Dependence and unobserved heterogeneity: overdispersion

• The model underlying loglinear models is Poisson regression. A Poisson distribution has
  • a variance equal to its mean, and
  • the expectation that the events being counted are independent.

• If we are missing something important in the table, such as a relevant variable (unobserved heterogeneity) or some form of dependence (e.g., individuals contributing multiple events) the variance of the counts may be greater than their mean. The effect of this is that standard errors will be underestimated by the normal assumptions.
• One strategy is to scale the standard errors by $\sqrt{X^2 / df}$ (making the confidence interval wider), or to divide $G^2$ by $X^2 / df$.

• Another approach is to use robust standard error estimates: this is a general strategy available in some programs, for instance Stata (lookup robust).

• Examining parameter estimates and their standard errors is an alternative way of deciding whether they contribute to the model's fit: either a conventional z-test ( $\vert\beta/ASE\vert > 2$), or a Wald test ($(\beta/ASE)^2$ as a $\chi ^2$ variable with $df = 1$). ($ASE$ stands for asymptotic standard error.)

Logistic Regression Unit 1