# 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 $\mu\,\!$ 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.