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The log-linear model is a form of Poisson regression that allows for the analysis of multi-way contingency tables.

In Poisson regression, we can model the response using the (canonical) log link:

$$\mu(x) = e^{\alpha + \beta x}$$

In this way, we get a log-linear model:

$$\log(\mu(x)) = \alpha + \beta x$$

For an $I \times J$ contingency table sampled under the Poisson sampling scheme, with covariate dependent cell means and independent margins, we have:

$$\mu_{ij} = \mu\alpha_i\beta_j$$

With $\sum_i \alpha_i = \sum_{j} \beta_j = 1$, which can be written, under a reparameterization, as::

$$\log(\mu_{ij}) = \lambda + \lambda_i^X + \lambda _j^Y$$

Where $\lambda_i^X$ ($\alpha_i$) and $\lambda_j^Y$ ($\beta_j$) specify row and column marginal distributions given total count, and $\lambda$ specifies the mean of the total count ($\mu = \sum_{i,j} \mu_{ij}$).

Extensions of this model include interaction terms, $\lambda_{ij}^{XY}$ and other dimensions $\lambda{k}^Z$.