A workaround for using linear models (rather than Tobit) with censored data?

Question

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.

Answer 1

You could consider using a two-part mixed-effects model for semi-continuous data. This combines a mixed-effects logistic regression for the dichotomous outcome $\mbox{I}(x_1 = 0)$ with a linear mixed model for the logarithm positive responses $\log(x_1)$ for $x_1 > 0$.

This model can be fitted, for example, in the GLMMadaptive package in R. For an example, check here.

Answer 2

Thank you for the additional explanation in the comments about your outcome. They help in thinking about a way to move forward. I will first focus on that issue before sharing why I find your solution to move to a linear model problematic.

1. With regards to $x_1$ and using a tobit model, it is critical to note that tobit regression works under the assumption that the underlying variable has a continuous scale, theoretically ranging from negative to positive. However, you do not observe some piece of that range (called truncation). With your time variable, a negative value does not make sense, so tobit regression doesn't apply. See this thread. You need a different generalized linear modeling approach.

2. For modeling $x_i$, you have a couple of possibilities given the presence of all those 0s. In both cases, you would need to move to the generalized linear modeling framework to accomodate your strictly positive dependent variable. One possibility would be a gamma GLM model. The other possibility to consider is the class of zero-inflated models combined with a gamma GLM. These models have a separate model for the 0 vs. > 0 case and then a model for the values > 0.

3. Finally, your proposed solution for adding $x_1$, $x_2$, and $x_3$ together as a new dependent variable and then running a linear model w/ covariates $x_2$ and $x_3$ included is problematic from my perspective. You are proposing to include as independent variables in your regression the very same variables that go into forming your dependent variable. An analogous situation would be a case where the outcome was the sum of 3 conceptually related items from a Likert-scaled survey and then in a regression model predicting that sum score I include 2 of those three items as predictors. We would expect the two items to be highly correlated with the sum score, so much so that we would have trouble finding other covariates powerful enough to explain what is leftover. And what exactly is it that is leftover?

By combining the items you create a new construct, and you need to convince readers that the construct is meaningful. You are trying to use a mathematical trick to justify using a linear model and I'm not convinced that trick works as you think it does. And, at the same time, it poses all sorts of interpretation problems. I would be interested to know if others have different thoughts about this issue, but from my perspective it is problematic.

Edit: In general, my suggestion is to build a model for the variable you are actually interested in understanding rather than trying to manipulate that variable in such a way that you sort of retain its original properties. This also means that if the variable were amenable to a known transformation that maintained its properties, you could do that, allowing you to utilize a linear model for the transformed version of the outcome. For example, the log transformation for the >0 values mentioned in @Dimitris Rizopoulos' response to your post.