Reliability HotWire

Issue 42, August 2004

Reliability Basics

Likelihood Ratio Confidence Bounds

In Weibull++, there are several methods available for calculating confidence bounds: Fisher Matrix, beta binomial and likelihood ratio. In this month's issue of the Hotwire, we will present the basic theory behind likelihood ratio (LR) confidence bounds. Conceptually, this method is a great deal simpler than that of the Fisher Matrix; however, this does not mean that the results are of any less value. In fact, the LR method is often preferred over the FM method in situations where there are smaller sample sizes.

Likelihood ratio confidence bounds are based on the equation:

 (1)

where:

• L() is the likelihood function for unknown parameter vector
• L() is the likelihood function calculated at the estimated vector
• is the chi-squared statistic with probability and k degrees of freedom, where k is the number of quantities jointly estimated

If is the confidence level, then = for 2-sided confidence bounds and = (2 - 1) for 1-sided confidence bounds. If X is a continuous random variable with pdf, then:

where are k unknown constant parameters that need to be estimated. Conduct an experiment and obtain R independent observations, , which correspond in the case of life data analysis to failure times. The likelihood function is given by:

The maximum likelihood estimators (MLE) of are obtained by maximizing L. These are represented by the L() term in the denominator of the ratio in Eqn. (1). Since the values of the data points are known and the values of the parameter estimates have been calculated using MLE methods, the only unknown term in Eqn. (1) is the L() term in the numerator of the ratio. It remains to find the values of the unknown parameter vector that satisfy Eqn. (1). For distributions that have two parameters, the values of these two parameters can be varied in order to satisfy Eqn. (1). The values of the parameters that satisfy this equation will change based on the desired confidence level , but at a given value of there is only a certain region of values for and for which Eqn. (1) holds true. This region can be represented graphically as a contour plot, which is given in the following example figure.

The region of the contour plot essentially represents a cross-section of the likelihood function surface that satisfies the conditions of Eqn. (1).

Note on Contour Plots in Weibull++

Contour plots can be used for comparing data sets. Consider two data sets (e.g. old and new design) where the engineer would like to determine if the two designs are significantly different and at what confidence. By plotting the contour plots of each data set in a multiple plot (the same distribution must be fitted to each data set), one can determine the confidence at which the two sets are significantly different. If, for example, there is no overlap (i.e. the two plots do not intersect) between the two 90% contours, then the two data sets are significantly different with a 90% confidence. If there is an overlap between the two 95% contours, then the two designs are NOT significantly different at the 95% confidence level. An example of non-intersecting contours is shown next. The Additional Reliability Analysis Tools chapter in the Life Data Analysis online reference discusses comparing data sets.

Additional information on Likelihood Ratio Confidence Bounds on Time (Type 1) and Likelihood Ratio Confidence Bounds on Reliability (Type 2) can also be found in the Life Data Analysis online reference.