Analysis of Censored Data

So far we have discussed parameter estimation methods for complete data only. We will expand on that approach in this section by including the maximum likelihood estimation method for right censored data. The method is based on the same principles covered previously but modified to take into account the fact that some of the data is censored.

MLE Analysis of Right Censored Data

The maximum likelihood method is, at this point, by far the most appropriate analysis method for censored data. It is versatile and applicable to most accelerated life testing models. When performing maximum likelihood analysis, the likelihood function needs to be expanded to take into account the suspended items. A big advantage of using MLE when dealing with censored data is that each suspension term is included in the likelihood function. Thus, the estimates of the parameters are obtained from consideration of the entire population. Using MLE properties, confidence bounds can be obtained which also account for all the suspension terms. In the case of suspensions and where x is a continuous random variable with pdf and cdf:

where are the k unknown parameters which need to be estimated from R observed failures at (T1, V1), (T2, V2)... (TR, VR) and M observed suspensions at where VR is the Rth stress level corresponding to the Rth observed failure and is the Mth stress level corresponding to the Mth observed suspension. The likelihood function is then formulated as follows:

and the parameters are solved by maximizing this equation. In most cases, no closed form solution exists for this maximum, or for the parameters.

Example 3

Example 1 was repeated using MLE with the following results:

In the individual analysis (probability plotting) the betas were averaged in order to estimate a common shape parameter. This introduced further uncertainties into the analysis. However, in this case (using MLE) the parameter β was estimated from the entire data set.

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