Specifying Confidence Values
The results from Weibull analysis are estimates based on the observed lifetimes of a very small sample. Because the sample size is generally very limited, uncertainty about the results exist. Thus, the degree of confidence, which is a measure of statistical precision, can be used to gauge the accuracy of the resulting analysis. Specified prior to looking at the data points and performing the Weibull analysis, the degree of confidence is a percentage value that is entered. The higher the percentage value, the higher the desired confidence of the results.
A confidence interval is used to show the range within which the true analysis value is expected to fall a certain percentage of the time (the degree of confidence). The confidence interval quantifies the uncertainty due to sampling error by expressing the confidence that a specific interval contains the quantity of interest. Whether a specific interval actually contains the quantity of interest, however, is unknown.
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Assurance refers to when the value entered for the degree of confidence is equal to the reliability.
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Confidence intervals can have either one or two bounds. The type of confidence bound selected is dependent upon the application. One-sided bounds are used to indicate that the quantity of interest is above the lower bound or below the upper bound with a specific confidence. A one-sided lower bound is used when predicting reliability. A one-sided upper bound is used for predicting the percentage of components failing under warranty.
Two-sided bounds are used to indicate that the quantity of interest is contained within the bounds with a specific confidence. Two-sided bounds are used for predicting the parameters of a distribution. Confidence interval calculations can be used on all distributions and parameter estimation methods. To find the confidence interval, the confidence method, the type of confidence interval and the degree of confidence must all be assigned. Table 7-7 describes the confidence methods available in most Weibull software.
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Confidence Method
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Description
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Modified Fisher Matrix
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Produces almost instantaneous results with reasonable accuracy when 10 or more failures are included in the sample. This method assumes B-lives for input percentages are normally distributed and produces a full plot (extrapolated bounds). The Modified Fisher Matrix method is considered the best confidence method when rank regression is the selected parameter estimation method for larger samples with few suspensions. The Modified Fisher Matrix method has various versions that include:
• Gumbel Truncated–The original (unmodified) Fisher Matrix method uses some of the Gumbel terms but does not use all of the second-partial derivative terms. It also has significant small sample bias. Although this has no effect when MLE is the estimation method, differences in solution parameters are significant when rank regression is the estimation method.
• Weibull Full–This Fisher Matrix method is significantly biased for rank regression and small samples. It uses all second-partial derivative terms.
• Gumbel Full–This Fisher Matrix method is based upon all of the Gumbel terms and is less biased for smaller samples. Consequently, it is considered the standard Fisher Matrix method.
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Likelihood Ratio
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When MLE or MMLE is the selected parameter estimation method, the likelihood ratio can be used to compare designs for significant differences, compensating for the small sample bias so often found in life data. The likelihood ratio method produces a full plot (extrapolated bounds) and provides the amount of differences between two data sets. In addition to comparing a new design to an old design, the likelihood ratio method can be used to compare supplier A against supplier B, application C against application D, etc.. It is accurate when 30 or more failures are included in the sample and is the best practice for data with suspensions. However, this method takes significant computer time and the results are almost identical to the Fisher Matrix method, which are calculated almost instantaneously.
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Beta-Binomial
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This method is evaluated at each occurrence point and is best used for determining bounds for probit analysis. Although beta-binomial bounds give more conservative results, they require more calculation time than the Fisher Matrix method.
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Monte Carlo
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This method is a special technique for simulation based upon the pivotal statistic method. Made possible only by today’s fast computers, Monte Carlo simulation is used as a prediction tool and can provide a reference for analytical techniques. When used for generating confidences, Monte Carlo simulation generates random data samples to add to existing data sets with very few data points so that more accurate correlation p-values, confidence limits for B-lives and parameters can be generated. Producing generally conservative results, the Monte Carlo method is considered the best practice for confidence estimation for distributions without exact derivations. Because Monte Carlo simulations are performed for each confidence point, this method requires a great deal of calculation time. Unless the confidence seed value is kept the same, recalculating for the same conditions produces slightly different results each time, giving an indication of actual variability. Monte Carlo simulation is the recommended method for generating confidence intervals for data sets with 10 or fewer data points or for data sets with random suspensions.
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Greenwood’s Variance
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This method is best used for determining bounds for Kaplan-Meier models, which are described in Related Quantitative Models.
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Although increasing the sample size can reduce uncertainty, testing more units to failure can be very costly and even impossible in cases that risk safety. A more cost-effective method of reducing sample uncertainty is to employ prior experience with the subject failure mode. If a Weibull library has been built, the Weibull probability plots can be reviewed for the failure modes of the current design prior to starting a new design. In addition to probability plots, the ideal Weibull library contains failure analysis and corrective analysis reports from a FRACAS (Failure Reporting, Analysis and Corrective Action System), root cause analyses, statements indicating how designs or processes could be changed to avoid a failure mode in the future, materials laboratory analyses, failure modes and effects analyses (FMEAs), fault tree analyses and all other related reports. The WeiBayes distribution, which requires only one parameter, can then use an entered slope value based on engineering experience and the Weibull probability plots from earlier designs. For small samples, defining the slope for the WeiBayes distribution can reduce uncertainty by factors of two or three.