How do you specify the variance function in nbreg to coincide with Cameron and Trivedi’s (Regression Analysis of Count Data, page 62) NB1 and NB2 variance functions?

What is the difference between the models fit using nbreg, dispersion(mean) and nbreg, dispersion(constant)?

Stata’s nbreg model comes in two flavors: the default mean dispersion (or equivalently nbreg, dispersion(mean) and the constant dispersion nbreg, dispersion(constant). In short,

        . nbreg, dispersion(mean)    (or just plain nbreg)

corresponds to Cameron and Trivedi’s NB2 variance function, while

        . nbreg, dispersion(constant)

corresponds to NB1.

To see why, let’s do the variance calculations ourselves. The negative binomial model is the hierarchical model y_i | g_i, where g_i is gamma distributed and

        y_i | g_i ~ Poisson(g_i)

That is, the conditional mean and variance of y_i given g_i is merely g_i.

For nbreg, dispersion(mean), with mu_i = exp(xb_i) (xb is the linear predictor),

        g_i ~ Gamma(1/alpha, alpha*mu_i)

where I define the Gamma(a,b) distribution as that having mean ab and variance ab^2, and alpha is an ancillary parameter to be estimated from the data.

Naturally,

        E(y_i) = E{E(y_i | g_i)}
	       = E(g_i)
	       = (1/alpha)*alpha*mu_i 
	       = mu_i

The variance of y_i is

        Var(y_i) = E{Var(y_i | g_i)} + Var{E(y_i | g_i)}
                 = E(g_i) + Var(g_i)   
  		 = 1/alpha*(alpha*mu_i) + (1/alpha)*(alpha*mu_i)^2
 		 = mu_i + alpha*mu_i^2
		 = mu_i * (1 + alpha*mu_i)

This corresponds to Cameron and Trivedi’s equation (3.13) and thus corresponds to the NB2 model in their terminology. The dispersion for this model is (1 + alpha*mu_i), which depends on mu_i, hence the moniker “mean dispersion”.

By comparison, nbreg, dispersion(constant) has the distribution of g_i as

        g_i ~ Gamma(mu_i/delta, delta)

where delta is the ancillary parameter. I could have easily called this alpha and not delta, but nbreg uses delta to make the distinction between both models clearer.

Here E(y_i) = mu_i as well, and the variance of y_i is

        Var(y_i) = E{Var(y_i | g_i)} + Var{E(y_i | g_i)}
                 = E(g_i) + Var(g_i)
                 = (mu_i/delta)*delta + (mu_i/delta)*delta^2
                 = mu_i + mu_i*delta
                 = mu_i * (1 + delta)

which (except for calling it delta instead of alpha) corresponds to Cameron and Trivedi’s equation (3.11), and hence the NB1 model. For this model, the dispersion is (1 + delta) and thus is constant over all observations.

For both models, the dispersion is greater than one. This is why nbreg serves its purpose of modeling data that exhibit dispersion beyond that which can be handled using Poisson regression, which has dispersion set to 1.