Notes on Poisson Regression and Some Extensions

The probability function for zero-inflated count variable Y z can be specified as,⎧⎨p + (1 − p)e −µ if y = 0Pr(Y z = y|p, µ) =⎩(1 − p) e−µ µ yy!if y > 0(2)Unlike the Poisson distribution which is determined by a single parameter, µ, the ZIP distributionis determined by two parameters µ (i.e., the parameter for the standard Poisson distribution) andp (i.e., the probability of being an inflated zero). The mean and variance of a zero-inflated countvariable are, respectively,E(Y z ) = µ(1 − p)andvar(Y z ) = µ(1 − p)(1 + µp),which shows the nature of the overdispersion via the mean to variance ratio when count data arecharacterized by an excess of 0’s. The variance is (1 + µp) times the mean.Estimation.We can model the inflated-zero probability as a function of covariates,and the Poisson rates as,The likelihood function can then be written as,p i = exp(x′ i α)1 + exp(x ′ i α),µ i = exp(x ′ iβ).log L =n∑log p i + (1 − p i ) exp(µ i ) + log{(1 − p i ) exp(−µ i )} + yµ i − log Γ(y i + 1). (3)i=1Interpretation. Interpretation of the results from this model is straightforward. We caninterpret the parameters in the inflated portion as effects on the log odds of being zero. 2 Theother parameters have the usual Poisson regression interpretation. Using results from the nullmodel and applying the formulas above for the mean and variance, we get ̂µ = e 0.829 (0.846),which is 1.94 for the mean, where ̂α = −1.7069, so p = 0.153. Using the formula above, thevariance is 2.621. The mean is exactly reproduced by the model, the variance is closer to theestimate of 2.79 from the raw data than was the estimate from the Poisson regression model.Example.Zero-inflated poisson regression Number of obs = 1496Nonzero obs = 1140Zero obs = 356Inflation model = logit LR chi2(6) = 180.92Log likelihood = -2468.552 Prob > chi2 = 0.00002 Other probability distributions for zero-inflation are possible.8