8.3.2.6 __Tests for Validity of
the Assumption Of A Theoretical Reliability Parameter Distribution__

The validity of many statistical techniques used in the
calculation, analysis, or prediction of reliability parameters depends on the
distribution of the failure times. Many techniques are based on specific
assumptions about the probability distribution and are often sensitive to
departures from the assumed distributions. That is, if the actual distribution
differs from that assumed, these methods sometimes yield seriously wrong
results. Therefore, in order to determine whether or not certain techniques
are applicable to a particular situation, some judgment must be made as to the
underlying probability distribution of the failure times.

As was discussed in Section 8.3.1, some
theoretical reliability functions, such as those based on the exponential,
normal, lognormal, and Weibull distributions will plot as straight lines on
special types of graph paper. This is the simplest procedure and should be
used as a "first cut" in determining the underlying distribution. Plot the
failure data on the appropriate graph paper for the assumed underlying distribution; “eyeball” it, and
if it quite closely approximates a straight line, you are home
free.

If it cannot be determined visually
that the reliability function follows a straight line when plotted on special
graph paper, then one must resort to the application of analytical
“goodness-of-fit” tests.

The two goodness-of-fit tests
described in this section assume a null hypothesis, i.e., the sample is from
the assumed distribution. Then a statistic, evaluated from the sample data, is
calculated and looked-up in a table that shows how “lucky” or “unlucky” the
sample. The luck is determined by the size of the two-sided tail area. If that
tail is very small (you were very unlucky if the null hypothesis is true), the
null hypothesis (there is no difference between the actual and the assumed
distributions) is rejected. Otherwise, the null hypothesis is accepted, i.e.,
the actual distribution could easily have generated that set of data (within
the range of the data); the test says nothing about the behavior of the
distribution outside the range of the data.

Goodness-of-fit tests are
statistical tests, not engineering tests. No matter what the distribution or
what the test, it is possible to take a sample small enough so that virtually
no distribution will be rejected, or large enough so that virtually every
distribution will be rejected.

Thus, while a method for small
sample sizes is presented as well as one for large sample sizes, it is a fact
of life that must be accepted that tests based on small samples are simply not
very powerful (power is the probability of rejecting the null hypothesis where
it, indeed, is incorrect). Therefore, the methodology is presented here for
completeness, but very likely a more logical approach is to first make an
assumption regarding the failure distribution based on engineering judgment or
on historical data or on knowledge of the failure characteristics of similar
parts. Once the failure distribution has been assumed the test can be
performed for goodness-of-fit for that particular distribution. If the
hypothesized distribution is shown not to fit, it is quite certain that the
assumed distribution was not the one from which the samples were selected. If,
however, the goodness-of-fit test shows that the data could have come from the
hypothesized distribution, then it is virtually certain that tests for fit to
other distributions would yield like results.

In summary then, it must be
realized that the tests presented in the next two sections have limitations.
The only cure for these limitations is a larger number of observations. If
this proves uneconomical or not feasible from the standpoint of the test time
required to generate the desired number of failures or the cost of testing ,
or some other practical constraint, then the only alternative is to use the
results of small sample size analyses with proper discretion.