## Estimating Conway-Maxwell-Poisson Regression in SAS

Conway-Maxwell-Poisson (CMP) regression is a flexible way to model frequency outcomes with both under-dispersion and over-dispersion. In SAS, CMP regression can be estimated with COUNTREG procedure directly or with NLMIXED procedure by specifying the likelihood function. However, the use of NLMIXED procedure is extremely cumbersome in that we need to estimate a standard Poisson regression and then use estimated parameters as initial values parameter estimates for the CMP regression.

In the example below, we will show how to employ GLIMMIX procedure to estimate a CMP regression by providing both the log likelihood function and the variance function in terms of the expected mean.

proc glimmix data = mylib.credit_count; model majordrg = age acadmos minordrg ownrent / link = log solution; _nu = 1 / exp(_phi_); _variance_ = (1 / _nu) / ((_mu_) ** (1 / _nu)); _z = 0; do i = 0 to 100; _z = _z + (_mu_ ** i) / fact(i) ** _nu; end; _prob = (_mu_ ** majordrg) / (fact(majordrg) ** _nu) * (_z ** (-1)); _logl_ = log(_prob); run;

Since the scale parameter **_phi_** is strictly above 0, the function **1 / exp(_phi_)** in the line #3 is to ensure the **Nu** parameter bounded between 0 and 1.

In addition, the DO loop is to calculate the normalization constant **Z** such that the PMF would sum up to 1. As there is no closed form for the calculation of **Z**, we need to calculate it numerically at the cost of a longer computing time.

Other implicit advantages of GLIMMIX procedure over NLMIXED procedure include the unnecessity to provide initiate values of parameter estimates and a shorter computing time.