. 2010 Sep 10;87(3):325–340. doi: 10.1016/j.ajhg.2010.07.021

Table 2.

Equivalence between Log-Linear Models and Logistic Models for a Three-Way Table with Binary Response Variable Y

Log-Linear Model

Logistic Model

MLE of μ_ijk

Block independence model (M_B):

\log μ_{i j k} = λ + λ_{i}^{X_{p}} + λ_{j}^{X_{q}} + λ_{k}^{Y} + λ_{j}^{X_{p} X_{q}}

β₀

\frac{n_{i j} . n .. k}{n}

Partial independence model (M_P):

\log μ_{i j k} = λ + λ_{i}^{X_{p}} + λ_{j}^{X_{q}} + λ_{k}^{Y} + λ_{i j}^{X_{p} X_{q}} + λ_{i k}^{X_{p} Y}

β_{0} + β_{i}^{X_{p}}

\frac{n_{i} . k^{n} . j k}{n_{.. k}}

Homogeneous association model (M_H):

\log μ_{i j k} = λ + λ_{i}^{X_{p}} + λ_{j}^{X_{q}} + λ_{k}^{Y} + λ_{i j}^{X_{p} X_{q}} + λ_{i k}^{X_{p} Y} + λ_{j k}^{X_{q} Y}

β_{0} + β_{i}^{X_{p}} + β_{j}^{X_{q}}

iterative estimation

Saturated model (M_S):

\log μ_{i j k} = λ + λ_{i}^{X_{p}} + λ_{j}^{X_{q}} + λ_{k}^{Y} + λ_{i j}^{X_{p} X_{q}} + λ_{i k}^{X_{p} Y} + λ_{j k}^{X_{q} Y} + λ_{i j k}^{X_{p} X_{q} Y}

β_{0} + β_{i}^{X_{p}} + β_{j}^{X_{q}} + β_{i j}^{X_{p} X_{q}}

n_ijk

The models M_B and M_P are used in the discussion of the difference between the test of interactions and the test of associations. The details of these two models are provided in the Appendix.