Chapter 20 Planning Accelerated Life Tests William Q. Meeker and Luis A. Escobar Iowa State University and Louisiana State University 20 - 1 Copyright 1998-2008 W. Q. Meeker and L. A. Escobar. Based on the authors’ text Statistical Methods for Reliability Data, John Wiley & Sons Inc. 1998. December 14, 2015 8h 10min Possible Reasons for Conducting an Accelerated Test Accelerated tests (ATs) are used for different purposes. These include: • ATs designed to identify failure modes and other weaknesses in product design. • ATs for improving reliability • ATs to assess the durability of materials and components. # 20 - 3 • ATs to monitor and audit a production process to identify changes in design or process that might have a seriously negative effect on product reliability. Model and Assumptions " log(t) − µ σ • Failure-time distribution is loglocation-scale Pr(T ≤ t) = F (t; µ, σ) = Φ 11605 . temp ◦C + 273.15 • µ = µ(x) = β0 + β1x, where x= • σ does not depend on the experimental variables. • Units tested simultaneously until censoring time tc. • Observations statistically independent. 20 - 5 Planning Accelerated Life Tests Chapter 20 Objectives • Outline reasons and practical issues in planning ALTs. • Describe criteria for ALT planning. • Illustrate how to evaluate the properties of ALTs. • Describe methods of constructing and choosing among ALT plans ◮ One-variable plans. ◮ Two-variable plans. 20 - 2 • Present guidelines for developing practical ALT plans with good statistical properties. Motivation/Example Reliability Assessment of an Adhesive Bond 20 - 4 • Need: Estimate of the B10 of failure-time distribution at 50◦C (expect ≥ 10 years). • Constraints ◮ 300 test units. ◮ 6 months for testing. • 50◦C test expected to yield little relevant data. Assumed Planning Information for the Adhesive Bond Experiment The objective is finding a test plan to estimate B10 with good precision. • Weibull failure-time distribution with same shape parameter at each level of temperature σ and location scale parameter µ(x) = β0 + β1x, where x is ◦C in the Arrhenius scale. • .1% failing in 6 months at 50◦C. • 90% failing in 6 months at 120◦C. Result: Defines failure probability in 6 months at all levels of temperature. If σ is given also, defines all model parameters. 20 - 6 Temp ◦C 50 110 130 150 Failure Probability pi 0.001 0.60 1.00 1.00 60 100 100 Expected Number Failing E(ri) Engineers’ Originally Proposed Test Plan for the Adhesive Bond 100 100 100 Allocation Proportion Number πi ni 1/3 1/3 1/3 20 - 7 Critique of Engineers’ Original Proposed Plan • Arrhenius model in doubt at high temperatures (above 120◦C). • Question ability to extrapolate to 50◦C. • Data much above the B10 are of limited value. Suggestion for improvement: Simulation of Engineers’ Modified Traditional ALT Plan 60 80 90 100 Degrees C on Arrhenius scale 70 120 Precision factors R for quantile estimates at 50 Degrees C R( 0.1 quantile)= 2.288 R( 0.5 quantile)= 2.484 R(Ea)= 1.165 Levels = 80,100,120 Degrees C, n=100,100,100 Censor time=183,183,183, parameters= -16.74,0.7265,0.5999 50 Results based on 300 simulations Lines shown for 50 simulations 40 10% 20 - 11 20 - 9 • Test at lower more realistic temperatures (even if only small fraction will fail). 20 10 5 200 100 50 2000 1000 500 20000 10000 5000 50000 • Larger allocation to lower temperatures. Days 10 10 10 10 10 10 5 4 3 2 1 0 40 60 100 Degrees C 80 120 140 20 - 8 160 10% 5% 50% Adhesive Bond Engineers’ Originally Proposed Test Plan n = 300, πi = 1/3 at each 110◦C, 130◦C, 150◦C Temp ◦C 50 80 100 120 " .04 .29 .90 Failure Probability pi 4 29 90 Expected Number Failing E(ri) ∂g(θ ) ∂θ b) • Asymptotic variance of g(θ b = Avar[g(θ)] #′ θ " # ∂g(θ ) . ∂θ 20 - 10 20 - 12 • Use Monte Carlo simulation. Specific results, provides picture of data, requires much computer time. Simple to compute (with software) and general results. Σb • Properties depend on test plan, model and (unknown) parameter values. Need planning values. Assume inferences needed on a function g(θ ) (one-to-one and all the first derivatives with respect to the elements of θ exist, and are continuous). Methods of Evaluating Test Plan Properties For this plan and the Weibull-Arrhenius model, Ase[log(tb.1(50))] = .4167. The asymptotic precision factor for a 95% confidence interval of t.1(50) is R = exp(1.96 × Ase) = 2.26. Allocation Proportion Number πi ni 0 1/3 100 1/3 100 1/3 100 Engineers’ Modified Traditional ALT Plan with a Maximum Test Temperature of 120◦C Days Statistically Optimum Plan for the Adhesive Bond • Objective: Estimate B10 at 50◦C with minimum variance. • Constraint: Maximum testing temperature of 120◦C. • Inputs: Failure probabilities pU = .001 and pH = .90. 20 - 13 Adhesive Bond Weibull Distribution Statistically Optimum Plan 5 4 3 2 60 80 100 120 140 20 - 15 160 10% 5% 50% Allocations: πLow = .71 at 95◦C, πHigh = .29 at 120◦C 10 10 10 10 1 0 40 Degrees C Expected Number Failing E(ri) 38 78 1.0 0.8 0.6 0.4 0.2 0.4 0.6 ξL .2 .1 .5 1 0.8 2 1.0 20 - 14 Contour Plot Showing log10{Avar[log(tb.1)]/ min Avar[log(tb.1)]} as Function of ξL, πL to Find the Optimum ALT Plan πL 0.2 0.0 0.0 60 80 90 100 Degrees C on Arrhenius scale 70 120 Precision factors R for quantile estimates at 50 Degrees C R( 0.1 quantile)= 2.103 R( 0.5 quantile)= 2.309 R(Ea)= 1.155 Levels = 95,120 Degrees C, n=212,88 Censor time=183,183, parameters= -16.74,0.7265,0.5999 50 Results based on 300 simulations Lines shown for 50 simulations 40 10% 20 - 16 Simulation of the Weibull Distribution Statistically Optimum Plan 50000 20000 10000 5000 2000 1000 500 200 100 50 20 10 5 10 10 10 10 10 10 5 4 3 2 1 0 40 60 100 Degrees C 80 120 140 20 - 18 160 10% 5% 50% Allocations: πLow = .74 at 78◦C, πHigh = .26 at 120◦C Adhesive Bond Lognormal Distribution Statistically Optimum Plan Days Days 10 10 ni Failure Probability pi .001 .18 .90 Weibull Distribution Statistically Optimum Plan πi 213 87 Allocation Proportion Number .71 .29 20 - 17 For this plan and the Weibull-Arrhenius model, Ase[log(tb.1(50))] = .3794 50 95 120 Temp ◦C Days Temp ◦C 50 78 120 Failure Probability pi .001 .13 .90 30 69 Expected Number Failing E(ri) Lognormal Distribution Statistically Optimum Plan 233 77 Allocation Proportion Number πi ni .74 .26 20 - 19 For this plan and the Lognormal-Arrhenius model, Ase[log(tb.1(50))] = .2002 Want a Plan That 20 - 21 • Meets practical constraints and is intuitively appealing. • Is robust to deviations from assumed inputs. • Has reasonably good statistical properties. Types of Accelerated Life Test Plans • Optimum plans—Maximize statistical precision. • Traditional plans—Equal spacing and allocation; may be inefficient. • Optimized (best) compromise plans—require at least 3 levels of the accelerating variable (e.g., 20% constrained at middle) and optimize lower level and allocation. 20 - 23 Critique of the Statistically Optimum Plan • Still too much temperature extrapolation (to 50◦C). • Only two levels of temperature. • Optimum Weibull and lognormal plans quite different ◮ 95◦C and 120◦C for Weibull versus. ◮ 78◦C and 120◦C for lognormal. 20 - 20 In general, optimum plans not robust to model departures. Criteria for Test Planning Subject to constraints in time, sample size and ranges of experimental variables, • Minimize Var[log(tbp)] under the assumed model. • Maximize the determinant of the Fisher information matrix. • Minimize Var[log(tbp)] under more general or higher-order model(s) (for robustness). 20 - 22 • Control the expected number of failures at each experimental condition (since a small expected number of failures at critical experimental conditions suggests potential for a failed experiment). General Guidelines for Planning ALTs (Suggested from Optimum Plan Theory) • Choose the highest level of the accelerating variable to be as high as possible. • Lowest level of the accelerating variable can be optimized. • Allocate more units to lower levels of the accelerating variable. • Test-plan properties and optimum plans depend on unknown inputs. 20 - 24 Practical Guidelines for Compromise ALT Plans • Use three or four levels of the accelerating variable. • Limit high level of the accelerating variable to maximum reasonable condition. • Reduce lowest level of the accelerating variable (to minimize extrapolation)—subject to seeing some action. 80 100 120 140 50% 10% 5% 160 20 - 25 • Allocate more units to lower levels of the accelerating variable. • Use statistically optimum plan as a starting point. • Evaluate plans in various meaningful ways. 5 4 3 2 1 0 60 Adhesive Bond Adjusted Compromise Weibull ALT Plan πLow = .52, πMid = .20, πHigh = .28 10 10 10 10 10 10 40 Degrees C 20 - 27 156 60 84 Allocation Proportion Number πi ni .52 .20 .28 Failure Probability pi .001 .03 .24 .90 5 14 76 Expected Number Failing E(ri) Adjusted Compromise Weibull ALT Plan for the Adhesive Bond (20% Constrained Allocation at Middle) Temp ◦C 50 78 98 120 Simulation of the Adhesive Bond Compromise Weibull ALT Plan Levels = 78,98,120 Degrees C, n=155,60,84 Censor time=183,183,183, parameters= -16.74,0.7265,0.5999 80 90 100 120 Precision factors R for quantile estimates at 50 Degrees C R( 0.1 quantile)= 2.381 R( 0.5 quantile)= 2.645 R(Ea)= 1.177 70 Degrees C on Arrhenius scale 60 Results based on 300 simulations Lines shown for 50 simulations 50 10% 20 - 28 20 - 26 For this plan with the Weibull-Arrhenius model, Ase[log(tb.1(50))] = .4375. 50000 20000 10000 5000 2000 1000 500 200 100 50 20 10 5 40 Basic Issue 2: Allocation of Test Units ◮ Tends to equalize the number of failures at experimental conditions. 20 - 30 • Need to constrain a certain percentage of units to the middle level of the accelerating variable. ◮ Suggested by statistically optimum plan. • Extrapolation in time (assumed failure-time distribution). 20 - 29 • Low level of the acceleration variable—expect to extrapolate in time. • Middle and high levels of the acceleration variable—expect to interpolate in time. ◮ Testing more units near the use conditions is intuitively appealing. • Allocate more test units to low rather than high levels of the accelerating variable. Days Suggested Plan: • Extrapolation in the acceleration variable (assumed temperaturetime relationship). Need to Balance: Basic Issue 1: Choose Levels of Accelerating Variables Days • Constraints on test positions (instead of test units): Consider replacement after 100p% failures at each level of accelerating variable. Generalizations and Comments • Increases asymptotic variance of estimator of B10 at 50◦C by 33% (if assumptions are correct). • Continue tests at each level of accelerating variable until at least 100p% units have failed. Properties of Compromise ALT Plans Relative to Statistically Optimum Plans However it also, • Include some tests at the use conditions. • Need to quantify robustness. 20 - 32 • Desire to estimate reliability (instead of a quantile) at use conditions. • Fine tune with computer evaluation and/or simulation of user-suggested plans. • Reduces low test temperature to 78◦C (from 95◦C). • Uses three levels of accelerating variable, instead of two levels. • Is more robust to departures from assumptions and uncertain inputs. 20 - 31 Choosing Experimental Variable Definition to Minimize Interaction Effects • Guidance on variable definition and possible transformation of the response and the experimental models should, as much as possible, be taken from mechanistic models. • Care should be used in defining experimental variables. • Moderate increases in two accelerating variables may be safer than using a large amount of a single accelerating variable. • Proper choice can reduce the occurrence or importance of statistical interactions. ALT with Two or More Variables • There may be interest in assessing the effect of nonaccelerating variables. Size (mm) • • • 200 100 • • • • • • Volts 300 • • 500 • Comparison of Experimental Layout with Volts/mm Versus Size and Volts Versus Size 20 - 36 20 - 34 • Knowledge from mechanistic models is also useful for planning experiments. • Models without statistical interactions simplify modeling, interpretation, explanation, and experimental design. 20 - 33 • There may be interest in assessing joint effects of two more accelerating variables. Examples of Choosing Experimental Variable Definition to Minimize Interaction Effects • • • 150 Size (mm) • For humidity testing of corrosion mechanism use RH and temperature (not vapor pressure and temperature) • 100 • • 3.0 Voltage Stress (Volts/mm) 50 2.5 1.5 1.0 • For testing dielectrics, use size and volts stress (e.g., mm and volts/mm instead of mm and volts) • For light exposure, use aperture and total light energy (not aperture and exposure time) • To evaluate the adequacy of large-sample approximations with censored data, use % failing and expected number failing (not % failing and sample size). 20 - 35 3.0 2.0 2.5 2.0 1.5 1.0 • 50 100 • • • • 300 • Pr(T ≤ t) = F (t; µ, σ) = Φ • • " • • 300 • • • 200 • 100 Voltage Stress (Volts/mm) # log(t) − µ . σ Multiple Variable ALT Model and Assumptions Volts 200 • • 3.0 • 2.5 1.0 1.5 2.0 3.0 • σ does not depend on the experimental variables. • Units tested simultaneously until censoring time tc. 0.0 -6 0.2 p=.1 0.6 Standardized Factor 1 0.4 p=.01 140 0.8 180 p=.5 p=.99 160 1.0 200 p=1 1.2 Insulation ALT 3 × 3 VPM × THICK Factorial Test Plan -0.2 -6 p=3.2x10 pUU=1.8x10 120 Electrical Stress (volts per mm) 100 20 - 37 20 - 39 20 - 41 Standardized Factor 2 • Observations statistically independent. 0.4 0.35 0.3 0.25 0.2 0.15 80 -2 -1 0 1 • µ = µ(x) is a function of the accelerating (or other experimental) variables. 400 Comparison of Experimental Layout with Volts versus Size and Volts/mm versus Size • Size (mm) 2.5 2.0 1.5 1.0 • Failure-time distribution Size (mm) Thickness (cm) Insulation ALT From Chapter 6 of Nelson (1990) and Escobar and Meeker (1995) # 20 - 40 20 - 38 • Engineers needed rapid assessment of insulation life at use conditions. • 1000/10000 hours available for testing. • 170 test units available for testing. • Possible experimental variables: ◮ VPM (Volts/mm) [accelerating]. ◮ THICK (cm) [nonaccelerating]. ◮ TEMP (◦C) [accelerating]. Models Used in Examples µ = β0 + β1 log(VPM) " 11605 temp ◦C + 273.15 µ = β0 + β1 log(VPM) + β2 log(THICK) µ = β0 + β1 log(VPM) + β2 σ constant. The ALT Design Problem • Design test plan to estimate life at the use conditions of VPMU = 80 volts/mm, THICKU = 0.266 cm, TEMPU = 120 ◦C. • Interest centers on a quantile h i in lower tail of life distribution, tp = exp µ(xU ) + Φ−1(p)σ . • Need to choose levels of the accelerating variable(s) x1, . . . , xk and allocations π1, · · · , πk to those conditions. Equal allocation can be a poor choice. 20 - 42 Multi-Variable Experimental Region • Maximum levels for all variables: VPMH = 200 volts/mm THICKH = 0.355 cm TEMPH = 260 ◦C. • Explicit minimum levels for all experimental variables: VPMA = 80volts/mm THICKA = 0.163cm TEMPA = 120◦C (also stricter implicit limits for VPM and TEMP). 20 - 43 • May need to restrict highest combinations of accelerating variables; e.g., constrain by equal failure-probability line (by using a maximum failure probability constraint p∗ or equivalently a standardized censored failure time ζ ∗ constraint). Degenerate and Nondegenerate Test Plans to Estimate tp Degenerate plans: • Test all units at xU . • Test two (or more) combinations of the experimental variables on a line with slope s passing through xU . Nondegenerate practical plans: 20 - 45 • Test at three (or more) noncollinear combinations of the experimental variables in the plane. Splitting Degenerate Plans • It is possible to split a degenerate plan into a nondegenerate optimum test plan (maintaining optimum Var[log(tbp)]). • Use secondary criteria to chose best split plan. • Split xi = (x1i, x2i)′ with allocation πi into xi1 = (x1i1, x2i1)′ and xi2 = (x1i2, x2i2)′ with allocations πi1 and πi2 (where πi1 + πi2 = πi) 0.4 0.35 0.3 0.25 0.2 0.15 0.15 0.2 0.25 0.3 0.35 0.4 -0.2 0.0 -6 0.2 0.4 0.6 140 0.8 160 p=.33 Standardized Factor 1 120 Electrical Stress (volts per mm) 100 180 1.0 200 p=1 1.2 Insulation ALT VPM × THICK Optimum Test Plan pUU=1.8x10 -0.2 pUU=1.8x10 0.0 -6 80 0.2 0.4 0.6 140 0.8 160 p=.28 Standardized Factor 1 120 Electrical Stress (volts per mm) 100 180 p=1 200 1.0 1.2 Insulation ALT VPM × THICK Optimum Test Plan with p∗/ζ ∗ constraint 20 - 44 20 - 46 • Optimum degenerate plan corresponds to a single-variable optimum. • Some units tested at highest levels of accelerating variables. • Otherwise there is at least one optimum degenerate test plan in the x1 × x2 plane. • When acceleration does not help sufficiently, it is optimum to test all units at the use conditions. Optimum Degenerate Plan: Technical Results 80 1 0 -1 -2 Standardized Factor 2 20 - 48 Standardized Factor 2 1 0 -1 -2 πi1xi1 + πi2xi2 = πixi. • Can introduce a p∗ constraint [or a ζ ∗ constraint where p∗ = Φ(ζ ∗)]. 20 - 47 • Can also split compromise plans and maintain Var[log(tbp)]. Thickness (cm) Thickness (cm) 0.0 -6 pUU=1.8x10 80 140 160 p=.33 Standardized Factor 1 0.5 120 180 p=.99 Electrical Stress (volts per mm) 100 0.2 0.4 p=.5 p=.99 0.6 Standardized Factor 1 p=.1 0.8 160 1.0 200 220 180 1.0 200 p=1 p=1 Insulation ALT VPM × TEMP 3 × 3 Factorial Test Plan 0.0 p=.01 140 1.2 Insulation ALT VPM × THICK 20% Compromise Test Plan with p∗/ζ ∗ constraint 0.4 0.3 0.35 0.25 -0.2 p=.00016 120 Electrical Stress (volts per mm) 100 0.2 0.4 0.6 Standardized Factor 1 0.8 1.0 200 p=1 1.2 20 - 49 20 - 51 Plan |F | Interaction Model 0.0 ∞ ∞ 0.0 0.0 0.0 1.2 × 10−5 80.1 7.3 × 10−4 ∞ 1.7 × 10−5 145 V[log(b tp )] Comparison of Test Plans and Properties for the VPM×THICK ALT No Interaction Model |F | 80.1 0.0 138 2.4 × 10−3 Optimum degenerate No ζ ∗ 131 1.6 × 10−3 0.0 144 Optimum split No ζ ∗ 131 9710 V[log(b tp )] Optimum degenerate ζ ∗ = 2.5454 0.0 1.2 × 10−4 3×3 Factorial from Nelson (1990) Optimum split ζ ∗ = 2.5454 96.1 102 0.2 0.4 0.6 Standardized Factor 1 0.8 160 1.0 200 p=1 20 - 50 7.0 × 10−3 0.0 p=.39 140 180 Insulation ALT VPM × TEMP Optimum Test Plan 96.1 1.2 20% Compromise degenerate ζ ∗ = 4.04 -0.2 20% Compromise split ζ ∗ = 4.04 260 240 220 200 180 160 120 Electrical Stress (volts per mm) 140 -06 pUU=1.8x10 80 |F | Interaction Model V[log(b tp )] Comparison of Test Plan Properties for the VPM×TEMP ALT |F | No Interaction Model 20 - 52 50.5 1.3 × 10−3 0.0 1613 ∞ ∞ 3.0 × 10−6 V[log(b tp )] 50.5 0.0 430 0.0 2.7 × 10−6 Optimum degenerate No ζ ∗ 54.7 2.0 × 10−3 5768 349 Optimum split No ζ ∗ 54.7 0.0 324 1.7 × 10−3 20% Compromise degenerate No ζ ∗ 77.7 1.2 × 10−3 77.3 20% Compromise split No ζ ∗ 77.7 0.0 20% Compromise degenerate ζ ∗ = 5.0 20 - 54 1.7 × 10−6 0.0 0.0 20% Compromise split ζ ∗ = 5.0 3×3 Factorial Adapted from Nelson (1990) Plan 120 100 1.2 0.2 0.15 260 240 220 200 180 -6 pUU=1.8x10 80 0.0 p=1 160 Standardized Factor 2 1.0 0.8 0.6 0.2 0.4 0.0 160 140 120 -0.2 p=.36 140 180 Insulation ALT VPM × TEMP 20% Compromise Test Plan with p∗/ζ ∗ constraint 260 240 220 200 180 160 120 Electrical Stress (volts per mm) 140 -06 pUU=1.8x10 100 120 80 20 - 53 Standardized Factor 2 -0.2 1 0 1.2 1.0 0.8 0.6 0.2 0.4 0.0 -0.2 Temperature (degrees C) Standardized Factor 2 Standardized Factor 2 -1 -2 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 Thickness (cm) Temperature (degrees C) Temperature (degrees C) Extensions of Results to Other Problems • With one accelerating and several other regular experimental variables, replicate single-variable ALT at each combination of the regular experimental variables. • Can use a fractional factorial for the regular experimental variables. • If the approximate effect of a regular experimental variable is known, can tilt factorial to improve precision. • With two or more accelerating variables, our results show how to tilt the traditional factorial plans to restrict extrapolation and maintain statistical efficiency. 20 - 55