Issue 16, June 2002

Verifying the Assumption of a Constant Shape Parameter in Accelerated Life Testing Data

It is a generally accepted practice to assume a constant shape parameter across the different stress levels (independent of stress) when analyzing data from an accelerated life test. For the Weibull distribution, the shape parameter is Beta, β, and for the lognormal distribution, the shape parameter is the log-std, σ_T'. This implies that the unit/component will fail in the same manner across different stress levels. But how can you validate this assumption? Does the unit/component actually behave in this manner? There are multiple ways that you could test the validity of this assumption, such as a likelihood ratio test or a maximum ratio test. However, this article will employ a graphical method utilizing ReliaSoft's ALTA 6 and Weibull++ 6 software.

Example
Consider an accelerated temperature test where a total of 20 units were tested to failure. Eight units were tested at 406K, six units each were tested at 436K and 466K. The data set is presented in Table 1.

Table 1: Accelerated Life Test Data

Since temperature was used as the stimulus during the test, the Arrhenius-Weibull relationship will be selected to model the data. The data can be analyzed in ReliaSoft's ALTA 6 software using the Arrhenius-Weibull relationship, as shown in Figure 1. The parameters were estimated using maximum likelihood estimation (MLE).

Figure 1: Data analyzed using ALTA 6

For this data set, the shape parameter, β, was estimated to be equal to 2.6953. Now that the shape parameter has been estimated, the assumption of a constant shape parameter can be validated using Weibull++ 6 and the contour plot. The data entered into ALTA 6 can be transferred to Weibull++ 6 by selecting Weibull++ 6 from the Tools menu. The data loaded into Weibull++ is shown in Figure 2.

Figure 2: Accelerated test data entered into Weibull++ 6

The data set must now be separated based on each stress level. This can be accomplished using the Batch Auto Run feature of Weibull++ 6. The Batch Auto Run will create a data sheet for each stress level and automatically calculate the parameters for each data set. Since the Weibull distribution was selected as the life distribution when estimating the parameters in ALTA 6, the Weibull distribution will again be used to estimate the parameters in Weibull++ 6. In addition, maximum likelihood estimation (MLE) will also be used. Figure 3 shows that three additional data sheets have been added to the Folio (one for each stress level).

Figure 3: Data separated based on each stress level

Once the Batch Auto Run process has been completed, then a contour plot of the three separated data sets can be used to test the assumption of a constant shape parameter. Figure 4 displays the contour plot of the data from each of the stress levels.

 Figure 4: Contour plot of accelerated test data using Weibull++ 6

The shape on the right is generated based on the data at 406K, the shape in the middle is generated based on the data at 436K and the shape on the left is generated based on the data at 466K. The contour plot in Weibull++ 6 displays the slice(s) of the joined confidence bound region of the estimated parameters (in this case, β and η). The slice(s) are generated based on the given confidence level(s). In this example, only the 90% confidence level was used to limit the number of contours on the plot so that the graph is easier to read. As can be seen from the plot, each of the contours for each of the stress levels are contained within the same Beta space. From this, one can conclude that the assumption of a constant shape parameter is valid.

It is also possible to check this assumption by viewing the Probability-Weibull plot in ALTA 6. Figure 5 displays the Probability-Weibull plot for all of the accelerated life test data.

Figure 5: Probability-Weibull plot of accelerated life test data

From the plot you can see that the slope of the points are following the slope of the lines at each stress level. This also indicates that the assumption of a constant shape parameter is valid. 

Contour Plot Where the Shape Parameter is Not Constant
It is also possible that the contour plot might look like the one shown in Figure 6.

Figure 6: Contour plot where the constant shape parameter assumption is not valid

Figure 6 shows an example of the contours that are not contained within the same Beta space. This indicates that the shape parameter, β, is not constant across the different stress levels. This might be due to a different failure mode that appeared during the test or it is possible the test was not set up properly. Failure analysis may need to be conducted to determine the reason for the difference.

Conclusion
This example contained 20 samples and the results were fairly obvious, but you might not be as lucky. Sample size will play a large part in determining how well (or not so well) the shape parameter can be assumed constant across different stress levels. If a certain stress level contains only a couple of data points, then it is possible that the points may not follow the slope of the line on a Probability-Weibull plot. But with only a couple of points, it may be hard to say whether or not this violates the assumption of a constant shape parameter.

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