397 22. Accelerated Life Test Models and Data Analysis...

397

Accelerated Lif22. Accelerated Life Test Models and Data Analysis

Today’s consumers demand high quality andreliability in the products they buy. Accelerated lifetests (ALT) are commonly used by manufacturersduring product design to obtain reliabilityinformation on components and subsystems in atimely manner. The results obtained at high levelsof the accelerating variables are then extrapolatedto provide information about the product lifeunder normal use conditions.

The introduction and Section 22.1 describe thebackground and motivations for using acceleratedtesting. Sections 22.2 and 22.3 discuss statisticalmodels for describing lifetime distributions inALT. Commonly used ALT models have two parts:(a) a statistical distribution at fixed levels ofthe accelerated variable(s); and (b) a functionalrelationship between distribution parametersand the accelerating variable(s). We describerelationships for accelerating variables, such asuse rate, temperature, voltage, and voltage rate.We also discuss practical guidelines and potentialproblems in using ALT models. Section 22.4describes and illustrates a strategy for analyzingALT data. Both graphical and numerical methodsare discussed for fitting an ALT model to data andfor assessing its fit. These methods are thoroughlyillustrated by fitting an ALT model with a singleaccelerating variable to data obtained from anactual ALT experiment. Extrapolation of the resultsat accelerated levels to normal use levels isalso discussed. Section 22.5 presents statisticalanalysis of a wider variety of ALT data types thatare encountered in practice. In particular, theexamples involve ALTs with interval censoringand with two or more accelerating variables.Section 22.6 discusses practical considerationsfor interpreting statistical analysis of ALT data.This Section emphasizes the important role ofcareful planning of ALT to produce useful results.Section 22.7 discusses other kinds of acceleratedtests often conducted in practice. Brief descriptionsof each and specific applications in industry arealso provided. Section 22.8 reviews some of thepotential pitfalls the practitioner of accelerated

22.1 Accelerated Tests ................................. 39822.1.1 Types of Accelerated Tests........... 39822.1.2 Methods of Acceleration ............. 39922.1.3 Choosing

an Accelerated Life Test Model .... 399

22.2 Life Distributions ................................. 40022.2.1 The Lognormal Distribution ........ 40022.2.2 The Weibull Distribution............. 400

22.3 Acceleration Models ............................. 40022.3.1 Scale-Accelerated Lifetime Model 40122.3.2 Accelerating Product Use Rate ..... 40122.3.3 Models

for Temperature Acceleration ...... 40122.3.4 Models for Voltage

and Voltage–Stress Acceleration.. 40322.3.5 Models

for Two-or-More-VariableAcceleration ............................. 405

22.3.6 Guidelines and Issuesfor Using Acceleration Models ..... 407

22.4 Analysis of Accelerated Life Test Data .... 40722.4.1 Strategy for ALT Data Analysis...... 40722.4.2 Data Analysis

with One Accelerating Variable ... 408

22.5 Further Examples ................................ 41222.5.1 Analysis

of Interval Censored ALT Data ...... 41322.5.2 Analysis of Data

From a Laminate Panel ALT ......... 41422.5.3 Analysis of ALT Data with Two

or More Explanatory Variables .... 416

22.6 Practical Considerationsfor Interpreting the Analysis of ALT Data 421

22.7 Other Kinds of ATs ............................... 42122.7.1 Continuous Product Operation

Accelerated Tests ....................... 42222.7.2 Highly Accelerated Life Tests ....... 422

testing may face. These are described withinpractical situations, and strategies for avoidingthem are presented. Section 22.9 lists somecomputer software packages that are useful foranalyzing ALT data.

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398 Part C Reliability Models and Survival Analysis

22.7.3 Environmental Stress Tests ......... 42222.7.4 Burn-In ................................... 42222.7.5 Environmental Stress Screening .. 422

22.8 Some Pitfalls of Accelerated Testing ...... 42322.8.1 Failure Behavior Changes at High

Levels of Accelerating Variables ... 42322.8.2 Assessing Estimation Variability .. 42322.8.3 Degradation and Failure

Measured in Different Time Scales 424

22.8.4 Masked Failure Modes ............... 42422.8.5 Differences Between Product

and Environmental Conditionsin Laboratoryand Field Conditions.................. 424

22.9 Computer Softwarefor Analyzing ALT Data ......................... 424

References .................................................. 425

Rapid developments in technology, consumer demandfor highly reliable products, and competitive marketshave placed pressure on manufacturers to deliver prod-ucts with high quality and reliability. Similarly, thereis strong pressure to shorten the concept-to-market cy-cle time, subject to the constraint of having a productwith high reliability. Achieving these goals requiresmore attention to reliability early in the product de-sign process and has lead to such programs as designfor reliability and design for six sigma. As a result,there has been an increased need for upstream testsof product materials and components, often followedby subsequent testing of subsystems and even completesystems.

Customers may expect products to work properlyfor years or even decades. For instance, a refrigeratormight be expected to work without failure for at least10 years. Testing under normal operating conditions forreasonable lengths of time is unlikely to result in any fail-ures and thus little information about product reliability.Hence, manufacturers perform accelerated tests (AT) ontheir products by exposing them to harsher conditions to

generate failures and useful reliability information morequickly.

Depending on the nature of the product, life testsare accelerated by increasing product use rate or expos-ing the product to higher levels of accelerating variablessuch as temperature, pressure, and voltage. It is custom-ary to fit a statistical model to the AT data and thenextrapolate to use conditions to characterize the pro-duct’s long-term performance. It is desirable that thestatistical model for ATs be based on physical/chemicaltheory that can be used to justify the extrapolation. Oper-ationally, however, detailed knowledge of theory relatingthe accelerating variables to life is not always avail-able and the model is chosen on the basis of previousexperience in similar situations.

Due to space constraints, we provide a limited dis-cussion and description of the models and methods usedin AT. For those actually involved in AT and for thosewho want more information, we highly recommendNelson [22.1]. Other useful books with informationthat will complement this chapter include Tobias andTrindade [22.2] and Meeker and Escobar [22.3].

22.1 Accelerated Tests

22.1.1 Types of Accelerated Tests

During product design, manufacturers perform ex-periments to obtain timely information on materialproperties and component durability. Other experimentsinvolving prototype systems and subsystems are usedto help make product and process design decisionsthat will improve product robustness. Experiments arealso run to support decision making in production pro-cess design. Tests during the production stage includecertification of components, burn-in tests, and tests de-signed to monitor the production process over time.For further discussion of these issues, see Meeker andHamada [22.4] and Meeker and Escobar [22.5]. Of-

ten these tests must be accelerated to obtain timelyinformation.

ATs can be characterized by the nature of the re-sponse variable in the test (i. e., what can be measuredor observed, relative to reliability):

• Accelerated Life Tests ALTThe response in an ALT is related to the lifetime ofthe product. Often ALT data are right-censored be-cause the test is stopped before all units fail. In othercases, the ALT response is interval-censored becausefailures are discovered at particular inspection times.• Accelerated Repeated Measures DegradationTests ARMDT

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Accelerated Life Test Models and Data Analysis 22.1 Accelerated Tests 399

In an ARMDT, one measures degradation on a sam-ple of units at different points in time. In general,each unit provides several degradation measure-ments. The degradation response could be actualchemical or physical degradation or performancedegradation (e.g., drop in power output).• Accelerated Destructive Degradation TestsADDTAn ADDT is similar to an ARMDT, except that themeasurements are destructive, so one can obtain onlyone observation per test unit.

These different kinds of ATs can be closely relatedbecause they can involve the same underlying physi-cal/chemical mechanisms for failure and models for ac-celeration. They are different, however, in that differentkinds of statistical models and analyses are performedbecause of the differences in the kind of response.

This chapter focuses on analyses of data fromALTs. See Meeker and Escobar ([22.3], Chapts. 13 and21) for further discussion of ARMDTs and see Nel-son ([22.1], Chapt. 11) and Meeker, Escobar, Kugler,and Kramer [22.6] for models, methods, and examplespertaining to ADDTs.

22.1.2 Methods of Acceleration

There are different methods of accelerating tests to in-duce product failures more quickly. These methods varydepending on the nature of the product or material beingtested.

• Accelerate the Product Use RateThis method is appropriate for products that areordinarily not in continuous use. For example, themedian life of a bearing for a certain washing ma-chine agitator is 12 years, based on an assumed userate of eight loads per week. If the machine is testedat 112 loads per week (16 per day), the median lifeis reduced to roughly 10 months.• Accelerate Product AgingChanging environmental conditions (e.g., increasinghumidity or temperature) can be used to increase therate of chemical degradation of products such asinsulating materials and adhesive bonds.• Accelerate by Increasing Product StressIncreasing stress (e.g., voltage or pressure) on a spec-imen will generally cause it to fail sooner.

It is also possible to accelerate product failures by usingcombinations of these accelerating variables. For exam-ple, in fatigue testing one uses higher cycling rates andhigher than usual levels of stress. In electro-chemical

reactions, increasing voltage will also increase the rateof chemical change. Putting higher voltage stress on aninsulating material may generate heat that will acceler-ate chemical change. In all types of acceleration, careshould be taken to make sure that the underlying mech-anisms and the resulting failure modes in an AT are thesame as those that will affect the product in actual use.

22.1.3 Choosingan Accelerated Life Test Model

The task of finding an ALT model can be divided intotwo steps:

1. Choose an appropriate statistical distribution to de-scribe lifetime at fixed levels of the acceleratingvariable(s). Typically the same distribution is usedat all levels of stress, as would be suggested bythe commonly used scale-accelerated failure-time,SAFT, model. Probability plots (i. e., plotting a non-parametric estimate on special distribution-specificprobability scales) are used to help identify an ap-propriate distribution.

2. Choose a model to describe the relationship betweenthe lifetime distributions and the accelerating vari-ables. It is best if the selected model is based onphysical or chemical theory, empirical considera-tions, or both.

These two steps are discussed in detail in Sects. 22.2and 22.3, respectively.

As will be discussed in subsequent sections, thecombination of probability plotting and maximum like-lihood provides the basic statistical tools for analyzingmost ALT data. Probability plots provide a convenient,intuitive method to display ALT data and assess agree-ment with proposed models for the data. These methodsare described in detail in Nelson ([22.1], Chapt. 3) andMeeker and Escobar ([22.3], Chapts. 6 and 19). Max-imum likelihood (ML) is the most common method offitting an ALT model to data because it allows for cen-sored data and because ML estimates have desirablestatistical properties. The ML estimates of model pa-rameters are those parameter values that maximize theprobability of the observed data for the proposed model.The theory of ML methods has been thoroughly ex-plored in the literature. Nelson ([22.1], Chapt. 5) andMeeker and Escobar ([22.3], Chapts. 8 and 19) discussML methods relevant to ALT data analysis. Both prob-ability plotting and maximum likelihood methods arenow widely available in commercial statistical software,as described in Sect. 22.9.

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22.2 Life Distributions

In this chapter, the symbol T will denote lifetime fordevices, components, systems, etc. Then, it will be as-sumed that T is a positive continuous random variable.The cumulative distribution function CDF for T is theprobability of failure by time t (or the fraction failing bytime t) and one writes F(t) = Pr(T ≤ t). The probabilitydensity function PDF f (t) is the derivative of the CDF.That is, f (t) = dF(t)/dt. Selecting a probability distri-bution for lifetime is equivalent to specifying either F(t)or f (t).

This section describes log-location-scale probabilitydistributions. Important members of this family includethe popular Weibull and lognormal distributions. Othermembers of the family are described in Chapt. 4 ofMeeker and Escobar [22.3]. Distributions that are otherthan log-location-scale (which could also be used inALT modeling) are described in Chapt. 5 of Meeker andEscobar [22.3].

A random variable Y has a location-scale distributionif its CDF can be written as

F(y; µ, σ) = Pr(Y ≤ y) = Φ

(y −µ

σ

),

where µ is a location parameter, σ is a scale parameter,and Φ does not depend on any unknown parameters. Inmany reliability applications, it is assumed that log(T )has a location-scale distribution. Then T is said to havea log-location-scale distribution.

22.2.1 The Lognormal Distribution

Lifetime T has a lognormal distribution if its CDF andPDF are

F(t; µ, σ) = Φnor

(log(t)−µ

σ

),

f (t; µ, σ) = 1

σtφnor

(log(t)−µ

σ

), t > 0 ,

where Φnor and φnor are the standard normal (Gaussian)CDF and PDF, respectively. In particular,

φnor(z) = 1√2π

exp

(− z2

2

).

The parameters (µ, σ) are the mean and the standarddeviation of log(T ), respectively. Then [exp(µ), σ] are,respectively, the scale and shape parameters of T . Thelognormal p quantile is tp = exp[µ+Φ−1

nor(p) σ]. Notethat exp(µ) corresponds to the median lifetime. That is,t0.50 = exp(µ).

22.2.2 The Weibull Distribution

Lifetime T has a Weibull distribution if its CDF andPDF are

F(t; µ, σ) = Φsev

(log(t)−µ

σ

)= 1− exp

[−(

t

η

)β]

f (t; µ, σ) = 1

σtφsev

(log(t)−µ

σ

)= β

η

(t

η

)β−1

exp

[−(

t

η

)β]

, t > 0,

where Φsev and φsev are the standard smallest extremevalue SEV CDF and PDF defined by

Φsev(z) = 1− exp[− exp(z)]and φsev(z) = exp[z − exp(z)] ,

σ > 0, and −∞ < µ < ∞. Here (µ, σ) are the loca-tion and scale parameters for the distribution of log(T ).The expressions for the CDF and PDF above also useη = exp(µ) and β = 1/σ , the traditional Weibull scaleand shape parameters, respectively. The Weibull p quan-tile is tp = exp{µ+ log[− log(1− p)] σ} = η[− log(1−p)]β . Note that η is approximately the 0.63 quantile ofthe Weibull distribution.

The parameterization in terms of (µ, σ) is partic-ularly convenient for lifetime regression models and isused extensively in this chapter. The (η, β) parameteriza-tion is commonly used in the engineering and statisticalliterature.

22.3 Acceleration Models

fitting a model to data obtained at high levels of theaccelerating variables and extrapolating the results touse conditions levels. Ideally, this model should be de-

rived from physical or chemical theory and verifiedempirically to justify the extrapolation. If a physicalunderstanding of the failure mechanism is lacking, an

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Accelerated Life Test Models and Data Analysis 22.3 Acceleration Models 401

empirical model might be justified for extrapolation ifit is based on extensive experience with how failuremechanisms relate to accelerating variables.

This section discusses basic ideas of accelerationmodels and some physical considerations that lead tothese models. For more information on these models,see Nelson ([22.1], Chapt. 2) and Meeker and Esco-bar ([22.3], Chapt. 18).

22.3.1 Scale-Accelerated Lifetime Model

A simple, commonly used model used to characterizethe effect that explanatory variables x = (x1, . . . , xk)′have on lifetime T is the scale-accelerated failure-time(SAFT) model. Some of these explanatory variables areaccelerating, but others may just be of interest (e.g.,for product design optimization decisions). If xU de-notes the ordinary use conditions, under a SAFT model,lifetime at x, T (x), is scaled by a deterministic factorthat might depend on x and unknown fixed parameters.More specifically, T (x) = T (xU)/AF (x) where the ac-celeration factor AF (x) is a positive function of xsatisfying AF (xU) = 1. Lifetime is accelerated (decel-erated) when AF (x) > 1 [AF (x) < 1]. Some specialcases of these important SAFT models are discussed inthe following sections.

Observe that under a SAFT model, the probabilitythat failure under conditions x occurs at or before timet can be written as Pr[T (x) ≤ t] = Pr[T (xU) ≤ AF (x) ×t]. As described in Sect. 22.2, it is common practice toassume that the lifetime T (x) has a log-location-scaledistribution such as a lognormal or Weibull distributionin which µ is a function of the accelerating variable(s)and σ is constant (i. e., does not depend on x). Inthis case,

F(t; xU) = Pr[T (xU) ≤ t]

= Φ

(log(t)−µU

σ

), (22.1)

where Φ denotes a standard cumulative distributionfunction (e.g., standard normal or standard smallest ex-treme value) and µU is the location parameter for thedistribution of log[T (xU)]. Thus,

F(t; x) = Pr [T (x) ≤ t]

= Φ

(log(t)−{µU − log [AF (x)]}

σ

).

(22.2)

Note that T (x) also has a log-location-scale distri-bution with location parameter µ = µU − log[AF (x)]and a scale parameter σ that does not dependon x.

22.3.2 Accelerating Product Use Rate

Increasing the use rate can be an effective method ofacceleration for some products. In simple situationsthe cycles-to-failure distribution does not depend onthe cycling rate. In such situations reciprocity holds.Then the underlying model for lifetime is SAFT whereAF (UseRate) = UseRate/UseRateU is the factor bywhich the test is accelerated.

Use-rate acceleration may be appropriate for prod-ucts such as electrical relays and switches, papercopiers, and printers, and home appliances suchas toasters and washing machines. The manner inwhich the use rate is increased may depend onthe product. For example, Nelson ([22.1], page 16)states that failure of rolling bearings can be ac-celerated by running them at three or more timesthe normal speed. Johnston et al. [22.7] demon-strated that the cycles-to-failure of electrical insula-tion was shortened, approximately, by a factor ofAF (412) = 412/60 ≈ 6.87 when the applied AC volt-age in endurance tests was increased from 60 Hz to412 Hz.

ALTs with increased use rate attempt to simulateactual use. Thus other environmental factors should becontrolled to mimic actual use environments. If the cy-cling rate is too high, it can cause reciprocity breakdown.For example it may be necessary to have test units (suchas a toaster) cool down between cycles of operation.Dowling ([22.8], page 706) describes how increased cy-cle rate may affect the crack growth rate in per-cyclefatigue testing.

Reciprocity breakdown is known to occur, for ex-ample, for certain components in copying machineswhere components tend to last longer (in termsof cycles) when printing is done at higher rates.In such cases, the empirical power-rule relationshipAF (UseRate) = (UseRate/UseRateU)p is often used,where p can be estimated by testing at two or more userates.

22.3.3 Models for Temperature Acceleration

This section describes common models that are usedto describe the relationship between lifetime and tem-perature.

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22.3


The Arrhenius Relationshipfor Temperature Acceleration

The Arrhenius equation is widely used to relate the rateof a chemical reaction R to temperature temp. Thisrelationship can be written as

R(temp) = γ0 exp

( −Ea

kB × temp K

)

= γ0 exp

(−Ea × 11605

temp K

), (22.3)

where γ0 and the activation energy Ea are con-stants that depend on material properties and testmethods, kB = 8.6171 × 10−5 = 1/11605 is Boltzman-n’s constant in units of electron volts per ◦C, andtemp K = temp ◦C+273.15 is the temperature kelvin.The Arrhenius lifetime relationship is based on the viewthat failure occurs after there has been a critical amountof chemical reaction. Then when R(temp) is larger, thefailure will occur sooner (e.g., Klinger [22.9]). Empiricalobservations have suggested that the Arrhenius equa-tion provides an adequate description of the relationshipbetween product life and temperature in a variety of ap-plications such as integrated circuits (several differentkinds of failure modes), light-emitting diodes (LEDs),adhesive bonds, lubricants, incandescent bulb filaments,insulating tapes, and dielectric materials. It should benoted, however, that the nature of the failure mecha-nism may limit the range of temperature over which theArrhenius relationship is adequate.

Let tempU be the temperature at use conditions. Thenthe Arrhenius acceleration factor is

AF (temp, tempU, Ea) = R(temp)

R(tempU)

= exp

[Ea

(11605

tempU K− 11605

temp K

)].

For simplicity, the time-acceleration factor is sometimeswritten as AF (temp) instead of AF (temp, tempU, Ea).

Let tempLow < tempHigh be two temperature levelsand define the temperature differential factor TDF

TDF =(

11605

tempLow K− 11605

tempHigh K

). (22.4)

Then, one can write

AF (tempHigh, tempLow, Ea) = exp (Ea × TDF) .

(22.5)

Example 22.1: Adhesive-Bonded Power Element.Meeker and Hahn [22.10] describe an adhesive-bondedpower element that was designed for use at temp = 50 ◦Cand a life test of this element is to be conducted attemp = 120 ◦C. Suppose that experience with this prod-uct suggested that Ea = 0.5. Using (22.4) and (22.5)gives TDF = 6.39 and then AF (120) ≈ 24.41 givesthe acceleration factor for the chemical reaction whentesting the power element at 120 ◦C.

Note that the Arrhenius relationship (and the as-sumption that the failure is directly related to the amountof material reacting) implies that the acceleration modelis a SAFT model. If T (tempU) has a log-location-scaledistribution with parameters µU and σ , then it followsfrom (22.2) that T (temp) also has a log-location-scaledistribution with parameters µ = µU −β1x and σ , whereβ1 = Ea, x = TDF, and the TDF is computed for thetemperatures tempU < temp.

Nonlinear Degradation Path Modelsfor Reaction-Rate Acceleration

As discussed earlier, when failure is caused directly bydegradation, one can relate the distribution of degrada-tion to the distribution of failure. Suppose that failureoccurs when degradation reaches a certain critical levelDf. This section discusses the situation in which degra-dation follows a nonlinear path (e.g., the degradationpath will reach an asymptote because of a limited amountof material that is available to go into the reaction) overtime. The linear path case is discussed in the next section.

The amount of degradation at time t and temperaturetemp is given by

D (t; temp)

= D∞ ×{1− exp

[−RU ×AF (temp) × t] }

,

(22.6)

where RU is the rate of reaction rate at the usetemperature tempU. Note that RU × AF (temp) isthe rate reaction at temp and AF (temp) > 1 whentemp > tempU. Figure 22.1 shows the decrease instrength of adhesive bonds, as a function of time for dif-ferent temperatures. In this application, D∞ < 0 so that,for fixed temp, D(t; temp) is a decreasing function of tand failure occurs when D(t; temp) falls below Df, say50 N. In other applications, D∞ > 0, then D(t; temp)is increasing and failure occurs once D(t; temp) ex-ceeds Df. For both cases, the lifetime T (temp) at anylevel temp is given by T (temp) = T (tempU)/AF (temp)where T (tempU) = − (1/RU) log (1−Df/D∞). Notethat this has the form of a SAFT model. Sufficient

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0

–20

–40

–60

–80

0 20 40 60 80Weeks

Decrease in strength

25 °C

50 °C

60 °C

70 °C

Fig. 22.1 Nonlinear decreasing degradation path at differ-ent temperatures for a SAFT model

conditions for a degradation model to be SAFT aregiven by LuValle, Welsher and Svoboda [22.11] andKlinger [22.9]. Under a SAFT model and a log-location-scale distribution with parameters µ and σ for T (tempU),T (temp) at any temp also has a log-location-scale distri-bution with the same σ and

µ = µU − log[AF (temp)

]= β0 +β1x,

where x = 11605/(tempK), xU = 11605/(tempUK),β1 = Ea, β0 = µU −β1xU, and µU is the location pa-rameter of the distribution of log(T ) at use conditions.

Linear Degradation Path Modelsfor Reaction-Rate Acceleration

There are situations in which a linear model can beused to approximate degradation over time. For example,when RU ×AF (temp) × t in (22.6) is small enough sothat D(t) is small compared to D∞, then

D(t; temp) = D∞ ×{1− exp

[−RU ×AF (temp)

×t]}≈ D∞ ×RU ×AF (temp) × t = R+

U

×AF (temp) × t ,

where R+U = D∞ ×RU. There are also practical situa-

tions (e.g., wear of automobile tires) for which a linearpath adequately approximates degradation over time. Inthis case, D(t; temp) = R+

U ×AF (temp) × t where it isassumed that D(0; temp) = 0 and that R+

U ×AF (temp)is the degradation rate at condition temp.

Again, failure occurs when D(t; temp) reachesa critical level Df. The equation D(t; temp) = Df yieldsthe lifetime at temp, T (temp) = T (tempU)/AF (temp)which, again, is in SAFT form. Thus, if T (temp) has

a log-location-scale distribution, µ = β0 +β1x and σ

does not depend on x (where x has the same definitiongiven in Sect. 22.3.3).

22.3.4 Models for Voltageand Voltage–Stress Acceleration

Voltage or voltage stress can also be used to accel-erate degradation and hasten product failures. Voltagemeasures the amount of force needed to move an elec-tric charge between two points. Such a flow of chargesproduces an electrical current. Voltage stress measuresvoltage per unit of thickness of a dielectric. For dielectriccomponents (e.g., capacitors and insulators), chemicaldegradation reduces the dielectric strength over time.Also, stronger electric fields can accelerate the growth ofdiscontinuities, electrochemical reactions, or electrome-chanical changes that cause failure.

Example 22.2: Accelerated Life Test of Insulation forGenerator Armature Bars. Doganaksoy et al. [22.12]discuss an ALT for a new mica-based insulation de-sign for generator armature bars GAB. Degradation ofan organic binder in the insulation causes a decrease involtage strength and this was the primary cause of failurein the insulation. The insulation was designed for use ata voltage stress of 120 V/mm. Voltage-endurance testswere conducted on 15 electrodes at each of five acceler-ated voltage levels between 170 V/mm and 220 V/mm(i. e., a total of 75 electrodes). Each test was run for6480 h at which point 39 of the electrodes had notyet failed. Table 22.1 gives the data from these tests.The insulation engineers were interested in the 0.01and 0.05 quantiles of lifetime at the use condition of

Table 22.1 GAB insulation data

Voltage stress Lifetime(V/mm) (103 h)

170 15 censoreda

190 3.248, 4.052, 5.304, 12 censoreda

200 1.759, 3.645, 3.706, 3.726, 3.990, 5.153,6.368, 8 censoreda

210 1.401, 2.829, 2.941, 2.991, 3.311, 3.364,3.474, 4.902, 5.639, 6.021, 6.456,4 censoreda

220 0.401, 1.297, 1.342, 1.999, 2.075, 2.196,2.885, 3.019, 3.550, 3.566, 3.610, 3.659,3.687, 4.152, 5.572

a Units were censored at 6.480 103 h

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7

6

5

4

3

2

1

0170 180 190 200 210 220

Life (103 h)

Voltage stress (Volts /mm)

Fig. 22.2 GAB insulation data. Scatter plot of life versusvoltage. Censored observations are indicated by ∆

120 V/mm. Figure 22.2 plots the insulation lifetimesagainst voltage stress.

Inverse Power RelationshipThe inverse power relationship is frequently used to de-scribe the effect that stresses like voltage and pressurehave on lifetime. Voltage is used in the following dis-cussion. When the thickness of a dielectric material orinsulation is constant, voltage is proportional to voltagestress. Let volt denote voltage and let voltU be the volt-age at use conditions. The lifetime at stress level volt isgiven by

T (volt) = T (voltU)

AF (volt)=(

volt

voltU

)β1

T (voltU) ,

where β1, in general, is negative. The model has SAFTform with acceleration factor

AF (volt) = AF (volt, voltU, β1) = T (voltU)

T (volt)

=(

volt

voltU

)−β1

. (22.7)

If T (voltU) has a log-location-scale distribution withparameters µU and σ , then T (volt) also has a log-location-scale distribution with µ = β0 +β1x, wherexU = log(voltU), x = log(volt), β0 = µU −β1xU, and σ

does not depend on x.

Example 22.3: Time Acceleration for GAB Insulation.For the GAB insulation data in Example 22.2, an es-timate for β1 is β1 = −9 (methods for computing suchestimates are described and illustrated in Sect. 22.4). Re-call that the design voltage stress is voltU = 120 V/mm

and consider testing at volt = 210 V/mm. Thus, usingβ1 = β1, AF (210) = (210/120)9 ≈ 154. Thus by in-creasing voltage stress from 120 to 210 V/mm, oneestimates that lifetime is shortened by a factor of1/AF (210) ≈ 1/154 = 0.0065. Figure 22.3 plots AFversus volt for β1 = −7,−9,−11. Using direct compu-tations or from the plot, one obtains AF (210) ≈ 50 forβ1 = −7 and AF (210) ≈ 471 for β1 = −11.

Motivation for the Inverse Power RelationshipThe following description of a failure process that re-lates lifetime to pressure-like stresses, leading to theinverse power relationship, comes from Meeker andEscobar ([22.3], Sect. 18.4.3). Insulation units or spec-imens will have a characteristic dielectric strength D .This property varies from unit to unit and degrades overtime. When D degrades to the level of the applied volt-age stress volt, a failure-causing event (e.g., short circuitor a flash-over) is triggered. Suppose one can express thedielectric strength at time t by D(t) = δ0 × t1/β1 . LettingD(t) = volt and solving for t, T (volt) = (volt/δ0)

β1 . Theacceleration factor for comparing lifetimes at volt andvoltU is

AF (volt) = AF (volt, voltU, β1) = T (voltU)

T (volt)

=(

volt

voltU

)−β1

,

which agrees with (22.7).The inverse power relationship can be used to de-

scribe cases for which the accelerating voltage increasesthe degradation rate. For example, suppose that degra-

500500

200

100

50

20

10

5

2

1120 130 140 150 160 170 180 200190

Acceleration factor relative to 120 stress

Voltage stress

–7–9

–11

Inverse Power Rule Relationship

Fig. 22.3 Time-acceleration factor as a function of voltagestress and exponent −β1 = −7, −9,−11

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dation is more appropriately described by

D(t) = δ0 [R(volt) × t]1/γ1 ,

where R(volt) = γ0 exp[γ2 log(volt)

]. Again, if failure

occurs when Df = volt, then lifetime is

T (volt) = 1

R(volt)

(volt

δ0

)γ1

.

Then the acceleration factor is

AF (volt) = T (voltU)

T (volt)=(

volt

voltU

)γ2−γ1

which is an inverse power relationship with β1 =γ1 −γ2.

This discussion can be extended to other materialsand products such as filaments of light bulbs, capacitors,roller bearings, and ball bearings. The inverse powerrelationship has also been used with other acceleratingfactors such as loading stress, cycling rate, humidity, andpressure.

22.3.5 Models for Two-or-More-VariableAcceleration

Some ALTs use two or more accelerating variablesthat affect the degradation process. In comparison withsingle-variable ALTs, in some situations, the degrada-tion rate can be increased more effectively when twoor more variables are used to accelerate the test. ManyALTs use temperature in combination with another en-vironmental condition such as pressure, voltage, andhumidity. The rest of this section describes models forthese ALTs.

Extending the Arrhenius RelationshipRecall the Arrhenius equation (22.3), which gives therate of a chemical reaction as a function of temperature.Suppose that X represents a non-thermal acceleratingvariable. A model for reaction rate in this situation is

R(temp, X) = γ0 × exp

( −γ1

kB × temp K

)× exp

(γ2 X + γ3 X

kB × temp K

),

(22.8)

where the parameters γ1 = Ea (the effective activa-tion energy), γ0, γ2, and γ3 are specific properties ofthe failure-causing chemical reaction. In (22.8), the ra-tio γ3 X/(kB × temp K) represents possible interactionbetween the two accelerating variables. This model

could be further extended by adding appropriate fac-tors for other explanatory variables to the right-handside of (22.8).

Using a SAFT formulation, one can equivalentlyexpress the extended Arrhenius relationship (22.8) interms of the acceleration factor given by

AF (temp, X) = R(temp, X)

R(tempU, XU). (22.9)

When T (tempU, XU) has a log-location-scale distribu-tion, (22.1), (22.8) and (22.9) imply that T (temp, X) alsohas a log-location-scale distribution with

µ = µU − log[AF (temp, X)]= β0 +β1x1 +β2x2 +β3x1x2 , (22.10)

where β1 = Ea, β2 = −γ2, β3 = −γ3, x1 = 11605/

(temp K), x2 = X, β0 = µU−β1x1U−β2x2U−β3x1Ux2U,and σ does not depend on the explanatory variables(temp, X).

Temperature–Voltage Acceleration ModelsThere has been a variety of approaches that have beenused to model the combination of temperature andvoltage acceleration. For instance, Meeker and Esco-bar ([22.3], Sect. 17.7) analyzed data from a studyrelating voltage and temperature to the failure of glasscapacitors. They modeled the location parameter of loglifetime as a simple linear function of temp ◦C and volt.The extended Arrhenius relationship in Sect. 22.3.5 canalso be used with X = log(volt), as in Boyko and Ger-lach [22.13]. Klinger [22.14] modeled the Boyko andGerlach [22.13] data by including second-order termsfor both accelerating variables.

To derive the time-acceleration factor for the ex-tended Arrhenius relationship with temp and volt,one can follow steps analogous to those outlined inSect. 22.3.4. Using the dielectric strength (degradationpath) model at time t, D(t) = δ0

[R(temp, volt) × t

]1/γ1 .Using (22.8) with X = log(volt), we obtain

R(temp, volt) = γ0 × exp

( −Ea

kB × temp K

)× exp

[γ2 log(volt)+ γ3 log(volt)

kB × temp K

].

Again, failure occurs when the dielectric strengthcrosses the applied voltage stress, that is, D(t) = volt.This occurs at time

T (temp, volt) = 1

R(temp, volt)

(volt

δ0

)γ1

.

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From this, one computes

AF (temp, volt) = T (tempU, voltU)

T (temp, volt)= exp[Ea(x1U − x1)]

×

(volt

voltU

)γ2−γ1

× {exp[x1 log(volt)

−x1U log(voltU)]}γ3 .

where x1U = 11605/(tempU K) and x1 = 11605/

(temp K). When γ3 = 0, there is no interaction betweentemperature and voltage. In this case, AF (temp, volt)can be factored into two terms, one that involves temper-ature only and another term that involves voltage only.Thus, if there is no interaction, the contribution of tem-perature (voltage) to acceleration is the same at all levelsof voltage (levels of temperature).

Temperature–Current Density AccelerationModels

d’Heurle and Ho [22.15] and Ghate [22.16] studiedthe effect of increased current density (A/cm2) onelectromigration in microelectronic aluminum conduc-tors. High current densities cause atoms to move morerapidly, eventually causing extrusion or voids that leadto component failure. ATs for electromigration oftenuse increased current density and temperature to accel-erate the test. An extended Arrhenius relationship couldbe appropriate for such data. In particular, when T has

0.990.98

2 3 4 5 6 7 8 9 10

0.950.9

0.80.70.60.50.40.30.2

0.10.05

0.020.01

Proportion failing

Lifetime (103 h)

30V; 61 °C30V; 94 °C30V; 114 °C40V; 77 °C40V; 110 °C40V; 130°C

Fig. 22.4 LED device data. Lognormal multiple probabil-ity plot with lognormal ML fits for each combination ofvoltage/temperature

a log-location-scale distribution, then (22.10) applieswith x1 = 11605/temp K, x2 = log(current). The modelwith β3 = 0 (without interaction) is known as Black’sequation (Black [22.17]).

Example 22.4: Light Emitting Device (LED) Example.A degradation study on a light emitting device LED wasconducted to study the effect of current and tempera-ture on light output. A unit was said to have failed if itsoutput was reduced to 60% of its initial value. Two lev-els of current and six levels of temperature were used inthe test. Figure 22.4 is a probability plot of lifetimes foreach combination of current and temperature. The plotsuggests that the lognormal distribution is an appropri-ate distribution for lifetime and that increasing eithertemperature or current results in shorter device life.

Temperature–Humidity Acceleration ModelsRelative humidity is another environmental variablethat can be combined with temperature to acceleratecorrosion or other chemical reactions. Examples ofapplications include paints and coatings, electronic de-vices and electronic semiconductor parts, circuit boards,permalloy specimens, foods, and pharmaceuticals. Al-though most ALT models that include humidity werederived empirically, some humidity models have a phys-ical basis. For example, Gillen and Mead [22.18] andKlinger [22.19] studied kinetic models relating ag-ing with humidity. LuValle et al. [22.20] provideda physical basis for studying the effect of humid-ity, temperature, and voltage on the failure of circuitboards. See Boccaletti et al. [22.21], Nelson ([22.1],Chapt. 2) Joyce et al. [22.22], Peck [22.23], and Peckand Zierdt [22.24] for ALT applications involving tem-perature and humidity.

The extended Arrhenius relationship (22.10)) ap-plied to ALTs with temperature and humidity usesx1 = 11605/temp K, x2 = log(RH), and x3 = x1x2,where RH is a proportion denoting relative humidity.The case when β3 = 0 (no temperature–humidity inter-action) is known as Peck’s relationship and was usedby Peck [22.23] to study failures of epoxy packing.Klinger [22.19] suggested the term x2 = log[RH/(1−RH)] instead of log(RH). This alternative relationship isbased on a kinetic model for corrosion.

The Eyring ModelFundamental work on the relationship between reac-tion rates and temperature, in combination with othervariables, was done by Eyring in a number of impor-tant works (Eyring, Gladstones, and Laidler [22.25] and

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Accelerated Life Test Models and Data Analysis 22.4 Analysis of Accelerated Life Test Data 407

Eyring [22.26]). The Eyring model is the same as the ex-tended Arrhenius model described in this section, exceptthat it has an additional factor g(temp K) outside of theexponential in (22.8). Usually, g(temp K) = (temp K)p

is used. For the purposes of using a fitted accelerationmodel to predict life at use conditions, for any practi-cal values of the model parameters, this extra factor andestimating p along with Ea have no perceptible effecton the prediction. Thus the simpler Arrhenius model iswhat is generally used in practice. See Meeker and Es-cobar ([22.3], Sect. 18.3.2) and the above references formore information on the Eyring model.

22.3.6 Guidelines and Issuesfor Using Acceleration Models

The main objective of AT is to collect data at high lev-els of accelerating variables and extrapolate the resultsto use conditions. This section provides some guide-lines and lists some common issues that arise in fittingacceleration models to ALT data.

• Whenever possible, the acceleration model shouldhave a basis in physical or chemical theory. A care-ful review of previous theoretical and empiricalwork will often shed light on which environmen-tal factors and degradation processes are relevantto a failure mechanism. It is highly recommendedthat a group of experts oversee the entire projectof planning, collecting and analyzing AT data. Thisgroup should include experts with a clear under-standing of the mechanical, physical and chemicalnature of how the product degrades and eventu-ally fails. This team should also include someonewho is knowledgeable of the statistical aspects

of planning life tests and analyzing the resultingdata.• ATs often assume a relatively simple failure mech-anism or degradation process. It is possible athigh levels of an accelerating variable to intro-duce new failure mechanisms. Acceleration couldalso alter other environmental factors that affect thefailure mechanism. For example, Meeker and Esco-bar ([22.3], page 526) pointed out that ALT tests ofcircuit packs at higher temperatures reduced humid-ity, causing fewer failures than expected. Potentialscenarios like these should be considered for bothplanning and modeling ATs.• The main difficulty with extrapolation in ATs is thatthere is rarely sufficient data at lower levels of theaccelerating variable to verify that the accelerationmodel for the failure mechanism still holds at useconditions. Thus, a good test plan minimizes theamount of acceleration while providing a statisticallyefficient description of the failure mechanism at useconditions. There is a large amount of work that hasbeen done on ALT planning. For an overview of thiswork, see, for example, Nelson ([22.1], Chapt. 6)and Meeker and Escobar ([22.3], Chapt. 20 andSect. 22.5).• The statistical analysis of ALT data depends im-portantly on the acceleration model and distributionassumptions. A sensitivity analysis should be con-ducted to study how results (e.g., quantile and failureprobability estimates) vary with changes in the as-sumed model and distribution. The results of suchsensitivity analyses can help decision-makers under-stand whether model assumptions are conservativeor not.

22.4 Analysis of Accelerated Life Test Data

This section discusses the statistical analysis of ALTdata. First, we outline a useful strategy to followwhen choosing and fitting a model to ALT data andassessing the adequacy of the fit. Then we illus-trate the proposed strategy with several different ALTapplications having either one or two acceleratingvariables.

22.4.1 Strategy for ALT Data Analysis

This section outlines some guidelines for analyzing ALTdata. Suppose that groups of data were collected under

several individual conditions (levels or level combina-tions of the accelerating variables).

1. Make a scatter plot of the lifetimes versus the ex-planatory variable(s) to help in the identification ofan appropriate relationship or model.

2. For each individual condition, make a probabilityplot (Meeker and Escobar, [22.3], Chapts. 6 and 21)of the data under candidate distributions and obtainML estimates of the parameters. A distribution pro-vides a good fit if the plotted points fall roughlyalong a straight line. Check the constant-σ assump-

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tion visually by comparing the slopes of the lines inthe probability plots and statistically by comparingthe estimates of σ .

3. Fit a model constraining σ to be the same at eachlevel and compare with the unconstrained model inStep 3. This provides the basis for a formal test ofconstant σ .

4. Based on the previous steps, choose and fit andoverall model (distribution and acceleration model).

5. Compare the overall model fit in Step 4 to thecommon-σ model fit in Step 3. Large discrepanciesbetween these two models suggest an inadequate fitfor the overall model.

6. Use diagnostics such as residual plots to assess thefit of the overall model and to check the modelassumptions.

22.4.2 Data Analysiswith One Accelerating Variable

This section describes the analysis of several differentkinds of ALT data sets, following and illustrating theALT analysis strategy outlined in Sect. 22.4.1.

ALT Data Scatter PlotThe first step is to make a scatter plot of the data andstudy how the accelerating variable affects the lifetime.Use a different plotting symbol to indicate censoredobservations.

Example 22.5: Scatter Plot of the GAB InsulationData. Example 22.2 described a generator armature

0.98

0.950.9

0.8

0.2 0.5 1.0 2.0 5.0 10.0

0.70.60.50.40.30.2

0.10.05

0.02

Proportion failing

Lifetime (103 h)

190 V/mm200 V/mm210 V/mm220 V/mm

Fig. 22.5 GAB insulation data. Lognormal multiple proba-bility plot with lognormal ML fits for each level of voltagestress

bar insulation ALT. A scatter plot of the data, givenin Fig. 22.2, suggests a downward trend in lifetime asvoltage stress is increased. There was heavy censoringat the lower voltage stresses. In particular, 8 out of 12units failed at 190 V/mm and none of the 12 units failedat 170 V/mm.

Multiple Probability Plots at IndividualConditions of the Accelerating Variable

For each level (individual condition) of the acceleratingvariable(s), compute a nonparametric estimate of thefraction failing as a function of time and plot it on prob-ability paper for a suggested lifetime distribution. Thedistribution adequately describes the data if, for eachindividual condition, the points lie close to a straightline. ML can be used to fit a line through the points ateach level. A multiple probability plot showing thesenonparametric estimates and fitted lines for all of thelevels of the accelerating variable provides a conve-nient visual assessment of the constant-σ assumption.If the assumption is reasonable, the lines will be ap-proximately parallel. Repeat this for different lifetimedistributions.

Example 22.6: Probability Plots at Individual Condi-tions for the GAB Insulation Data. Figures 22.5 and 22.6give lognormal and Weibull multiple probability plots,respectively, for the GAB insulation data. There is noth-ing plotted for 170 V/mm because there were no failuresat this level of voltage stress. The points for each volt-age stress level fall roughly along a straight line, andthe lines appear to be reasonably parallel (but moreparallel with the Weibull distribution). It appears that

0.98

0.9

0.7

0.2 0.5 1.0 2.0 5.0 10.0

0.5

0.30.2

0.1

0.05

0.030.02

Proportion failing

Lifetime (103 h)

190 V/mm200 V/mm210 V/mm220 V/mm

Fig. 22.6 GAB insulation data. Weibull multiple probabil-ity plot with Weibull ML fits for each level of voltagestress

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Table 22.2 GAB insulation data. Weibull ML estimates for each voltage stress

Parameter ML estimate Standard error Normal-approximation95% confidence interval

190 µ 2.49 0.44 [1.63, 3.35]σ 0.42 0.23 [0.14, 1.26]

200 µ 2.07 0.20 [1.68, 2.45]σ 0.45 0.16 [0.23, 0.89]

210 µ 1.74 0.13 [1.49, 2.00]σ 0.44 0.11 [0.26, 0.73]

220 µ 1.17 0.11 [0.94, 1.39]σ 0.42 0.09 [0.28, 0.64]

The individual maximum log likelihoods were L190 = −12.31, L200 = −22.09, L210 = −27.54, and L220 = −25.05. The totallog likelihood for this model is L1 = −86.99

both the Weibull and the lognormal distributions pro-vide adequate descriptions of the GAB data. At eachlevel of voltage stress, the Weibull distribution is fittedto the data, and estimates of µ and σ were computedby using the method of ML. Table 22.2 gives the MLestimates of µ and σ for each level of voltage stress. Ta-ble 22.2 also provides approximate standard errors andnormal-approximation-based confidence intervals. Thelines drawn in Fig. 22.6 represent the ML estimates ofthe individual CDFs (fraction failing) at each level ofvoltage stress.

The ML estimates of σ for all levels of voltage stressare similar. The similarities are reflected in the near-parallel CDF lines drawn in Fig. 22.6. Although detailsare not shown here, the maximum total log likelihoodvalue achieved by fitting Weibull distributions with com-mon σ (the location parameters are allowed to float) toeach individual condition is L2 = −87.01 which is veryclose to the total log-likelihood L1 = −86.99 obtainedfrom fitting separate Weibull distributions. This suggeststhat it is reasonable to assume that σ is constant acrossvoltage stresses.

Maximum Likelihood Estimateof the ALT Model

The inverse power relationship is often used to re-late the lifetime distribution of a dielectric to voltage

Table 22.3 GAB insulation data. ML estimates for the inverse power relationship Weibull regression model


β0 53.39 9.46 [34.84, 71.93]β1 −9.68 1.77 [−13.14, −6.21]σ 0.44 0.06 [0.33, 0.58]

The maximum log likelihood for this model is L3 = −87.72

stress. Suppose that T (volt), the lifetime at volt, hasa log-location-scale distribution with parameters (µ, σ)and that σ does not depend on volt. Under the inversepower relationship, µ = β0 +β1x where x = log(volt).Fitting the model to data is accomplished by computingestimates for the parameters β0, β1, and σ .

Example 22.7: Maximum Likelihood Estimate of the In-verse Power Model for the GAB Insulation Data. Theresults of fitting the inverse power relationship Weibullmodel to the data are summarized in Table 22.3. A likeli-hood ratio test comparing with the constant-σ /floating-µmodel allows a formal assessment of whether the inversepower relationship is consistent with the data. Fitting theinverse power relationship with constant σ also allowsextrapolation to the use level of 120 V/mm.

Suppose that model B is obtained by imposing con-straints on the parameters of model A (e.g., the constraintthat the location parameters at different levels of volt-age stress are related by the inverse power relationship).For i = A, B, let Li be the maximum total log likeli-hood achieved by fitting model i to the data. If model Bis the correct model, in large samples, the likelihoodratio statistic W = 2 × (LA −LB) has a χ2

ν distributionwith degrees of freedom ν equal to the difference be-tween the number of estimated model parameters inthe two models. Large values of W indicate evidence

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against the constrained model model B, relative to theunconstrained model A. The difference is statisticallyimportant at the α level if W ≥ χ2

(1−α;ν) where χ2(1−α;ν)

is the (1−α) quantile of the χ2ν distribution.

For the GAB insulation data, model A is the (un-constrained) set of Weibull distributions with commonσ (five parameters), and model B is the (constrained)inverse power relationship Weibull model (three param-eters). Then LA = L2 = −87.01, LB = L3 = −87.72,and W = 1.42 < χ2

(0.95;2) = 5.99. Thus, the differencebetween the fits is not statistically important at the0.05 level and the data do not provide evidence againstmodel B.

The maximum log likelihood value attained withthe lognormal distribution is −89.52, which is closeto L3 = −87.72 for the Weibull distribution. Again, asnoted earlier with the probability plots, both the Weibulland lognormal models fit the data well.

Figure 22.7 is a probability plot showing the ML fitof the inverse power relationship Weibull model to thedata. It also gives the ML estimate of the fraction fail-ing at use conditions voltU = 120 V/mm. The dashedlines give pointwise normal-approximation 95% confi-dence intervals for the fraction failing at use conditions.Conversely, this plot can be used to obtain an estimateand an approximate confidence interval for specifiedquantiles at use conditions. Figure 22.8 is a scatterplot of the GAB data with ML estimates of the 0.1,0.5, and 0.9 quantiles at all levels of voltage stressesbetween 120 and 220 V/mm. The horizontal line indi-

0.98

10–2

0.70.5

0.30.2

0.1

0.50.30.2

0.1

0.003

0.001

0.005

10–1 100 101 102 103

0.9

Proportion failing

Lifetime (103 h)

170 V/mm190 V/mm200 V/mm210 V/mm220 V/mm120 V/mm

Fig. 22.7 GAB insulation data. Weibull multiple probabil-ity plot depicting the inverse power relationship Weibullregression model ML fit

cates the censoring time at 6480 h. Densities (smallestextreme-value densities because time is plotted on a logscale) are shown at the voltage stresses in the exper-iment as well as at the use condition of 120 V/mm.Figure 22.8 suggests that it is unlikely that one wouldobserve any failures below 170 V/mm during the ALTexperiment.

Assessing Statistical UncertaintyAn estimate of the variance–covariance matrix for theML estimates θ = (β0, β1, σ)′ is

∑θ=⎛⎜⎝ 89.52 −16.73 0.27

−16.73 3.13 −0.05

0.27 −0.05 0.004

⎞⎟⎠ . (22.11)

The elements of this matrix are used to computenormal-approximation confidence intervals, as de-scribed in Example 22.9. This matrix is usually available(at least as an option) as output from computer softwareand is useful for computing the variance of functions ofthe parameters (e.g., quantiles and fraction failing esti-mates) that may be of interest. In modern software forlife data analysis, however, such computations are easyto request and printing the variance–covariance matrixis often suppressed by default.

Diagnostics and Assessing Fitof ALT Model to Data

Plots like Figs.22.7 and 22.8 can be used to assessthe fit of a model to a set of data. Graphical analyses

104

103

102

101

100

10–1

100 110 130 150 170 190 210 240

Hours to failure (103 h)

Voltage stress on inverse power rule scale

90%50%10%

Fig. 22.8 GAB insulation data. Scatter plot showing hoursto failure versus voltage stress (log–log scales) and theinverse power relationship Weibull regression model fit.Censored observations are indicated by ∆

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of the standardized residuals, exp[log(t)− µ]/σ , pro-vide additional diagnostics for judging model adequacy.Residuals can be plotted against the order of the testruns, fitted values, and levels of explanatory variablesthat are or are not in the fitted model. It is necessaryto take into account the amount of censoring during in-terpretation. An overall assessment of the distributionalassumption can be obtained by using a probability plotof the residuals.

Example 22.8: Residuals Plots for the Inverse Power Re-lationship Weibull Model Fit to the GAB Insulation Data.Figure 22.9 plots residuals versus fitted values. The verti-cal axis shows the standardized residuals. The horizontallines are the 0.05, 0.50, and 0.95 quantiles of a standardexponential distribution (e.g., − log(1−0.95) = 2.996is the 0.95 quantile of this distribution). For the Weibulldistribution, the residuals should have an approximatestandard Weibull (µ = 0, σ = 1) distribution (which isleft-skewed). Thus, the Weibull distribution is appropri-ate if the residuals have a long lower tail. At 210 V/mm,where all of the test units failed, the residuals (firstcolumn of points in Fig. 22.9) indicate a left-skeweddistribution about the center line, indicating no lackof fit of the Weibull distribution. The interpretation ofthe residuals is not as simple for the levels 170, 190,200, and 210 V/mm (the last four columns of points inFig. 22.9) because all of these levels resulted in somecensored observations. The triangles (�) in Fig. 22.9indicate lower bounds for residuals for these censoredobservations (i. e., the actual unobserved residuals corre-sponding to these points would be larger than the plottedtriangles). With this in mind, there is no strong evidenceof lack of fit of the Weibull distribution at these levels.Figure 22.10 is a Weibull probability plot of the residu-als for all failures. The residuals fall close to a straightline, which suggests that there is no problem with theWeibull assumption.

Estimation at Use Conditionsand Sensitivity to Model Assumptions

It is important to quantify the statistical uncertainty inestimates from an ALT. Here, we illustrate the compu-tation of normal-approximation confidence intervals forquantiles. See Sect. 17.4.2 of Meeker and Escobar [22.3]for a more general discussion of the methodology. Thefollowing example also contains a sensitivity analysis ofthe estimates with respect to the assumed distribution.

Example 22.9: ML Estimates and Confidence Intervalsfor Quantiles for the GAB Insulation Data Life Distri-bution. For the GAB experiment, it was of interest to

5.000

2.000

1.000

0.500

0.200

0.1000.050

0.020

0.010

0.005

0.95

0.50

0.05

2 5 10 20 50

Standardized residuals

Fitted values

Fig. 22.9 GAB insulation data. Plot of standardized residuals versusfitted values for the inverse power relationship Weibull regressionmodel. Censored observations are indicated by ∆

0.0003

0.005 0.010 0.020 0.050 0.100 0.200 0.500 1.000 2.000 5.000

0.00050.001

0.0030.0050.010.02

0.050.10.20.30.50.70.90.980.999

Probability

Standardized residuals

Fig. 22.10 GAB insulation data. Weibull probability plot of thestandardized residuals from the inverse power relationship Weibullregression model fit

estimate the 0.01 and 0.05 quantiles of insulation life atthe 120 V/mm use condition. The ML estimate of the lo-cation parameter at 120 V/mm, computed from the MLestimates for the parameters in Table 22.3, is

µ = β0 + β1x = 53.39−9.68 log(120) = 7.07 .

The ML estimate of the log p quantile is

log( t p) = µ+ log[− log(1− p)] σ= 7.07+ log[− log(1− p)]× 0.44

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Table 22.4 GAB insulation data. Quantiles ML estimates at 120 V/mm

Distribution Quantile ML estimate Normal-approximation95% confidence interval

Weibull t01 154.6 × 103 h [26.4, 907.4] ×103 h

17.7 y [3.0, 103.6] y

t05 317.0 × 103 h [51.4, 1953.0] ×103 h

36.2 y [5.9, 223.0] y

Lognormal t01 254.4 × 103 h [43.4, 1492.9] ×103 h

29.1 y [5.0, 170.0] y

t05 390.8 × 103 h [63.4, 240.9] ×103 h

44.6 y [7.0, 27.5] y

with an estimated standard error selog(tp) =√

Varlog(tp)

=√

c′∑θc, where c = {1, log(volt), log[− log(1− p)]}′

and∑

θis given in (22.11). A 100(1−α)% normal-

approximation confidence interval for tp is

exp[log( t p)± z(1−α/2) × selog(tp)

]= t p exp

(±z(1−α/2) × selog(tp)

),

where zγ is the γ quantile of the standard nor-mal distribution. Table 22.4 gives the ML estimates

and 95% confidence intervals for the 0.01 and0.05 quantiles at 120 V/mm. the data that wasalmost as good as the fit with the Weibull distri-bution. Table 22.4 also includes quantile estimatesfor the lognormal distribution. Although the differ-ences between the Weibull and lognormal distributionpoint estimates may be judged to be large, theconfidence interval for the quantile under one dis-tribution contains the point estimate under theother. The lognormal distribution quantile estimatesare optimistic relative to those for the Weibulldistribution.

22.5 Further Examples

This section illustrates the use of the statistical methods,outlined in Sect. 22.4, to various other ALT applica-

Table 22.5 IC device data

Temperature (◦C) Time Bounds (h) Frequency CensoringLower Upper

150 788 1 Right150 2304 49 Right175 788 1 Right175 2304 49 Right200 788 1 Right200 2304 49 Right250 384 788 1 Interval250 788 1 Right250 1536 2304 6 Interval250 2304 42 Right300 96 192 1 Interval300 192 384 6 Interval300 384 788 20 Interval300 384 2 Right300 788 1536 16 Interval300 1536 5 Right

tions. The examples show the fit of ALT models tointerval-censored data and reliability data with two or

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Accelerated Life Test Models and Data Analysis 22.5 Further Examples 413

more explanatory variables. These examples illustratea wider (but not exhaustive) range of data types that arisein practice, provide more insight into how to developALT models, address the goals and objectives of the ex-periment, and study the effect of model assumptions onthese aims (sensitivity analyses).

22.5.1 Analysisof Interval Censored ALT Data

Interval-censored data are common in ALT studies. Con-straints on resources limit the frequency of inspectiontimes in tests. As a consequence, engineers often fail toobserve failures instantaneously, and inspection timesserve as bounds for lifetimes.

Example 22.10: Analysis of Integrated Circuit DeviceALT Data. Table 22.5 gives the results of an ALT foran integrated circuit (IC) device at five different levelsof junction temperature. Because the diagnostic testingof each device was expensive, each unit could only beinspected a few times during the ALT. Consequently,an observation was either interval-censored (bounds forlifetime were recorded) or right-censored (lower boundfor lifetime was recorded). The engineers chose inspec-tion times that were approximately equally spaced inlog time. Tests were conducted at 150, 175, 200, 250,and 300 ◦C on a total of 250 devices (50 at each level oftemperature).

The objectives of the ALT was to quantify the long-term reliability of the ICs and to estimate the effectiveactivation energy of the failure mechanism. In particular,the engineers involved in the study wanted to assess the

0.00550

Hours100 200 500 1000 2000 5000

0.010.02

0.05

0.1

0.20.30.40.50.60.70.8

0.9

Proportion failing

250°C300°C

Fig. 22.11 IC device data. Lognormal multiple probabilityplot with individual lognormal ML fits for each temperature

hazard rate and the fraction failing by 100 000 hours atthe use junction temperature of 100 ◦C.

Figure 22.11 is a lognormal probability plot of thedata at individual conditions. There are no plots for 150,175, 200 ◦C because there were no failures at these levelsof temperature. The ML results for estimating the log-normal parameters at individual temperatures are givenin Table 22.6. The fitted CDF lines drawn in the plot ap-pear to be reasonably parallel. Also, the ML estimatesof σ and corresponding confidence intervals do not indi-cate evidence against the constant-σ assumption acrosstemperature levels. Table 22.7 gives the ML results offitting an Arrhenius lognormal model to the IC devicedata. An estimate of the effective activation energy isEa = β1 = 0.94.

Figure 22.12 is a probability plot depicting the MLfit of the Arrhenius lognormal model to the data. Italso gives the ML estimate of the fraction failing attempU = 100 ◦C with pointwise normal-approximation95% confidence intervals. Figure 22.12 plots the MLestimates of the 0.1, 0.5, and 0.9 quantiles with thedata for temperatures between 100 and 300 ◦C. Eventhough there is a large amount of statistical uncertainty,Figs. 22.12 and 22.13 both suggest that it is very un-likely that failures would occur before 100 000 hours attempU = 100 ◦C.

Observing failures at two or more levels of theaccelerating variable is necessary to estimate the param-eters of the Arrhenius relationship. There are situationsin which the activation energy Ea is specified, based

0.0001

102 103 104 105 106 107 108

0.00050.0020.0050.010.020.0050.1

0.20.30.40.50.60.70.80.90.950.980.99

Fraction failing

Hours

300 °C

250200 175 150 100

Fig. 22.12 IC device data. Lognormal multiple probabilityplot depicting the Arrhenius lognormal regression modelML fit

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Table 22.6 IC device data. Lognormal ML estimates for each temperature

Temperature (◦C) Parameter ML estimate Standard error Normal-approximation95% confidence interval

250 µ 8.54 0.34 [7.87, 9.22]σ 0.74 0.25 [0.38, 1.43]

300 µ 6.58 0.20 [8.59, 9.36]σ 0.61 0.07 [0.48, 0.76]

The individual maximum log likelihoods were L250 = −26.11 and L300 = −63.18. The total log likelihood for this modelis L4 = −89.29

Table 22.7 IC device data. ML estimates for the Arrhenius lognormal regression model


β0 −12.45 1.95 [−16.26, −8.63]β1 0.94 0.09 [0.76, 1.12]σ 0.62 0.07 [0.50, 0.77]

The maximum log likelihood for this model is L5 = −89.45

108

107

106

105

104

103

102

50 100 150 200 250 300

Hours

Arrhenius scale (°C)

90%50%10%

Fig. 22.13 IC device data. Scatter-plot interval midpointsfailure versus ◦C (on an Arrhenius scale) and the Arrheniuslognormal regression model fit. Censored observations areindicated by ∆

on prior experience with the product, in which caseonly one temperature level with failures may sufficefor estimation. For example, MIL-STD-883 [22.27]gives guidelines for conducting reliability demonstra-tions tests when Ea is given.

Example 22.11: Analysis of the IC Device Data whenthe Activation Energy is Given. Figure 22.14 is simi-lar to Fig. 22.12, but obtained with Ea = 0.8 eV given.Now, the confidence intervals for the fraction failingat tempU = 100 ◦C are much narrower than when Eahad to be estimated. This difference reflects a tremen-dous, probably unreasonable, gain in precision. Becausethere will be doubt about the actual value of Ea, one

0.0001

102 103 104 105 106 107

0.0005

0.0020.0050.010.02

0.050.1

0.20.30.40.50.6

0.8

0.90.95

0.7

Fraction failing

Hours

300 °C

250200 175 150

100

Fig. 22.14 IC device data. Lognormal probability plotshowing the Arrhenius lognormal model ML estimates and95% confidence intervals for F(t) at 100 ◦C given Ea =0.8 eV

must exercise caution in interpreting such an analysis.Varying β1 = Ea over some plausible range of valuesprovides a useful sensitivity analysis. Also a Bayesiananalysis could be used when the uncertainty on Ea canbe specified by a probability distribution (see Meekerand Escobar [22.3], Sect. 22.2).

22.5.2 Analysis of DataFrom a Laminate Panel ALT

A sample of 125 circular-holed notched specimens ofa carbon eight-harness-satin/epoxy laminate panel were

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380

20 100 500 2000 10000 50000

370360350340330320310300290280

270

Stress on log scale

Thousands of cycles

Fig. 22.15 Laminate panel data. Log–log scatter plot of lifeversus stress

subjected to a cyclic four-point out-of-plane bending atseveral levels of stress, measured in MPa. The data arefrom Shimokawa and Hamaguchi [22.28]. In these tests,final specimen fracture and fiber fracture occurred si-multaneously. The lifetime of a specimen is the numberof cycles to fracture. Figure 22.15 is a log–log scat-ter plot of the data. Note that the response (thousandsof cycles) is plotted on the horizontal axis (the usualconvention in the fatigue-fracture literature). Ten speci-mens were right-censored (two at 280 MPa and eight at270 MPa). The goals of experiment were to estimate therelationship between life and stress (or the S–N curve)and a safe level of stress for a given life quantile. Fig-ure 22.16 is a multiple lognormal probability plot ofthe data. The lognormal distribution appears to be ap-propriate here. Figure 22.15 indicates a strong linearcorrelation between log life and log stress. These plotssuggest fitting a inverse power relationship lognormalmodel to the data.

Example 22.12: ML Estimate of the Inverse Power Re-lationship Lognormal Model for the Laminate PanelData. For the inverse power relationship lognormalmodel, the parameters are µ = β0 +β1x and σ , wherex = log(Stress). Table 22.8 gives the results of the MLestimation of β0, β1 and σ with the laminate panel data.

Table 22.8 Laminate panel data. ML estimates for the inverse power relationship lognormal regression model


β0 99.36 2.14 [95.16, 103.55]β1 −16.05 0.37 [−16.78, −15.32]σ 0.52 0.04 [0.46, 0.60]The maximum log likelihood for this model is L6 = −898.3

0.00005

0.00050.0020.0050.020.050.10.20.30.50.70.80.90.950.980.9950.998

101 102 103 104 105 106 107

Fraction failing

Thousands of cycles

380 MPa 340 300 280

270 197.9

Fig. 22.16 Laminate panel data. Lognormal probability plotshowing the inverse power relationship lognormal modelML fit and 95% confidence intervals for F(t) at 197.9 MPa

380

340

101 102 103 104 105 106 107

320300280260

240

220

200

180

Stress (MPa)

Thousands of cycles

10% 50% 90%

Fig. 22.17 Laminate panel data. Scatter-plot showing cyclesto failure versus MPa (on a log scale) and the inverse powerrelationship lognormal regression model ML fit. Censoredobservations are indicated by ∆

Estimates of the fraction failing at the individual stresslevels of the experiment are drawn in Fig. 22.16. Fig-ure 22.17 is a scatter plot of the data with ML estimatesof the 0.1, 0.5, and 0.9 quantiles. The confidence in-tervals for the life distribution at use conditions in this

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example are much narrower, when compared to thosefor the IC device example shown in Fig. 22.12. This isbecause there is less censoring and less relative extrap-olation in the inference for the laminate panel data. Themedian (0.50 quantile) life line in Fig. 22.17 providesthe estimate of the S–N curve that was of interest to theengineers.

Example 22.13: Estimate of a Safe-Level Stress for theLaminate Panel Data. The engineers wanted to esti-mate a safe level of stress at which no more than 10%of the population of laminate panels would fail before1 000 000 cycles. This involves determining the stresslevel at which the 0.10 quantile is 1 000 000 cycles.At any stress below this safe level of stress, the pop-ulation fraction failing before 1 000 000 cycles will bemore than 0.10. Using the inverse power relationshiplognormal model, the safe stress level is

safe stress = exp{[

log(106)−99.36−Φ−1

nor(0.1)

× 0.52]/

(−16.05)}

= 197.9 MPa.

Figure 22.16 plots an estimate of the fraction failing(rightmost line) at this safe stress level. As seen in thisplot, the stress 197.9 MPa is an extrapolation because theestimation is outside the range of the data. Of course,there is potential danger in making decisions based onthis estimate, especially if the inverse power relationshiplognormal model fails to hold below 270 MPa. Becausethe inverse power relationship is an empirical modeland its use is based on past experience, it is important toperform a sensitivity analysis by computing safe-levelestimates under alternative relationships. For example,the reciprocal relationship (x = 1/Stress) gives a modelmaximum log likelihood value of L7 = −892.1 whichis larger than that from the inverse power relationship(indicating that the reciprocal model provides a betterfit to the data). The estimate of the safe level of stressunder the reciprocal relationship is equal to 216 MPa.This is larger (less conservative) than that under the in-verse power relationship. Quantile estimates at a givenstress level are also less conservative under the recipro-cal relationship. That is, the estimates of life under thereciprocal relationship are longer than those under theinverse power relationship.

22.5.3 Analysis of ALT Data with Twoor More Explanatory Variables

The single-variable analyses in Sects. 22.4, 22.5.1,and 22.5.2 can be extended to two or more variables (ac-

10

9

8

7

6

5

4

360 70 80 90 100 110 120 130

Temperature (°C)

Hours to failure (103 h)

30 V40 V

Fig. 22.18 LED device data. Scatter-plot showing hours tofailure versus temperature with voltage indicated by differ-ent symbols. Censored observations are indicated by ∆

celerated or otherwise). For example, ATs with multipleaccelerating variables often use temperature in combi-nation with humidity, pressure, and/or voltage. The goalof an AT may be to compare new and old product de-signs under different humidity levels, using temperatureto accelerate the lifetimes. In this case, product design isa qualitative factor and the humidity range would mimicthat seen in the actual application. This section illustratesthe analysis of data from AT studies like these.

Example 22.14: Light Emitting Device Data with Accel-erated Current and Temperature. Recall the LED devicedata from Example 22.4. The engineers were interestedin the reliability of the LED at use conditions of 20 mAcurrent and 40 ◦C temperature. Figure 22.18 plots theLED device data versus temperature for different levelsof current. The plot suggests that failures occur earlierat higher levels of current or temperature.

The probability plot in Fig. 22.4 suggests that thelognormal distribution can be used to describe the life-times at each combination of current and temperature.Furthermore, because the lines are approximately paral-lel, it is reasonable to assume that σ is constant acrossindividual conditions.

The inverse power (for current) and Arrhenius (fortemperature) relationships are combined to model theeffect that current and temperature have on lifetime. Inparticular, suppose that LED lifetime at volt and temp◦Chas a lognormal distribution with parameters µ given by

Main effects without interaction model:

µ = β0 +β1x1 +β2x2 , (22.12)

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0.001

2 3 4 5 6 7 8 9 1011 12

0.0050.010.02

0.05

0.1

0.20.30.40.50.60.70.8

0.90.95

0.980.99

Proportion failing

Time (103 h)

30V; 61°C30V; 94°C30V; 114°C40V; 77°C40V; 110°C40V; 130°C

Fig. 22.19 LED device data. Multiple lognormal probabil-ity plot depicting the regression model (22.12) (withoutinteraction) ML fit

Main effects with interaction model:

µ = β0 +β1x1 +β2x2 +β3x1x2 , (22.13)

where x1 = log(volt), x2 = 11605/(temp K), and β2 =Ea, and σ does not depend on x. Fitting the model withinteraction to the data resulted in life estimates at useconditions that were shorter than those observed at themore stressful test conditions. The engineers were surethis was wrong. It turned out that the incorrect estimateswere caused by the interaction term in the model. Theinteraction term lead to nonlinearity in the response sur-face and resulted in a saddle point in the surface relatingµ to the transformed explanatory variables. Because thismodel gave nonsense results, it will not be consideredany further.

Table 22.9 LED device subset data. ML estimates for the lognormal regression models (22.12) and (22.13)

Model Parameter ML estimate Standard error Normal-approximation95% confidence interval

Main effects β0 1.33 0.28 [0.78, 1.89]without interaction β1 −0.46 0.06 [−0.58, −0.34]

β2 0.073 0.005 [0.063, 0.083]σ 0.11 0.07 [0.09, 0.12]

Main effects β0 13.54 4.20 [5.31, 21.78]with interaction β1 −3.96 1.21 [−6.33, −1.60]

β2 −0.31 0.13 [−0.57, −0.05]β3 0.11 0.04 [0.04, 0.18]σ 0.10 0.006 [0.09, 0.12]

The maximum log likelihood values are L8 = −173.5 (main effects without interaction) and L9 = −169.5 (main effects withinteraction model)

0.00005

2 4 6 8 10 1412 16 18

0.00050.0020.005

0.020.050.1

0.20.3

0.5

0.70.80.90.950.98

0.9950.998

Proportion failing

Time (103 h)

30V; 61 °C30V; 94 °C30V; 114 °C40V; 77 °C40V; 110 °C20V; 40°C

Fig. 22.20 LED device subset data. Multiple lognormal probabil-ity plot depicting the lognormal regression model (22.12) (withoutinteraction) ML fit. The data at (40 V, 130 ◦C) were omitted

The fit of the main-effects model without interactionreveals strong lack of fit at the high-level combinationof volt = 40 and temp◦C = 130. Figure 22.19 is a prob-ability plot depicting the ML estimates of the fractionfailing as a function of time at individual conditions.Observe that the true fraction failing as a function oftime is underestimated throughout the range of data atvolt = 40 and temp◦C = 130. If the main-effects modelwere physically appropriate, it appears that it may nothold when both volt and temp are at these high levels,probably because a new failure mechanism is excited atthose extreme conditions.

Table 22.9 provides the ML results for the inverse-power Arrhenius lognormal model after deleting the

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0.000001

2 4 6 8 10 1412 16 18

0.00001

0.00050.0020.01

0.050.10.2

0.4

0.80.90.950.980.9950.999

0.00005

0.6

0.9999

Proportion failing

Time (103 h)

30V; 61 °C30V; 94 °C30V; 114 °C40V; 77 °C40V; 110 °C20V; 40°C

Fig. 22.21 LED device subset data. Multiple lognormal probabilityplot lognormal regression model (22.12) (with interaction) ML fit.The data at (40 V, 130 ◦C) were omitted

data at volt = 40 V and temp◦C = 130 ◦C. Figure 22.20shows a probability plot of the LED device subset dataalong with ML estimates of fraction failing as a functionof time for the model without interaction.

5000

3000

1000

0

50 55 60 65 70

0 1000 2000 3000 4000 5000

Old

New

500 600 700 800 900 1000

900

700

500

70

65

60

55

50

New Old

Method

Temp

Stroke

kCycles

Fig. 22.22 Spring fatigue data. Scatter-plot matrix of the data

Figure 22.20 is a lognormal probability plot of theLED device subset data along with ML estimates of frac-tion failing as a function of time. It also includes the MLestimate of the fraction failing as a function of time atthe use conditions volt = 20 V and temp◦C = 40 ◦C (therightmost line). The dashed lines represent approximate95% pointwise confidence intervals for the fraction fail-ing as a function of time at these conditions. For thismodel there is much better agreement between the modeland the data. Figure 22.21 is a similar multiple prob-ability plot using the interaction model. Comparisonof the models with and without the interaction term,after deleting the (40 V, 130 ◦C) condition, indicatessome lack of fit for the model without interaction (com-pare the log likelihoods in Table 22.9). Predictions withthe interaction model, however, are very pessimisticand much worse than had been expected by the en-gineers (though not as easily dismissed as the withthe interaction model when fitted to the complete dataset). The estimates at use conditions are highly sen-sitive to the model choice because of the rather largeamount of extrapolation involved in computing the es-timates. The engineers were uncertain as to the correctmodel, but felt that almost certainly the truth wouldlie between the two different fits shown in Figs. 22.20and 22.21.

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Example 22.15: An Accelerated Life Test for a Springwith Three Explanatory Variables. Meeker, Escobar, andZayac [22.29] describe an experiment to compare oldand new processing methods for manufacturing a spring.The experimenters used three variables, namely, strokedisplacement (Stroke), processing method (Method),and temperature (temp). Springs were compressed cycli-cally and Stroke measured the distance that a spring wascompressed. Stroke was the accelerating variable. Life-time was measured in cycles to failure. The springswere designed to be used at Stroke= 20 mils (1 mil=1/1000 inch) and the nominal processing temperaturewas temp = 600 ◦F. The goals of the experiment were tocompare the old and new processing methods, to see iftemperature had an important effect on spring life, andto check if the B10 life (0.10 quantile) at use conditionswas at least 500 000 kilocycles.

Meeker, Escobar, and Zayac [22.29] give the resultsfrom a 2 × 2 × 3 factorial with factors Stroke (50 mil,60 mil, 70 mil), temp (500 ◦F, 1000 ◦F), and Method(Old, New). Each of the 12 factor combinations wasreplicated nine times for a total of 108 observations. Allnine observations at Stroke= 60, temp = 500 ◦F, andMethod= New were right-censored at 500 000 cycles.Figure 22.22 is a scatter-plot matrix of the Spring ALTdata. It appears that on average, spring lifetimes arelonger at shorter stroke, at lower temperature, and withthe new processing method, respectively.

Figure 22.23 gives Weibull probability plots ofthe data at individual conditions (factor combinations).The plot indicates that the Weibull distribution canbe used to describe the data. Moreover, the plottedlines showing the ML estimates of fraction failing asa function of time at each test condition suggest thatit is reasonable to assume that σ (or, equivalently, theWeibull shape parameter β) is constant. The likeli-hood ratio test for homogeneity of the Weibull shapeparameters provides no evidence of differences. Thus,one can assume constant σ across experimental factorcombinations.

Table 22.10 Spring fatigue data. ML estimates for the Weibull regression model

Parameter ML estimate Standard error Normal-approximation 95% confidence interval

β0 32.03 2.48 [27.16, 36.90]

β1 −5.51 0.59 [−6.66, −4.36]

β2 −0.000 88 0.000 27 [−0.0014, −0.000 35]

β3 −1.27 0.15 [−1.56, −0.98]

σ 0.57 0.054 [0.47, 0.69]

The maximum total log likelihood for this model is L10 = −625.8

0.98

0.9

0.7

0.5

0.3

0.2

0.1

0.05

0.03

5 10 20 50 100 200 500 1000 2000 5000

Proportion failing

Thousands of cycles

50 Stroke, 500 Temp; Old Method50 Stroke, 1000 Temp; New Method50 Stroke, 1000 Temp; Old Method60 Stroke, 500 Temp; New Method60 Stroke, 500 Temp; Old Method60 Stroke, 1000 Temp; New Method60 Stroke, 1000 Temp; Old Method70 Stroke, 500 Temp; New Method70 Stroke, 500 Temp; Old Method70 Stroke, 1000 Temp; New Method70 Stroke, 1000 Temp; Old Method

Fig. 22.23 Spring fatigue data. Weibull multiple probability plotwith individual Weibull ML fits for each test condition

The model for the location parameter µ was devel-oped as follows. Previous experience with testing similarsprings suggested taking a log transformation of Stroke.On the other hand, no transformation is applied to tempbecause there were no prior studies involving temp (notethat because temp is a processing temperature ratherthan a temperature driving a chemical reaction, thereis no suggestion from physical/chemical theory that theArrhenius relationship should be used). An indicatorvariable was used to represent the effect of the qualitativefactor Method. Thus,

µ = β0 +β1 log(Stroke)+β2 temp+β3 Method(22.14)

where Method = 0 for the New processing method andMethod = 1 for the Old method. Terms representingfactor interactions can be added to (22.14). Meekeret al. [22.29] extended (22.14) to include two-factorinteraction terms. The likelihood-ratio test suggested,

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0.001

0.0030.005

0.010.020.030.05

0.1

0.20.30.50.70.90.98

101 102 103 104 105 106 107

Proportion failing

Thousands of cycles

Stroke = 20,Temp = 600,Method =Old

Stroke = 20,Temp = 600,Method =New

Fig. 22.24 Spring fatigue data. Weibull multiple probability plot ofthe Weibull regression ML fit with estimates at use conditions

however, that there was little evidence for such interac-tions.

Table 22.10 gives the results of fitting the Weibullmodel with the location parameter given in (22.14) andconstant σ . For this fit, Fig. 22.24 is a probability plotshowing the ML estimates for fraction failing as a func-tion of time for each of the 12 factor combinations andfor the Old and New processing methods at use condi-tions of Stroke = 20 mils and temp = 600 ◦F. We notethe following:

• The estimate and confidence interval for β1, the co-efficient for Stroke, are negative, which suggeststhat, with temp and Method held constant, increas-ing Stroke shortens spring life, on average. A similaranalysis shows that increasing temperature decreasesthe life of the units.• The estimate for the coefficient β3 for Method isβ3 = −1.27 < 0 and the corresponding confidenceinterval excludes 0. Thus, there is evidence thatswitching from the Old to the New method length-

Table 22.11 Spring fatigue data. Quantiles ML estimates at (20 mil, 600 ◦F) for the Old and New processing methods

Processing Quantile ML estimate Standard error Normal-approximationmethod 95% confidence interval

Old 0.10 252 208 160 148 [72 654, 875 502]0.50 737 371 483 794 [203 801, 2 667 873]0.90 1 460 875 989 321 [387 409, 5 508 788]

New 0.10 900 221 612 067 [237 471, 3 412 619]0.50 2 631 947 1 855 063 [661 194, 10 476 722]0.90 5 214 400 3 794 441 [1 252 557, 21 707 557]

ens spring life, on average. In Fig. 22.24, observe thatthe fraction-failing estimate for the New method isto the right (suggesting longer lives) of the fractionfailing for the New method. On the average, springsunder the New method last 1/ exp(β3) = 3.56 timeslonger than those under the Old method. A 95% con-fidence interval for this factor is [2.66, 4.76] whichis obtained by taking the reciprocal of the antilogof the endpoints of the confidence interval for β3 inTable 22.10.

For a graphical comparison of quantile estimatesunder the two processing methods, draw horizontallines in Fig. 22.24 that intersect the New and Oldfraction-failing lines, and read off the estimates fromthe horizontal time axis. Estimates of the 0.10, 0.50and 0.90 quantiles for the two processing methods un-der use conditions are given in Table 22.11. This tablealso provides estimates of the standard errors of quantileestimators and 95% normal-approximation confidenceintervals. Observe that quantile estimates for the Newmethod are 3.57 times larger than those for the Oldmethod.

The design engineers wanted to know if the B10life at use conditions would exceed 500 000 kilocycles.Line segments are drawn in Fig. 22.24 to indicatethe B10 estimates. The Old method fails to sat-isfy this requirement because its B10 life estimate is252 208 kilocycles. The estimate for the New methodis 900 222 kilocycles, which does satisfy the require-ment. The lower endpoint of the confidence intervalis, however, below the 500 000 target. Thus, there issome doubt that the New method meets the reliabilityrequirement.

Example 22.16: Sensitivity Analysis for the Spring Fa-tigue Data. Meeker et al. [22.29] studied the effect thatvarying model assumptions has on the analysis of thespring fatigue data. In particular, they investigated thesensitivity of the estimate of B10 life to alternativetransformations of Stroke and temp, and the assumeddistribution. A summary of their findings follows.

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Accelerated Life Test Models and Data Analysis 22.7 Other Kinds of ATs 421

Meeker et al. [22.29] extended (22.14) to

µ = β0 +β1W+β2temp+β3Method , (22.15)

where

W =⎧⎨⎩

(Stroke)λ −1

λ, λ = 0

log(Stroke), λ = 0(22.16)

and where λ was allowed to range between −1 and 2.The transformation W is known as a Box–Cox trans-formation, and the linear (no transformation, λ = 1) andlog (λ = 0) relationships are special cases of this trans-formation. Meeker et al. [22.29] used a range of fixedvalues of λ and fitted by ML methods a Weibull distri-bution with µ in (22.15) and constant σ . They observedthat, as λ moved from −1 to 2, the ML estimate ofthe B10 life at use conditions decreased by a factorof between 103 and 104. In particular, switching fromλ = 0 to λ = 1 decreased the B10 life estimate from900 221 kilocycles to 84 925 kilocycles. From past ex-perience with the spring, the engineers thought that λ

should be close to 0 and almost certainly less than 1.Thus, a conservative approach for designing the prod-

uct using the spring as a component would be to assumethat λ = 1.

Using a similar sensitivity analysis with temperature,Meeker et al. [22.29] observed that the estimate of B10life was not as sensitive to temp as it was to Stroke.Increasing λ from −1 to 2 increased the B10 life estimateby no more than a factor of 1.25. They also explain thatthe data did not provide any information about whattransformation (λ value) would be best for model fitting(because there were only two levels of temperature inthe experiment).

Another sensitivity analysis involved fitting a log-normal (instead of Weibull) distribution with a locationparameter given by (22.15) and constant σ . For any fixedvalue of λ between −1 and 2, the lognormal distribu-tion gave more optimistic estimates of B10 life thanthe Weibull distribution. In particular, the lognormal es-timates of B10 life were about twice as large as theWeibull estimates. This is expected because, when bothdistributions are fitted to the same data, the Weibulldistribution is less optimistic when extrapolating intime.

22.6 Practical Considerations for Interpreting the Analysis of ALT Data

Extrapolation is an integral part of any ALT. As a re-sult, interpretation of statistical analysis of ALT data isoften a challenge. In this section, we discuss the mostimportant issues and suggest strategies to address them.

ALTs are most effective when predicting the life forwell-understood failure mechanisms for relatively sim-ple materials or components. ALTs are less useful orthere may be more restrictions for more complicatedproducts. There are also difficulties when there are dis-crepancies between laboratory and field conditions. Inparticular, when the ALT model generates failure modesthat are different from those seen in the field, the in-formation obtained from an ALT may be difficult orimpossible to interpret.

ALTs have a better chance to produce useful resultsif they are carefully planned with attention to obtaininginformation about possible failure modes from whichthe product or material might fail.

• Information from earlier experiences with productprototypes or similar products is important. Forexample, pilot studies can provide information onwhich environmental factors and failure modes arerelevant relative to customer or field use.• Knowledge of the physical or chemical nature ofproduct degradation and failure is crucial in provid-ing a basis for the statistical model to be fitted toALT data. This could provide information on howmuch extrapolation is justified.• An obvious way to limit the effect of extrapolationis to test at lower levels of the accelerating vari-ables. Doing this, however, results in fewer observedfailures, which translates into statistical estimateswith larger amounts of uncertainty. For one-variableALTs, Meeker and Hahn [22.10] provided practicalguidelines for choosing ALT plans that meet practi-cal constraints and have good statistical efficiency.

22.7 Other Kinds of ATs

Discussions up to this point pertain to ALTs where prod-ucts are tested at high levels of the accelerating variables

and extrapolating to assess life at use levels of these vari-ables. Of primary interest is the life distribution with

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respect to a few simple failure mechanisms. This sec-tion briefly discusses other types of ATs that are used inassessing or improving product reliability. See Meekerand Escobar ([22.3], Chapt. 19) and references thereinfor more detailed discussion of these tests and otherreferences.

22.7.1 Continuous Product OperationAccelerated Tests

In this type of AT, tests are performed on the completesystem. Such tests are useful because, in addition to test-ing individual components, the tests can catch problemswith component interfaces that cannot be tested in low-level ALTs. In these tests, a sample of product is testedon a more or less continuous basis to accelerate failuremechanisms. Applications include accelerated testing ofengines (e.g., in automobiles, motorcycles, lawn equip-ment, washing machines) and refrigerator compressors.Caution is taken to avoid damage to the system thatwould not occur in actual product use (e.g., overheatingor melted parts due to extreme temperatures). In auto-mobile engine testing, there is one test standard in whichthe engine is run continuously and another in which theengine is cycled on and off continuously. This is becausethe two different ATs will tend to generate different kindsof failures.

22.7.2 Highly Accelerated Life Tests

In highly accelerated life tests HALT, products are op-erated under extremely high levels of variables suchas temperature (−90 ◦C to 190 ◦C), vibration (up to60 Grms), and other stresses directly related to systemoperation. These tests are used to identify failure modesand determine product operational limits. It is generallyagreed that these tests provide little or no informationabout product life at use conditions. This is becausethe results of the tests usually lead to product designchanges. HALTs are typically performed on solid-stateelectronics, but, with proper equipment, they can be ap-plied to other products as well. Confer et al. [22.30]remarked that HALTs can be used to inspect lots ofincoming components and to screen products by burn-in. At extremely high levels of the stressing variables,the failure-mechanism model may no longer hold orthe system experiences some damage. Thus, precautionsmust be taken to avoid these effects. See, for example,McLean [22.31] and Hobbs [22.32] for more informationon HALT methods.

22.7.3 Environmental Stress Tests

These tests are closely related to HALTs in that themain objective is to identify reliability issues quickly(e.g., product design flaws) and address them while inthe product-development stage. Different authors havedifferent names for this type of test. One is stress–life(STRIFE) tests in which one or two prototypes areaggressively tested to produce failures quickly. Mostcommon STRIFE tests involve temperature and vibra-tion that are cycled in increasing amplitudes throughoutthe test. Nelson ([22.1], pages 37–39) refers to such testsas ‘elephant tests.’ He described tests of ceramic cook-ware exposed to cycles of high and low temperatureuntil failure. Failures occurring in these tests are oftendue to design flaws in the product. Thus, this type of testallows engineers to identify design flaws and fix themwhile in the design stage. Also, the engineers must deter-mine if failures due to these flaws would occur in actualto avoid unnecessary product redesign. Nelson ([22.1],pages 38–39) describes an example where a transformerwas redesigned to eliminate a failure mode that was dis-covered in an elephant test. Years later it was realizedthat none of the transformers with the old design everfailed from this failure mode; the expensive redesign hadbeen unnecessary.

Nelson ([22.1], Chapt. 1) and Meeker and Es-cobar ([22.3], pages 519–520) give more detaileddiscussions of issues related to environmental stresstests.

22.7.4 Burn-In

Early failures (infant mortality) are the most commonproblem with electronic products. Defective compo-nents, sometimes called freaks, comprise a small fractionof all products and are caused by manufacturing defects.Burn-in involves operating the units at higher that usuallevels of accelerating variables for a specific length oftime to screen out such defects. For example, burn-in ofproducts such as integrated circuits involve testing unitsat high levels of temperature and humidity. Units thatsurvive this burn-in period are then put into service. Thistype of test is expensive because it is essentially a 100%inspection of products. Also, the longer the burn-in pe-riod, the more defective items are identified. Tradeoffsbetween these and other considerations are studied todetermine the length of the burn-in period. Jensen andPetersen [22.33] describe some of the engineering issuesrelated to burn-in. Kuo et al. [22.34] discuss statisticaland economic issues of burn-in.

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Accelerated Life Test Models and Data Analysis 22.8 Some Pitfalls of Accelerated Testing 423

22.7.5 Environmental Stress Screening

Environmental stress screening (ESS) methods weredeveloped to improve traditional burn-in tests. It hasbeen suggested that ESS is less expensive and moreeffective in detecting defectives from the productpopulation. In ESS, products are tested at milder,but more complex, stress processes than in burn-in

tests. The methodology has been developed mostlyalong engineering considerations. The goal is still toscreen out defectives without causing damage to thesystems. It is widely used in consumer, industrial,and military electronics manufacturing. Nelson ([22.1],pages 39–40) provided references regarding ESS mil-itary standards and more in-depth discussions ofESS.

22.8 Some Pitfalls of Accelerated Testing

This section reviews potential difficulties one may facein the analysis and interpretation of ALT data. SeeMeeker and Escobar [22.35] for a more detailed dis-cussion of these pitfalls.

22.8.1 Failure Behavior Changes at HighLevels of Accelerating Variables

It is possible that failure mechanisms at high levels ofthe accelerating variables may not be the same as thoseat low levels. High levels of the accelerating variablescould induce failure modes that have not been identi-fied beforehand or which may not occur in normal useof the product. Such failure modes can be direct resultsof physical changes brought on by extreme stressing(e.g., at high temperatures product parts could warp ormelt). Also, even with the same failure mechanism, theassumed relationship between life and the acceleratingvariable(s) may not be valid at higher levels. For ex-ample, in a voltage-accelerated ALT, the relationshipbetween log(Life) and log(volt) may not be linear athigher levels of volt.

Previously unrecognized failure modes could still beaccounted for in the data analysis by regarding failurescaused by them as right-censored with respect to the fail-ure mode of primary interest. This approach, however,requires the assumption of independent failure modes.Nelson ([22.1], Chapt. 7) discusses methods for ana-lyzing failure data with competing failure modes, butthe failure modes at the higher levels of the acceler-ating variables restrict the amount of information thatwill be available on the mode of interest. If possible, theinvestigator should establish limits on the acceleratingvariables within which only the primary failure modeoccurs and the assumed ALT model holds.

In some applications there will be an effort to elimi-nate a particular failure mode that is seen only at higherlevels of stress. This would be done to make it possible

to study another failure mode of primary interest moreeffectively.

22.8.2 Assessing Estimation Variability

Estimates from ALTs will contain statistical uncer-tainty due to having only a limited amount of data.It is unsafe to base product-design decisions on pointestimates alone. Instead, conclusions from data analy-sis should account for statistical uncertainty by usingconfidence intervals. Such confidence intervals providea range of plausible values for quantities of interest.All of our examples have presented such confidence in-tervals. Recall the analysis of the GAB insulation datain Example 22.2. Estimates and corresponding point-wise 95% confidence intervals for the fraction failing atvoltU = 120 are shown in Fig. 22.7. The confidence in-terval for the probability of failure before 5 × 105 h is[0.0017, 0.94]. This interval is too wide to be of anypractical value. There are ways to reduce statistical un-certainty (reduce confidence interval width). In general,one could, for instance, increase the sample size, ob-serve more failures by testing at higher stress levels, orreduce the amount of extrapolation. One could also useprior information to fix the values of model parame-ters that would normally be estimated from data, if thereis good information about such parameters. Recall theanalyses of the IC device data in Examples 22.10 and22.11. In the first analysis, all model parameters, in-cluding the activation energy Ea, were estimated andpointwise confidence intervals for the fraction failing atuse condition temp =100 ◦C are given in Fig. 22.12. Fig-ure 22.14 is a similar plot for the second analysis withEa fixed at 0.8 eV, resulting in much narrower confi-dence intervals. Of course, conclusions from an analysislike that shown in Fig. 22.14 could be misleading if Eawere to be misspecified. Meeker and Escobar ([22.3],Sect. 22.2) discuss applications of Bayesian analysis

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where prior distributions are set up to reflect informationabout model parameters.

It is important to recognize that the uncertainty re-flected in confidence intervals does not include the error(bias) that results from fitting an incorrect model (andstrictly speaking, no statistical model is exactly correct).More extrapolation will exacerbate the problem. So, itis important to do sensitivity analysis to study the ef-fect that changes in the model assumptions will have onthe results. For example, one could study how estimatesand confidence intervals change when the form of thestress-life relationship and/or distribution is changed, aswas done in several of the examples in this chapter.

22.8.3 Degradation and Failure Measuredin Different Time Scales

In certain applications, lifetime is measured in differenttime scales (e.g., cycles or hours to failure) depend-ing on which factors affect the degradation process.Often, time scales are determined by how the prod-uct is actually used. Incandescent light bulb burn timedrives a filament-evaporation process, leading to even-tual failure. One way to speed up time is by increasingvoltage. The bulb is operated continuously and lifetimeis measured in hours of operation. Also, turning a lightbulb on and off produces mechanical shocks that cancause the initiation and growth of fatigue cracks withinthe filament. Bulb life in this case might be measuredin on–off cycles. Testing bulbs continuously at highervoltages does not simulate the on–off cycling effect.Thus, predictions based on a simple model with onlyone time scale will not accurately describe light bulbsthat fail through a combination of burn time and on–offcycles.

22.8.4 Masked Failure Modes

In some situations, higher levels of the accelerating vari-able can produce a new failure mode, not ordinarily seenat use conditions. This new failure mode can mask thefailure mode of interest. If the new failure mode is not

recognized (because engineers naively think they are thesame and do not check), the resulting conclusions willbe highly optimistic relative to the truth. See Meekerand Escobar ([22.3], Fig. 19.22) for an illustration andfurther discussion.

22.8.5 Differences Between Productand Environmental Conditionsin Laboratory and Field Conditions

For accurate prediction of failure behavior in the field,it is necessary that laboratory conditions be similar tothose in the field, and that units tested in the laboratorybe similar to products in actual use. Laboratory con-ditions are not always similar to field conditions. Forexample, in a life test of a circuit pack, accelerated lev-els of temperature resulted in lower humidity levels. Thisin turn inhibited corrosion, which was the primary causeof failure in the field. Thus, there were more failures inthe field than were predicted by the data analysis. Lab-oratory tests should have controlled both temperatureand humidity levels. Also, laboratory tests are generallycarefully controlled, while in actual use, the environmentmay be highly variable.

There are various points in the process from pro-totype development to product manufacture at whichproduct-building inconsistencies could occur. Techni-cians may build prototypes to be used in testing ina manner that is different from the way an assemblyline builds the final product. For example, products withepoxy have to be cured for a certain length of time inan oven. Uncured epoxy can be highly reactive and maycause corrosion and lead to premature failure. It is possi-ble that the units are effectively cured in the lab, but notin the factory assembly line. ALT tests will likely pre-dict failure behavior that is more optimistic than that inthe field. Raw materials and instruments could also vary.It is also possible, because of identified design flaws, tomodify the product design several times throughout theprocess. Any inconsistencies in how a product is builtwill most likely cause disagreements between laboratoryand field results.

22.9 Computer Software for Analyzing ALT Data

Nelson ([22.1], pages 237–240) provides a table listingsoftware packages for analysis of ALT data by maximumlikelihood methods (which has been updated in the 2004paperback edition). The table provides the packages’

features and specific capabilities. Commercially avail-able statistical software such as SAS JMP [22.36],SAS [22.37], MINITAB [22.38], and S-PLUS [22.39]have procedures specifically for ALT data analysis.

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Accelerated Life Test Models and Data Analysis References 425

ALTA [22.40] is a software package specifically for ALTanalysis.

All of the analyses in this chapter were done withS-PLUS Life Data Analysis (SPLIDA) by Meekerand Escobar [22.41]. SPLIDA runs on top of S-PLUS. Special features for some plots were createdby modifying SPLIDA functions and using S-PLUS

annotation tools. SPLIDA consists of special S-PLUSfunctions and GUIs, and is available for downloadfrom http://www.public.iastate.edu/˜splida/. The SPL-IDA download also contains files with all of the datasets from this chapter, as well as many other exam-ple data sets, including all examples from Meeker andEscobar [22.3] and many from Nelson [22.1].

References

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circular-holed notched specimens of a carboneight-harness-satin/epoxy laminate. In: Statis-tical Research on Fatigue and Fracture, Cur.Mater. Res., Vol. 2, ed. by T. Tanaka, S. Nishijima,M. Ichikawa (Elsevier Applied Science, London 1987)pp. 159–176

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22.36 SAS Institute Inc.: JMP User’s Guide, Version 5.1.1(SAS Institute, Cary 2004) http://www.jmp.com/

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Software: Changes andEnhancements, Release 8.2 (SAS Institute, Cary2001) http://www.sas.com/statistics/

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