Common Beta Hypothesis Test In Reliability Growth and Repairable Systems Analysis
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 6: Multiple Systems (Unknown Equivalent Times), Repairable and Fleet. As shown by Crow , suppose that K number of systems are under test. Each system has an intensity function given by Eqn. 1.
where q = 1, ... , K. You can compare the intensity functions of each of the systems by comparing the of each system. The CBH test tests the hypothesis, Ho, such that β1 = β2 =...= βK. 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:
Then for each system, assume that:
are conditionally distributed as independent Chi-Squared random variables with 2Mq degrees of freedom. When K = 2, you can test the null hypothesis, Ho, using the following statistic:
If Ho is true, then F equals and has conditionally an F-distribution with (2M1, 2M2) degrees of freedom. The critical value, F, can then be determined by referring to the Chi-Squared tables. Now, if K ≥ 2, then the likelihood ratio procedure  can be used to test the hypothesis β1 = β2 =...= βK. Consider the following statistic:
Calculate the statistic D, such that:
The statistic D is approximately distributed as a Chi-Squared random variable with (K - 1) degrees of freedom. Then, after calculating D, refer to the Chi-Squared tables with (K - 1) degrees of freedom to determine the critical points. Ho is true if the statistic D falls between the critical points.
Table 1 - Repairable system data
Given that the intensity function for the qth system is:
test the hypothesis that β1 = β2, while assuming a significance level equal to 0.05. Calculate and using Eqn. 2.
Using Eqn. 3, calculate the statistic F with a significance level of 0.05.
F = 2.0980
Since 1.2408 < 2.0980, the hypothesis that β1 = β2 is accepted at the 5% significance level. Now suppose instead it is desired to test the hypothesis that β1 = β2 = β3. Calculate the statistic D using Eqn. 4.
Using the Chi-Squared tables with K - 1 = 2 degrees of freedom, the critical values at the 2.5 and 97.5 percentiles are 0.1026 and 5.9915, respectively. Since 0.1026 < D < 5.9915, the hypothesis is accepted that β1 = β2 = β3 at the 5% significance level.
This analysis can be replicated in RGA 6. The data is entered in a Multiple Systems (Unknown Equivalent Times) data sheet by making the following selections in the Data Type Expert.
The data set is entered as follows:
The significance level can be specified in the User Setup (available from the File menu) by entering the required level in the Significance Level input box on the Data Folio page, as shown next.
After analyzing the data, the CBH test result (Passed or Failed) will be displayed in the Results area of the Data Folio Control Panel, as shown next.
More detailed CBH results can be found in the Results window, which can be accessed by selecting Show Other and then Statistical Tests from the Data menu.
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 the beta values are the same and that all the systems behaved similarly).
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.
Copyright 2006 ReliaSoft Corporation, ALL RIGHTS RESERVED