Template:Likelihood Ratio Confidence Bounds

Likelihood Ratio Confidence Bounds
Another method for calculating confidence bounds is the likelihood ratio bounds (LRB) method. Conceptually, this method is a great deal simpler than that of the Fisher matrix, although that does not mean that the results are of any less value. In fact, the LRB method is often preferred over the FM method in situations where there are smaller sample sizes. Likelihood ratio confidence bounds are based on the following likelihood ratio equation:


 * $$-2\cdot \text{ln}\left( \frac{L(\theta )}{L(\widehat{\theta })} \right)\ge \chi _{\alpha ;k}^{2}$$

where:
 * $$L(\theta )$$ is the likelihood function for the unknown parameter vector $$\theta $$
 * $$L(\widehat{\theta })$$ is the likelihood function calculated at the estimated vector $$\widehat{\theta }$$
 * $$\chi _{\alpha ;k}^{2}$$ is the chi-squared statistic with probability $$\alpha $$ and $$k$$ degrees of freedom, where $$k$$ is the number of quantities jointly estimated

If $$\delta $$ is the confidence level, then $$\alpha =\delta $$ for two-sided bounds and $$\alpha =(2\delta -1)$$ for one-sided. Recall from Chapter Basic Statistical Background that if $$x$$ is a continuous random variable with $$pdf$$:
 * $$f(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})$$,

where $${{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}$$ are $$k$$ unknown constant parameters that need to be estimated, one can conduct an experiment and obtain $$R$$ independent observations, $${{x}_{1}},$$ $${{x}_{2}},$$$$...,{{x}_{R}}$$, which correspond in the case of life data analysis to failure times. The likelihood function is given by:


 * $$L({{x}_{1}},{{x}_{2}},...,{{x}_{R}}|{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})=L=\underset{i=1}{\overset{R}{\mathop \prod }}\,f({{x}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})$$


 * $$i=1,2,...,R$$

The maximum likelihood estimators (MLE) of $${{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}},$$ are obtained by maximizing $$L.$$ These are represented by the $$L(\widehat{\theta })$$ term in the denominator of the ratio in the likelihood ratio equation. Since the values of the data points are known, and the values of the parameter estimates $$\widehat{\theta }$$ have been calculated using MLE methods, the only unknown term in the likelihood ratio equation is the $$L(\theta )$$ term in the numerator of the ratio. It remains to find the values of the unknown parameter vector $$\theta $$ that satisfy the likelihood ratio equation. For distributions that have two parameters, the values of these two parameters can be varied in order to satisfy the likelihood ratio equation. The values of the parameters that satisfy this equation will change based on the desired confidence level $$\delta ;$$ but at a given value of $$\delta $$ there is only a certain region of values for $${{\theta }_{1}}$$ and $${{\theta }_{2}}$$ for which the likelihood ratio equation holds true. This region can be represented graphically as a contour plot, an example of which is given in the following graphic.



The region of the contour plot essentially represents a cross-section of the likelihood function surface that satisfies the conditions of the likelihood ratio equation.

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. Chapter Comparing Life Data Sets discusses comparing data sets.



Confidence Bounds on the Parameters
The bounds on the parameters are calculated by finding the extreme values of the contour plot on each axis for a given confidence level. Since each axis represents the possible values of a given parameter, the boundaries of the contour plot represent the extreme values of the parameters that satisfy:


 * $$-2\cdot \text{ln}\left( \frac{L({{\theta }_{1}},{{\theta }_{2}})}{L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})} \right)=\chi _{\alpha ;1}^{2}$$

This equation can be rewritten as:


 * $$L({{\theta }_{1}},{{\theta }_{2}})=L({{\widehat{\theta }}_{1}},{{\widehat{\theta }}_{2}})\cdot {{e}^{\tfrac{-\chi _{\alpha ;1}^{2}}{2}}}$$

The task now becomes to find the values of the parameters $${{\theta }_{1}}$$ and $${{\theta }_{2}}$$ so that the equality in above the likelihood ratio equation is satisfied. Unfortunately, there is no closed-form solution, thus these values must be arrived at numerically. One method of doing this is to hold one parameter constant and iterate on the other until an acceptable solution is reached. This can prove to be rather tricky, since there will be two solutions for one parameter if the other is held constant. In situations such as these, it is best to begin the iterative calculations with values close to those of the MLE values, so as to ensure that one is not attempting to perform calculations outside of the region of the contour plot where no solution exists.

Example 1:

Confidence Bounds on Time (Type 1)
The manner in which the bounds on the time estimate for a given reliability are calculated is much the same as the manner in which the bounds on the parameters are calculated. The difference lies in the form of the likelihood functions that comprise the likelihood ratio. In the preceding section we used the standard form of the likelihood function, which was in terms of the parameters $${{\theta }_{1}}$$ and $${{\theta }_{2}}$$. In order to calculate the bounds on a time estimate, the likelihood function needs to be rewritten in terms of one parameter and time, so that the maximum and minimum values of the time can be observed as the parameter is varied. This process is best illustrated with an example.

Example 2:

Confidence Bounds on Reliability (Type 2)
The likelihood ratio bounds on a reliability estimate for a given time value are calculated in the same manner as were the bounds on time. The only difference is that the likelihood function must now be considered in terms of $$\beta $$ and $$R$$. The likelihood function is once again altered in the same way as before, only now $$R$$ is considered to be a parameter instead of $$t$$, since the value of $$t$$ must be specified in advance. Once again, this process is best illustrated with an example.

Example 3: