More dimensions: trend models

• The association summarised by a quantitative covariate may be in tables with dimensions greater than two.

• A linear-by-linear association may be fitted on two dimensions and allowed to vary across categories of a third (nominal) variable. Similarly an association with a single quantitative component could vary by a third variable.

  
  

  $$\log F_{ijk} = \lambda_0 + \lambda_i + \lambda_j + \lambda_k + \lambda_{ik} + \lambda_{jk} + \lambda_k\beta\times i\times j$$
  
  

  The interaction in $\lambda_k\beta ij$ constitutes a test whether the overall $IJ$ association described by $\beta ij$ is the same across all levels of the $K$ variable. If all the $\lambda_k$s are equal to one, the association pattern is not changing.

• Alternatively we can find models where we allow a full two-way association, but constrain the three-way association to be linear in the $K$ dimension:
  
  $$\log F_{ijk} = \lambda_0 + \lambda_i^A + \lambda_j^B + \lam... ..._{jk}^{BC}%% + \lambda_{ij}^{AB} + \lambda_{ij}^{AB}\times k$$
  
  

• The Payne, Payne and Heath paper in Dale and Davies, Analyzing Social and Political Change considers this model in detail. Where the $K$ dimension is something like time, these can be considered trend models, because they allow the association to change in a (log-)linear trending fashion over time.

• The interpretation is simple: there is an $IJ$ association, which changes over time. For instance, there may be over-population of the diagonal, but it is less in earlier times than later, and the change can be approximated by a log-linear effect.

• In terms of odds-ratios, they are on the whole significantly different from one, but each one changes in a log-linear fashion across time (towards or away from one).

• Other models are available which constrain all the odds ratios to move in the same direction (e.g., Xie's ``unidiff'' model): these test whether the association is weakening or strengthening, whereas the trend model simply allows the odds ratios to change

• Fitting a trend model is straightforward, since it is not different from the 2-D linear models we have considered:

  compute time = c.
  genlog a b c with time
    /print=est/plot=none
    /design= a b c 
             a by b   a by c   b by c
             a by b by time
  

  The interpretation is that a varies across time, b varies across time, there is an association between a and b, and this association evolves log-linearly over time.

• Extensions of this model by imputing values to time, or by estimating it log-multiplicatively, are straightforward.

Loglinear Analysis Unit 6