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NIST/SEMATECH 8.4.2.2: Arrhenius-lognormal maximum likelihood fit of a three-temperature test

ToolAccelerated Life
SourceNIST/SEMATECH (2012). "Maximum likelihood (MLE) estimation." e-Handbook of Statistical Methods. Sections 8.4.2.1 and 8.4.2.2.
LevelGraduate

Problem

Components were tested for 1,000 hours at 85, 105 and 125 °C (100, 50 and 25 units) and 5, 35 and 24 failed at the recorded times. Fit a lognormal life distribution with an Arrhenius dependence on temperature by maximum likelihood, give the activation energy and the common slope with their standard errors, test the Arrhenius model against separate fits of the three cells by a likelihood-ratio test, and estimate the probability of failure after 100,000 hours at a 25 °C use temperature.

Inputs to Enter

  • Life-stress model Arrhenius, distribution lognormal, temperatures in °C (the defaults)
  • 85 °C: failures at 401, 428, 695, 725, 738 h and 95 units suspended at 1,000 h
  • 105 °C: 35 failures (171 … 926 h, the handbook's list) and 15 suspended at 1,000 h
  • 125 °C: 24 failures (24 … 696 h) and 1 suspended at 1,000 h
  • Use temperature 25 °C, time of interest 100,000 h
Open the Accelerated Life tool (loaded with this example)

Answers

  • Separate cells: ln T50 = 8.891, 6.470, 5.33 with σ = 1.21, 0.715, 0.81 and log-likelihoods −53.4, −265.2, −156.5 (sum −475.1)
  • Arrhenius MLE: ΔH = 0.863 eV (s.e. 0.0761), σ = 0.772 (s.e. of ln σ 0.0928), ln A = −19.906 (s.e. 2.3204), log-likelihood −476.7
  • Likelihood-ratio test of the Arrhenius model: −2 ln λ = 3.2 on 3 degrees of freedom (p = 0.363), the model is not rejected
  • CDF at 25 °C and 100,000 h ≈ 0.003 (handbook rounding); tool median life at 25 °C 872,994 h (scipy cross-check)
  • Handbook graphical method (8.4.2.1) for comparison: ΔH = 0.808, σ ≈ 0.74, acceleration 85 → 125 °C 13.9

Notes

The tool's default inputs are this example; press Submit to reproduce it. The handbook's quoted acceleration factor of 3,708 to 25 °C does not follow from its own ΔH (2,693 does) and is not reproduced.

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