Reliability HotWire

Issue 101, July 2009

Reliability Basics

Individual and Joint Parameter Bounds in Weibull++

Weibull++ can provide both individual and joint parameter bounds for a given data set. This article briefly explains the difference between these bounds with respect to their calculation and application. The article also provides step-by-step guidance on how to obtain these bounds using Weibull++.

Confidence Bounds in Weibull++
In Weibull++, there are several choices to obtain confidence bounds for a given data set based on the method used to calculate these bounds. For bounds on individual parameters, Weibull++ provides Fisher matrix bounds, likelihood ratio bounds, beta binomial bounds and Bayesian confidence bounds. Joint parameter bounds in Weibull++ are presented as contour plots.

Individual Parameter Bounds
Individual parameter bounds are used to evaluate uncertainty in terms of the expected (or mean) values of the parameters.

(A) Fisher Matrix Bounds
Fisher Matrix bounds are used widely in many statistical applications. These bounds are calculated using the Fisher information matrix. The inverse of the Fisher information matrix yields the variance-covariance matrix, which provides the variance of the parameter. The bounds on the parameters are then calculated using the following equations:

Lower bound = Lower bound equation

Upper bound = Upper bound equation

where:

  • is the estimate of mean value of the parameter θ.
  • Var() is the variance of the parameter.
  • α = 1 - CL, where CL is the confidence level.
  • zα/2 is the standard normal statistic.

Parameters that do not take negative values are assumed to follow the lognormal distribution and the following equations are used to obtain the confidence bounds:

Lower bound = Lower bound equation

Upper bound = Upper Bound Equation

Interested readers can find further details on the procedure in [1].

(B) Likelihood Ratio Bounds 
For data sets with very few data points, Fisher matrix bounds are not sufficiently conservative. The likelihood ratio method produces results that are more conservative and consequently more suitable in such cases. (For data sets with larger numbers of data points, there is not a significant difference in the results of these two methods.) Likelihood ratio bounds are calculated using the likelihood function as follows:

Likelihood Ratio Bounds Equation

where:

  • L(θ) is the likelihood function for the unknown parameter θ.
  • L() is the likelihood function calculated at the estimated parameter value .
  • α = 1 - CL, where CL is the confidence level.
  • Chi-Squared Statistic is the Chi-Squared statistic with k degrees of freedom, where k is the number of quantities jointly estimated.

In the calculations of the likelihood ratio bounds on individual parameters, only one degree of freedom (k = 1) is used in the Chi-Squared Statistic statistic. This is due to the fact that these calculations provide results for a single confidence region. For details, refer to [2].

(C) Beta Binomial Bounds
The beta-binomial method of confidence bounds calculation is a non-parametric approach to confidence bounds calculations that involves the use of rank tables. In Weibull++, these bounds are available only with mixed-Weibull distributions. Details on the calculation of these bounds are available in [3].

(D) Bayesian Confidence Bounds
This method of estimating confidence bounds is based on the Bayes theorem. These confidence bounds rely on a different school of thought in statistical analysis, where prior information is combined with sample data in order to make inferences about model parameters and their functions. Details on the calculation of these bounds are available in [4].

Obtaining Individual Parameter Bounds in Weibull++
For individual parameter bounds, the user can choose the confidence bound calculation method in one of two ways:

  • On the Main page of the Standard Folio Control Panel, the Settings area contains shortcuts that allow the user to cycle through available settings. The user can click the Confidence Bounds Method setting, circled in Figure 1, until the desired option is displayed.
  • On the Analysis tab of the Standard Folio Control Panel, the user can select the desired method from all available confidence bounds methods, as shown in Figure 2. Note that any methods that do not apply to the specific analysis are unavailable.

Selecting the Confidence Bounds Method on the Main Page of the Control Panel
Figure 1: Selecting the Confidence Bounds Method on the Main Page of the Control Panel

Selecting the Confidence Bounds Method on the Analysis Page of the Control Panel
Figure 2: Selecting the Confidence Bounds Method on the Analysis Page of the Control Panel

Once the data set has been analyzed, confidence bounds for individual parameters can be calculated using the Quick Calculation pad, accessed by clicking the QCP icon.

QCP Icon

For demonstration purposes, we have analyzed the times-to-failure data shown in Figure 1 using the two-parameter Weibull distribution and the likelihood ratio confidence bound method. We then access the QCP. The first step before calculating parameter bounds in the QCP is to choose the desired confidence level and the type of confidence bound on the Confidence Bounds page of the QCP, as shown in Figure 3. Here, we have chosen two-sided 90% confidence bounds.

Setting Confidence Level and Type of Confidence Bounds in the QCP
Figure 3: Setting Confidence Level and Type of Confidence Bounds in the QCP

We can then calculate the parameter bounds by clicking the Calculate Bounds button on the Parameter Bounds page, as shown next.

The Parameter Bounds Page in the QCP
Figure 4: The Parameter Bounds Page in the QCP

Joint Parameter Bounds or Contour Plots
Joint parameter bounds or contour plots are used to compare two data sets and graphically perform hypothesis testing at a specific confidence level in order to evaluate whether the two data sets are statistically different. These plots are calculated by using the joint region for the selected distribution’s parameters. In the case of the Weibull distribution, the contour plots show the joint region of the parameters Beta and Eta at a specific confidence level. The equation used to calculate the contour plots is the same as the one used to obtain likelihood ratio bounds on individual parameters. However, the calculations that go into the generation of the contour plots use the Chi-Squared Statistic statistic with two degrees of freedom (k = 2) in order to include the joint confidence region for both the parameters.

Obtaining Contour Plots in Weibull++
The user can create contour plots by clicking the Plot icon and then choosing Contour Plot in the Plot Type drop-down list on the Plot Sheet Control Panel.

Plot Icon

In the Contours Setup window that appears, we select the 90% confidence level, as shown in Figure 5. (Additional confidence levels can be plotted simultaneously, if desired.)

Contours Setup Window
Figure 5: Contours Setup Window

The resulting contour plot is shown in Figure 6. 

Contour Plot for Weibull Distribution Parameters Beta and Eta Based on Data from Figure 1
Figure 6: Contour Plot for Weibull Distribution Parameters Beta and Eta Based on Data from Figure 1

This plot shows the joint region of the parameters Beta and Eta at the 90% confidence level for this particular data set. To compare data sets, we can create a contour plot that includes both data sets in a MultiPlot (added by choosing Project > Add Additional Plot > Add MultiPlot and then selecting the data sets for inclusion). We can then evaluate the difference between the data sets based on the overlap of the contours. For more detail on this, see [5].

Conclusion
In reliability engineering, it is important to be able to obtain bounds around parameters in order to get a sense of the variability around the mean estimates of the parameters. We call these bounds individual parameter bounds. Note that the parameter bounds become narrower with more test samples and more complete data (failures rather than suspensions).

It is equally important to be able to compare two different data sets and evaluate whether they are statistically different at a specific confidence level. Contour plots provide a graphical method that can help us visually evaluate this hypothesis test. Contour plots give a joint region for the parameters of the underlying lifetime distribution.

While similar, individual and joint parameter bounds yield different results because of the difference in the degrees of freedom that are examined in the two cases. They are both powerful tools in the hands of the reliability engineer. Here we showed how to obtain these bounds, provided a discussion of their differences, and explained what answers each one can provide.

References
[1] ReliaSoft Corporation, Life Data Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2005.
[2] http://www.weibull.com/hotwire/issue18/relbasics18.htm
[3] http://www.weibull.com/hotwire/issue48/relbasics48.htm
[4] http://www.weibull.com/hotwire/issue60/relbasics60.htm
[5] http://www.weibull.com/hotwire/issue19/relbasics19.htm

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