Categorical Data Analysis Using The SAS® System, 2nd Edition

Appendix A: Equivalence of the Loglinear and Poisson Regression Models

Suppose n = (n₁, n₂,...,n_s)′ denotes a vector of independent Poisson variables and let μ = (μ₁, μ₂,...,μ_s)′ denote the corresponding vector of expected values. Suppose variation among the elements of μ can be described with the loglinear model

μ = exp(X_pβ_p)

where X_p = [1, X] is an (s × (t + 1)) matrix of known coefficients with full rank (t + 1) ≤ s and β_p = [β₀,β′]′ is a (t + 1) vector of unknown coefficients. The likelihood function for n is

[View full size image]

where ,