A STATISTICAL TEST FOR MONOTONIC AND NON-MONOTONIC TREND IN REPAIRABLE SYSTEMS Jan Terje Kvaløy Department of Mathematics and Science, Stavanger University College, P.O. Box 2557 Ullandhaug, N-491 Stavanger, Norway, Email: jan.t.kvaloy@tn.his.no Bo Lindqvist, Håkon Malmedal Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway, Email: bo@math.ntnu.no; hakonma@stud.math.ntnu.no ABSTRACT A statistical test for trend in recurrent event data, for instance failure data from a repairable system, based on the general null hypothesis of a renewal process and with power against both monotonic and nonmonotonic trends is proposed. Simulations show that the test has favorable qualities as a general test for trend by having good power properties against both monotonic and nonmonotonic alternatives. 1 INTRODUCTION In recurrent event data it is often of interest to detect possible systematic alterations in the pattern of events. An example is a repairable system for which it is important to detect possible systematic changes in the pattern of failures. Such changes can for instance be caused by various aging effects or reliability growth. We say that there is a trend in the pattern of failures if the inter-failure times tend to alter in some systematic way, which means that the inter-failure times are not identically distributed. By using statistical trend tests it is possible to decide whether such an alteration is statistically significant or not. In this paper we shall focus on the repairable system example, but the methods presented are relevant for any kind of recurrent event data which can be modeled by the same statistical models as used for repairable systems in this paper. A trend in the pattern of failures of a repairable system can be either monotonic, corresponding to an improving or deteriorating system, or non-monotonic, corresponding to for instance cyclic variations or a so called bathtub trend characterizing a system going through the three phases infant illness, useful life and wear out. The purpose of the paper is to present a general test for trend with power against both monotonic and nonmonotonic trends. Common statistical models for systems without trend are the homogeneous Poisson process (HPP) and the renewal process (RP). The HPP, assuming independent identically exponentially distributed interarrival times, is a special case of the far more general RP, assuming independent identically distributed interarrival times. For both models there exist a number of tests with power against monotonic trends but few tests with power against nonmonotonic trends. A test with power both against monotonic and nonmonotonic trends based on an HPP model, called the Anderson-Darling test for trend, was studied by Kvaløy & Lindqvist [1]. The problem, however, with this test and many other tests based on the null hypothesis of an HPP is that even if the null hypothesis is rejected, it does not necessarily indicate a trend, just that the process is not an HPP. An RP may for instance cause rejection (Lindqvist et al. [2]).
Elvebakk [3] has studied how the Anderson-Darling test for trend could be generalized to a test of the null hypothesis RP by use of resampling techniques. In the present paper we discuss how the Anderson-Darling test for trend can be directly generalized to a test of the null hypothesis RP, without using resampling techniques. This yields a straightforwardly calculated test, which is a generally applicable test for trend with good power properties both against monotonic and non-monotonic trends. We also show that the well known Lewis-Robinson test (Lewis & Robinson [4]) follows from a similar derivation taking the same starting point, and we compare the generalized Anderson-Darling test for trend to the Lewis-Robinson test and the test ( [5]) in a simulation study. 2 MODELS AND TESTS 2.1 Notation and terminology In this paper attention is restricted to one system which is observed from time t =. The successive failure times are denoted T 1,T 2,... and the corresponding interfailure or interarrival times are denoted X 1,X 2,... where X i = T i T i 1, i =1, 2,... (1) Another way to represent the same information is by the counting process representation N(t) = number of failures in (,t]. (2) A system is said to be failure censored if it is observed until a given number n of failures has occurred, and time censored if it is observed until a given time τ. We say that a system inhibits no trend if the marginal distributions of all interarrival times are identical, otherwise there is a trend. If the expected length of the interarrival times is monotonically increasing or decreasing with time, corresponding to an improving or a deteriorating system, there is a monotone trend (a decreasing or an increasing trend), otherwise the trend is nonmonotone. 2.2 Renewal process The stochastic process T 1,T 2,... is an RP if the interarrival times X 1,X 2,... are independent and identically distributed. The conditional intensity of an RP given the history F t up to, but not including time t, can be written γ(t F t )=z(t T N(t ) ) (3) where z(t) is the hazard rate of the distribution of the interarrival times and T N(t ) is the last failure time strictly before time t. 2.3 Trend-renewal process The trend-renewal process (TRP) is presented by Lindqvist et al. [6], and is a generalization of the RP to a process with trend. The RP, HPP and nonhomogenous Poisson process (NHPP) are all special cases of this process. Let λ(t) be a nonnegative function, called the trend function, defined for t andlet Λ(t) = t λ(u)du. Then the process T 1,T 2,... is a TRP if Λ(T 1 ), Λ(T 2 ),... is an RP. The conditional intensity of a TRP may be written (Lindqvist et al. [6]) γ(t F t )=z(λ(t) Λ(T N(t ) ))λ(t) (4)
where z(t) is the hazard rate of the distribution of the underlying RP. In other words the intensity depends on both the age of the system and the time since last failure. The NHPP follows as the special case when the RP part of the TRP is an HPP with intensity 1. In this case the hazard rate z(t) 1 and thus γ(t F t )=λ(t). The special case of an RP follows when λ(t) is constant, and the HPP follows when both λ(t) and z(t) are constant. 2.4 Generalized Anderson-Darling test for trend The construction of the generalized Anderson-Darling test for trend is based on the independent increment property of the RP and Donsker s theorem. Let ξ 1,...,ξ n be n independent identically distributed random variables with E(ξ i )= and Var(ξ i )=σ 2,let[u] denote the integer part of u, letz j = ξ 1 + + ξ j and define Q n (t) = Z [nt] σ nt [nt] + n σ n ξ [nt]+1 (5) for t 1. Then according to Donsker s theorem (e.g Billingsley [7]) Q d n W,where d denotes convergence in distribution and W is a Brownian motion. Now consider a failure censored RP with interarrival times X 1,...,X n. Define ξ i = X i µ where µ = E(X i )andletz j = ξ 1 + + ξ j = j i=1 X i jµ = T j jµ. It follows from Donsker s theorem that V d n,µ,σ W where V n,µ,σ (t) = T [nt] [nt]µ σ n + nt [nt] σ n (X [nt]+1 µ) (6) and σ 2 = Var(X i ). Further defining W (t) =W (t) tw (1), which is a Brownian bridge, we will have that Vn,σ d W,where n X Vn,σ(t) =V n,µ,σ (t) tv n,µ,σ (1) = σ (T [nt] +(nt [nt]) X [nt]+1 t) 2 (7) T n T n where X = T n /n. Notice that µ cancels in (7). Replacing σ by a consistent estimator ˆσ, we conclude that V n,ˆσ d W. It follows from the above that under the null hypothesis of no trend, Vn,ˆσ (t) will be approximately a Brownian bridge. On the other hand, if there is a trend in the data, V n,ˆσ (t) is likely to deviate from the Brownian bridge. Thus tests for trend can be based on measures of deviation from a Brownian bridge of Vn,ˆσ (t). A suggestion is to use the Anderson-Darling type statistic GAD = 1 (Vn,ˆσ (t))2 1 t(1 t) dt d (W (t)) 2 dt. (8) t(1 t) The test based on this statistic will be called the generalized Anderson-Darling test for trend, and the limit distribution is the usual limit distribution of the Anderson-Darling statistic (Anderson & Darling [8], [9]). It is easily realized that the statistic will have sensitivity against deviations from the Brownian bridge caused both by monotonic and nonmonotonic trends. Straightforward but tedious calculations show that GAD = n X 2 [ n q 2 ˆσ 2 i ln( i i 1 )+(q i + r i ) 2 ln( n i +1 ] r2 i n i n ) (9) i=1
where q i =(T i ix i )/T n and r i = nx i /T n 1 Various consistent estimators of σ 2 can be used. Rather than using the usual estimator, S 2 = n i=1 (X i X) 2 /(n 1), we recommend using the estimator ˆσ 2 = n 1 1 (X i+1 X i ) 2. (1) 2(n 1) i=1 This estimator is consistent under the null hypothesis of independent and identically distributed interarrival times, but tends to be smaller than S 2 in cases with positive dependence between subsequent interarrival times. Thus using this variance estimator yields a test statistic with larger power against alternatives like monotonic and bathtub shaped trends than using the estimator S 2. Finally, empirical explorations have shown that the small sample level properties of the test can be improved by multiplying the test statistic by an (asymptotically negligible) factor (1 4/n) (Malmedal [1]). Thus the final form of the proposed test statistic is 2 (n 4) X GAD = ˆσ 2 [ n qi 2 ln( i i=1 i 1 )+(q i + r i ) 2 ln( n i +1 n i r2 i n ) where q i =(T i ix i )/T n, r i = nx i /T n 1andˆσ 2 is calculated from (1). Using the limit distribution of the test statistic (Anderson & Darling [8], [9]), the null hypothesis is rejected on the 5% level if GAD > 2.492. For time censored data we propose conditioning on the observed number of failures, and treating the data as if they were data from a failure censored system. 2.5 The Lewis-Robinson test Other trend tests can be constructed by following the same approach but using other measures of deviation from the Brownian bridge of Vn,ˆσ (t). A possibility is for instance to consider the statistic 1 1 Vn,ˆσ (t)dt d W (t)dt (12) where 1 W (t)dt is normally distributed with expectation and variance 1/12. This statistic will primarily have power against deviations from a Brownian bridge caused by monotonic trends. By scaling the test statistic to be asymptotically standard normally distributed, straightforward calculations yield the test statistic 1 LR Vn,ˆσ = (t) dt = X n 1 i=1 T i n 1T 2 n 12 ˆσ T n. (13) n 12 This test statistic is a variant of the classical Lewis-Robinson test statistic (Lewis & Robinson [4]), often written LR = X n 1 i=1 T i n 1T 2 n (14) S T n 1 n 12 For time censored data n 1 is replaced by n and T n by τ. The original derivation by Lewis & Robinson [4] differs from the derivation presented here by taking a different starting point and using other arguments to prove the asymptotic normality. ] (11)
2.6 The test A classical trend test is the rank test developed by [5]. This is a test of the null hypothesis RP versus the alternative of a monotonic trend. The test statistic is computed by counting the number of reverse arrangements, M, among the interarrival times X 1,...,X n. We have a reverse arrangement if X i <X j for i<j,andm is thus M = n 1 n i=1 j=i+1 I(X i <X j ) (15) where I(A) = 1 or according to whether the event A occurs or not. Under the null hypothesis M is approximately normally distributed with expectation n(n 1)/4 and variance (2n 3 +3n 2 5n)/72 for n 1. For n<1 there exist tables. 3 SIMULATION STUDY In this section the generalized Anderson-Darling test for trend (11), the Lewis-Robinson test (14) and the test (15) are compared in a simulation study. Rejection probabilities are estimated by simulating 1 data sets for each choice of model and parameter values, and recording the relative number of rejections of each test. In the curves of (Fig. 2) and (Fig. 4) the rejection probabilities are simulated and plotted in a number of points, with straight lines drawn between the points. Let ˆp denote the estimated rejection probability. Then the standard deviation of ˆp is ˆp(1 ˆp)/1.16. All simulations are done in C++. The nominal significance level has been set to 5%, and only simulations of failure censored processes are reported. Generating data T 1,T 2,... from a TRP is straightforward. Recall that the process T 1,T 2,... is a TRP if Λ(T 1 ), Λ(T 2 ),... is an RP. Then failure times T 1,T 2,... from a TRP are generated by first generating arrival times θ 1,θ 2,... from the relevant RP and transforming these to T 1 =Λ 1 (θ 1 ),T 2 =Λ 1 (θ 2 ),... Arrival times of the RP are generated by simulating interarrival times θ i θ i 1 from the RP-distribution. 3.1 Level properties First the level properties of the tests are studied by generating datasets from an HPP with intensity 1 (i.e. the interarrival times are independent and identically exponentially distributed with expectation 1). In (Fig. 1) the simulated level of the tests for samples of size 4 to 8 are reported. The plots in (Fig. 1) show that all tests have satisfying level properties for samples of sizes larger that around 15. For very small sample sizes all.9 Generalized Anderson Darling.9 Lewis Robinson.9.8.8.8.7.7.7.6.5.4.3.6.5.4.3.6.5.4.3.2.2.2.1.1.1 1 2 3 4 5 6 7 8 Sample size 1 2 3 4 5 6 7 8 Sample size 1 2 3 4 5 6 7 8 Sample size Figure 1: Simulations of HPPs with intensity 1 and sample sizes from 4 to 8.
tests tend to be conservative and the generalized Anderson-Darling test for trend is the most conservative test. For the test the normal approximation has been used for all sample sizes. The level properties of this test for samples smaller than 1 are better if critical values from tables are used. 3.2 Power properties 3.2.1 Monotonic trend Datasets with a monotonic trend are generated by simulating data from TRPs with the underlying RP distribution being Weibull and the trend function λ(t), governing the trend of the process, being on the power law form λ(t) =bt b 1. The intensity of the process can then be written γ(t F t )=αβ(t b TN(t ) b )β 1 bt b 1. (16) For samples of size 25 the rejection probability as a function of b is simulated, where b<1 corresponds to a decreasing trend, b = 1 corresponds to no trend and b>1 corresponds to an increasing trend. This is repeated for two different values of the shape parameter β of 1 Overdispersed TRP 1 Underdispersed TRP.9.9.8.8.7.6.5.4.3.7.6.5.4.3 GAD LR.2.1 GAD LR.5 1 1.5 2 2.5 3 b.2.1.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Figure 2: Simulations of TRPs with underlying Weibull RPs with shape parameters respectively.75 (overdispersed TRP) and 1.5 (underdispersed TRP), trend function λ(t) =bt b 1 and samples of size 25. b the underlying Weibull distribution, β =.75 and β = 1.5, corresponding respectively to a process which is overdispersed or underdispersed relative to an NHPP. The scale parameter α is set to 1. The results are displayed in (Fig. 2). We see in (Fig. 2) that the generalized Anderson-Darling test for trend and the Lewis- Robinson test have fairly similar power properties, with the generalized Anderson-Darling test being a bit more powerful than the Lewis-Robinson against increasing trend and opposite against decreasing trend. The test is clearly less powerful than the other tests against increasing trend but is the most powerful test against decreasing trend in the overdispersed case. 3.2.2 Nonmonotonic trend Datasets with a bathtub trend are generated by simulating data from TRPs with trend function on the form displayed in (Fig. 3). Here d represents the average of the function λ(t) over the interval [,τ]. The degree of bathtub shape can be expressed by the parameter c, withc = corresponding to a horizontal line (no trend).
(t) a d b c e fi e fi t Figure 3: Bathtub-shaped trend function. For samples of size 25 and 75, the rejection probability as a function of c is simulated with τ set to a value which implies that the expected number of failures in [,τ] becomes respectively approximately 25 and approximately 75. The parameter e is adjusted according to c to make the expected number of failures in each phase (decreasing, no, increasing trend) equal. The shape parameter of the underlying Weibull distribution is set to β = 1.5 (underdispersed). The results are displayed in (Fig. 4). 1.9.8.7.6.5.4.3 25 failures GAD LR 1.9.8.7.6.5.4.3 75 failures GAD LR.2.2.1.1.1.2.3.4.5.6 c.1.2.3.4.5.6 Figure 4: Simulations of TRPs with underlying Weibull RPs with shape parameter 1.5 (underdispersed TRP), trend function displayed in (Fig. 3) and samples of size respectively 25 and 75. We see in (Fig. 4) that the generalized Anderson-Darling test for trend has clearly better power properties than the other tests in these cases. Also note that the generalized Anderson-Darling test for trend improves substantially with increasing sample size while the other tests do not improve much. Similar patterns are seen if we consider overdispersed TRPs (Malmedal [1]). One difference seen for overdispersed TRPs is that none of the tests are particular powerful for small sample sizes due to the larger variance of the underlying RP (Malmedal [1]). 3.3 Discussion In the simulations presented the generalized Anderson-Darling test for trend is the best test against increasing and bathtub shaped trend, while the other tests are slightly better against decreasing trend. It is not surprising that the generalized Anderson-Darling test is better than the other tests against bathtub shaped trend since this test, contrary to c
the other tests, is constructed to have power against both nonmonotonic alternatives as well as monotonic ones. The fact that the generalized Anderson-Darling test also performs at least as good as the other tests against monotonic alternatives makes this test a very attractive choice as a trend test for general use. The only drawback with the generalized Anderson-Darling test for trend (11) is that it is too conservative for very small sample sizes. In such cases other level correcting factors than 1 4/n could be used. Another alternative is to use a bootstrap version of the test (Elvebakk [3]; Malmedal [1]). The bootstrap test has good level properties even for small samples and it also turns out that bootstrapping in many cases improves the power properties of the test, in some cases substantially (Malmedal [1]). 4 CONCLUSION A test for trend based on the general null hypothesis of an RP which has power against both monotonic and nonmonotonic trends has been presented. This test has good overall properties compared to existing trend tests and is recommended for general use. For very small sample sizes it is recommended to use a resample version of the test. ACKNOWLEDGEMENT We would like to thank associate professor Michael Kosorok for introducing us to how the Anderson-Darling test could be generalized from the null hypothesis HPP to RP by using the independent increment property of the problem. REFERENCES [1] Kvaløy, J. T. & Lindqvist, B. H., TTT-based Tests for Trend in Repairable Systems Data, Reliability Engineering and System Safety, 6, (1998), 13-28. [2] Lindqvist, B. H., Kjønstad, G. A. & Meland, N., Testing for trend in repairable systems data, Proceedings of ESREL 94, La Baule, France, May 3 - June 3, 1994. [3] Elvebakk, G., Analysis of Repairable Systems Data: Statistical Inference for a Class of Models Involving Renewals, Heterogeneity and Time Trends, PhD thesis, Department of Mathematical Sciences, Norwegian University of Science and Technology, 1999. [4] Lewis, P. A. & Robinson, D. W., Testing for a monotone trend in a modulated renewal process, in F. Proschan & R.J Serfling (eds), Reliability and Biometry, 163-182, SIAM, Philadelphia, 1974. [5], H. B., Nonparametric tests against trend, Econometrica, 13, (1945), 245-259. [6] Lindqvist, B. H., Elvebakk, G. & Heggland, K., The Trend-Renewal Process for Statistical Analysis of Repairable Systems, submitted for publication, 1999. [7] Billingsley, P., Convergence of Probability Measures, John Wiley, New York, 1968. [8] Anderson, T. W. & Darling, D. A., Asymptotic theory of certain goodness of fit criteria based on stochastic processes, Annals of Mathematical Statistics, 23, (1952), 193-212. [9] Anderson, T. W. & Darling, D. A., A test of goodness of fit, Journal of the American Statistical Association, 49, (1954), 765-769. [1] Malmedal, H., Trend testing, Diploma Thesis, (In Norwegian), Department of Mathematical Sciences, Norwegian University of Science and Technology, 21.