  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)).
