# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 1 Introduction
### Assessment of Recent Work  
Several scientific domains other than reliability engineering have used power analysis for years. Here, the focus often is on the evaluation of statistically significant effects in relation to the sample size and the determination of rules for termination in the event of a low probability of success in a study (see, e.g., [3; 4; 5; 6]). In quality engineering, statistical power analysis is used in order to efficiently deal with lot acceptance tests and production parameter assessment [7]. In the reliability domain and within the scope of life testing of technical products, a holistic approach for power analyses has not been established for successful demonstration of reliability targets considering all possible test types. The uncertainty of the tested sample is assessed primarily by means of the confidence level, which solely addresses the type I statistical error [8]. Several tested samples of the same size result in different confidence intervals due to sampling variability [9]. Therefore, it is not sufficient to calculate the sample size required to demonstrate the reliability target solely by means of the confidence level; e.g., see [8]. A simple transfer of these power analyses to the life testing framework is challenging, as state-of-the-art analytic power analyses are only permissible for normally distributed residuals [1]. This does not apply to life data, which are usually Weibull or log-normally distributed [9]. Further challenges in life testing are the various test types and configurations, such as time- or failure-censored tests with or without load acceleration. For accelerated life tests, a life-stress model must be determined to demonstrate reliability for field load. It is possible to carry out life tests at different system levels, as well, such as at the component or subsystem level, which brings additional challenges. These aspects must be considered for the application of power analyses when planning reliability tests. The metric for the assessment of the accuracy must contain information as to whether a life test is capable of demonstrating a required reliability \(R_{\text{r}}\) with required lifetime \(t_{\text{r}}\) and confidence level \(C_{\text{r}}\). Due to the randomness of the process and the sole frequentist nature of reliability targets, it is necessary to deal with probabilities, and statistical power (as a statistical variable) must therefore be transferred to the domain of reliability test planning in order to be interpreted as the probability of a successful reliability test. The challenges are the generalization of the target variable in order for different test strategies to be dealt with and compared to each other, and performing the calculation in an efficient way. In practical applications, success run tests (SR tests, called zero-failure tests or failure-free tests) are often used for reliability demonstration because the required sample size and reliability can easily be obtained using the binomial distribution [10]. Due to the equation only representing a significance test, only the confidence level \(C\) is considered. Over the past decades, there has been great effort to reduce the required sample size for the SR test by using [PERSON] rule [11], as its testing effort is quite high, especially for high reliability requirements, and even higher if failures occur during testing. Although the concept of a reliability demonstration has to be based on a frequentist view, due to the confidence level which is required and to be demonstrated [PERSON] rule continues to be used for aggregation of additional information. However, the demonstration is carried out solely using the frequentist confidence level instead of a credibility interval or other related concepts. Methods described by [PERSON] and [PERSON] [12; 13], [PERSON] and [PERSON] [14], [PERSON] et al. [15; 16], [PERSON] [17; 18; 19], [PERSON] and [PERSON] [20], and [PERSON] et al. [21] use reliability information from similar products such as predecessors.