Clarification about definitions associated with "censored values"

Modeling
1

I’m trying to understand how to use censored values in modeling with Stan. I have three questions that would be very helpful to have clarified.

  1. Right vs. left censoring. In the censored data part of the user guide, it states

Since Stan does not allow unknown values in its arrays or matrices, the censored values must be represented explicitly, as in the following right-censored case.

Then, it defines a set of y_cens values that are required to be larger than a previously defined value U

real<lower=U> y_cens[N_cens];

Since they are larger than a value on the left, I’m confused why these values are considered to be right censored. Could you please explain how left and right censoring are defined?

  1. I’m interested to understand how censoring is implemented by Stan. In a sampling statement such as the following, how does Stan assure that the sampled parameter values are, in this case, larger than U? Is some form of rejection sampling applied?:
model {
...
 y_cens ~ normal(mu, sigma);
}
  1. Following the notation above, with U as the endpoint used for censoring, is it possible to define variable censors for an array of parameters? If it is possible, how would it be programmed in Stan? Something like the following:
...
transformed data {
...
array[3] real Us = {1.1,5.2,6.3}; //arbitrary values used for example... perhaps generated by some other function
...
}
...
transformed parameters {
...
array[3] real<lower=Us> t_O;
...
}
...
model {
t_O ~ <some_pdf>(...)
}

In my particular problem, I’ve got a set of observed data points, t, and a set of associated latent variables, t_O. Each t_{O_i} must be less than the corresponding t_i. How would I implement this?

Thank you!

2

Right-censoring is when the right-hand tail of the population is censored. y_cens are these censored values. So y_cens should be larger than some value.

Sampling statements don’t actually cause Stan to draw a random sample from the distribution on the right hand side, they just increment the target density by the appropriate lpdf. Stan assures that the sampled parameter values are larger than U by constructing an unconstrained parameter under-the-hood, and taking y_cens to be U plus the exponential of this unconstrained parameter, and then adding the relevant Jacobian adjustment to the target. The code tells Stan to do this in the parameters block where y_cens is declared with a lower-bound constraint.

Recent versions of Stan (but maybe not 2.21, which is still the Rstan version on CRAN?) allow you to pass vectors of bounds to <lower>. However, you need to declare the relevant parameter in the parameters block, not in the transformed_parameters block.

1 Like
3

Thank you! This is really interesting and helps clarify things. Good to know that more recent versions of Stan accept vectors as constraints. This should help a lot!