Collapsibility

• With tabulated data, it is clear that adding a new variable to the table changes the dataset. The number of cells rises by a factor of $n$, the number of categories in the new variable, and the counts are spread out over this larger number of cells.
• Because of the risk of sparsity, and loss of parsimony, we may seek ways to exclude variables from the modelling process. But there are two ways of excluding variables: dropping them from the model (e.g., the /DESIGN statement) and not using them in constructing the table.
• The criteria for excluding a variable from a model are clear: its exclusion does not raise $G^2$ significantly. When can we take bigger step of excluding it from the table? That is, when can we collapse the table along one variable?

• In general we can collapse a table where the variable has no significant interaction with any other terms in the model. For instance, in an A*B*C table, if C has no significant interaction with the A*B association, then the marginal association is not different from the conditional association, and we lose no important information in modelling the marginal A*B table. As demonstrated earlier () if there is association the marginal table may be entirely misleading.

• If one variable is clearly understood as a dependent variable, it may be appropriate to collapse the table along dimensions not related to the dependent variable, even when interactions are present between that dimension and other independent variables.

• We use an example from Lindsey (p32ff) which he draws from Fingleton: a survey of shoppers, in particular a 3-way table of shopping trips by size of town visited (small, medium, large), mode of transport (walk, bus, car) and frequency (often, seldom). (See http://teaching.sociology.ul.ie/~brendan/CDA/datasets/travel.sps.)

• If we consider frequency as the dependent variable, we can think in terms of a logit model: independence in this framework is given by /DESIGN freq mode*size. The model doesn't fit, with $G^2$ of 132.9 for 8. Adding a freq*size term drops the $G^2$ by about 50 for 2 df, which is a significant improvement. Adding the freq*mode term instead drops the deviance to 9.02 for 6 df: this is a far more important interaction. The combined model with both interactions has a $G^2$ of 4.14 for 4 df: adding freq*size at this stage is only marginally significant, $p=0.09$.

• So we ask the question: should we exclude size from both the model statement and the table? First we fit the model without the freq*size interaction on the uncollapsed table:

  genlog freq size mode
    /print=est/plot=none
    /design = freq size mode mode*size.
  

  and then collapse the table:

  genlog freq mode
    /print=est/plot=none
    /design = freq mode.
  

  We find the fit changing from 132.9 for 8 df to 123.9 for 2 df, $p=0.17$. Not only does the fit not improve, but we are facing the opposite problem of sparsity, a table that is too small to fit interesting tables to (only $2\times3$). Therefore we retain size as a classifying variable.