Statistical Reliability: Class No. 6

Data Analysis Methods:
Right Censoring
Product-Limit Estimator

Let A1, A2, A3, and A4 be nested events with A1 being the smallest and A4 being the largest. Show that

P(A1)=P(A1|A2)P(A2|A3)P(A3|A4)P(A4)

Show that in Example 2.2 on Page 43 of your textbook the Product Limit estimator is the same as the empirical survivor function
Analyze the rat diet data supplied in class. Write a professional quality report. Attach a cover letter directed at management. The report itself should be directed at the scientists or engineers who are neither management nor statisticians but are technically oriented. The report should have:
- Summary and Conclusions
- Data Analysis
- Appendix with the raw data and other things that interfere with the smooth flow of the Data Analysis section of the report.
  (Due October 1, 1997)

Go to the bottom of this page to download the data for this homework

Hints:

Are the three groups - low fat, saturated fat, unsaturated fat, significantly different? What test are you using? If two different tests lead to different significance levels can you explain the reason?
Avoid vague language such as "they appear to be different." Instead use language such as: "there is statistical evidence (alpha=.05) to indicate that the three groups are different."
Are the mortality (hazard) rates constant, decreasing, or increasing for each group? Why can you use JMP's exponential plot to find this information out? Why is this information useful?
What distribution fits each group best? Why? Why is this information useful?
Avoid language implying causation such as: "Saturated fat causes tumors." Instead, use language such as: "In this study the x group had a significantly lower mortality rate than the y group."
Use JMP's help menu to learn about the properties of the statistical procedures you are using.

A simple probability formula for nested sets
How the P-L estimator works: Calculation of the P-L estimator from first principles.
General P-L estimator formula
Standard error of the P-L estimator at a particular time point: Greenwood's formula
Confidence intervals for probability of surviving pass a particular time point
Estimates and confidence intervals for the cumulative hazard function based on the P-L estimator
Improved plotting point for the P-L estimator
Q-Q plots based on the P-L estimator
- Weibull
- lognormal
- Exponential
Example: Strength of weathered braided cord
There is an extension of the P-L estimator due to Turnbull that can also handle left-censored and interval-censored data. (See References below.)

Formula (2.14) on Page 39 of your textbook for the standard error applies when the estimate of the survivor function is based on the empirical survivor function . Recall that the books defines the empirical survivor function only when there is no censoring or there is Type I and II right censoring.
If you have arbitrary right censoring then one uses the Product-Limit estimator to estimate the survivor function. In this case the formula for the standard error of the estimator is given by Greenwood's formula (Formula 2.21 on Page 45 of your textbook.) not Formula (2.14).

Why is it important to have a non-parametric estimate of the survivor function?
What is the relationship between the empirical survivor function and the P-L estimator of the survivor function
Why is right censoring the most common type of censoring?
What is the difference between individual and simultaneous confidence intervals?
What is the difference between Type I and Type II censoring? Why are these two types of right censoring given so much attention?
Why do we need parametric models for reliability data? Why not simply work with the empirical survivor function or the P-L estimator of the survivor function?

Statistical Methods for Survival Data Analysis (1980) Elisa T. Lee. Lifetime Learning Publications. Belmont, California. (The rat data comes from this book)
Nonparametric Estimation of a Survivorship Function with Doubly Censored Data. (1974) Turnbull, B. W. Journal of the American Statistical Society, 69, 169-73.
The Empirical Distribution Function with Arbitrary Grouped, Censored, and Truncated Data. (1976) Turnbull, B. W. Journal of the Royal Statistical Society B, 38, 618-26.

Data on the strengths of 48 pieces of weathered braided cord. (Example 2.3, Page 46 of your textbook)

(Instructions for importing a text file into JMP)