How to truncate Negative Binomial (log parametrization)

[Philip Wong]

Hi all, first time poster here.

I have a question about the truncation of a negative binomial variable.

Below you can find my model which is part of a hurdle model therefore I only want to model strictly positive counts, for which I need a truncation at 1.

Please take a look at the statement in bold.

data {

  int<lower=1> I; //number of individuals

  int<lower=1> N; //number of observations for all individuals over time

  int<lower=1> L; //number of parameters for mu

  matrix[I,L] X; //predictor variables for mu parameter

  int<lower=1,upper=12> y[N]; //counts

  int mu_idx[N]; //index indicating what parameter should be used for each observation

}

parameters {

  real beta_zero;

  vector[L] beta;

  real<lower=0> phi;

} 

model {

  vector[I] mu_param;

  real mu[N];

  beta_zero ~ normal(1,2);

  beta ~ normal(0,3);

  phi ~ normal(2,0.5);

  mu_param <- beta_zero + X * beta;

  mu <- to_array_1d(mu_param[mu_idx]);

  y ~ neg_binomial_2_log(mu,phi);

  for (n in 1:N){

  increment_log_prob(log1m_exp(neg_binomial_2_log_log(0,mu[n],phi))); //should be a truncation at 1

  }

}

The model works fine, I'm just wondering whether I should use 0 or 1 in the statement in bold if I want to truncate my distribution at 1?

Also if you have any other tips on which I can improve my model it is greatly appreciated.

Thanks in advance!

Philip

[Bob Carpenter]

I'd recommend upgrading to 2.11 and using our new syntax!
I'd declare mu to be the type you want rather than trying
to convert a vector to array.

This

> log1m_exp(neg_binomial_2_log_log(0,mu[n],phi)))

is just the log of 1 minus the neg binomial evaluated
at 0, which gives you truncation below at 1 inclusive.

The bigger point is that negative binomial goes above
12, but your data doesn't. Could it have? If not, negative
binomial is probably the wrong model here.

- Bob

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