## 1 Introduction  
Cognitive behavioral therapy (CBT) can be useful for various mental health problems including depression, anxiety, substance use disorders, etc. Digital cognitive behavioral therapy (dCBT)--administered either as stand-alone or as an adjunct treatment augmenting traditional sessions with a human therapist--is a potentially promising way to deliver CBT by means of digital technologies such as smartphones. Potential merits of the dCBT include accessibility and cost-efficiency (e.g., real time coaching and support can be provided remotely instead of face-to-face visits), more consistent quality of treatment delivery (e.g., psychological therapy is provided by clinically validated computer programs), and personalization of treatment (e.g., the course of therapy is tailored to an individual patient's goals, availability, and engagement) [1, 2]. Fundamentally, clinical investigation of an experimental dCBT should follow a similar path to clinical trials of other therapeutic interventions [3]. The randomized controlled trial (RCT) is the most rigorous research design to obtain evidence in support of clinical efficacy of a dCBT product. However, such trials pose some unique challenges. First, many dCBTs are complex interventions with multiple ingredients that are applied based on a dynamic feedback loop to optimize the treatment for an individual patient. Second,it may be very difficult to define an optimal \"dose\" or \"exposure\" of a dCBT to attain the desired therapeutic effect [4]. Third, while engagement with dCBT is viewed as an essential component of this treatment modality [5], it is quite challenging to quantify and build reliable statistical models relating engagement with clinical efficacy. For instance, \"more time using a digital therapeutic\" does not necessarily translate into \"better treatment effect\" [6]. Fourth, high heterogeneity in the usage/engagement patterns may potentially confound treatment effects. For instance, if patients are randomized to treatments that do not meet their expectations, this may affect the patients' motivation to engage and comply with the treatments, which, in turn, may impact the study outcomes [7]. Engagement may be strongly correlated with efficacy of an intervention. On the other hand, engagement by itself is not sufficient; for instance, study participants may enjoy using the app and provide very positive feedback, but this may not directly translate into clinical benefit. Acquiring important data on engagement and properly accounting for it in the analysis may be essential for valid statistical inference. In this paper, we investigate some statistical issues for a 1:1 RCT comparing efficacy of an experimental CBT intervention for a mental health disorder versus control. For illustration purposes, we assume that participants in both experimental and control groups receive some background standard of care medication throughout the study. Participants in the experimental group (intervention condition) have sessions with a human therapist at which they learn some novel cognitive and behavioral strategies to overcome difficulties in their daily lives, and, in addition, they receive a smartphone app that augments the sessions by providing personalized tips on disease management, e.g., through serious games, educational videos, etc., that are expected to magnify therapeutic benefit. Therefore, in the experimental group the intervention has two components: novel cognitive behavioral therapy delivered during sessions with a human therapist (CBT) and a self-administered digital cognitive behavioral therapy (dCBT). Participants in the control group just have regular sessions with a therapist (without being exposed to a novel CBT) and use a sham app (without active therapeutic ingredients of a psychosocial intervention) on their smartphones. We assume the primary endpoint \(Y\) represents some measure of clinical efficacy, e.g., change from baseline to the end of treatment period in Montgomery-Asberg Depression Rating Scale (MADRS) [8] total score in depression.