Example of Nonparametric Distribution Analysis (Arbitrary Censoring)

A reliability engineer wants to assess the reliability of a new type of muffler and to estimate the proportion of warranty claims that can be expected with a 50,000-mile warranty. The engineer collects failure data on both the old type and the new type of mufflers. Mufflers were inspected for failure every 10,000 miles.

The engineer records the number of failures for each 10,000-mile interval. Therefore, the data are arbitrarily censored. The engineer uses Nonparametric Distribution Analysis (Arbitrary Censoring) to determine the probability of failure for various mileage intervals, and to estimate the percentage of mufflers that will survive until at least 50,000 miles. The engineer also wants to validate corresponding results that were obtained using a parametric analysis:

  1. Open the sample data, MufflerReliability.MTW.
  2. Choose Stat > Reliability/Survival > Distribution Analysis (Arbitrary Censoring) > Nonparametric Distribution Analysis.
  3. In Start variables, enter StartOld StartNew.
  4. In End variables, enter EndOld EndNew.
  5. In Frequency columns (optional), enter FreqOld FreqNew.
  6. Click OK.

Interpret the results

Using the Turnbull Estimates table, the engineer can determine the probability of failure at various mileage intervals. For the old type of mufflers, approximately 19.3% of the mufflers are expected to fail between 50,000 and 60,000 miles. For the new type of mufflers, approximately 10.3% are expected to fail between 50,000 and 60,000 miles.

The engineer can also determine what proportion of the mufflers are expected to survive at least 50,000 miles. For the old mufflers, the probability of surviving past 50,000 miles is approximately 75.3%. For the new mufflers, the probability of surviving past 50,000 miles is approximately 95.4%. These probabilities are consistent with the results that the engineer obtained using a parametric analysis with a Weibull distribution.

Distribution Analysis, Start = StartOld and End = EndOld

Variable Start: StartOld End: EndOld Frequency: FreqOld
Censoring Censoring Information Count Right censored value 83 Interval censored value 965 Left censored value 1
Turnbull Estimates Interval Probability Standard Lower Upper of Failure Error * 10000 0.000953 0.0009528 10000 20000 0.005720 0.0023284 20000 30000 0.026692 0.0049766 30000 40000 0.075310 0.0081477 40000 50000 0.138227 0.0106563 50000 60000 0.192564 0.0121746 60000 70000 0.228789 0.0129693 70000 80000 0.135367 0.0105629 80000 90000 0.117255 0.0099333 90000 * 0.079123 *
Table of Survival Probabilities Survival Standard 95.0% Normal CI Time Probability Error Lower Upper 10000 0.999047 0.0009528 0.997179 1.00000 20000 0.993327 0.0025137 0.988400 0.99825 30000 0.966635 0.0055448 0.955767 0.97750 40000 0.891325 0.0096094 0.872491 0.91016 50000 0.753098 0.0133137 0.727004 0.77919 60000 0.560534 0.0153241 0.530499 0.59057 70000 0.331745 0.0145374 0.303252 0.36024 80000 0.196378 0.0122655 0.172338 0.22042 90000 0.079123 0.0083342 0.062788 0.09546

Distribution Analysis, Start = StartNew and End = EndNew

Variable Start: StartNew End: EndNew Frequency: FreqNew * NOTE * 8 cases were used * NOTE * 2 cases contained missing values or was a case with zero frequency.
Censoring Censoring Information Count Right censored value 210 Interval censored value 839
Turnbull Estimates Interval Probability Standard Lower Upper of Failure Error 20000 30000 0.002860 0.0016488 30000 40000 0.010486 0.0031451 40000 50000 0.032412 0.0054678 50000 60000 0.102955 0.0093830 60000 70000 0.170639 0.0116151 70000 80000 0.248808 0.0133481 80000 90000 0.231649 0.0130259 90000 * 0.200191 *
Table of Survival Probabilities Survival Standard 95.0% Normal CI Time Probability Error Lower Upper 30000 0.997140 0.0016488 0.993909 1.00000 40000 0.986654 0.0035430 0.979710 0.99360 50000 0.954242 0.0064517 0.941597 0.96689 60000 0.851287 0.0109856 0.829756 0.87282 70000 0.680648 0.0143949 0.652435 0.70886 80000 0.431840 0.0152936 0.401865 0.46181 90000 0.200191 0.0123546 0.175976 0.22441
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