Fitting models to square tables

• Square tables have only two dimensions, so only the following basic models can be fitted:
  • Grand mean: $F_{ij}=\tau_0$
  • Equiprobability on one dimension: $F_{ij}=\tau_0\tau_i$ and $F_{ij}=\tau_0\tau_j$
  • Independence: $F_{ij}=\tau_0\tau_i\tau_j$
  • Association: $F_{ij}=\tau_0\tau_i\tau_j\tau_{ij}$

• The association model is the saturated model, and the independence model is usually a poor fit for square tables: we need to model a reduced form of the association.

• Simplest case is the mover-stayer model: there are two sorts of cases, those who move, for whom there is independence (all destinations equally likely) and those who stay.

• This is also known as quasi-independence because it means independence everywhere except the diagonal.

  NB: some movers may end up in the same place by chance, so the diagonal will include both movers and stayers. Off the diagonal there are only movers, and there is independence off the diagonal.

• To fit: drop the diagonal and fit independence on the rest of the table:
  1. compute a variable equal to zero if the case is on the diagonal and equal to one if not:

     compute offdiag = VAR1<>VAR2.
     

     where VAR1 and VAR2 are the classifying variables.
  2. Weight the table according to this variable:

     genlog VAR1 VAR2
       /cstructure=offdiag 
     . . .
     

     or Statistics Loglinear General and put offdiag into the Cell Structure box.
  3. Fit it:

     genlog VAR1 VAR2
       /cstructure=offdiag 
       /design = VAR1 VAR2.
     

• Why is it called `mover-stayer'?
  • It assumes there are two types of people, those who have a non-zero probability of moving, and those who will not move. By weighting out the diagonal, we exclude the stayers, and can then test the movers for independence.

• Weighting out the diagonal is equivalent to fitting it exactly by including cell-specific terms. Weighting out reduces the degrees of freedom by dropping data points; fitting cell-specific terms uses up the same degrees of freedom. The deviance is the same in both cases because the cells, if fitted exactly, have residuals of zero.
• This fact is useful for comparing pairs of models (i.e., weighting out is equivalent to certain parameters being included so we can establish whether certain pairs of models are nested).

• We will get a clearer idea what fitting cell-specific terms means in the following sections.