Here’s an overview of a distribution-free approach commonly called the Kaplan-Meier (K-M) Product Limit Reliability Estimator.

There are no assumptions about underlying distributions. And, K-M works with datasets with or without censored data. We do need to know when failures or losses (items removed from the evaluation or test other than as a failure. Censored items).

K-M provides an estimate for the reliability function or CDF. K-M is a non-parametric method. It is conservative and not defined beyond the last point of data, failure or loss.

## Calculation K-M estimates

Use the following steps to calculate K-M estimates:

- Order the actual failure times from t
_{1}through t_{r}, where there are r failures. - With each t
_{i}, pair with the number n_{i}, with n_{i}equal to the number of operating units just before the i-th failure occurred at time t_{i}. - Estimate R( t
_{1}) by (n_{1}– 1) / n_{1} - Estimate R(t
_{i}) by R( t_{i – 1}) × (n_{i}– 1) / n_{i} - Estimate the CDF F(t
_{i}) by 1 – R( t_{i})

Note: For censored units only count them up to the last actual failure time before the were removed.

## A Simple Example

Let’s say we have 20 units on test for 200 hours. 6 have failed at times 10, 32, 56, 98, 122, and 181 hours. Plus four units were removed for other experiments at 50, 100, 125, and 150 hours. The remaining 10 units ran until the test ended at 200 hours.

**Step 1.** Order the times to failure.

t_{i} | Failure time |
---|---|

1 | 10 |

2 | 32 |

3 | 56 |

4 | 98 |

5 | 122 |

6 | 181 |

**Step 2.** Count surviving units just prior to failure at t_{i}

Remember to remove censored units (losses) at next failure time.

t_{i} | Failure time | n_{i} |
---|---|---|

1 | 10 | 20 |

2 | 32 | 19 |

3 | 56 | 17 |

4 | 98 | 16 |

5 | 122 | 14 |

6 | 181 | 11 |

**Step 3.** Estimate R(t_{1})

$$ \large\displaystyle R\left( {{t}_{1}} \right)=\frac{{{n}_{1}}-1}{{{n}_{1}}}=\frac{19}{20}=0.95$$

**Step 4.** Estimate each R(t_{i})

Where

$$ \large\displaystyle R\left( {{t}_{i}} \right)=R\left( {{t}_{i-1}} \right)\times \frac{{{n}_{i}}-1}{{{n}_{i}}}$$

t_{i} | Failure time | n_{i} | R(t_{i}) |
---|---|---|---|

1 | 10 | 20 | 0.95 |

2 | 32 | 19 | 0.90 |

3 | 56 | 17 | 0.847 |

4 | 98 | 16 | 0.794 |

5 | 122 | 14 | 0.737 |

6 | 181 | 11 | 0.670 |

Beyond 181 hours the reliability, R, is 0.607.

**Step 5.** Estimate CDF if desired

Recall F(t_{i}) = 1 – R (t_{i})

Related:

Censored Data and CDF Plotting Points (article)

Confidence Limits (article)

Kruskal-Wallis Test (article)

luckynp85 says

Hi Fred,

One doubt.

For 2nd failure, R(t2) = R(t2-1)*(n2-1)/n2 = 0.95*18/19 = 0.9 and not 0.947?

Am I missing something?

Regards,

Laxman

Fred Schenkelberg says

good catch and I’ve updated the calculations

Cheers,

Fred