Common Beta Hypothesis Test In Reliability Growth and Repairable Systems Analysis

Common Beta Hypothesis Test In Reliability Growth and Repairable Systems Analysis

When conducting a reliability growth or a repairable systems analysis in RGA 6 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 6: Multiple Systems (Unknown Equivalent Times), Repairable and Fleet. As shown by Crow [1], 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

$\beta _{q}$ of each system. The CBH test tests the hypothesis, H_o, such that β₁ = β₂ =...= 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

$\tilde{\beta}_{q}$ denote the conditional maximum likelihood estimate of $\beta _{q}$ , which is given by:

are conditionally distributed as independent Chi-Squared random variables with 2M_q degrees of freedom. When K = 2, you can test the null hypothesis, H_o, using the following statistic:

If H_o 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 [1] can be used to test the hypothesis β₁ = β₂ =...= β_K. Consider the following statistic:

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. H_o is true if the statistic D falls between the critical points.

	System 1	System 2	System 3
Start	0	0	0
End	2000	2000	2000
Failures	1.2	1.4	0.3
	55.6	35	32.6
	72.7	46.8	33.4
	111.9	65.9	241.7
	121.9	181.1	396.2
	303.6	712.6	444.4
	326.9	1005.7	480.8
	1568.4	1029.9	588.9
	1913.5	1675.7	1043.9
		1787.5	1136.1
		1867	1288.1
			1408.1
			1439.4
			1604.8

test the hypothesis that β₁ = β₂, while assuming a significance level equal to 0.05. Calculate $\tilde{\beta}_{1}$ and $\tilde{\beta}_{2}$ using Eqn. 2.

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 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.