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Dataplot Vol 1 Auxiliary Chapter

RELIABILITY TRENDS TEST

Name:
    RELIABILITY TRENDS TEST
Type:
    Analysis Command
Purpose:
    Compute tests to determine whether or not the repair times of a system show significant trend. The trend may be either an improvement or a degradation.

    Specifically, it computes the following three tests

    1. The reverse arrangement test.
    2. The military handbook test.
    3. The Laplace test.
Description:
    Determining if repair times exhibit a trend is necessary for choosing an appropriate reliability model. These formal tests for trend in repair times are a compliment to the Duane plot for detecting trend.

    Each of these tests will be described separately.

    1. The Reverse Arrangement Test

      Given r repairs, T1, T2, ...., Tr, the interarrival times I2=R2-T1, I3=T3-T2, ...., Ir=Tr-Tr-1 and the censoring time Tend > Tr, we calculate how many instances we have of a later interarrival time being strictly greater than an earlier interarrival time. These are called reversals. Too many reversals indicates a significant improving trend and too few reversals indicates a significant degradation trend. More formally,

      1. Count a reversal every time Ij < Ik for some j and k with j < k.

      2. Compute the total number of reversals, R.

      3. For r repair times, the maximum possible number of reversals is r(r-1)/2.

      4. If there are no trends, the expected number of reversals is r(r-1)/4.

      5. For r > 12, the following approximation can be used to determine if the number of reversals is statistically significant.

          z = R - C1 + 0.5/SQRT(C2) where
 C1 = r(r-1)/4 and
 C2 = (2r+5)(r-1)r/72

        The test statistic is compared to a standard normal distribution. That is, if |z| > 1.96, we have statistical significance at the 95% level.

        For r <= 12, tabled values are used.

      The advantage of this test is that it is simple and it makes no assumptions about a model for the possible trend.

    2. The Military Handbook Test

      Given r repairs, T1, T2, ...., Tr and the censoring time Tend > Tr, we calculate the test statistic

        T = 2*SUM(ln(T(end)/T(i))) where the summation is from i=1 to r

      This test statistic follows a chi-square distribution with 2*r degrees of freedom.

      This test is recommended for the case when the choice is between no trend and a non-homogeneous Poisson process (NHPP) power law (Duane) model.

    3. The Laplace Test

      Given r repairs, T1, T2, ...., Tr and the censoring time Tend > Tr, we calculate the test statistic

        z = SQRT(12*r)*SUM(T(i) - T(end)/2)/r*T(end) where the summation is from i=1 to r

      This test statistic follows a standard normal distribution.

      This test is recommended for the case when the choice is between no trend and a non-homogeneous Poisson process (NHPP) exponential law model.

    If you have internet access, you can enter the following command to see a discussion of the power law model

      WEB HANDBOOK NHPP POWER LAW
Syntax:
    RELIABILITY TREND TEST <y> <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    RELIABILITY TREND TEST Y
    RELIABILITY TREND TEST Y SUBSET TAG > 0
Note:
    These commands require a censoring time. Enter the following command before the RELIABILITY TREND TEST command to specify the censoring time:

      LET TEND = <censor time>
Default:
    None
Synonyms:
    None
Related Commands:
    DUANE PLOT = Generate a Duane plot.
    POWER LAW RANDOM NUMBERS = Generate random number from a NHPP power law model.
    INTERARRIVAL TIME = Compute interarrival times of a variable.
Applications:
    Reliability
Implementation Date:
    1998/5
Program:
    skip 25
    read hahn.dat y tag
    .
    LET TEND = 135000
    RELIABILITY TREND TEST Y SUBSET TAG = 1

    The following output is generated:

    REVERSE ARRANGEMENTS TEST:
          NUMBER OF FAILURE TIMES               =       37
          OBSERVED NUMBER OF REVERSALS          =      215
          MAXIMUM POSSIBLE NUMBER OF REVERSALS  =      666
          EXPECTED NUMBER OF REVERSALS          =      333
          VALUE OF TEST STATISTIC               =  -21.40790
          90% CRITICAL VALUE FOR NO IMPROVEMENT = 1.282
          95% CRITICAL VALUE FOR NO IMPROVEMEMT = 1.645
          99% CRITICAL VALUE FOR NO IMPROVEMENT = 2.33
     
     
    MILITARY HANDBOOK TEST:
          NUMBER OF FAILURE TIMES               =       37
          CHI-SQUARE TEST STATISTIC VALUE       =    35.72037
          CHI-SQUARE TEST STATISTIC CDF VALUE   =   0.4925163E-04
     
          IMPROVEMENT TEST
          HYPOTHESIS     ACCEPTANCE INTERVAL    CONCLUSION
          NO TREND          (0.000,0.900)         ACCEPT
          NO TREND          (0.000,0.950)         ACCEPT
          NO TREND          (0.000,0.990)         ACCEPT
     
     
          DEGRADATION TEST
          HYPOTHESIS     ACCEPTANCE INTERVAL    CONCLUSION
          NO TREND          (0.100,1)             REJECT
          NO TREND          (0.050,1)             REJECT
          NO TREND          (0.010,1)             REJECT
     
     
     
    LAPLACE TEST:
          NUMBER OF FAILURE TIMES               =       37
          NORMAL TEST STATISTIC VALUE       =    3.416969
          NORMAL TEST STATISTIC CDF VALUE   =   0.9996833
     
          IMPROVEMENT TEST
          HYPOTHESIS     ACCEPTANCE INTERVAL    CONCLUSION
          NO TREND          (0.000,0.900)         REJECT
          NO TREND          (0.000,0.950)         REJECT
          NO TREND          (0.000,0.990)         REJECT
     
     
          DEGRADATION TEST
          HYPOTHESIS     ACCEPTANCE INTERVAL    CONCLUSION
          NO TREND          (0.100,1)             ACCEPT
          NO TREND          (0.050,1)             ACCEPT
          NO TREND          (0.010,1)             ACCEPT
        

    Date created: 6/5/2001
    Last updated: 4/4/2003
    Please email comments on this WWW page to alan.heckert@nist.gov.