## 1 Introduction
In the experimental group, participants may engage differentially with the dCBT. For simplicity, we assume that \(X=\) the percentage of successfully completed dCBT trainings at the end of the treatment period provides a meaningful quantification of individual engagement with the dCBT. While the utility of \"completion rate\" as a measure of engagement for digital mental health interventions has been well documented [9; 10; 11; 12], we acknowledge that this is only one way to measure engagement. In practice, other, more elaborate metrics may be considered. However, in our opinion, a simple metric such as \"completion rate\" is rather informative and can be used as a starting point for developing statistical models. In other words, we assume that engagement \(X\) is measured on a scale 0 to 1, where 0 means 0% completed trainings, 0.6 means 60% completed trainings, etc. By contrast, in the control group, the sham application has no active therapeutic ingredients of a psychosocial intervention, and therefore, the engagement is a structural zero for every participant in the control group. To illustrate the latter point, consider, for example, a recent randomized, sham-controlled clinical trial of a smartphone-based application as an adjunct to the standard-of-care in schizophrenia [6]. Participants in the experimental group were exposed for a period of 12 weeks to an app that was designed as a self-management tool in schizophrenia. The users could engage with the app by prompt or on demand, and the app provided interactive, cognitive and behavior exercises that were hypothesized to improve schizophrenia symptoms. Participants in the sham group were exposed for 12 weeks to an app that was similar in appearance to the experimental app, but which did not deliver any active therapeutic content (it only sent periodic prompts to open the app, in which case a digital clock timer was displayed). The sham control arm was chosen to account for the nonspecific effects of engagement with a smartphone. Clearly, in this case the engagement defined as \"exercise completion rate\" can be quantified only in the experimental group, and it can be regarded as zero in the control group. For the considered setting (1:1 RCT; primary endpoint \(Y\) is changed from the baseline in MADRS total score at the end of the treatment period; engagement \(X\) is an important covariate for subjects in the experimental group but not in the control group; no missing data in the study) we address the following research questions:  
1. What is the proper approach to the statistical analysis of such a trial? 2. How do we compare treatment effects accounting for different levels of engagement? 3. How should we perform a sample size planning for such a trial given that engagement patterns are unknown upfront? The results presented here should aid trial statisticians in developing statistical analysis plans and in facilitating important discussions with clinical investigators at the study planning stage. The article is organized as follows. In Section 2, we describe the trial setup, assumptions for the data generating mechanism, and two approaches to data analysis following the trial--analysis of covariance (ANCOVA) and a two-sample \(t\)-test. In Section 3, we give an illustrative example of how to analyze data from such an experiment. In Section 4, we present results from a simulation study to evaluate the power of statistical tests. In Section 5, we discuss the optimization of the trial design and considerations for the sample size planning. Finally, Section 6 provides a summary and a discussion of the future work.