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Accelerated Stress Testing and Reliability Conference

Design and Analysis of Accelerated Life Tests

Rong Pan

Arizona State University

[email protected]

ASTR 2016, Sep 28 - 30, Pensacola, FL January-4-17 1


Agenda

• Characteristics of lifetime data – Lifetime distributions

– Analysis of censored lifetime data

• Experimental designs – Factorial designs

– Critical issues in planning ALTs

• ALT data analysis – Weibull regression

– Data analysis using Software

– Optimal ALT planning

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Session Topic

Introduction

Part I Characteristics of lifetime data

• Lifetime distributions

• Analysis of censored lifetime data

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Features of Lifetime Distribution

Failure data from electrical appliance test (Lawless, p.7. Attr. Nelson (1970))

Variable: cycles to failure

• Nonnegative

• Right (positively) skewed

• Some long life observations

Normal distribution is not a good idea!

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Exponential Distribution

• The simplest lifetime distribution – One parameter

or

• Constant failure rate (constant mean-time-to-failure, MTTF)

• Memoryless property – Regardless of past experience, the chance of failure

in future is the same.

• Closure property – System’s failure time is still exponential, if its

components’ failure times are exponential and they are in a series configuration.

)exp()|( ttf )exp(1

)|(

t

tf

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Understanding Hazard Function

• Define a hazard function

• Instantaneous failure

• Is a function of time

– For the exponential distribution,

t

tTttTtth t

)|Pr(lim)( 0

)(th

)(

)()(

tR

tfth

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Bathtub Curve

• Empirical hazard function

• Three stages

– Infant mortality

– Mature

– Wear-out

• Any distribution good at representing any or all of these stages?

Time

Failure rate

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Weibull Distribution

1

)(

tth

tttf exp)(

1

ttR exp)(

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• When hazard function is a power function of time

• Two parameter

– Characteristic life, scale parameter

– Shape parameter

• Relationship with exponential distribution

– When the shape parameter is 1


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Lognormal Distribution

• From normal to lognormal and vice versa

– If T has a lognormal distribution, then log(T) has a normal distribution

– If X has a normal distribution, then exp(X) has a lognormal distribution

• Median failure time

– Log(t50) is a robust estimate of the location parameter of lognormal distribution

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Reliability Data

• Failure time censoring

– Right censoring

– Left censoring

– Interval censoring

• Data from reliability tests

– Type-I censoring (time censoring)

– Type-II censoring (failure censoring)

– Read-outs (multiple censoring)

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Example

Low-cycle fatigue test of nickel super alloy (Meeker & Escobar (1998), p. 638, attr. Nelson (1990), p. 272)

kCycles Cens

211.626 1

200.027 1

57.923 0

155 1

13.949 1

112.968 0

152.68 1

156.725 1

138.114 0

56.723 1

121.075 1

122.372 0

112.002 1

43.331 1

12.076 1

13.181 1

18.067 1

21.3 1

15.616 1

13.03 1

8.489 1

12.434 1

9.75 1

11.865 1

6.705 1

5.733 1 ASTR 2016, Pan 12


Parametric Distribution Models

• Likelihood function – Exact failure time

• Failure density function

– Right censoring time • Reliability function

– Left censoring time • Failure function

– Interval censoring time • Difference of failure functions

• Maximizing likelihood function – Exponential distribution

)(tf

)(tR

)(tF

)()( 1 ii tFtF

r

TTTMTTF

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Parametric Distribution Models

• Choose a distribution model

– Fit multiple distribution models

– Criteria (smaller the better) • Negative log-likelihood values

• AIC (Akaike’s information criterion)

• BIC (Bayesian information criterion)

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Model Selection using JMP

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Summary

• Three important Lifetime distributions

– Exponential

– Weibull

– Lognormal

• Failure time censoring

– Unavoidable in life tests

– Affects likelihood function

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Session Topic

Part II Experimental designs

• Factorial designs

• Critical issues in planning ALTs

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Design of Experiments

• Design of experiments have been used for reliability problems

– More than one accelerating factor

– An accelerating factor as well as process variables

– Robust design (noise factors)

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All experiments are designed

– some are not very good

A well-designed experiment:

• Efficient use of resources

• Identify/estimate

• important main effects (x1, x2, …)

• Interactions among factors (x1x2, x1x3, …)

• Higher-order terms (i.e., x2 )

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• Experiment developed to study the effect of two factors on the time to failure of glass capacitors (adapted from Zelen (1959))

• Two factors of interest:

– Temperature (degrees C)

• Range of 170 to 180

– Applied Voltage (volts) • Range of 200 to 350

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What we could do:

• Select several levels of both factors :

– Voltage: 200, 225, 250, 275, 300, 325, 350

• (Range of 200 to 350 volts)

– Temperature: 170, 175, 180 • (Range of 170 to 180 degrees C)

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Voltage: 200

Temperature: 170 175 180

Suppose 180ᵒ provided the best response.

Temperature: 180

Voltage 200 225 250 275 300 325 350

Suppose we find:

• Temperature:

180

• Voltage: 225

What could be wrong with this? Experimenting one factor at a time is generally a bad idea. - Too many runs - Missing interaction

effects - May not find the

true optimal 22


If the true responses were …

Voltage

TEMP

170 175 180

200 850 865 1250

225 706 782 943

250 645 803 835

275 843 940 997

300 1450 1350 1102

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A Better Way…

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This time, instead of five or six levels, let’s designate a “low” value and a “high” value (based on the ranges we are interested in investigating).

Temp voltage

Actual Coded Actual Coded

Low 170 -1 200 -1

High 180 1 350 1

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A Better Way…

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• Temp = Temperature, V = Voltage, y = life of the glass capacitor.

• Objective: Determine if temperature is significant; if voltage is significant; if there is a significant interaction.

• Goal: maximize the response, life of the capacitor.

Factors Replicate

Temp V I II Total

-1 -1 520 900 1420

-1 1 267 347 614

1 -1 1065 1087 1152

1 1 250 435 685

This is a

22

factorial

design

with two

replicates.

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Main Effects

Plot for

Temperature

and Voltage

(response is

Average Life)

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Interaction

Plot for

Temperature

and Voltage

(response is

Average Life)

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Contour

Plot for

Average

Life

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23 Factorial Design

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• Response: Lifetime of rolling ball bearings

• Three factors, each at two levels

– Inner Ring Heat Treatment

– Outer Ring Osculation

– Cage Design

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General 2k Factorial Design

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• As the number of factors (k) increases, the size of the design gets quite large. • 26 = 64 runs • 210 = 1024 runs • 215 = 32768 runs

• Situations with 10 or more factors are not that

uncommon. • But rarely are all the factors significant. • We need smaller alternatives to the full factorial design.

• Implement fractions of the full factorials (fractional factorials).

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What about Design?

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• Factorial, fractional factorial, Plackett-Burman designs, combined array designs etc. can be very useful for any of the problems we encounter in reliability testing.

• Constrained or unusual design region? – Use optimal designs based on some criteria (D-

optimality, or optimality at some use condition). – Current studies have shown that optimal designs can

be small, efficient, and cover the design region of interest.

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Optimal Designs?

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• Design optimality – Original work by Kiefer (1950s, 1960s) – Variance oriented criteria – Last 30 years - development of computer algorithms

for design construction

• Often called “Alphabetic Optimality” ‒ D-optimal, A-optimal, G-optimal ‒ Determine points in the experimental design that

will minimize variances/covariances associated with regression coefficients for a model of interest.

‒ Measured by “efficiencies”; D-efficiency, A-efficiency…

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Accelerated Life Test

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• Unacceptable cost of life testing • Too long testing time • Too few or no failure at normal operating conditions

• Units are tested at severer-than-normal stress conditions, the failure mechanisms are accelerated and more failures occur • For studying failures and failure mechanisms • For predicting product reliability

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Challenges

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• Failure time data are often censored

• Right censoring

• Interval censoring

• Failure time distribution is typically not normal distribution

• Exponential/Weibull

• Lognormal

• Extrapolating results

• Use condition is outside of the region of test conditions

• Model-based extrapolation is needed

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Suggestions for ALT Planning

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• Understand the purpose of an ALT experiment

• Anticipate the impact of data censoring

• Assess the robustness of a test plan to assumptions

• Be aware of the danger of model extrapolation

• Plan for the uncertainty of use condition

• Incorporate practical engineering/management constraints into the plan

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What about Statistical Analysis?

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• In DOE, Typically assume the responses are normally distributed

• Carry out standard t-tests and analysis of variance to identify important factors and interactions – then build a model.

• Interest is on the mean response

• In life testing: – Response is often skewed, can consist of censored

data

– Interest is on percentiles

– Standard analysis techniques often do not apply.

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What about Statistical Analysis?

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• One approach to deal with skewed or nonnormal responses: – Transform the data, make it normal (quasi-normal)

• Transformations don’t work on censored data!

• Nonnormal responses? – Weibull regression

– Use generalized linear models to fit the response. • Can handle censored data

• Can handle count or binomial data

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Session Topic

Part III ALT data analysis

• Weibull regression

• Data analysis using Software

• Optimal ALT planning

Questions

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Log-Location-Scale Distribution

• Both Weibull distribution and lognormal distribution belong to the log-location-scale distribution families – Log (Weibull variable) ~ Smallest Extreme Value (SEV)

distribution

– Log (lognormal variable) ~ Normal distribution

• The location parameter is modeled by a linear model – Log(characteristic life)

– Log(median life)

• To estimate regression coefficients, the maximum likelihood method is used

kk xx ...log 110

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Weibull Regression

• SEV distribution

• The linear model

• Other type of parameterization,

• Maximum likelihood method – Complete data

– Censored data

( ) 1 exp exp( ) ,Z

F z z z

loglog

TZ

/1

kxxk

*

1

** ...log10

0 1 1log

k kx x

),( Weibull

jj *

where

and

and

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( ) 1 exp exp( ) ,Z

F z z z


Physical Acceleration Model

• Arrhenius model

– Thermal stress

• Inverse power model

– Voltage, current, pressure, humidity, etc.

• Erying model

– Thermal stress with another stress factor

• Coffin-Mason model

– Thermal cycling

• Log-linear relationship

– Natural stress variables,

/50

aE KTt Ae

VAt

150

/50

aE KTt Ae V

)(max

50 TeAftKT

Ea

KTs

11 Vs log2

2211050log sst

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Statistical Acceleration Model

• Increase environmental stresses to fail the product earlier – Stress factors: temperature, voltage, current, pressure,

humidity, radiation, loading cycle, etc.

• To quantify effects of these factors, a regression model is needed – Log-location-scale regression model is common

– Proportional hazard model is another alternative

• With the regression model, the failure distribution under the use stress condition can be inferred

• Check model assumptions

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Example: Rolling Ball Bearings

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Example: Rolling Ball Bearings

• Cage design seems to have no effect on the lifetime.

• The 2x2 design with replicates is shown at right.

• There seems to be an interaction between Inner Ring Heat Treatment and Outer Ring Osculation

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R Output of Exponential Regression for Rolling Ball Bearings

Value Std. Error z p

(Intercept) 3.4924 0.354 9.878 5.19e-23

IHRT 0.4217 0.374 1.127 2.60e-01

ORO 0.4648 0.375 1.239 2.15e-01

Cage.Design 0.0316 0.355 0.089 9.29e-01

Scale fixed at 1

Exponential distribution

Loglik(model)= -35.9 Loglik(intercept only)= -37.9

Chisq= 3.97 on 3 degrees of freedom, p= 0.26

Number of Newton-Raphson Iterations: 3



(Intercept) 3.4924 0.354 9.878 5.19e-23

IHRT 0.4217 0.374 1.127 2.60e-01

ORO 0.4648 0.375 1.239 2.15e-01

Cage.Design 0.0316 0.355 0.089 9.29e-01

Parameter Estimates

Standard Error of the Estimates



(Intercept) 3.4924 0.354 9.878 5.19e-23

IHRT 0.4217 0.374 1.127 2.60e-01=0.260

ORO 0.4648 0.375 1.239 2.15e-01=0.215

Cage.Design 0.0316 0.355 0.089 9.29e-01=0.929

Scale fixed at 1

Exponential distribution

Loglik(model)= -35.9 Loglik(intercept only)= -37.9

Chisq= 3.97 on 3 degrees of freedom, p= 0.26

Number of Newton-Raphson Iterations: 3

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R Output of Exponential Regression for Rolling Ball Bearings

Nothing is significant!

• Either none of the factors are significant, or the exponential distribution is a bad fit.

• With no significant main effects, there is no reason to fit a model with interaction.


R Output for Weibull Regression Analysis of Rolling Ball Bearings


(Intercept) 3.6080 0.113 31.981 2.03e-224

IHRT 0.4412 0.164 2.695 7.03e-03

ORO 0.4454 0.165 2.701 6.92e-03

Cage.Design 0.0995 0.111 0.893 3.72e-01

Log(scale) -1.1924 0.300 -3.979 6.91e-05

Scale= 0.303

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Now, IHRT and ORO are significant and Cage.Design is not.


Using JMP for ALT Data Analysis

• One regressor Analyze ->

Reliability and Survival ->

Fit Life by X

• More than one regressor Analyze ->

Reliability and Survival ->

Fit Parametric Survival

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Using JMP for ALT Data Analysis (cont.)

• Fit parametric regression – Survival distribution

parameterized by a general linear model

– Censoring can be handled

– Both location parameter and scale parameter could be modeled

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Another Example

Nickel-base superalloy (Meeker and Escobar, 1998, Table C.12)

• Curvilinear relationship between pseudostress and lifetime

• Non-constant variance

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1st order model

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Accelerated Stress Testing and Reliability Conference 2nd order model

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With a scale parameter model

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Using JMP for Planning ALTs

DOE ->

Accelerated Life Test Design

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Introducing an R Package

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• What you see when you start R

• Install and run R packages

– Select a CRAN mirror

– Install the package you need

– Load the package

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ALTopt Package

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• Create D-, U- and I-optimal designs for accelerated life testing with right censoring or interval censoring – altopt.ic(): Create optimal test plans with interval

censoring – altopt.rc(): Create optimal test plans with right censoring

• Evaluate a given test plan

– alteval.ic(): Evaluate a test plan with interval censoring – alteval.rc(): Evaluate a test plan with right censoring

• Provides several graphing functions, including

contour plot, Fraction of Use Space (FUS) plot and Variance Dispersion of Use Space (VDUS) plot

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ALTopt Demo

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• Example: – Design <- altopt.ic("U", 100, 30, 5, 2, 1, formula = ~ x1 + x2 +

x1:x2, coef = c(0, -4.086, -1.476, 0.01), useCond = c(1.758, 3.159))

– pv.contour.ic(Design$opt.design.rounded, x1, x2, 30, 5, 2, 1, formula = ~ x1 + x2 + x1:x2, coef = c(0, -4.086, -1.476, 0.01), useCond = c(1.758, 3.159))

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References

• Wayne B. Nelson (2005a). A bibliography of accelerated test plans, IEEE Transactions on Reliability, vol. 54, pp. 194-197.

• Wayne B. Nelson (2005b). A bibliography of accelerated test plans part II-references, IEEE Transactions on Reliability, vol. 54, pp. 370-373.

• Wayne B. Nelson (2015). An updated bibliography of accelerated life tests, Proceedings of the Reliability and Maintainability Symposium, pp. 177-182.

• William Q. Meeker and Luis A. Escobar (1998). Pitfalls of accelerated testing, IEEE Transactions on Reliability, vol. 47, no. 2, pp. 114-118.

• William Q. Meeker, Georgios Sarakakis, and Atheanasios Gerokostopoulos (2013). More pitfalls of accelerated tests, Journal of Quality Technology, vol. 45, pp. 1-11.

• William Q. Meeker and Luis A. Escobar (1998). Statistical Methods for Reliability Data. Wiley.

• Steven E. Rigdon, Brandon R. Englert, Issac A. Lawson, Connie M. Borror, Douglas C. Montgomery and Rong Pan (2012). Experiments for reliability achievement, Quality Engineering, vol. 25, no. 1, pp. 54-72.

• Eric M. Monroe and Rong Pan (2008). Experimental design considerations for accelerated life tests for nonlinear constraints and censoring, Journal of Quality Technology, vol. 40, no. 4, pp. 355-367.

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Q&A

Thank You!

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Design and Analysis of Accelerated Life Tests - ieee … and... · Design and Analysis of Accelerated Life Tests ... ASTR 2016, Pan •Failure time data are often censored ... What