Counts and Rates
Two-group sample size planning for a zero-inflated Poisson (ZIP) regression using the expected-information Wald approach described by Williamson, Lin, Lyles & Hightower (2007). Specify the excess-zero probabilities and Poisson means in the two groups, then solve for the number of subjects needed to achieve the requested power.
The model has two linked components. The excess-zero probability is modeled as logit(πi) = β0 + β1xi, while the Poisson mean is modeled as log(λi) = γ0 + γ1xi. For a binary group indicator, the two group-specific probabilities and means determine the four regression parameters.
The zero-inflated mean is (1−π)λ, and the variance is (1−π)λ(1+πλ). The model therefore allows more zeros than the ordinary Poisson distribution while retaining a Poisson count component.
For each group, the calculator evaluates the expected Fisher information for the parameter vector θ = (β0, β1, γ0, γ1). For a planned sample of n0 subjects in Group 0 and n1 in Group 1, the total information is the weighted sum of the two group information matrices.
Here H selects the regression coefficient(s) in the requested hypothesis and Δ = Hθ is the corresponding alternative effect. Under the Wald approximation, the test statistic follows a noncentral chi-square distribution with degrees of freedom equal to the number of tested coefficients. The calculator increases the Group 0 sample size until the requested power is reached, while applying the specified Group 1:Group 0 allocation ratio.
For a reproducible numerical check of the implementation, use π0 = 0.20, π1 = 0.30, λ0 = 2.00, λ1 = 3.00, two-sided α = 0.05, target power = 0.80, equal allocation, and the joint test of β1 and γ1. The implementation returns 87 subjects per group (174 total), with achieved power approximately 80.37%. This is the fixed numerical validation target used for the calculator implementation.
Williamson et al. (2007) illustrate the method using weekly mosquito counts from 492 households in Western Kenya. Their fitted ZIP model gave approximately π0 = 0.569, π1 = 0.564, λ0 = 3.113, and λ1 = 3.693. Using the expected-information Wald method, they reported 505 subjects per group for 80% power for the joint test, 165,000 per group for the zero-inflation coefficient alone, and 419 per group for the count coefficient alone. Their displayed parameter estimates are rounded; therefore a reconstruction using only the printed rounded values can differ slightly from the published sample sizes.
Williamson, J. M., Lin, H.-M., Lyles, R. H., & Hightower, A. W. (2007). Power Calculations for ZIP and ZINB Models. Journal of Data Science, 5(4), 519–534. DOI: 10.6339/JDS.2007.05(4).360.
Channouf, N., Fredette, M., & MacGibbon, B. (2014). Power and sample size calculations for Poisson and zero-inflated Poisson regression models. Computational Statistics & Data Analysis, 72, 241–251. DOI: 10.1016/j.csda.2013.09.029.
The calculator is intended for planning two-group ZIP regression designs. More complex covariate structures, unequal covariate distributions, clustering, repeated measures, and likelihood-ratio-test planning may require a specialized simulation or matrix-based implementation.