Stat 567:Statistical Reliability
Twenty First Class (November 24, 1996)

Weibull Regression/
Model Fitting Guided by Physical Theory

Class Objectives:

• Learn to use JMP's Nonlinear fit platform with the types of nonlinear models that arise from physical theory.

Homework Assignments:

• Use JMP to analyze the data on the failure stresses of fiber bundles found in Table 4.2, Page 82 of your textbook. Follow the analysis done in class for the single fibers. Write enough so that I can understand your analysis
• Use JMP to analyze the data on the failure stresses of Tungsten Carbide Alloy found in Table 4.3, Page 88 of your textbook. Write enough so that I can understand your analysis

Class Outline and Main Points:

• Weibull Regression
• Review: JMP's Nonlinear Fit platform
  • Object function for Weibull distribution MLE
    • Loss function
    • Model function when the location parameter of log data is a linear function of load - Weibull Regression case
• Reducing the dimension of MLE numerical optimization
  • Setting selected partial derivatives of the log-likelihood equal to zero
  • Finding relationships among some of the MLEs of the parameters
• Models of failure stresses of fibers as a function of length based on a weak-link assumption and generalizations
  • Model 0: Weak-link model with Weibull baseline distribution
  • Model 1: As Model 0, but failure stresses decrease with length at a slower rate than predicted by the weak-link model
  • Model 2: As Model 1, but in addition assume that the Weibull shape parameter may depend on fiber length
  • Model 3: Unrelated Weibull distributions are assumed at each fiber length
• Models 0 and 3 as natural reference models for judging model fit
• JMP's Nonlinear Fit platform
  • Loss function for Model 0
  • Loss function for Model 1
  • Loss function for Model 2
• Finding good initial estimates for JMP's Nonlinear Fit platform
• Using JMP's Nonlinear Fit platform for Model 3
  • Additivity of maximized log-likelihoods (-SSE)
• Likelihood ratio tests based on the SSE Nonlinear Fit platform output
  • Use of the platform's option of restricting parameter values

Study Questions

• Why is it good to have reference models when testing model adequacy?
• Why is SSE the negative of the log likelihood in ML estimation with JMP's Nonlinear Fit platform?

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