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:

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

Old Mufflers

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 of Failure	Standard Error
Lower	Upper	Probability of Failure	Standard 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 Probability	Standard Error	95.0% Normal CI
Time	Survival Probability	Standard 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

New Mufflers

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 of Failure	Standard Error
Lower	Upper	Probability of Failure	Standard 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 Probability	Standard Error	95.0% Normal CI
Time	Survival Probability	Standard 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