Reliability HotWire |
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Reliability Basics |
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When conducting a reliability growth or a repairable systems analysis in RGA on data consisting of multiple systems, you expect that each of the systems performed in a similar manner. In particular, you would expect the interarrival rate of the failures across the systems to be fairly consistent. A statistical test, called the Common Beta Hypothesis (CBH), is available to test for this assumption.
This test is important because it warns the analyst about the possibility that the systems, which are assumed to be similar, are in fact different (i.e., have different behavior). If the test fails to support the assumption, the analyst should investigate the reasons for difference between the multiple systems and possibly split the data into different separate analyses. Note that in repairable analysis or reliability growth, the systems need to be in the same configuration all the time. In other words, if a fix or improvement is implemented in one system it should be implemented across all the systems to keep the consistency between the various systems and to avoid creating a non-homogenous sample. The Common Beta Hypothesis test tests whether other differences are present among the systems. This article describes this test and provides an example to illustrate how the test can be applied.
The Common Beta Hypothesis test is
applicable to the following data types available in
RGA: Multiple
Systems (Concurrent Operating Times), Repairable and Fleet. As shown
by Crow [1], suppose that
(1)
where β_{1} = β_{2} =...= β_{K}.
In other words, the CBH tests for the hypothesis that the interarrival
rate of the failures across the systems are fairly consistent.
Let denote the conditional maximum likelihood estimate of , which is given by:
(2)
where: -
*K*= 1 -
*M*=_{q}*N*if data on the_{q}*q*^{th}^{ }system is time terminated or*M*if data on the_{q}= (N_{q}- 1)*q*system is failure terminated (^{th }*N*is the number of failures on the_{q}*q*system)^{th } -
*X*is the_{iq}*i*time-to-failure on the^{th}*q*system^{th}
Then for each system, assume that:
are conditionally distributed as independent
chi-squared random variables with K = 2, you can test the null hypothesis,
H, using the following statistic:_{o}
(3)
If F equals
and
has conditionally an F-distribution with (2M)
degrees of freedom. The critical value, _{1}, 2M_{2}F, can then be determined
by referring to the chi-squared tables. Now, if K ≥ 2, then
the likelihood ratio procedure [1] can be used to test the hypothesis
β=_{1} β=_{2} ...= β.
Consider the following statistic:_{K}
where:
Also, let:
Calculate the statistic
The statistic D falls between the critical points.
Given that the intensity function
for the
test the hypothesis that βwhile assuming a significance level
equal to 0.05. Calculate
and
using Eqn. 2._{2},
Then:
Using Eqn. 3, calculate the statistic
Since 1.2408 < 2.0980, the hypothesis
that βis accepted at
the 5% significance level. Now suppose instead it is desired to
test the hypothesis that _{2 }β=_{1} β=_{2}
β_{3}. Calculate the statistic D
using Eqn. 4.
Using the chi-squared tables with β=_{2} β_{3} at the
5% significance level.
This analysis can be replicated in
RGA. The significance level
that will be used in new folios can be specified in the Application
Setup window by choosing the desired level from the
The data is entered in a Multiple Systems (Concurrent Operating Times) data sheet by making the following selections in the Data Sheet Setup window.
The data set is entered as follows:
After analyzing the data, the CBH
test result (Passed or Failed) will be displayed in the
More detailed CBH results can be found
in the Results window, which can be accessed by choosing
The Results window shown above tells
us that 0.1026 < D < 5.9915, therefore the data passed the CBH test
(i.e., it can be assumed that
1. Crow, L.H., "Reliability Analysis for Complex, Repairable Systems in Reliability and Biometry," SIAM, ed. by Proschan and R. J. Serfling, Philadelphia, Pennsylvania, pp. 379-410, 1974. |
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