# Why does log-linear analysis seem to ignore the Poisson regression equidispersion assumption?

As far as I understand it, log-linear analysis is based on the use of a Poisson regression. This is what I understood from various online resources, like this online tutorial or this text whose intro says: "In this chapter we study the application of Poisson regression models to the analysis of contingency tables. This is perhaps one of the most popular applications of log-linear models [...]".

The wikipedia article on log-linear analysis lists three assumptions for log-linear analysis:

1. The observations are independent and random;
2. Observed frequencies are normally distributed about expected frequencies over repeated samples [...]
3. The logarithm of the expected value of the response variable is a linear combination of the explanatory variables. [...]

However, unless I misread it, it does not mention the assumption that the mean should equal the variance (aka equidispersion), as usually assumed when using a Poisson regression:

Mean=Variance By definition, the mean of a Poisson random variable must be equal to its variance.

What am I missing?

Can I simply ignore the equidispersion assumption when using a Poisson regression for a log-linear analysis? Isn't there a risk, for example, that coefficients detected as significant will actually be non-significant -or vice versa?

Or is it implied that I should use an alternative when the equidispersion assumption is not met, like negative binomial or generalized Poisson regressions?

Thanks,