What is the relationship between Cox regression and Tobit regression?

To handle censored data, I see that some researchers use censored regression methods, like Tobit regression, some use classic survival analysis models, like Cox regression.

I know that Cox regression and Tobit regression are two different models from the perspective of math.

My questions: What are the pros and cons of these two methods? What problems are they good at solving, respectively? Do they have different assumptions?

Abbreviated Model Descriptions

The Cox model is a survival model that cleverly models the hazard ratios through the observed ranks of the data, without needing to make an assumption of the underlying baseline distribution, but still requires the proportional hazards assumption.

The Tobit model is essentially standard linear regression, except that it can also handle censored data. The assumed distribution is then normal.

Pros and Cons

Cox Model:

Pro: Don't need to make assumption about baseline distribution. This is very important for survival analysis: time-to-event data tends to be very not normal, often with extremely heavy right tails. Additionally, by only considering the rank of the data, you have a model that is more robust to the expected outliers.

Cons: Can be very difficult to interpret coefficient effects.

Tobit Model:

Pro: Simple extension of a model most analysts are already familiar with to allow for censoring, i.e. if all your data were observed and appropriate for linear regression (with one caveat mentioned in Cons section), then it would be appropriate to use a Tobit model.

Cons: Requires the assumption of linear effects and gaussian errors. In some applications, this is totally appropriate, but time-to-event data (i.e. survival analysis) rarely fits that criteria. Also, it's worth noting that the Tobit model is more sensitive to the normality assumption than vanilla linear regression.

• The (exponentiated) Cox regression coefficients are interpreted as a hazard ratio (HR), where a hazard is an instantaneous risk of an outcome's occurrence. I agree somewhat it can be difficult to explain, although in the health sciences we often interpret HRs as risk ratios which is appropriate when the outcome is rare. – AdamO Mar 16 '18 at 22:28

Neither a normally distributed error term nor a linear link would be an adequate choice for modeling time-to-event outcomes in most circumstances. The distribution of failure times tend to skew right by a large degree.

For models with no censoring, most of the books on failure time analyses discuss parametric models. These are exponential, Gamma, or Weibull maximum likelihood procedures. Log transforming the event time could justify the application of a linear regression model, and thus the Tobit model could have some applicability for parametric models of lognormal data with censoring. The rationale for lognormal regression models for time-to-event data seems dubious in my opinion: normally distributed data arise as the sums of millions of unmeasured factors contribute to an outcome. Exponential and Weibull models, conversely, are probability models that have been discussed in more detail, derived as solutions to differential equations for Martingale processes, and are summarized by simple hazard functions.

The Cox model does not bother with the distribution of failure time. It is semiparametric, and thus works for a general class of parametric models provided the hazards are proportional. The Cox model uses a partial likelihood to rank risk-sets: groups of people at risk of the disease at each outcome, and evaluates a ratio of likelihoods according to an arbitrary baseline hazard function. Censored observations are simply dropped from subsequent analyses. Most agree it makes the full use of the data while assuming as little as possible about what the underlying distribution is/is not.

• How does the tobit model relate to parametric survival model? Is the Tobit model equivalent to a log-normal parametric survival model? – Munichong May 7 '18 at 17:41