In reality, failure data are often collected under diffract operational conditions (covariates), leading to heterogeneity among the data. Heterogeneity can be classified as observed and unobserved heterogeneity. Un-observed heterogeneity is the effect of unknown, unrecorded, or missing covariates. In most reliability studies, the effect of unobserved covariates is neglected. This may lead to inaccurate reliability modeling, and consequently, wrong operation and maintenance decisions. There is a lack of a systematic approach to model the unobserved covariate in reliability analysis. This paper aims to present a framework for reliability analysis in the presence of unobserved and observed covariates. Here, the unobserved covariates will be analyzed using frailty models. A case study will illustrate the application of the framework.

实际上，故障数据通常是在衍射操作条件（协变量）下收集的，导致数据之间存在异质性。异质性可分为观察到的和未观察到的异质性。未观察到的异质性是未知、未记录或缺失协变量的影响。在大多数可靠性研究中，未观察到的协变量的影响被忽略。这可能会导致不准确的可靠性建模，从而导致错误的操作和维护决策。缺乏对可靠性分析中未观察到的协变量进行建模的系统方法。本文旨在提出一个在存在未观察到和观察到的协变量的情况下进行可靠性分析的框架。在这里，未观察到的协变量将使用衰弱模型进行分析。案例研究将说明该框架的应用。