 Reliability Basics

# Relating Distribution Parameters to Real-World Applications

For managers and engineers who are new to reliability, it can be difficult to relate the parameters of a statistical distribution to their real-life reliability requirements. Consider, for example, the fact that the eta parameter of the Weibull distribution is the time by which about 63.2% of the units in a population are expected to fail. In practice, this value may not be particularly useful. When looking at warranties, for instance, the maximum percentage of units that can fail during a warranty period is often 10% or less. For this reason, it can be easier to understand and explain reliability concepts in terms of metrics other than those that statisticians typically use. This article presents an example of constructing a contour plot for the Weibull distribution using the parameter beta and the time at which 10% of the units in the population are expected to fail (i.e., the B10 life).

Joe the reliability engineer was approached by his boss with a new problem. For years, the company management had been focused on one reliability requirement—the B10 life of their product. Joe’s boss really liked the contour plots available in Weibull++, which show the possible combinations of two parameters of a distribution that could fit a particular data set. The standard contour plot for a Weibull distribution, for example, shows the possible values of beta for each possible value of eta at one or more specified confidence levels (e.g., at a 90% confidence level, there is a 90% chance that the actual parameters of the distribution fall somewhere within or on the boundary of the contour area). Joe's boss asked if, instead of looking at a plot of the parameters for a Weibull distribution, he could see a contour plot of the shape parameter, beta, and the B10 life. Eager to please his boss, Joe read the Reliability HotWire article "Individual and Joint Parameter Bounds" (July 2009) to help him get started.

Joe knew that the first thing he needed to do was to recast the likelihood function in terms of BX life instead of the scale parameter, eta. He started with the equation for the unreliability for a Weibull distribution: where:

• Q is unreliability.
• t is time.
• beta and eta are the Weibull shape and scale parameters, respectively.

He solved the equation for eta: Since the BX life is the time by which X percent of the units in a population are expected to fail, Joe substituted X/100 for Q and BX for t. This resulted in the following expression for eta in terms of beta, X and BX: Since the data set that Joe’s boss was the most interested in consisted of specimens that had all been run to failure, Joe substituted the above expression for eta into the likelihood function for a Weibull distribution with exact times to failure. After a few lines of algebra, he obtained the following expression for the likelihood function for any percentile, X, in terms of beta, BX and time: where N is the number of exact times to failure in the data set and the ti values represent those times to failure. As a quick check, Joe noted that when X=1-1/e≈63.2% and BX=eta, the above equation reduces to the familiar equation for the likelihood function for a complete data set in terms of beta, eta and time. Since his boss specifically asked about a contour plot using B10 life, Joe substituted in X=10 to get the likelihood function in terms of beta, B10 and time. Referring again to the previous Reliability HotWire article on individual and joint parameter bounds, Joe noted that since he was creating a contour plot, he was constructing joint parameter bounds. Therefore, he used the Chi-Squared statistic with 2 degrees of freedom: where:

• L is the likelihood function given above in terms of beta, B10 and time
• is the maximum likelihood value
• is the Chi-Squared statistic with a significance level of α and 2 degrees of freedom.

Rearranging this equation, Joe obtained the expression for the likelihood value corresponding to the beta vs. B10 contour as: From this equation, the beta vs. B10 contour can be obtained by choosing specified values of beta in the likelihood function and iteratively solving for the corresponding two values of B10, or vice versa.

As a test of his method, Joe chose a simple data set with the following times to failure: 10, 20, 30, 40, 50. He entered this data set into Weibull++ and calculated parameters using MLE. He found that the LK value was -20.1840, which he converted to a maximum likelihood value of 1.715E-9 using a General Spreadsheet. (Note: For small data sets with exact times to failure, RRX is the preferred analysis method, but in this case Joe needed to use MLE in order to obtain the maximum likelihood value to generate his special contour plot.) He also viewed the resulting beta vs. eta contour plot shown in Figure 1. Figure 1 - Traditional contour plot (beta vs. eta) at a 90% confidence level

Next, Joe chose a 90% confidence level for his special contour plot, which is equivalent to a 10% significance level. This information allowed him to compute the likelihood value corresponding to the beta vs. B10 contour as: Joe substituted this value into the likelihood function in terms of B10 life to obtain the following equation, which he used to find two solutions to B10 life for different values of beta. Using the fact that beta on the beta vs. eta contour plot ranged from 0.89 to 4.55, Joe constructed the following table:

Table 1 - Solutions to the likelihood unction for various values of beta

 Beta B10 Lower B10 Upper 0.89 2.0717 2.706 0.9 2.0068 2.9814 1 2.1006 5.0724 1.5 4.3349 13.493 2 7.153 18.8639 2.5 10.0745 22.0319 3 12.9525 23.861 3.5 15.769 24.8318 4 18.6178 25.1297 4.5 22.1183 24.309 4.55 22.7773 23.897

Then, he put this information into a general spreadsheet in Weibull++ and made a scatter plot to obtain the beta vs. B10 contour plot shown in Figure 2. For more information on generating scatter plots in Weibull++, please see the Reliability HotWire article "Using the Chart Wizard to Create a Custom Plot" (February 2010). Figure 2 - Beta vs. B10 contour plot at a 90% confidence level

This contour plot allowed Joe and his boss to see how the beta parameter for the data set relates to the B10 life, which is easier to understand in "real-world" terms than eta. Now Joe can create his special contour plots to compare data sets and perform hypothesis tests in terms that his boss and his other colleagues can relate to the company reliability goals. 