I have a left censored dependent variable where many of the observations have a value of zero. The data is clustered (multiple measurements over time for each person). I initially decided to use a Tobit model with random effects to model my outcome variable. But I am currently thinking of using a multilevel linear model with random effects instead (for various reasons that I don't want to discuss in this post).
Normally, a multilevel linear model is not suitable for modeling censored outcome variables. However, I thought of a trick that might fix the issue, but I wanted to seek the community's feedback to check whether my rationale is correct. Before I discuss the proposed solution, I'll give some background about the study's setting. The main variable of interest represents a specific component of service time in a large service organization. This component of service time is optional (i.e., workers are supposed to perform it only when needed). This is why a lot of the observations had a zero value for this component. Let's call the component $x_1$. There are two more components of service time that are performed by workers on regular basis: $x_2$ and $x_3$. These components are not censored.
My idea is to construct a new variable $z_1$ that represents total service time (i.e., $z_1 = x_1 + x_2 + x_3$). This new variable is not censored, so I can use multilevel linear models to regress $z_1$ on the independent variables of interest. But I am interested in the effects of those IVs on $x_1$ rather than $z_1$. So I'll control for $x_2$ and $x_3$ in the model to examine the effects of the IVs on $x_1$. In other words, my model would be: $z_1 = \beta_0 + \beta_1 x_2 + \beta_2 x_3 + W \Gamma + \epsilon$ where $W$ is a vector of IVs of interest. My rationale is that if I control for $x_2$ and $x_3$, then the only remaining service component in $z_1$ is $x_1$, which means that $\Gamma$ represents the effects of the IVs on $x_1$ rather than $z_1$. At the same time, I can use the multilevel linear model instead of a Tobit model since $z_1$ is not censored.
Is this approach correct? Did I miss anything?
Edit: Made a correction to the subscripts of the model parameters.