8.
Assessing Product Reliability
8.2. Assumptions/Prerequisites 8.2.3. How can you test reliability model assumptions?


Likelihood Ratio Tests are a powerful, very general method of testing model assumptions. However, they require special software, not always readily available.  Likelihood functions for reliability
data are described in
Section 4. Two ways we use likelihood functions to choose models or verify/validate
assumptions are:
1. Calculate the maximum likelihood of the sample data based on an assumed distribution model (the maximum occurs when unknown parameters are replaced by their maximum likelihood estimates). Repeat this calculation for other candidate distribution models that also appear to fit the data (based on probability plots). If all the models have the same number of unknown parameters, and there is no convincing reason to choose one particular model over another based on the failure mechanism or previous successful analyses, then pick the model with the largest likelihood value. 2. Many model assumptions can be viewed as putting restrictions on the parameters in a likelihood expression that effectively reduce the total number of unknown parameters. Some common examples are: 

Examples where assumptions can be tested by the Likelihood Ratio Test 
i) It is suspected that a type of data, typically modeled by a Weibull distribution, can be fit adequately by an exponential model. The exponential distribution is a special case of the Weibull, with the shape parameter \(\gamma\) set to 1. If we write the Weibull likelihood function for the data, the exponential model likelihood function is obtained by setting \(\gamma\) to 1, and the number of unknown parameters has been reduced from two to one.Clearly, we could come up with many more examples like these three, for which an important assumption can be restated as a reduction or restriction on the number of parameters used to formulate the likelihood function of the data. In all these cases, there is a simple and very useful way to test whether the assumption is consistent with the data. The Likelihood Ratio Test Procedure 

Details of the Likelihood Ratio Test procedure
In general, calculations are difficult and need to be built into the software you use 
Let \(L_1\)
be the maximum value
of the likelihood of the data without the additional assumption. In other
words, \(L_1\)
is the likelihood of the data with all the
parameters unrestricted and maximum likelihood estimates substituted for
these parameters.
Let \(L_0\) be the maximum value of the likelihood when the parameters are restricted (and reduced in number) based on the assumption. Assume \(k\) parameters were lost (i.e., \(L_0\) has \(k\) less parameters than \(L_1\)). Form the ratio \(\lambda = L_0 / L_1\). This ratio is always between 0 and 1 and the less likely the assumption is, the smaller \(\lambda\) will be. This can be quantified at a given confidence level as follows:
