# Competing Failure Modes

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Competing Failure Modes |

This example validates the competing failure mode calculations in Weibull++ standard folios.

Reference Case

The data set is from Table 15.1 on page 383 in the book *Statistical Methods for Reliability Data* by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.

Data

State F/S | Time to F/S | Failure Mode |
---|---|---|

F | 275 | w |

F | 13 | s |

F | 147 | w |

F | 23 | s |

F | 181 | w |

F | 30 | s |

F | 65 | s |

F | 10 | s |

S | 300 | |

F | 173 | s |

F | 106 | s |

S | 300 | |

S | 300 | |

F | 212 | w |

S | 300 | |

S | 300 | |

S | 300 | |

F | 2 | s |

F | 261 | s |

F | 293 | w |

F | 88 | s |

F | 247 | s |

F | 28 | s |

F | 143 | s |

S | 300 | |

F | 23 | s |

S | 300 | |

F | 80 | s |

F | 245 | w |

F | 266 | w |

Result

In the book, parameters [math]\mu\,\![/math] and [math]\sigma\,\![/math] are used for the Weibull distribution. They are defined by [math]\mu = ln(\eta)\,\![/math] and [math]\sigma = \frac{1}{\beta}\,\![/math]. The results are:

- For failure mode s, the log-likelihood value is -101.36.
- For failure mode s, [math]\mu_{s}\,\![/math] = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
- For failure mode s, [math]\sigma_{s}\,\![/math] = 1.49 and its approximated 95% confidence interval are [0.94, 2.36].
- For failure mode w, the log-likelihood value is -47.16.
- For failure mode w, [math]\mu_{w}\,\![/math] = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
- For failure mode w, [math]\sigma_{s}\,\![/math] = 0.23 and its approximated 95% confidence interval are [0.12, 0.44].

Results in Weibull++

- The following picture shows the ML estimates and the variance covariance matrix for each failure mode.

- The following picture shows the 95% confidence intervals for the parameters of each failure mode.

- In terms of [math]\mu\,\![/math] and [math]\sigma\,\![/math], the results are:

- For failure mode s, [math]\mu_{s} = ln(\eta_{s})\,\![/math] = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
- For failure mode s, [math]\sigma_{s} = \frac{1}{\beta_{s}}\,\![/math] = 1.49 and its approximated 95% confidence interval are [0.94, 2.36].
- For failure mode w, [math]\mu_{w} = ln(\eta_{w})\,\![/math] = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
- For failure mode w, [math]\sigma_{s} = \frac{1}{\beta_{s}}\,\![/math] = 0.23 and its approximated 95% confidence interval are [0.12, 0.44].

The above results are exactly the same as the results in the book.