Interactions

• Interactions work in a similar manner, but the way they are constrained is a little more complicated.
• For a 2-way interaction, e.g., $\lambda_{ij}^{AB}$ there is a non-zero effect for every $ij$ combination, except where $i=1$ or $j=1$ (or whatever the reference category is):
  
  $$\lambda_{i1}^{AB} = \lambda_{1j}^{AB} = 0$$
  
  

• This means the saturated model looks like this:
  

  	1	2	3
  1	$\lambda_0$	$\lambda_0 + \lambda_2^B$	$\lambda_0 + \lambda_3^B$
  2	$\lambda_0 + \lambda_2^A$	$\lambda_0 + \lambda_2^A + \lambda_2^B + \lambda_{22}^{AB}$	$\lambda_0 + \lambda_2^A + \lambda_3^B + \lambda_{23}^{AB}$
  3	$\lambda_0 + \lambda_3^A$	$\lambda_0 + \lambda_3^A + \lambda_2^B + \lambda_{32}^{AB}$	$\lambda_0 + \lambda_3^A + \lambda_3^B + \lambda_{33}^{AB}$

Loglinear Analysis Unit 3