Stat 567:Statistical Reliability
Sixteenth Class (October 29, 1997)

Analysis of Accelerated Life Tests

Class Objectives:

• Learn to analyze data from accelerated life tests
• Learn about the Power Law stress function
• Learn about the Arrhenius stress function
• Learn to extrapolate to ambient conditions

Homework Assignments (Due Monday, November 10):

• Read the class notes and the web summary for this class. In particular, note that there is a great deal of material in the web summary (and its links) that complements the class lecture. Make sure that you understand how we used JMP to obtain the values of maximized log-likelihoods.
• Reproduce the last page of the Nonlinear Fit JMP platform analysis of the insulating fluid data presented in class. Calculate the variance-covariance data.
• Calculate an approximate 95% confidence interval for b10 at 26 kilovolts for the insulating fluid data. (Assume the model used in class.)
• What happens if you try to estimate the mediam life at 20 kilovolts?
• Calculate an approximate 95% confidence interval for the mediam at 30 kilovolts for the insulating fluid data. (Assume the model used in class.)
• Do a likelihood ratio test for equal sigma versus unequal sigmas similar to the one done is Fit a Regression Model but do not exclude the data at 32 Kilovolts
• Extra credit: Can you improve on the confidence intervals above by getting likelihood ratio confidence intervals?

Class Outline and Main Points:

• Accelerated Life Tests (ALT)
  • Using stress to accelerate failures
    • Advantages
      • Quick results
      • More precise estimates if model is correct
    • Disadvantages
      • Need to model the relationship of stress and life
      • Need to understand the physics of failure
      • Model is difficult to check
      • An incorrect model leads to misleading results
  • Key assumption:
    • All failure modes are affected by the accelerating stress the same way
    • ALT are most successful when there is a single or predominant failure mode
  • Precautions
    • FMA (Failure Mode Analysis) should be used to check that the failures at high stress are the same as the failures at design stress
    • Some testing should be done at design stress
• Inverse power stress function
  • Used when voltage is the accelerating stress
  • Acceleration factor
• Arrhenius Stress function
  • Used when temperature is the accelerating stress
  • Acceleration factor
  • Relationship of regression parameters to physical quantities
• Data - Accelerated life test: insulating fluid
  • The accelerating stress is voltage
  • The inverse power relation is used to model the effect of the stress
  • A zero failure code denotes a failure
  • A one failure code denotes a removal time

• Analysis of Accelerated Life Test Data:
  • See how to use JMP to Fit a Regression Model to this life test data
• Using the model
  • Delta Method
    • To obtain standard errors and confidence intervals for quantities of engineering interest one can use the Delta method. As you should recall, to use the Delta Method one needs to know the variance-covariance matrix of the model parameters. (See how to use JMP to obtain the variance-covariance matrix of the model parameters.)
  • Quantiles
    • See Pages 57-58 of your textbook to review how to use the Delta Method to obtain standard errors for Weibull and lognormal quantiles in the single sample case. There the estimated parameters are sigma and mu and the function used in conjunction with the Delta Method is:

• When we are doing Weibull or lognormal regression with a single regressor the function to use is:

where s=log(Voltage) for the insulating fluid data set. Notice that we have simply substituted mu for the model function of mu. For example, for voltage level 28 KV the value of s is log(28).

• Survival probabilities
  • Weibull regression
    • See Pages 66 and 68 of your textbook (or your class notes) to see how to use the Delta method to obtain standard errors and confidence intervals for Weibull survival probabilities in the single sample case. The modification necessary to use the Delta Method in Weibull regression should be clear from the discussion above about quantiles.
  • Lognormal regression
    • Since the normal cumulative distribution function or survival function cannot be expressed in closed form in the lognormal case using the Delta Method cannot be done using simple analytic calculations. Numerical techniques need to be used to calculated the partial derivatives associated with the Delta Method.

Study Questions

• What is the physical basis of the Arrhenius relation?
• For what kind of stress is the Arrhenius relation used for?
• Name one kind of stress for which the Inverse Power relation is used
• For the insulating fluid data, how was JMP used to obtain the maximized log-likelihood obtained for the model that assumed a separate lognormal at each voltage level?

SAS JMP files (Mac) of classroom examples and homework

• Insulating fluid accelerated life test data

References

• Accelerated Testing: Statistical Models, Test Plans, and Data Analyses. (1990) Wayne Nelson. Wiley-Interscience. (Encyclopedic discussion of the different physical models and (practical) statistical methods uses to analyze data from accelerated life tests. Don't fail to consult this reference if you are asked to analyze data from an accelerated life test.)

Acknowledgements

I like to thank Jeff Hooper and Sig Amster for first introducing me to the ideas of accelerated life tests.

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