 Solution

Introduce a latent indicator $Z_i=1$ for a structural zero. In this parametrization $P(Z_i=1)={\pi }_i$, and conditional on $Z_i=0$ the response has a Poisson distribution with mean ${\mu }_i$. The zero-inflated Poisson regression usesIts observed probability mass function is ${\pi }_i+(1-{\pi }_i)e^{-{\mu }_i}$ at zero and $(1-{\pi }_i)e^{-{\mu }_i}{\mu }_i^{y_i}/y_i!$ at a positive count. The zero-mixing probability here is explicitly ${\pi }_i$; it is distinct from the weight of the count component.

For the expectation-maximization algorithm, start with positive ${\mu }_i$ and $0<{\pi }_i<1$. At iteration $k$, the E-step computesThis distinguishes structural zeros from Poisson-generated zeros. Up to terms independent of the new parameters, the expected complete-data log-likelihood isIt separates into a fractional-response logistic fit and a weighted Poisson fit. Because treatment is binary, each component is saturated over its two treatment groups, so the M-step has closed forms.

For $j=0,1$, let $I_j=\{i:x_i=j\}$, $n_j=|I_j|$, $S_j={\sum }_{i\in I_j}{\tau }_i$, and $C_j={\sum }_{i\in I_j}{y}_i$. The M-step score equations giveHere $\sum (1-{\tau }_i)y_i=C_j$, since every positive count has ${\tau }_i=0$. The explicit coefficient updates areIterate the E- and M-steps until the observed log-likelihood and parameter estimates stabilize. This is the EM algorithm for zero-inflated Poisson regression with explicit binary-group updates; merely naming two regression routines would not supply these expressions. Both treatment groups must be represented for both contrasts to be identifiable. Zero fitted group means or endpoint mixing probabilities are boundary solutions, interpreted through limits of the log or logit coefficients. As with other mixture models, multiple starts help distinguish competing stationary solutions; EM increases the likelihood but does not guarantee a global maximum from an arbitrary start.