Log-multiplicative models

• Log-multiplicative models are analogous to models which fit quantitive scores to summarise interaction. They differ in that the values of the score variable(s) are to be estimated by the model. This makes them
  • no longer loglinear; and
  • no longer directly estimable in the GLM framework: iteration is necessary.

• Where the score models assume that the effect comes from a dimension that we have measured (e.g., the actual value of the dose) logmultiplicative models assume that the effect can be represented on an unobserved dimension, with the actual locations on this dimension to be estimated. They are thus more general.

• The equation of a log-multiplicative model estimating one scale is
  
  $$\log F_{ij} = \lambda_0 + \lambda_i^A + \lambda_j^B + \alpha_i \zeta_j$$
  
  
  or
  
  $$F_{ij} = \tau_0\tau_i^A\tau_j^B e^{\alpha_i \zeta_j}$$
  
  
  Because of the multiplication of $\alpha_i$ and $\zeta$ it is no longer a linear model in the log format. $\alpha$ is a slope variable, with a value for each row (in this case), and $\zeta$ is the scale, a continuous variable with a separate (estimated) value for each column.

• This corresponds closely to the model where we apply a known scale:
  
  $$\log F_{ij} = \lambda_0 + \lambda_i + \lambda_j + \beta_i j$$
  
  

• Where both row and column effects are estimated, the model is Goodman's RC log-multiplicative model:

  
  

  $$\log F_{ij} = \lambda_0 + \lambda_i^A + \lambda_j^B + \phi\mu_i\nu_j$$
  
  

• $\phi$ is a scaling variable, $\mu_i$ and $\nu_j$ are the row and column effects.

• Estimation of these models is possible in GLIM using macros available in Lindsey's Modelling Frequency and Count Data, or by using the program lEM (available free via http://cwis.kub.nl/~fsw_1/mto/).