statisticalrethinkin..

12.5. VARIABLE PROCESS: ZERO-INFLATED OUTCOMES 333measurement error, (2) no birds present. Let p be the probability of an error. Let λ be thePoisson mean of actual bird densities. So the likelihood of observing a zero is:Pr(0|p, λ) = Pr(error|p) + Pr(no error|p) Pr(0|λ)= p + (1 − p) exp(−λ)And the likelihood of a non-zero value y is:Pr(y|p, λ) = Pr(no error|p) Pr(y|λ)= (1 − p) λy exp(−λ)y!So define ZIPoisson as the distribution above, with parameters p (probability of a zero) andλ (mean of Poisson) to describe it’s shape. en a zero-inflated Poisson regression takes theform:y i ∼ ZIPoisson(p i , λ i )logit(p i ) = α p + β p x ilog λ i = α λ + β λ x iNotice that there are two linear models and two link functions, one for each process in theZIPoisson. e parameters of the linear models differ, because any predictor such as x maybe associated differently with each component.To fit to data, this’ll work:m12.12