Stat 567:Statistical Reliability
Twentieth Class (November 19, 1997)

Proportional Hazard Model and
Proportional Odds Model

Class Objectives:

• Learn the definition and properties of the proportional hazard model
• Learn the definition and properties of the proportional odds model
• Preview where we are going in reliability data regression
• Learn how to use JMP to fit the proportional hazards model

Homework Assignments:

• Reproduce the last JMP output of the links from this class summary

Class Outline and Main Points:

Proportional Hazards Model

• Definition of the proportional hazards model
  • In terms of hazards functions
  • Origins of the name
  • Baseline hazard
  • Interpretation
• A result useful for diagnostics plots
  • The proportional hazards model is invariant under monotone functions of time
• Survivor function formulation of the proportional hazards model
• Risk ratios
  • Definition
  • Interpretation
• Theorem:
  • A model that is both an accelerated life model and a proportional hazards model must have a Weibull baseline distribution function. The converse is also true.
    • Derivation
  • When the baseline is Weibull both characterization of the model are useful
    • Diagnostic plots developed for either model can be used
• Diagnostic plots for grouped data
  • Plots of the log integrated hazards versus time at the different stresses should be vertically shifted copies of each other, and the horizontal scale can be transformed with a monotone transfomation, e.g., log t, for a tidier plot
    • Weibull baseline leads to parallel Weibull plots
• Fitting the proportional hazards model using SAS's JMP.

Proportional Odds Model

• Definition of proportional odds model
  • In terms of survivor functions
  • Interpretation
• Relationships between ratios of hazard functions and ratios of survivor functions
  • As time goes from zero to infinity there is a diminishing influence of the stress on the hazard function
    • Maybe the stress only affects some items in the populations and as these items fail the remaining components are unaffected by the stress
    • Importance on understanding the physical assumptions that a model implicitly makes
• A result useful for diagnostics plots
  • The proportional odds model is invariant under monotone functions of time
• Diagnostics plots for grouped data
  • Plots at the different stresses of the log (distribution/survival) versus time should be vertically shifted copies of each other, and the horizontal scale can be transformed, e.g., log t, for a tidier plot.

(Brief Discussion of Other Life Data Regression Models)

Study Questions

• Can you think of situations where each of the following models should apply?
  • Accelerated life
  • Proportional hazards
  • Proportional odds
• Why do we frequently start with a Weibull baseline when doing reliability data regression?
• Suppose that we find that the Weibull plots at the different stresses are not parallel. Can the data satisfy a proportional hazards model?
• If a model is both accelerated life and proportional hazards, what is its baseline distribution?

SAS JMP files (Mac) of classroom examples and homework

• Data on the failure stresses of single carbon fiber. (Table 4.1, Page 81 of your textbook)

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