Accelerated Life

Accelerated Life

Life-Stress Models, Joint Maximum Likelihood, Acceleration Factors and the Tests

What an accelerated life test does

A product that lasts years at its use condition cannot be tested to failure in a development schedule. An accelerated life test runs units at higher stress (temperature, voltage, humidity, load) so they fail sooner, and a physical model of how life depends on the stress carries the result back to the use condition. Two things are assumed and both are checked by the analysis: the failure mechanism is the same at every level (so the shape of the life distribution does not change with stress), and the chosen life-stress relationship holds over the range from the test levels to use. The extrapolation is only as good as those two assumptions; the page therefore reports the likelihood-ratio tests and the probability plot beside the use-life numbers.

The data

One unit per line: the stress level first (temperature, or temperature and relative humidity for the two-stress models), then the time, then F for a failure or S for a suspension (a unit removed or still running unfailed at that time), and an optional count for identical lines. Suspensions are not discarded: they enter the likelihood through their survival probability, which is what makes the maximum-likelihood method preferable to plotting methods when a test ends before every unit has failed. Units that fail by a different mechanism should be entered as suspensions at their failure time. Temperatures are converted to kelvin inside the tool (°C + 273.15); humidity is a fraction, a value above 1 is read as a percentage.

Life-stress relationships

Every model is written for the life measure L(S): the Weibull scale η, the lognormal median or the exponential mean at stress S. Each is linear in ln L against a transform of the stress, which is why the life-stress plot uses that transform on its axis.

• Arrhenius (temperature, from the rate equation of a chemical process): L(T) = C exp(B/T), T in kelvin. B = E_a/k_B where E_a is the activation energy in electron-volts and k_B = 8.617333 × 10⁻⁵ eV/K, so ln L is a straight line against 1/T with slope B. Typical activation energies run from about 0.3 eV (some electromigration and intermetallic mechanisms) to over 1 eV (corrosion, some dielectric breakdowns).
• Inverse power law (voltage, mechanical load, cycling amplitude): L(V) = 1/(K Vⁿ); ln L = −ln K − n ln V, a straight line on log-log paper. The acceleration factor between two stresses is (V_A/V_u)ⁿ.
• Eyring (temperature, from the theory of rate processes): L(T) = (1/T) exp(−(A − B/T)). The extra 1/T factor in front makes it a slightly curved line on the Arrhenius plot; with the sign convention of the ReliaSoft reference, B again equals E_a/k_B.
• Temperature-humidity (the Eyring-type form with 1/RH): L(T, RH) = A exp(φ/T + b/RH), humidity as a fraction. φ = E_a/k_B.
• Peck (the humidity term as a power): L(T, RH) = A RH⁻ⁿ exp(B/T), Peck's model for moisture-driven failure of plastic-encapsulated devices (his 1986 fit: n ≈ 2.7, E_a ≈ 0.79 eV). The acceleration factor is (RH_A/RH_u)ⁿ exp[B (1/T_u − 1/T_A)].

Life distributions and the common-shape assumption

The life at a fixed stress follows a Weibull distribution F(t) = 1 − exp[−(t/η)^β], a lognormal distribution (ln t normal with mean ln t₅₀ and standard deviation σ) or an exponential distribution (constant hazard, mean m). In the accelerated model the scale (η, t₅₀ or m) is the life-stress function L(S) and the shape (β or σ) is the same at every level: the stress compresses the time axis but does not change the form of the distribution. On probability paper the levels then plot as parallel lines, and the acceleration factor AF = L(use)/L(level) is the same for every percentile. All three cases are one log-location-scale regression, ln t = ln L(S) + σ ε, with ε standard smallest-extreme-value (Weibull, σ = 1/β), standard normal (lognormal) or smallest-extreme-value with σ fixed at 1 (exponential).

Joint maximum likelihood

The likelihood is the product over all units of the density at the failure time (failures) or the survival probability at the suspension time (suspensions), each evaluated at the unit's own stress with ln L(S) in place of the location. It is maximized over the model coefficients and the common shape at once, so every level contributes to every parameter with the weight its information deserves (a level with many failures determines the slope and the shape more than a level with few). The tool starts from the separate fit of each level (a weighted regression of their locations on the stress transform), standardizes the stress columns and takes Newton steps with a line search until the gradient vanishes. The estimates are invariant to the parameterization; the natural constants (C, K, A) are recovered from the intercept.

Standard errors come from the observed information: the negative Hessian of the log-likelihood at the maximum, inverted, is the large-sample covariance matrix of the estimates (Nelson 1990 Chapter 5; Meeker and Escobar 1998 Chapter 19). Bounds on a coefficient are estimate ± K_CL s.e.; bounds on a positive quantity (C, K, A, β, σ, a life) are taken on the log scale, estimate × exp(± K_CL s.e. of its logarithm), which keeps them positive and is the usual normal-approximation (Fisher-matrix) interval of the reliability software. For a life t_p at use, ln t_p = ln L(use) + σ w_p with w_p = ln(−ln(1 − p)) (Weibull, exponential) or z_p (lognormal); its variance is g^T V g with g the gradient of ln t_p with respect to the parameters (the delta method), which includes the covariance between the slope and the intercept that an extrapolation far from the test levels depends on. The same construction through the standardized argument gives the bounds on R(t). These intervals are approximate: with fewer than about ten failures they are narrower than the truth, and a likelihood-ratio or bootstrap interval would be wider.

Acceleration factors and life at use

AF from a test level to use is exp[ln L(use) − ln L(level)], the ratio of the lives at the two conditions, and does not depend on the distribution: a unit at the test level ages AF times faster, so t hours of test stand for AF × t hours of use. The page tabulates AF for every level, the model's scale, median, B10 and B1 at every level, and at use the median, the mean (η Γ(1 + 1/β), t₅₀ exp(σ²/2) or m), B1, B10 and a chosen B-life with their bounds, and the reliability at an entered mission time. B10 is the time by which 10 % of the population has failed, the usual design life quantity; the mean is dominated by the long tail and is rarely the number to design to.

Likelihood-ratio tests

Three nested models are compared on the levels that have at least two failures (a level needs two to be fitted on its own): (1) every level with its own location and shape, (2) separate locations with one common shape, and (3) the life-stress model with the common shape. Twice the difference of the maximized log-likelihoods, −2 ln λ, is approximately chi-square with the difference in parameter count as degrees of freedom when the simpler model is true. The first comparison tests the common shape (the same failure mechanism at every level); the second tests the life-stress relation (the locations lie on the model's line); the two together reproduce the single test of the NIST handbook (its Arrhenius example gives 3.2 on 3 degrees of freedom, p = 0.36, so the Arrhenius-lognormal model with one σ is acceptable). A small p-value means the separate fits are significantly better than the model: either the shape changes with stress or the relation is wrong over the tested range, and the extrapolation to use is not supported by the data. With only as many levels as model coefficients the relation cannot be tested (it fits the level locations exactly) and only the common-shape test is reported. Levels without an own fit (no failures, or one) are left out of all three likelihoods for the test; the main fit uses them.

Reading the figures

Figure 1 is the probability plot of all levels on one sheet: the plotting positions are Bernard's median ranks of Johnson's adjusted ranks (the usual treatment of suspensions on probability paper), and the lines are the joint model's, with one common slope and the locations from the life-stress relation. Points that follow their line at every level support the model; a level whose points bend, or whose slope visibly differs, is the one the tests react to, often a second failure mode appearing at the highest stress. Figure 2 is the life-stress plot, median life on a log axis against the model's stress transform (1000/T for the temperature models, the stress itself on a log axis for the inverse power law): the own-fit medians of the levels, the fitted line and the use condition at the end of the extrapolation. The distance from the last test level to the use point shows how far the data are being stretched; the bounds on the use life grow with that distance.

Limits

Constant stress only (no step or ramp profiles, no cumulative-damage model); exact failure times (inspection-interval data should be entered at the interval midpoint or the interval end, which biases the fit slightly); one failure mechanism (units failing by another mechanism are suspensions for this one); the two-stress models have no interaction term; the exponential is a special case worth using only when the hazard is known to be constant. The default case is the three-temperature Arrhenius-lognormal example of the NIST/SEMATECH e-Handbook (Sections 8.4.2.1 and 8.4.2.2), which prints the per-cell fits, the pooled activation energy 0.863 eV, σ 0.772, the log-likelihood −476.7 and the likelihood-ratio statistic 3.2 that the page reproduces.