Falta de planificación efectiva

3.8

Concluding remarks

This chapter has presented statistical methods for ALT using the Arrhenius-Weibull model under constant stress testing, using theory of imprecise probability [9, 10], where the imprecision results from the use of the likelihood ratio test [63]. The proposed method applies a classical test for comparison of the survival distribu- tions at different stress levels. The observations at the increased stress levels were transformed to interval-valued observations at the normal stress level, by developing imprecision in the link function of the Arrhenius model via the likelihood ratio test between the pairwise stress levels. Using the Arrhenius model, we linked the data at different stress levels to the normal stress level, after which NPI can be used at normal stress level. We found that using an interval of values for the parameter in the link function between different stress levels enabled us to achieve a greater level of robustness than if we were to use a single point for the parameter. Using the proposed approach, the intervals [γ, γ] for the parameter γ for the link function have adequate imprecision if the model fits well. However, the intervals [γ, γ] for the parameter γ for the link function get wider if the model fits poorly. The latter can happen if the model assumptions are not fully correct, for example, using some mis- specification cases. However, if we have huge imprecision, the remaining inferences are probably of no use at all. Therefore, it will be a strong recommendation to do more detailed modelling or sample more data. We also comment on this in Chapter 6. Regarding the choice of the values of the factors for assessing robustness of the methods, we only show that any suggested form of misspecification can be included in simulations to then study the level of robustness.

The application of our novel method in this chapter assumed that we have failure times observed at the normal stress levelK0. The assumption of having failure data at the normal stress levelK0 may not be realistic in real world applications. In this case, we can apply our method to a higher stress level than the normal stress level

K0. The combined data at that level can then be transformed all together to the normal stress levelK0. Investigating this is a topic for future research.

Usually, events of interest in reliability and survival analysis are failure times [19, 25], such data often includes right-censored observations. The A(n) assumption

3.8. Concluding remarks 74

cannot handle right-censored observations, and demands fully observed data. Coolen and Yan [21] presented a generalization ofA(n), called rc-A(n), which is suitable for NPI with right-censored data, as discussed in Section 2.4. This method can be used at the second step in our approach if there are right-censored data, and the likelihood ratio test can just be applied in the first step. So generalizing this method to data including right-censored observations is straightforward, which will be illustrated in the next chapter.

We have considered the power-Weibull model to illustrate the use of our method without the assumption of equal shape parameter for the Weibull distribution at all stress levels too, and that other lifetime distributions and link functions can be applied, as long as the transformation of the failure times from higher stress levels to the normal stress level is monotonously increasing function. The approach presented in this chapter with the assumption of the Weibull failure time distributions at each stress level can be deleted and using nonparametric hypothesis tests instead, which will be illustrated in Chapter 4. In that chapter, we will also explain why we use pairwise tests instead of one test on all stress levels simultaneously.

Chapter 4

Imprecise inference based on the

log-rank test

4.1

Introduction

In this chapter, we develop a new imprecise statistical method for ALT data related to the method introduced in Chapter 3. There, we assumed an explicit model at each stress level, namely the Weibull failure time distribution. In this chapter, we do not assume a failure time distribution at each stress level, only a specific parametric link function between the levels.

In this chapter, we assume the Arrhenius model for the analysis of ALT with failure data under a constant level of stress also used in Section 3.3. If the Arrhenius model provides a realistic link between the different stress levels, then the observa- tions transformed from the increased stress levels to the normal stress level should be indistinguishable. For clarity, according to this model, an observationti _{at stress}

level i, subject to stress Ki, can be transformed to an observation at the normal

stress level K0, by the equation

ti→0 =tiexp γ K0 − γ Ki , (4.1.1)

where Ki is the accelerated temperature at level i (Kelvin), K0 is the normal tem- perature at level 0 (Kelvin), andγ is the parameter of the Arrhenius model.

4.1. Introduction 76

Note that in this chaper there is no distribution assumed at each stress level, hence we replace the likelihood ratio test used in Chapter 3, by the log-rank test. In this chapter, we propose a new log-rank test based method for predictive inference on a future unit functioning at the normal stress level. Testing the equality of the survival distribution of two or more independent groups, as we do in this chapter, is possible by using a nonparametric statistical test. There are several nonparametric test procedures that can be used to test the equality of the survival distributions; a popular one is the log-rank test [46, 65]. We use the log-rank test to find the intervals of values of the parameter γ of the Arrhenius link function for which we do not reject the null hypothesis that two groups of failure data, possibly including right-censored observations are, derived from the same underlying distribution. This can be interpreted such that, for such values ofγ, the combined data at stress level

K0 are well mixed. This interval of values of the parameter γ of the Arrhenius link function is used to transform the data from the increased stress levels to the normal stress level. Then, the ultimate aim is inference at the normal stress level. We consider nonparametric predictive inference at the normal stress level combined with the Arrhenius model linking observations at different stress levels. Note we also assume that we have failure data at the normal stress level, as discussed in Chapter 3. Note also that this method follows the same procedure as in Chapter 3, except that we use a different classical hypothesis test because we do not assume the Weibull failure time distribution at each stress level.

This chapter is organized as follows. Section 4.2 introduces the main idea of im- precise predictive inference based on ALT and the log-rank test. The main novelty of the approach in this chapter is that by using a classical nonparametric test, we do not need to assume a parametric failure time distribution at each stress level. This should make the method more widely applicable than the method presented in Chapter 3. In Section 4.3 we explain why we do not use a single log-rank test on all stress levels. Section 4.4 illustrates our method in seven examples using sim- ulated data and data from the literature. Section 4.5 presents results of simulation studies that investigate the performance of the proposed method using the Arrhe- nius link function. Section 4.6 presents results of simulation studies of robustness