Tobit model explanation

Question

We have 100 participants in two groups, $n=50$ in each group. We used an assessment of ability of basic functioning at 4 time-points. The assessment comprises 6 questions, each scored 0 – 5. We do not have individual scores for each question, just total scores that range from 0 – 30. Higher scores indicate better functioning. The problem is that the assessment is very basic and has a significant ceiling effect. Results are very negatively skewed. The majority of participants scored close to 30, especially at the 3 follow-up time-points. It is likely that not all of the participants who scored at the upper limits are truly equal in ability: some of the participants were just about scoring 30 and others scored 30 with ease and would score much higher if it were possible and so the data are censored from above.

I want to compare the two groups and over time but obviously this is very difficult given the nature of the results. Transformations of any kind make no difference. I have been advised that the Tobit model is the best equipped for this assessment and I can run the analysis in R using examples from Arne Henningen’s paper, Estimating censored regression models in R using the censReg package.

However, I have only a basic knowledge of statistics and have found information on the Tobit model to be quite complicated. I need to be able to explain this model in plain language and I cannot find a plain language, nuts and bolts explanation as to what the Tobit model actually does and how. Can anyone explain the Tobit model or point me in the direction of a readable reference without complicated statistical and mathematical explanations?

Extremely grateful for any help

Answer 1

The wiki describes the Tobit model as follows:

$$y_i = \begin{cases} y_i^* &\text{if} \quad y_i^* > 0 \\ \ 0 &\text{if} \quad y_i^* \le 0 \end{cases}$$

$$y_i^* = \beta x_i + u_i$$

$$u_i \sim N(0,\sigma^2)$$

I will adapt the above model with to your context and offer a plain english interpretation of the equations which may be helpful.

$$y_i = \begin{cases}\ y_i^* &\text{if} \quad y_i^* \le 30 \\ 30 &\text{if} \quad y_i^* > 30 \end{cases}$$

$$y_i^* = \beta x_i + u_i$$

$$u_i \sim N(0,\sigma^2)$$

In the above set of equations, $y_i^*$ represents a subject's ability. Thus, the first set of equations state the following:

1. Our measurements of ability is cut-off on the higher side at 30 (i.e., we capture the ceiling effect). In other words, if a person's ability is greater than 30 then our measurement instrument fails to record the actual value but instead records 30 for that person. Note that the model states $y_i = 30 \quad \text{if} \quad y_i^* > 30$.

2. If on the other hand a person's ability is less than 30 then our measurement instrument is capable of recording the actual measurement. Note that the model states $y_i = y_i^* \quad \text{if} \quad y_i^* \le 30$.

3. We model the ability, $y_i^*$, as a linear function of our covariates $x_i$ and an associated error term to capture noise.

I hope that is helpful. If some aspect is not clear feel free to ask in the comments.

Answer 2

There's an article by Berk in the 1983 edition of American Sociological Review (3rd issue) - that's how I learned about censoring. The explanation is specifically about selection bias but is absolutely relevant to your issue. Selection bias as Berk discusses is just censoring via the sample selection process, in your case the censoring is a result of an insensitive instrument. There's some nice charts that show you exactly how you can expect your regression line to be biased when Y is censored in different ways. In general the article is logical and intuitive rather than mathematical (yes I treat them as separate, preferring the former). Tobit is discussed as one solution to the problem.

More generally, it sounds like tobit is the right tool for the job at hand. Basically, the way it works is by estimating the probability of being censored and then incorporating that into the equation predicting the score. There is another approach proposed by Heckman using probit and the inverse mills' ratio which is basically the same thing but allows you to have different variables predicting the likelihood of censoring and the score on the test - obviously that would not be apposite for the situation you have.

One other recommendation - you might consider a hierarchical tobit model where observations are nested within individuals. This would correctly account for the tendency for errors to be associated within individuals. Or if you don't use a hierarchical model, at least be sure to adjust your standard errors for the clustering of the observations within individuals. I know this all can be done in Stata and am confident R with all its versatility can do it too.. but as an avid Stata user I can't provide you with any guidance about how to go about it in R.