In this paper, we propose an optimal sequential procedure for the early detection of potential side effects resulting from the administration of some treatment (e.g. a vaccine, say). The results presented here extend previous results obtained in Wang and Boukai (2024) who study the single side effect case to the case of two (or more) side effects. While the sequential procedure we employ, simultaneously monitors several of the treatment's side effects, the $(\alpha, \beta)$-optimal test we propose does not require any information about the inter-correlation between these potential side effects. However, in all of the subsequent analyses, including the derivations of the exact expressions of the Average Sample Number (ASN), the Power function, and the properties of the post-test (or post-detection) estimators, we accounted specifically, for the correlation between the potential side effects. In the real-life application (such as post-marketing surveillance), the number of available observations is large enough to justify asymptotic analyses of the sequential procedure (testing and post-detection estimation) properties. Accordingly, we also derive the consistency and asymptotic normality of our post-test estimators; results which enable us to also provide (asymptotic, post-detection) confidence intervals for the probabilities of various side-effects. Moreover, to compare two specific side effects, their relative risk plays an important role. We derive the distribution of the estimated relative risk in the asymptotic framework to provide appropriate inference. To illustrate the theoretical results presented, we provide two detailed examples based on the data of side effects on COVID-19 vaccine collected in Nigeria (see Nigeria (see Ilori et al. (2022)).

翻译：本文提出了一种最优序贯程序，用于早期检测某种治疗（如疫苗）实施后可能产生的潜在副作用。本文的研究成果将Wang和Boukai（2024）针对单一副作用情况的结果扩展至两种（或多种）副作用情形。尽管我们采用的序贯程序能够同时监测治疗的多种副作用，但所提出的(α, β)-最优检验无需依赖这些潜在副作用之间的相互关联信息。然而，在后续的所有分析中——包括平均样本量（ASN）精确表达式的推导、功效函数以及检测后（即发现后）估计量的性质——我们特别考虑了潜在副作用之间的相关性。在实际应用（如上市后监测）中，可获取的观测数量足够大，足以支持对序贯程序（包括检测与检测后估计）性质进行渐近分析。据此，我们还推导了检测后估计量的一致性及渐近正态性；这些结果使我们能够为各种副作用的概率提供（渐近的、检测后的）置信区间。此外，在比较两种特定副作用时，其相对风险起着重要作用。我们在渐近框架下推导了估计相对风险的分布，以提供适当的推断。为说明提出的理论结果，我们基于尼日利亚收集的COVID-19疫苗副作用数据（见Ilori等（2022））提供了两个详细示例。