Randomized clinical trials (RCTs) are considered the gold standard for providing causal evidence for differential treatment effects. However, results from RCTs may not be representative when potential participants refuse to be randomized or are excluded due to having a preference for which treatment they receive. If trial designs do not allow for patient treatment preferences, trials can suffer in accrual, adherence, retention, and external validity of results. Thus, there is interest surrounding clinical trial designs that incorporate patient treatment preferences. We propose a Partially Randomized, Patient Preference, Sequential, Multiple Assignment, Randomized Trial (PRPP-SMART) which combines a Partially Randomized, Patient Preference (PRPP) trial design with a Sequential, Multiple Assignment, Randomized Trial (SMART) design. This novel PRPP-SMART design is a multi-stage clinical trial design where, at each stage, patients either receive their preferred treatment or are randomized if they do not express a preference. Following an introduction of the novel PRPP-SMART design in Chapter 1, Chapter 2 proposes a two-stage PRPP-SMART with binary end-of-stage outcomes and develops Bayesian and frequentist weighted and replicated regression models (WRRMs) to estimate the outcome rates of the embedded dynamic treatment regimens (DTRs) in the PRPP-SMART design. DTRs consist of sequences of treatments that serve as tailored decision rules, guiding how to treat a patient throughout their course of care. Our WRRMs use data from both randomized and non-randomized participants. We compare our method to a more traditional PRPP analysis which only considers participants randomized to treatment. With regard to the traditional approach, our Bayesian and frequentist WRRMs produce more efficient DTR effect estimates with negligible bias despite the inclusion of non-randomized participants in the analysis. In Chapter 3, we develop an alternative Bayesian method to estimate DTR effects from a PRPP-SMART with binary outcomes, the Bayesian Joint Stage Model (BJSM), which leverages information-sharing between participants who receive their preferred treatment and those who are randomized. Additionally, the BJSM simultaneously estimates the stage-specific main treatment effects (i.e., the distinct effects of initial-stage treatments and second-stage treatments) within the same model used to estimate DTR effects. We compare our BJSM method to the Bayesian and frequentist WRRMs developed in Chapter 2 and demonstrate that our BJSM provides more efficient DTR effect estimates with negligible bias, along with minimally biased main treatment effects. In Chapter 4, we develop sample size and power calculations for a two-stage PRPP-SMART design with binary end-of-stage outcomes and utilize the BJSM method proposed in Chapter 3 to construct our calculations. We extend the Bayesian "set of the best" sample size methodology, previously used in traditional SMARTs without treatment preference, to the PRPP-SMART setting. This approach powers trials based on selecting a set of the most effective DTRs rather than using pairwise comparisons. Through simulations, we show that the required sample sizes derived using this approach achieve the desired statistical power.