Concentration-based confidence intervals for U-statistics

Concentration inequalities have become increasingly popular in machine learning, probability, and statistical research. Using concentration inequalities, one can construct confidence intervals (CIs) for many quantities of interest. Unfortunately, many of these CIs require the knowledge of population variances, which are generally unknown, making these CIs impractical for numerical application. However, recent results regarding the simultaneous bounding of the probabilities of quantities of interest and their variances have permitted the construction of empirical CIs, where variances are replaced by their sample estimators. Among these new results are two-sided empirical CIs for U-statistics, which are useful for the construction of CIs for a rich class of parameters. In this article, we derive a number of new one-sided empirical CIs for U-statistics and their variances. We show that our one-sided CIs can be used to construct tighter two-sided CIs for U-statistics, than those currently reported. We also demonstrate how our CIs can be used to construct new empirical CIs for the mean, which provide tighter bounds than currently known CIs for the same number of observations, under various settings

Comparison of Dynamic Treatment Regimes with An Ordinal Outcome

Sequential multiple assignment randomized trials (SMART) are used to develop optimal treatment strategies for patients based on their medical 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. 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 medical practice; for example, the quality of life (poor, moderate, good) is an ordinal variable. 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 results related to $dGOR$ and derive the asymptotic properties of it's estimate. We derive the required sample size formula. Then, we extend the proposed methodology to a $K$-stage SMART. Finally, we discuss some alternative ways to estimate $dGOR$ using concordant-discordant pairs and multi-sample $U$-statistic. 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 STAR*D, a multistage 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 method accessible to other researchers and practitioners