RELIABILITY TRENDS TEST

RELIABILITY TRENDS TEST

Analysis Command

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.

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, ...., T_r, the interarrival times I₂=R₂-T₁, I₃=T₃-T₂, ...., I_r=T_r-T_r-1 and the censoring time T_end > T_r, 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 I_j < I_k 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.
      

      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, ...., T_r and the censoring time T_end > T_r, we calculate the test statistic

   

   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, ...., T_r and the censoring time T_end > T_r, we calculate the test statistic

   

   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

RELIABILITY TREND TEST <y> <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

RELIABILITY TREND TEST Y
RELIABILITY TREND TEST Y SUBSET TAG > 0

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>

None

None

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.

Reliability

1998/5

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