Browsing by Subject "Conditional error probabilities"

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    Early Detection of Treatment's Side Effects: A Sequential Approach
    (2025-05) Wang, Jiayue; Boukai, Ben; Li, Fang; Peng, Hanxiang; Sarkar, Jyotirmoy
    With the emergence and spread of infectious diseases with pandemic potential, such as COVID-19, the urgency for vaccine development has led to unprecedented compressed and accelerated schedules that shortened the standard development timeline. To monitor the potential side effect(s) of the vaccine during the (initial) vaccination campaign, we developed an optimal sequential test that allows for the early detection of potential side effect(s). This test employs a rule to stop the vaccination process once the observed number of side effect incidents exceeds a certain (pre-determined) threshold. The optimality of the proposed sequential test is justified when compared with the (α,β) optimality of the non-randomized fixed-sample Uniformly Most Powerful (UMP) test. Firstly, we construct an optimal sequential procedure for the case of a single side effect. We study the properties of the sequential test and derive the exact expressions of the Average Sample Number (ASN) curve of the stopping time (and its variance) via the regularized incomplete beta function. Additionally, we derive the asymptotic distribution of the relative ’savings’ in ASN as compared to fixed sample size. Moreover, we construct the post-test parameter estimate and studied its sampling properties, including its asymptotic behavior under local-type alternatives. These limiting behavior results are the consistency and asymptotic normality of the post-test parameter estimator. We conclude this part with a small simulation study illustrating the asymptotic performance of the point and interval estimation and provide a detailed example, based on COVID-19 side effect data (see Beatty et al. (2021)) of our suggested testing procedure. Next, we propose an optimal sequential procedure for the case of two (or more) side effects. While the sequential procedure we employ, simultaneously monitors several of the treatment’s side effects, the (α,β)-optimal test we propose does not require any information about the inter-correlation between these potential side effects to obtain the optimal sample size and critical values. However, in all of the subsequent analyses, including the derivations of the exact expressions of the 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. 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 Ilori et al. (2022)). Finally, we consider the Bayesian framework and construct a Bayesian Sequential testing procedure to test the Relative Risk between two specific treatments based on the binary data obtained from the two-arm clinical trial. We noticed that the optimal sequential test mentioned in the first part can be utilized here to test Relative Risk as a one-sided hypothesis testing procedure. To apply the optimal sequential test more straightforward, we introduce the Bayesian framework into our analysis. Since by the Stopping Rule Principle (SRP), the posterior probabilities remain unaffected by the stopping rule used to reach that point with accumulated data. Accordingly, using the Bayesian test, we obtain the corresponding decision on each data point without affecting by the optimal stopping rule as in classical sequential test. Moreover, with the modified Bayesian test, we obtain the corresponding conditional error probabilities on each data point. Specifically, we utilized the data from Silva, Kulldorff, and Katherine Yih (2020) to analyze the results obtained from two tests under several different priors and make the conclusion.