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Sequential multiple assignment randomized trials (SMART) are used to construct data-driven optimal treatment strategies for patients based on their treatment and covariate histories in different branches of medical and behavioral sciences where a sequence of treatments are given to the patients; such sequential treatment strategies are often called dynamic treatment regimes (DTR). In the existing literature, the majority of the analysis methodologies for SMART studies assume a continuous primary outcome. However, ordinal outcomes are also quite common in clinical practice; for example, the quality of life is often measured in an ordinal scale (e.g., poor, moderate, good). In this work, first, we develop the notion of dynamic generalized odds-ratio (dGOR) to compare two dynamic treatment regimes embedded in a 2-stage SMART with an ordinal outcome. We propose a likelihood-based approach to estimate dGOR from SMART data. Next, we discuss some combinatorial properties of dGOR and derive the asymptotic properties of its estimate. We discuss some alternative ways to estimate dGOR using concordant-discordant pairs and multi-sample U-statistic. Then, we extend the proposed methodology to a K-stage SMART. Furthermore, we propose a basic policy search algorithm that uses dGOR to find an optimal DTR within a finite class. A simulation study shows the performance of the estimated dGOR in terms of the estimated power corresponding to the derived sample size. We analyze data from Sequenced Treatment Alternatives to Relieve Depression (STAR*D), a multi-stage randomized clinical trial for treating major depression, to illustrate the proposed methodology. A freely available online tool using R statistical software is provided to make the proposed methodology accessible to other researchers and practitioners. This is a joint work with Bibhas Chakraborty, Duke-NUS Medical School.
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