## Dispersion Models

In the last week, I’ve read an interesting article “Dispersion Models in Regression Analysis” by Peter Song (http://www.pakjs.com/journals/25%284%29/25%284%299.pdf), which describes a new class of models more general than classic generalized linear models based on the error distribution.

A dispersion model can be defined by two parameters, a location parameter **mu** and a dispersion parameter **sigma ^ 2**, and has a very general form of probability function formulated as:

**p(y, mu, sigma ^ 2) = {2 * pi * sigma ^ 2 * V(.)} ^ -0.5 * exp{-1 / (2 * sigma ^ 2) * D(.)}**

where the variance function V(.) and the deviance function D(.) varies by distributions. For instance, in a poisson model,

**D(.) = 2 * (y * log(y / mu) – y + mu)**

**V(.) = mu **

Below is a piece of SAS code estimating a Poisson with both the error distribution assumption and the dispersion assumption.

data one; do i = 1 to 1000; x = ranuni(i); y = ranpoi(i, exp(2 + x * 2 + rannor(1) * 0.1)); output; end; run; *** fit a poisson model with classic GLM ***; proc nlmixed data = one tech = trureg; parms b0 = 0 b1 = 0; mu = exp(b0 + b1 * x); ll = -mu + y * log(mu) - log(fact(y)); model y ~ general(ll); run; /* Fit Statistics -2 Log Likelihood 6118.0 AIC (smaller is better) 6122.0 AICC (smaller is better) 6122.0 BIC (smaller is better) 6131.8 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > |t| Alpha Lower Upper Gradient b0 2.0024 0.01757 1000 113.95 <.0001 0.05 1.9679 2.0369 5.746E-9 b1 1.9883 0.02518 1000 78.96 <.0001 0.05 1.9388 2.0377 1.773E-9 */ *** fit a poisson model with dispersion probability ***; *** proposed by Jorgensen in 1987 ***; proc nlmixed data = one tech = trureg; parms b0 = 0 b1 = 0 s2 = 1; mu = exp(b0 + b1 * x); d = 2 * (y * log(y / mu) - y + mu); v = y; lh = (2 * constant('pi') * s2 * v) ** (-0.5) * exp(-(2 * s2) ** (-1) * d); ll = log(lh); model y ~ general(ll); run; /* Fit Statistics -2 Log Likelihood 6066.2 AIC (smaller is better) 6072.2 AICC (smaller is better) 6072.2 BIC (smaller is better) 6086.9 Parameter Estimates Standard Parameter Estimate Error DF t Value Pr > |t| Alpha Lower Upper Gradient b0 2.0024 0.02015 1000 99.37 <.0001 0.05 1.9629 2.0420 2.675E-6 b1 1.9883 0.02888 1000 68.86 <.0001 0.05 1.9316 2.0449 1.903E-6 s2 1.3150 0.05881 1000 22.36 <.0001 0.05 1.1996 1.4304 -0.00002 */

Please note that although both methods yield the same parameter estimates, there are slight differences in standard errors and therefore t-values. In addition, despite one more parameter estimated in the model, AIC / BIC are even lower in the dispersion model.