The Complete Data Case

Abstract

Let be Y a binary outcome variable, X₁ a covariate with categories 1,…, J and X₂ a covariate with categories 1,…, K. In a logistic model we assume

$$P\left ( Y =1|X_{1}=j,X_{2}=k\right )=\Lambda \left ( \beta_{0} +\beta_{1j} +\beta_{2k} \right )=\mu_{jk} \left ( \beta \right )\;\;\;(2.1)$$

with parameter restrictions β₁₁ = 0 and β₂₁ = 0. Λ(x):= 1/(1 + e^−x) denotes the logistic function. We consider the covariates as random variables, and parametrize their joint distribution by

$$\begin{matrix} P\left ( X_{1}= j\right )=:\tau_{j} \;\;\textup{and}\\ P\left (X_{2}= k| X_{1}=j\right ):=\pi_{k|j} \end{matrix}$$

.