Tests of Comparison

It is often desirable to be able to compare two sets of accelerated 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 ALTA.

Simple Plotting

One popular graphical method for making this determination involves plotting the data at a given stress level with confidence bounds and seeing whether the bounds overlap or separate at the point of interest. One could also perform the same comparison, at the point of interest, utilizing the Quick Calculation Pad and compare the exact results generated by that utility.

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 multiple plot feature in ALTA.

Estimating P[t2 t1] Using the Comparison Wizard

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:

(1)

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 0.50. If the probability is equal to 0.50, that is equivalent to saying that 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 “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.

ALTA's Comparison Wizard allows you to perform such calculations. The comparison is performed at the given use stress levels of each data set. Eqn. (1) can then be expressed as,

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.