tobit.Rmd

# Tobit Regression


The following is a simple demonstration of tobit regression via maximum
likelihood. The issue is one where data is censored such that while we observe
the value, it is not the true value, which would extend beyond the range of the
observed data. This is very commonly seen in cases where the dependent variable
has been given some arbitrary cutoff at the lower or upper end of the range,
often resulting in floor or ceiling effects respectively. The conceptual idea is
that we are interested in modeling the underlying latent variable that would not
have such restriction if it was actually observed.


## Censoring with an Upper Limit

### Data Setup 

Data regards academic aptitude (GRE scores) with will be modeled using reading
and math test scores, as well as the type of program the student is enrolled in
(academic, general, or vocational).  See
[this](https://stats.idre.ucla.edu/r/dae/tobit-models/) for an applied example
and more detail.

```{r tobit-setup}
library(tidyverse)

acad_apt = read_csv("https://stats.idre.ucla.edu/stat/data/tobit.csv") %>%
  mutate(prog = factor(prog, labels = c('acad', 'general', 'vocational')))
```


Setup data and initial values.

```{r tobit-initial}
initmod = lm(apt ~ read + math + prog, data = acad_apt)
X = model.matrix(initmod)
init = c(coef(initmod), log_sigma = log(summary(initmod)$sigma))
```


### Function

```{r tobit-ll}
tobit_ll <- function(par, X, y, ul = -Inf, ll = Inf) {
  
  # this function only takes a lower OR upper limit
  
  # parameters
  sigma = exp(par[length(par)]) 
  beta  = par[-length(par)]
  
  # create indicator depending on chosen limit
  if (!is.infinite(ll)) {
    limit = ll
    indicator = y > ll
  } else {
    limit = ul
    indicator = y < ul
  }
  
  # linear predictor
  lp = X %*% beta
  
  # log likelihood
  ll = sum(indicator * log((1/sigma)*dnorm((y-lp)/sigma)) ) + 
    sum((1-indicator) * log(pnorm((lp-limit)/sigma, lower = is.infinite(ll))))
  
  -ll
}
```

### Estimation

Estimate via <span class="func" style = "">optim</span>.

```{r tobit-estimate}
fit_tobit = optim(
  par = init,
  tobit_ll,
  y  = acad_apt$apt,
  X  = X,
  ul = 800,
  method  = 'BFGS',
  control = list(maxit = 2000, reltol = 1e-15)
)


# this would be more akin to the default Stata default approach
# optim(
#   par = init,
#   tobit_ll,
#   y = acad_apt$apt,
#   X = X,
#   ul = 800,
#   control = list(maxit = 16000, reltol = 1e-15)
# )
```



### Comparison

Compare to <span class="pack" style = "">AER</span> package tobit function.

```{r tobit-compare-aer}
library(survival)

fit_aer = AER::tobit(
  apt ~ read + math + prog,
  data = acad_apt,
  left = -Inf,
  right = 800
)
```


```{r tobit-compare-aer-show, echo=FALSE}
rbind(
  fit_tobit = c(
    fit_tobit$par[1:5],
    sigma   = exp(fit_tobit$par[6]),
    logLike = -fit_tobit$value
  ),
  AER = c(coef(fit_aer), fit_aer$scale, logLik(fit_aer))
) %>% 
  kable_df()
```


```{r tobit-compare-aer}
```


<span class="pack" style = "">AER</span> is actually just using <span
class="func" style = "">survreg</span> from the <span class="pack" style =
"">survival</span> package. Survival models are usually for modeling time to
some event, e.g. death in medical studies, and the censoring comes from the fact
that the observed event does not occur for some people. Like our tobit function,
an indicator is needed to denote who is or isn't censored. In survival models,
the indicator is for the event itself, and means they are NOT censored.  So
we'll reverse the indicator used in the tobit function for <span class="func"
style = "">survreg</span>.

```{r tobit-surv}
fit_surv = survreg(Surv(apt, apt < 800, type = 'right') ~ read + math + prog,
                   data = acad_apt,
                   dist = 'gaussian')
```


Compare all results.

```{r tobit-compare-surv, echo=FALSE}
rbind(
  fit_tobit = c(
    fit_tobit$par[1:5],
    sigma = exp(fit_tobit$par[6]),
    logLike = -fit_tobit$value
  ),
  AER      = c(coef(fit_aer), fit_aer$scale, logLik(fit_aer)),
  survival = c(coef(fit_surv), fit_surv$scale, logLik(fit_surv))
) %>% 
  kable_df()
```





## Censoring with a Lower Limit

Create a censored data situation for the low end.  The scale itself would be censored for anyone scoring a 200, but that basically doesn't happen. In this data, 15 are less than a score of 500, so we'll do that.

```{r tobit-ll-setup}
acad_apt = acad_apt %>%
  mutate(apt2 = apt,
         apt2 = if_else(apt2 < 500, 500, apt2))
```

Estimate and use <span class="pack" style = "">AER</span> for comparison.

```{r tobit-ll-est}
fit_tobit = optim(
  par = init,
  tobit_ll,
  y  = acad_apt$apt2,
  X  = X,
  ll = 400,
  method  = 'BFGS',
  control = list(maxit = 2000, reltol = 1e-15)
)

fit_aer = AER::tobit(apt2 ~ read + math + prog,
                     data = acad_apt,
                     left = 400)
```


### Comparison

```{r tobit-ll-compare, echo=FALSE}
rbind(
  fit_tobit = c(
    fit_tobit$par[1:5],
    sigma   = exp(fit_tobit$par[6]),
    logLike = -fit_tobit$value
  ),
  AER = c(coef(fit_aer), fit_aer$scale, logLik(fit_aer))
) %>% 
  kable_df()
```

## Source

Original code available at https://github.com/m-clark/Miscellaneous-R-Code/blob/master/ModelFitting/tobit.R