Three major failure data trend tests are reviewed in this section. These tests may be used to determine whether or not a trend exists in the failure times of a data set and include the reverse arrangement test, the military handbook test and the Laplace trend test. There are several alternative trend tests, such as the Lewis-Robinson test [10] and the generalised Anderson- Darling test [143], but these alternatives are not as popular as the three trend test reviewed here [171].
3.6.1 The reverse arrangement test
The reverse arrangement test (RAT) is a simple method for testing for the presence of a trend in a data set without making any assumptions about what kind of model the trend will follow [167, 171]. The RAT is related to Kendall’s so-called tau test which was first introduced in 1938 [124] and may be used to distinguish between independent and identically distributed interarrival
times on the one hand, and a monotonic trend on the other [204].
Consider a data set with n − 1 interarrival times, T1, . . . , Tn−1. A reversal is defined as a later
observed instance that is strictly greater than an earlier observed instance [167, 171]. A reversal therefore occurs each time Ti < Tj for i = 1, 2, . . . , n − 2, and j = i + 1, . . . , n − 1. Kendall
demonstrated that the distribution of the total number of reversals of a system approaches a normal distribution rapidly as n increases [124]. More specifically, the random variable
Z = R − (n − 1)(n − 2) 4 + 0.5 r (n − 1)(n − 2)(2(n − 1) + 5) 72
is approximately normally distributed with a mean of 0 and a standard deviation of 1 for large values of n, where R is the total number of reversals and n − 1 denotes the total number of observed failures. This normal distribution approximation is accurate for values of n greater than 12 [204]. In fact, at a significance level β, the upper critical value for the distribution is
Rβn−1,u= zcrit r (n − 1)(n − 2)(2(n − 1) + 5) 72 + (n − 1)(n − 2) 4 − 0.5
for n-values larger than 12, where zcrit is the corresponding z-value of the required significance
level β. The lower corresponding critical value Rn−1,`β is Rβn−1,`= Rn−1,max− Rβn−1,u,
where Rn−1,max = (n − 1)(n − 2)/2. For n-values at most 12, the corresponding upper and lower
critical values are given in Table 3.1.
Table 3.1: Critical values of the random variable R for n-values less than or equal to 12. These are the upper and lower significant levels for the reverse arrangement test.
Single-sided lower Single-sided upper Sample significance level significance level
size (n) 1% 5% 10% 10% 5% 1% 4 0 0 6 6 5 0 1 1 9 9 10 6 1 2 3 12 13 14 7 2 4 5 16 17 19 8 4 6 8 20 22 24 9 6 9 11 25 27 30 10 9 12 14 31 33 36 11 12 16 18 37 39 43 12 16 20 23 43 46 50
3.6.2 The military handbook test
The military handbook test (MHT) was first introduced in 1981 by the United States Department of Defence [209] and is generally the best method to employ when deciding between a Non- homogeneous poisson process (NHPP) power law model, also known as the Duane model, and
no trend in failure data emanating from a system [171, 173]. The MHT method tests the hypothesis that no trend exists in the data set, based on the chi-square distribution. The relevant chi-square statistic is
χ22(n−1) = 2 n−1 X i=1 lnQend Ti ,
where Qend denotes the discrete observation time (measured in global units of time) at the end
of the observation period which may be an observed failure time or merely the system lifetime to date, as illustrated in Figure 3.6. Furthermore, Ti denotes the discrete failure time (also
measured in global units of time) of the i-th failure as before, and n − 1 again denotes the total number of observed failure events [167, 171].
0 Qend
T1 T2 . . . Tn−1
Figure 3.6: A graphical representation of the parameter Qend.
This statistic is compared to the percentiles of the chi-square distribution with 2(n − 1) degrees of freedom. In the case where χ22(n−1) > χ2crit,β, at a significance level of β, evidence exists against the null hypothesis of no trend present in the data, in which case the data set may be classified as emanating from a repairable system and exhibiting an improving trend. In the case where χ22(n−1)< χ2crit,1−β, however, evidence also exists that a trend is present in the data, in which case the data set is classified as emanating from a repairable system and exhibiting a decreasing trend [209]. Finally, in the case where χ2crit,β < χ22(n−1) < χ2crit,1−β, there is not enough evidence to reject the null hypothesis of no trend and the data is classified as emanating from a non-repairable system [171]. A graphical representation of the outcomes of the MHT is shown in Figure 3.7 at a significance level of β.
χ2 crit,1−β χ2crit,β Reliability degradation Reliability improvement No trend
Figure 3.7: Possible outcomes of the military handbook test at significance level β.
3.6.3 The Laplace test
The Laplace trend test, also known as the centroid test, is most often used for the purpose of identifying a trend in a data set of failure times [47] and is generally the most accurate trend test when deciding between an NHPP following an exponential law and no trend. It was developed by De Laplace in 1773 to test whether or not comets originate in our solar system [47]. The Laplace trend test is carried out to test the hypothesis that a trend does not exist among a set of interarrival failure data and can determine whether the system from which the data emanate exhibits improvement, deterioration or whether there is no trend in the data set. The test statistic for the hypothesis test is
U = Pn−2
i=1(Ti− Tn−12 ) ×p12(n − 2)
(n − 2) × Tn−1
where Ti denotes the discrete failure time (measured in global units of time) of the i-th failure
and n − 1 denotes the total number of observed failure events. In the case where the observation period is not ended at the instant when a failure is observed, the Laplace trend test statistic is
U = Pn−1
i=1(Ti−Qend2 ) ×p12(n − 1)
(n − 1) × Qend
,
where Qend again denotes the time (again measured in global units of time) at the end of
the observation period, as shown in Figure 3.6. This test statistic approximates a standard normal distribution, and so the critical value Z for the hypothesis test is obtained from the standard normal table at a given significance level β. If U ≥ Z1−β/2, then evidence exists of
strong reliability degradation of the system from which the failure data emanate, whereas if U ≤ Zβ/2, evidence exists of strong reliability improvement. In both these cases there is enough
evidence to reject the hypothesis that there does not exist a trend in the data set, in which case the set is classified as data emanating from a repairable system. Finally, in the case where Zβ/2 < U < Z1−β/2, there is not enough evidence to reject the hypothesis that there exists no
trend in the data, in which case the data set is referred to as non-committal and classified as emanating from a non-repairable system [179]. A graphical representation of the outcomes of the Laplace trend test is shown in Figure 3.8 at a significance level of β.
Zcrit,β 0 Zcrit,1−β Reliability improvement Reliability degradation No trend
Figure 3.8: Possible outcomes of the Laplace trend test at significance level β.