8.2.3.3.

Likelihood ratio tests

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.

ii) Assume we have \(n\) cells of data from an acceleration test, with each cell having a different operating temperature. We assume a lognormal population model applies in every cell. Without an acceleration model assumption, the likelihood of the experimental data would be the product of the likelihoods from each cell and there would be \(2n\) unknown parameters (a different \(T_{50}\) and \(\sigma\) for each cell). If we assume an Arrhenius model applies, the total number of parameters drops from \(2n\) to just 3, the single common \(\sigma\) and the Arrhenius \(A\) and \(\Delta H\) parameters. This acceleration assumption "saves" \((2n-3)\) parameters.

iii) We life test samples of product from two vendors. The product is known to have a failure mechanism modeled by the Weibull distribution, and we want to know whether there is a difference in reliability between the vendors. The unrestricted likelihood of the data is the product of the two likelihoods, with 4 unknown parameters (the shape and characteristic life for each vendor population). If, however, we assume no difference between vendors, the likelihood reduces to having only two unknown parameters (the common shape and the common characteristic life). Two parameters are "lost" by the assumption of "no difference".

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:

1. Calculate \(\chi^2 = -2 \mbox{ ln } \lambda\). The smaller \(\lambda\) is, the larger \(\chi^2\) will be.
2. We can tell when \(\chi^2\) is significantly large by comparing it to the \(100(1-\alpha)\) percentile point of a Chi-Square distribution with degrees of freedom. \(\chi^2\) has an approximate Chi-Square distribution with \(k\) degrees of freedom and the approximation is usually good, even for small sample sizes.
3. The likelihood ratio test computes \(\chi^2\) and rejects the assumption if \(\chi^2\) is larger than a Chi-Square percentile with \(k\) degrees of freedom, where the percentile corresponds to the confidence level chosen by the analyst.

Note: While Likelihood Ratio test procedures are very useful and widely applicable, the computations are difficult to perform by hand, especially for censored data, and appropriate software is necessary.