The Linear-by-Linear model

• An extension of this approach to two dimensions is the well-known `linear-by-linear association model'. (Agresti p182ff.)
• This takes the form
  
  $$\log F_{ij} = \lambda_0 + \lambda_i + \lambda_j + \beta ij$$
  
  

• In this model the association is described by a single number, $\beta$, which is multiplied by the product of the row number and column number:
  

  1  2  3  4  5
  2  4  6  8 10
  3  6  9 12 15
  4  8 12 16 20
  5 10 15 20 25
  

• $\beta$'s sign is related to the direction of the association: if higher values of I are associated with higher values of J, it will be positive, and vice versa.

• The model effectively means all `adjacent' odds ratios have the same value, $e^\beta$.

• Independence is given by $\beta=0$, $e^\beta=1$, i.e., the odds ratios are all 1.0.

• To fit this model we simply create a new variable that is the product of the column and row numbers:

  compute assoc = aopfamc*eopfamc.
  

  and add it to the model as a covariate:

  genlog aopfamc eopfamc with assoc 
   /print=est/plot=none
   /design= aopfamc eopfamc assoc .