Stat 567:Statistical Reliability
Fifteenth Class (October 22, 1997)

Probability (PP) and
Stabilized Probability (SP) Plots

Class Objectives:

• Discuss the techniques available for selecting between two models
• Discuss the techniques available for checking goodness of fit.
• Learn to use PP and SP plots to check model adequacy

Homework Assignments:

• Use JMP's Nonlinear Fit platform to find a likelihood ratio confidence interval for the mediam of the ball bearing data.
• Use the Weibull versus Lognormal tests with ball bearing data.
• Use JMP's Nonlinear Fit platform to fit a log normal to the cord data
• Use the SAS JMP software to obtain the PP and SP plots for the cord data of Example 2.3 on Page 46 of your textbook. Hand these plots in. Use the JMP screen shots below as a guide to obtaining these plots.
• At the end of Chapter 3 in you textbook the confidence interval for S(53) of the cord data obtained by way of the PL estimator (0.54, 0.83) is compared to the confidence interval for S(53) obtained using a Weibull fit and the Delta method (0.58, 0.76). Show the calculations leading to these confidence intervals. Which interval do you prefer and why?
• For extra credit: Can you obtain a LR confidence interval for S(53) of the cord data assuming a Weibull fit?

Class Outline and Main Points:

• Likelihood ratio tests for model selection require nested models, e.g.,
  • Exponential versus Weibull
  • Exponential versus gamma
• What to do if two models are not nested, e.g. Weibull versus lognormal?
  • Use Q-Q plots if possible
  • Fit a model that includes both models as special cases
    • Generalized gamma includes, exponential, Weibull, Gamma, and log-normal as special cases
    • Mixture model
    • Hope only one model is rejected if favor of the more comprehensive model
  • Bozdogan's ICOM procedure
  • Test based on maximized likelihood for models that are not nested have low power to discriminate
  • Some estimated quantities have similar values for competing models, e.g., early quantiles.
  • A test for Weibull versus lognormal
• Determining goodness-of-fit
  • Pearson chi-square goodness-of-fit test is possibly invalid since there is no alternative hypothesis
  • Better to use graphical methods
• Plots that can be used when the model is not in a location/scale family
  • Probability (PP) plot
    • Unlike the QQ plot it does not require a location/scale family
    • A defect: extreme points have the lowest variability
  • Stabilized probability (SP) plot
    • Approximately stabilizes the variance of the plotted points
    • Justification of arcsine-square root transformation
  • Cord data example (Example 2.3, Page 46 of your textbook)
    • QQ plot
    • PP plot
    • SP plot
• Using a Weibull fit to improve the estimate of S(53) in the cord strength data of Example 2.3, Page 46 of your textbook. (Before we estimated S(53) using the P-L estimator.)

Using SAS's JMP software to construct PP and S-P plots

The are two major steps to obtaining PP and SP plots in JMP. We illustrate them with the cord data of Example 2.3, Page 46 of your textbook.

A. Obtain the following table:

B. Use the Fit Y by X option on the Analyze menu to obtain the plots.

• Plotting the columns Failure by Weibull F gives the PP plot
• Platting the columns SP X-axis by SP Y-axis gives the SP plot

Below are some points to keep in mind when using JMP to obtain the table above.

1. Enter the cord data in the first two columns of the table above. (This is the JMP file that you can download below.) Use the Kaplan-Meier platform and save the estimates show in the picture below:

2. Eliminate all columns except Time, Censor, and Failure from the resulting table.

3. Define the column Weibull F using the following formula:

(Notice that the numbers in this formula are the MLEs of the location and scale parameters of the Gumbel fit of the log data.)

4. Define the column SP X-axis using the following formula:

5. Define the column SP Y-axis using the following formula:

6. Exclude all censored times so that only failure times remain. This is necessary since the PP and SP plots use only the failure times as plotting points.

7. Obtain the PP and SP plots using the Fit Y by X platform:

Study Questions

• Why is it that in some situations one cannot use the Q-Q plot but can use PP and SP plots?
• How does the SP plot improves on the PP plot?
• Confidence intervals for survival probabilities can be calculated
  • Directly from the PL estimator and its corresponding formula for standard errors
  • By first fitting a parametric model such as the Weibull, finding the MLEs an their variance-covariance matrix, and finally using the Delta method
• What are the advantages and disadvantages of each approach relative to the other?

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