## 2 Statistical Modeling and Some Theoretical Results  
We assume equal (1:1) RCT design for which an even number of \(n\) subjects are randomized between the experimental (\(E\)) and the control (\(C\)) conditions. Let \(\delta_{i}=1\) (or 0), if the \(i\)th participant is randomized to group \(E\) (or \(C\)). With equal allocation design, the group sample sizes are \(n_{E}=\sum_{i=1}^{n}\delta_{i}=n/2\) and \(n_{C}=\sum_{i=1}^{n}\left(1-\delta_{i}\right)=n/2\). For the \(i\)th participant, we acquire data \((Y_{i},X_{i},\delta_{i})\)\((i=1,\ldots,n)\), where \(Y_{i}\) is the response, \(\delta_{i}\) is the treatment assignment indicator, and \(X_{i}\) is the measure of engagement (applicable only to the experimental group). The responses \(Y_{i}\), conditional on \(X_{i}=x_{i}\) and \(\delta_{i}\), are assumed to satisfy the following statistical model:  
\[Y_{i}=\delta_{i}\mu_{E}+(1-\delta_{i})\mu_{C}+\gamma x_{i}\delta_{i}+\varepsilon _{i},\ i=1,\ldots,n, \tag{1}\]  
where \(\mu_{E}\) and \(\mu_{C}\) are the treatment effects for group \(E\) and group \(C\), respectively, \(\gamma\) is a linear regression slope, and \(\varepsilon_{i}\)'s are independent and identically distributed (i.i.d.) measurement errors, assumed to be normally distributed with zero mean and variance \(\sigma^{2}\), i.e., \(\varepsilon_{i}\sim N(0,\sigma^{2})\). The parameters in Equation (1) can be interpreted as follows: \(\mu_{E}\) is the effect of the novel CBT alone, i.e., this is the mean effect for a subject in the experimental group who had CBT sessions (required by the study design) and never used the dCBT (engagement level \(x=0\)). On the other hand, if a subject in the experimental group had both CBT and engaged with the dCBT (\(x>0\)), then the mean response for this subject is \(\mathrm{E}(Y)=\mu_{E}+\gamma x\), which can be magnified or decreased compared to \(\mu_{E}\) according to the values of \(x\) and \(\gamma\). For subjects in the control group, \(\mu_{C}\) is the mean effect due to the control intervention (a combination of the standard CBT and the sham app). In the described setting, several estimands may be of interest:  
1. The difference \(\Delta_{0}=\mu_{E}-\mu_{C}\), which is the contrast between the novel CBT alone and the control intervention. 2. The difference \(\Delta_{x}=(\mu_{E}+\gamma x)-\mu_{C}\), which is the contrast between the novel CBT + dCBT engaged at the level \(X=x\) and the control intervention. The estimand (I) is a special case of estimand (II) with \(x=0\). In practice, estimand (II) may be more relevant because it provides information on a combined effect of novel CBT + dCBT for a chosen level of \(x\) that is deemed important to the investigator. Oneparticularly interesting case is \(\Delta_{\overline{x}_{E}}=(\mu_{E}+\gamma\overline{x}_{E})-\mu_{C}\), which is the contrast between novel CBT + dCBT at the average level of engagement observed in the trial and the control condition. An investigator may wish to answer to different research questions in the study, such as: (i) Is the difference \(\Delta_{0}=\mu_{E}-\mu_{C}\) significant? (ii) Is the effect of engagement (the linear slope \(\gamma\)) significant? (iii) Is the difference \(\Delta_{x}=(\mu_{E}+\gamma x)-\mu_{C}\) for a pre-specified value of \(x\) significant? The choice of the primary research question is very important because it will, among other considerations, drive the choice of the sample size for the study. The answers to questions (i)-(iii) will carry different implications for development. For instance:  
* If both \(\Delta_{0}\) and \(\gamma\) are significantly different from zero, then the novel CBT is deemed efficacious, and its effect can be magnified or decreased by the individual engagement with the dCBT. * If \(\Delta_{0}\) is significantly different from zero but \(\gamma\) is not, then the novel CBT is deemed efficacious but the engagement with the dCBT is not helpful for synergizing this effect.