Outlying observations are frequently encountered in a wide spectrum of scientific domains, posing significant challenges for the generalizability of statistical models and the reproducibility of downstream analysis. These observations can be identified through influential diagnosis, which refers to the detection of observations that are unduly influential on diverse facets of statistical inference. To date, methods for identifying observations influencing the choice of a stochastically selected submodel have been underdeveloped, especially in the high-dimensional setting where the number of predictors p exceeds the sample size n. Recently we proposed an improved diagnostic measure to handle this setting. However, its distributional properties and approximations have not yet been explored. To address this shortcoming, the notion of exchangeability is revived, and used to determine the exact finite- and large-sample distributions of our assessment metric. This forms the foundation for the introduction of both parametric and non-parametric approaches for its approximation and the establishment of thresholds for diagnosis. The resulting framework is extended to logistic regression models, followed by a simulation study conducted to assess the performance of various detection procedures. Finally the framework is applied to data from an fMRI study of thermal pain, with the goal of identifying outlying subjects that could distort the formulation of statistical models using functional brain activity in predicting physical pain ratings. Both linear and logistic regression models are used to demonstrate the benefits of detection and compare the performances of different detection procedures. In particular, two additional influential observations are identified, which are not discovered by previous studies.

翻译：异常观测值在广泛的科学领域中频繁出现，对统计模型的普适性和下游分析的可重复性构成了重大挑战。这些观测值可通过影响诊断进行识别，即检测对统计推断各方面产生不当影响的观测值。迄今为止，用于识别影响随机选择子模型选择的观测值的方法尚不完善，尤其是在预测变量数量p超过样本量n的高维场景中。最近我们提出了一种改进的诊断度量来处理这一场景。然而，其分布特性与近似方法尚未得到充分探索。为弥补这一不足，本文重新引入可交换性概念，并利用其确定评估指标的精确有限样本与大样本分布。这为引入参数化和非参数化近似方法以及建立诊断阈值奠定了基础。所得框架被扩展至逻辑回归模型，随后通过模拟研究评估各种检测程序的性能。最后，该框架应用于一项热痛觉功能磁共振成像研究数据，旨在识别可能扭曲利用脑功能活动预测物理疼痛评分的统计模型构建的异常受试者。研究同时使用线性回归和逻辑回归模型来论证检测的益处，并比较不同检测程序的性能。特别地，研究识别出两个先前未发现的额外影响观测值。