It is often desirable to be able to compare two sets of reliability or life data in order to determine which of the data sets has a more favorable life distribution. The units from which the data are obtained could either be from two alternate designs, alternate manufacturers or alternate lots or assembly lines. Many methods are available in statistical literature for doing this when the units come from a complete sample, i.e. a sample with no censoring. This process becomes a little more difficult when dealing with data sets that have censoring, or when trying to compare two data sets that have different distributions. In general, the problem boils down to that of being able to determine any statistically significant difference between the two samples of potentially censored data from two possibly different populations. This section discusses some of the methods that are applicable to censored data, and are available in Weibull++.
This section includes the following subsections:
One popular graphical method for performing tests of comparison involves plotting the data with confidence bounds and seeing whether the bounds overlap, or separate, at the point of interest. (Note: One could also perform the same comparison at the point of interest utilizing the Quick Calculation Pad (QCP) and obtaining exact results.) This can be effective for comparisons at a given point in time or a given reliability level, but it is difficult to assess the overall behavior of the two distributions, as the confidence bounds may overlap at some points and be far apart at others. This can be easily done using the MultiPlot feature in Weibull++.
To determine whether two data sets are significantly different and at what confidence level, one can utilize the contour plots provided in Weibull++. By overlaying two contour plots from two different data sets (analyzed using the same distribution) at the same confidence level, one can visually assess whether the data sets are significantly different at that confidence level if there is no overlap on the contours. The disadvantage of this method is that the same distribution must be fitted to both data sets.
The following data represent the times-to-failure for a product. Certain modifications were made to this product in order to improve its reliability. Reliability engineers are trying to determine whether the improvements were significant in improving the reliability.
At what significant level can the engineers claim that the two designs are different? (Note: Note that there is more data available for the old design. This means that there will be less uncertainty on the estimates for the old design. When using the likelihood contours as a comparison, the differences in sample size between different sets and therefore the uncertainty of the estimates are taken into account.)
For both data sets the two-parameter Weibull distribution best fits the data. The contour plots were generated and overlaid on a MultiPlot.
From this plot, it can be seen that there is an overlap at the 95% confidence level and that there is no overlap at the 90% confidence level. It can then be concluded that the new design is better at the 90% confidence level. (Note: Note that if more precision were required one could repeat this for 91%, 92%, etc.)
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Using the Comparison WizardAnother methodology, suggested by Gerald G. Brown and Herbert C. Rutemiller, for performing tests of comparison is to estimate the probability of whether the times-to-failure of one population are better or worse than the times-to-failure of the second. The equation used to estimate this probability is given by:
(31)
where f1(t) is the pdf of the first distribution and R2(t) is the reliability function of the second distribution. The evaluation of the superior data set is based on whether this probability is smaller or greater than 0.5. If the probability is equal to 0.5, that is equivalent to saying the two distributions are identical.
If given two alternate designs with life test data, where X and Y represent the life test data from two different populations, and if we simply wanted to choose the component at time t with the higher reliability, one choice would be to select the component with the higher reliability at time t. However, if we wanted to design a product as long-lived as possible, we would want to calculate the probability that the entire distribution of one product is better than the other and choose X or Y when this probability is above or below 0.50 respectively.
The statement that the probability that X is greater or equal to Y can be interpreted as follows:
If P = 0.50, then the statement is equivalent to saying that both X and Y are equal.
If P < 0.50 or, for example, P = 0.10, then the statement is equivalent to saying that P = 1 - 0.10 = 0.90, or Y is better than X with a 90% probability. (Note: Or 90% of the lives of Y will be greater than the lives of X.)
Weibull++'s Comparison Wizard allows you to perform such calculations. The disadvantage of this method is that the sample sizes are not taken into account, thus one should avoid using this method of comparison when the sample sizes are different.
Using the data presented in Example 14, use Weibull++'s Comparison Wizard to estimate the probability that units from the new design will outlast units from the old design.
After entering Example 14's data sets into two different folios (or data sheets within the same folio) and analyzing them (in this case, the two data sets were analyzed using the two-parameter distribution and MLE), select Tests of Comparison from the Tools menu to access the Comparison Wizard. Specify the data sets that are being compared:
Click Compare. The result of the comparison is displayed in the next window.
See Also:
Additional Reliability Analysis Tools
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