Goodness of fit

• The standard test for 2-way independence uses the Pearson $X^2$ with expected values defined as $E_{ij}=\frac{n_{i+}n_{+j}}{n_{++}}$.

• This also applies to models other than independence: we can use the expected value under any model.

• In loglinear modelling estimation is by maximum likelihood methods which generate their own analogue of $X^2$, the `deviance' or $G^2$:
  
  $$G^2 = 2 \Sigma O_{ij} \log \frac{O_{ij}}{E_{ij}} = -2 [L_m - L_s]$$
  
  

• This is usually very close in value to Pearson's $X^2$, but has some mathematical properties which make it preferable.

• The interpretation is the same: if the $G^2$ is higher than the critical level of the theoretical $\chi ^2$ distribution, the model does not fit.