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The current body of research on developing optimal treatment strategies often places emphasis on intention-to-treat analyses, which fail to take into account the compliance behavior of individuals. Methods based on instrumental variables have been developed to determine optimal treatment strategies in the presence of endogeneity. However, these existing methods are not applicable when there are two active treatment options and the average causal effects of the treatments cannot be identified using a binary instrument. In order to address this limitation, we present a procedure that can identify an optimal treatment strategy and the corresponding value function as a function of a vector of sensitivity parameters. Additionally, we derive the canonical gradient of the target parameter and propose a multiply robust classification-based estimator for the optimal treatment strategy. Through simulations, we demonstrate the practical need for and usefulness of our proposed method. We apply our method to a randomized trial on Adaptive Treatment for Alcohol and Cocaine Dependence.

Existing methods in estimating the mean outcome under a given dynamic treatment regime rely on intention-to-treat analyses which estimate the effect of following a certain dynamic treatment regime regardless of compliance behavior of patients. There are two major concerns with intention-to-treat analyses: (1) the estimated effects are often biased toward the null effect; (2) the results are not generalizable and reproducible due to the potential differential compliance behavior. These are particularly problematic in settings with high level of non-compliance such as substance use disorder treatments. Our work is motivated by the Adaptive Treatment for Alcohol and Cocaine Dependence study (ENGAGE), which is a multi-stage trial that aimed to construct optimal treatment strategies to engage patients in therapy. Due to the relatively low level of compliance in this trial, intention-to-treat analyses essentially estimate the effect of being randomized to a certain treatment sequence which is not of interest. We fill this important gap by defining the target parameter as the mean outcome under a dynamic treatment regime given potential compliance strata. We propose a flexible non-parametric Bayesian approach, which consists of a Gaussian copula model for the potential compliances, and a Dirichlet process mixture model for the potential outcomes. Our simulations highlight the need for and usefulness of this approach in practice and illustrate the robustness of our estimator in non-linear and non-Gaussian settings.

Dynamic treatment regimens (DTRs), also known as treatment algorithms or adaptive interventions, play an increasingly important role in many health domains. DTRs are motivated to address the unique and changing needs of individuals by delivering the type of treatment needed, when needed, while minimizing unnecessary treatment. Practically, a DTR is a sequence of decision rules that specify, for each of several points in time, how available information about the individual's status and progress should be used in practice to decide which treatment (e.g., type or intensity) to deliver. The sequential multiple assignment randomized trial (SMART) is an experimental design widely used to empirically inform the development of DTRs. Sample size planning resources for SMARTs have been developed for continuous, binary, and survival outcomes. However, an important gap exists in sample size estimation methodology for SMARTs with longitudinal count outcomes. Further, in many health domains, count data are overdispersed - having variance greater than their mean. We propose a Monte Carlo-based approach to sample size estimation applicable to many types of longitudinal outcomes and provide a case study with longitudinal overdispersed count outcomes. A SMART for engaging alcohol and cocaine-dependent patients in treatment is used as motivation., Comment: Key words: dynamic treatment regimen (DTR); longitudinal outcome; overdispersed count; sample size estimation; sequential multiple assignment randomized trial (SMART)

One of the main goals of sequential, multiple assignment, randomized trials (SMART) is to find the most efficacious design embedded dynamic treatment regimes. The analysis method known as multiple comparisons with the best (MCB) allows comparison between dynamic treatment regimes and identification of a set of optimal regimes in the frequentist setting for continuous outcomes, thereby, directly addressing the main goal of a SMART. In this paper, we develop a Bayesian generalization to MCB for SMARTs with binary outcomes. Furthermore, we show how to choose the sample size so that the inferior embedded DTRs are screened out with a specified power. We compare log-odds between different DTRs using their exact distribution without relying on asymptotic normality in either the analysis or the power calculation. We conduct extensive simulation studies under two SMART designs and illustrate our method's application to the Adaptive Treatment for Alcohol and Cocaine Dependence (ENGAGE) trial., Comment: 20 pages, 5 figures, 3 tables

The cyclical and heterogeneous nature of many substance use disorders highlights the need to adapt the type or the dose of treatment to accommodate the specific and changing needs of individuals. The Adaptive Treatment for Alcohol and Cocaine Dependence study (ENGAGE) is a multi-stage randomized trial that aimed to provide longitudinal data for constructing treatment strategies to improve patients' engagement in therapy. However, the high rate of noncompliance and lack of analytic tools to account for noncompliance have impeded researchers from using the data to achieve the main goal of the trial. We overcome this issue by defining our target parameter as the mean outcome under different treatment strategies for given potential compliance strata and propose a Bayesian semiparametric model to estimate this quantity. While it adds substantial complexities to the analysis, one important feature of our work is that we consider partial rather than binary compliance classes which is more relevant in longitudinal studies. We assess the performance of our method through comprehensive simulation studies. We illustrate its application on the ENGAGE study and demonstrate that the optimal treatment strategy depends on compliance strata., Comment: 31 pages, 8 figures, 13 tables

Q-learning is a regression-based approach that is widely used to formalize the development of an optimal dynamic treatment strategy. Finite dimensional working models are typically used to estimate certain nuisance parameters, and misspecification of these working models can result in residual confounding and/or efficiency loss. We propose a robust Q-learning approach which allows estimating such nuisance parameters using data-adaptive techniques. We study the asymptotic behavior of our estimators and provide simulation studies that highlight the need for and usefulness of the proposed method in practice. We use the data from the "Extending Treatment Effectiveness of Naltrexone" multi-stage randomized trial to illustrate our proposed methods.

Clinicians and researchers alike are increasingly interested in how best to personalize interventions. A dynamic treatment regimen (DTR) is a sequence of pre-specified decision rules which can be used to guide the delivery of a sequence of treatments or interventions that are tailored to the changing needs of the individual. The sequential multiple-assignment randomized trial (SMART) is a research tool which allows for the construction of effective DTRs. We derive easy-to-use formulae for computing the total sample size for three common two-stage SMART designs in which the primary aim is to compare mean end-of-study outcomes for two embedded DTRs which recommend different first-stage treatments. The formulae are derived in the context of a regression model which leverages information from a longitudinal outcome collected over the entire study. We show that the sample size formula for a SMART can be written as the product of the sample size formula for a standard two-arm randomized trial, a deflation factor that accounts for the increased statistical efficiency resulting from a longitudinal analysis, and an inflation factor that accounts for the design of a SMART. The SMART design inflation factor is typically a function of the anticipated probability of response to first-stage treatment. We review modeling and estimation for DTR effect analyses using a longitudinal outcome from a SMART, as well as the estimation of standard errors. We also present estimators for the covariance matrix for a variety of common working correlation structures. Methods are motivated using the ENGAGE study, a SMART aimed at developing a DTR for increasing motivation to attend treatments among alcohol- and cocaine-dependent patients., Comment: Supplementary material at https://osf.io/q7zv8/ This version updates funding information not previously included. Text remains the final accepted version of the manuscript in Statistical Methods in Medical Research

Sequential, multiple assignment, randomized trial (SMART) designs have become increasingly popular in the field of precision medicine by providing a means for comparing sequences of treatments tailored to the individual patient, i.e., dynamic treatment regime (DTR). The construction of evidence-based DTRs promises a replacement to adhoc one-size-fits-all decisions pervasive in patient care. However, there are substantial statistical challenges in sizing SMART designs due to the complex correlation structure between the DTRs embedded in the design. Since the primary goal of SMARTs is the construction of an optimal DTR, investigators are interested in sizing SMARTs based on the ability to screen out DTRs inferior to the optimal DTR by a given amount which cannot be done using existing methods. In this paper, we fill this gap by developing a rigorous power analysis framework that leverages multiple comparisons with the best methodology. Our method employs Monte Carlo simulation in order to compute the minimum number of individuals to enroll in an arbitrary SMART. We will evaluate our method through extensive simulation studies. We will illustrate our method by retrospectively computing the power in the Extending Treatment Effectiveness of Naltrexone SMART study.