Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 61, March 2006

Reliability Basics

Common Beta Hypothesis Test in Reliability Growth and Repairable Systems Analysis

[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.]

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 K number of systems are under test. Each system has an intensity function given by Eqn. 1.

 

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

 

(2)

 

where:

  • K = 1

  • Mq = Nq if data on the qth system is time terminated or Mq = (Nq - 1) if data on the qth system is failure terminated (Nq is the number of failures on the qth system)

  • Xiq is the ith time-to-failure on the qth system

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:

 

(3)

 

If Ho is true, then F equals MATHand 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 =...= βK. Consider the following statistic:



 

where:



Also, let:



 

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.

 

Example
Consider the data in the following table.

 

Table 1 - Repairable system data

 

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

 

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 $\tilde{\beta}_{1}$ and $\tilde{\beta}_{2}$ using Eqn. 2.



Then:

 

 

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.


D = 0.5260

 

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. 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 Default Significance Level drop-down list on the RGA Standard Folios page, as shown next.

 

 

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 Results area of the folio's control panel, as shown next.

 

 

More detailed CBH results can be found in the Results window, which can be accessed by choosing Growth Data > Analysis > Statistical Test Report.


 

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).

 

Reference:

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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