# Competing Failure Modes

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Competing Failure Modes |

This example compares the competing failure mode calculations.

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 $\mu \,\!$ and $\sigma \,\!$ are used for the Weibull distribution. They are defined by $\mu =ln(\eta )\,\!$ and $\sigma ={\frac {1}{\beta }}\,\!$. The results are:

- For failure mode s, the log-likelihood value is -101.36.
- For failure mode s, $\mu _{s}\,\!$ = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
- For failure mode s, $\sigma _{s}\,\!$ = 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, $\mu _{w}\,\!$ = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
- For failure mode w, $\sigma _{s}\,\!$ = 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 $\mu \,\!$ and $\sigma \,\!$, the results are:

- For failure mode s, $\mu _{s}=ln(\eta _{s})\,\!$ = 6.11 and its approximated 95% confidence interval are [5.27, 6.95].
- For failure mode s, $\sigma _{s}={\frac {1}{\beta _{s}}}\,\!$ = 1.49 and its approximated 95% confidence interval are [0.94, 2.36].
- For failure mode w, $\mu _{w}=ln(\eta _{w})\,\!$ = 5.83 and its approximated 95% confidence interval are [5.62, 6.04].
- For failure mode w, $\sigma _{s}={\frac {1}{\beta _{s}}}\,\!$ = 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.