Optimal Sequential Procedure for Early Detection of Multiple Side Effects

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)).