Comparing Life Data Sets

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 data sets could be from two alternate designs, manufacturers, lots, assembly lines, etc. 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 available in Weibull++ that are applicable to censored data.

=Simple Plotting= One popular graphical method for making this determination involves plotting the data with confidence bounds and seeing whether the bounds overlap or separate at the point of interest. This can be easily done using the Overlay Plot feature in Weibull++. This approach 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 because the confidence bounds may overlap at some points and be far apart at others.

=Using Contour Plots= 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 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.

Critical Confidence Level
The Critical Confidence Level feature is an option in the Contours Setup window for overlay plots in Weibull++. To activate the utility, select the Plot Critical Level check box, as shown in the figure below.



This feature determines the confidence level at which the contour plots of two data sets meet at a single point. This is the minimum confidence level at which the contour plots of the two different data sets overlap. At any confidence level below this minimum confidence level, the contour plots of the two data sets will not overlap and there will be a statistically significant difference between the two populations at that level. For the two data sets in the example above, the critical confidence level 94.243%. This can be seen in the figure below.



The picture shows the contour plot at a confidence level of 90% and at the critical level of 94.243%. Please notice that due to the calculation resolution and plot precision, sometimes at the calculated critical level, the two curves may still look like they have a little bit of overlap or separation.

=Using the Life Comparison Tool= Another methodology, suggested by Gerald G. Brown and Herbert C. Rutemiller, 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:


 * $$P\left[ {{t}_{2}}\ge {{t}_{1}} \right]=\mathop{}_{0}^{\infty }{{f}_{1}}(t)\cdot {{R}_{2}}(t)\cdot dt$$

where $${{f}_{1}}(t)\,\!$$  is the  pdf  of the first distribution and  $${{R}_{2}}(t)\,\!$$  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 than 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.

Weibull++'s Life Comparison tool 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.