Other models

• Using this technique of partitioning the two-way association by fitting an extra variable to the model (without making this variable a classifying variable!) we can fit a lot of other models, some ad hoc and some expressing particular theoretical expectations.
• One is the `distance' model: if we feel there is an order in the categories, such that moves to adjacent categories are more likely than moves to further categories we can create a distance variable to take account of this:

  compute dist = 1+abs(aopfama-eopfama).
  
  tabu aopfama eopfama dist.
  
  +-------+-------+-------+-------+-------+
  |  1.00 |  2.00 |  3.00 |  4.00 |  5.00 |
  |  2.00 |  1.00 |  2.00 |  3.00 |  4.00 |
  |  3.00 |  2.00 |  1.00 |  2.00 |  3.00 |
  |  4.00 |  3.00 |  2.00 |  1.00 |  2.00 |
  |  5.00 |  4.00 |  3.00 |  2.00 |  1.00 |
  +-------+-------+-------+-------+-------+
  

• To fit this we first make a set of dummy variables and include them as covariates, with the row and column variables:

  factor dist ds 5 .
  genlog aopfama eopfama with ds2 ds3 ds4 ds5
   /print=none/plot=none
   /design = aopfama eopfama ds2 ds3 ds4 ds5.
  

• This implies that moving is more difficult the farther you have to go. The parameter estimates for the dummy variables should confirm this by monotonically decreasing (the farther you are from the diagonal the lower the expected frequency).

• An example of an ad hoc usage would be to take account of a single cell that has a high residual. We can define a variable with the value 1 for that cell and 0 otherwise, and add it to the model. This fits the cell exactly and has the same effect of weighting out the cell.

• A more efficient case is where you can group a number of cells and give them one parameter: then you're saying those cells are under-/over-populated for a single reason. For instance, when tabulating opinion at two times, you may feel that there are some people with very strong opinions who will always opt for strongly agree or strongly disagree and other people with no connection between their opinion at both times. If this is the case, the top-left and bottom-right corners of the table will be overpopulated and a model that doesn't take account of this will show positive residuals in these cells. We can fit a single term to take account of both:

  compute corner=0.
  if a=1 and e=1 corner = 1.
  if a=5 and e=5 corner = 1.
  

• It is important to realise that when we start looking at residuals and fitting models to take account of them the normal conditions of hypothesis testing no longer hold. We are instead trying to model or `smooth' the data, and we must be aware of the different nature of our findings.