˆ 2 TTT θL,γ = 2 χ2r+2,γ

• This also holds for r = 0, i.e., no failures at all over exposure period

Weibull Reliability Analysis—FWS-5/1999—68 Weibull Shortcut Methods for Known β

• T ∼W(α, β) (Weibull) ⇒ T β ∼E(θ) Exponential with mean θ = αβ

• As 100γ% lower conﬁdence bound for α compute  1/β  ×  2 TTT(β)   αˆL,γ =  2  χf,γ where for complete or type II censored samples we take f =2r and β ··· β − β TTT = T(1) + + T(r) +(n r)T(r) and for multiply right censored data we take f =2r +2and

β ··· β TTT = T1 + + Tn .

• Sensitivity should be studied by repeating analyses for several β’s

Weibull Reliability Analysis—FWS-5/1999—69 A- and B-Allowables, ABVAL Program

• My first Weibull work led to the program ABVAL (1983) supporting Cecil Parsons and Ron Zabora w.r.t. MIL-HDBK-5 • ABVAL computes A- and B-Allowables based on samples from the 3-parameter Weibull distribution. – The A-Allowable is a 95% lower confidence bound for 99% of the population, i.e., for the .01-quantile. – The B-Allowable is a 95% lower confidence bound for 90% of the population, i.e., for the .10-quantile. • Hybrid approach of using Mann-Fertig threshold estimate combined with maximum likelihood estimates for α and β, took lot of simulation and tuning and is very specialized in its capabilities • It is now a method in MIL-HDBK-5, I still maintain software

Weibull Reliability Analysis—FWS-5/1999—70 Selected References

• Abernethy, R.B. (1996), The New Weibull Handbook, 2nd edition, Reliability Analysis Center 800-526-4802 • Bain, L.J. (1978), Statistical Analysis of Reliability and Life-Testing Models, Marcel Dekker, New York • Kececioglu, D. (1993/4), Reliability & Life Testing Handbook Vol. 1 & 2, Prentice Hall PTR, Upper Saddle River, NJ. • Lawless, J.F. (1982), Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York • Mann, N.R., Schafer, R.E., and Singpurwalla, N.D. (1974), Methods for Statistical Analysis of Reliability and Life Data,JohnWiley& Sons, New York ⇒ Meeker, W.Q. and Escobar, L.A. (1998), Statistical Methods for Reliability Data,JohnWiley&Sons,NewYork

Weibull Reliability Analysis—FWS-5/1999—71 • Meeker, W.Q. and Hahn, G.J. (1985), Volume 10: How to Plan an Accelerated Life Test—Some Practical Guidelines,AmericanSociety for Quality Control • Nelson, W. (1982), Applied Life Data Analysis,JohnWiley&Sons, New York — (1985), “Weibull analysis of reliability data with few or no failures,” Journal of Quality Technology 17, 140-146 — (1990), Accelerated Testing, Statistical Models, Test Plans and Data Analyses, John Wiley & Sons, New York • Scholz, F.W. (1994), “Weibull and Gumbel distribution exact conﬁdence bounds.” BCSTECH-94-019 (RAP) — (1996) “Maximum likelihood estimation for type I censored Weibull data including covariates.” ISSTECH-96-022 — (1997), “Conﬁdence bounds for type I censored Weibull data including covariates.” SSGTECH-97-025 (WEIBREG) The Boeing Company, P.O. Box 3707, MS-7L-22, Seattle WA 98124-2207

Weibull Reliability Analysis—FWS-5/1999—72