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.

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.

Example:

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:


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

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, then it is equivalent to saying that the two distributions are identical.

Sometimes we may need to compare the life when one of the distributions is truncated. For example, if the random variable from the first distribution is truncated with a range of [L, U}, then the comparison with the truncated distribution should be used. For details, please see Stress-Strength Analysis.

Consider two product designs where X and Y represent the life test data from two different populations. If we simply wanted to determine which component has a higher reliability, we would simply compare the reliability estimates of both components at a time $$t\,\!$$. But if we wanted to determine which product will have a longer life, we would want to calculate the probability that the distribution of one product is better than the other. Using the equation given above, the probability that X is greater than or equal to Y can be interpreted as follows:


 * If $$P=0.50\,\!$$, then lives of both X and Y are equal.
 * If $$P<0.50\,\!$$ or, for example, $$P=0.10\,\!$$, then $$P=1-0.10=0.90\,\!$$, or Y is better than X with a 90% probability.

Example: