Parameter estimates

• When we fit models we get a great deal of output (from SPSS in particular) and most of it is useful sometimes. We will consider three aspects in particular:
  • overall fit
  • fitted values and residuals, and
  • parameter estimates.

• We will consider parameter estimates first.

• For convenience, consider a 1-dimensional table. This allows two simple models:
  
  $$\log F_i = \lambda_0$$
  
  
  and
  
  $$\log F_i = \lambda_0 + \lambda_i$$
  
  
  • The former is the grand-mean or equiprobability model
  • the latter is the saturated model, because we have only one dimension.

• These are mathematical statements of the model: what do they look like in principle?

• The grand mean model estimates just a constant, the intercept in a poisson regression. This constant is to be interpreted as the log of the expected count in all $I$ cells.

• The saturated model is written as if it has $I+1$ terms, where $I$ is the number of categories the variable has. Since we have only $I$ pieces of information we cannot determine all $I+1$ values, so we impose a simple constraint.

• Other methods are possible, but the typical approach is to set one parameter, often that for the first category, to zero. All other parameters consequently take a relative form: the effect of being in category $i$ relative to category 1.

• SPSS uses the last category as the reference category.

• Thus for all (say 5) cells in the 1-D table we have the following arrangement of estimated effects:

  
  

  $$\log F_1 = \lambda_0 + \lambda_1 = \lambda_0$$
  
  
  
  
  $$\log F_2 = \lambda_0 + \lambda_2$$
  
  
  
  
  $$\log F_3 = \lambda_0 + \lambda_3$$
  
  
  
  
  $$\log F_4 = \lambda_0 + \lambda_4$$
  
  
  
  
  $$\log F_5 = \lambda_0 + \lambda_5$$
  
  

• In the case of the housing variable in smalltab.sav the saturated model generates the following parameter estimates:

  Parameter Estimates
  
                                                 Asymptotic 95% CI
  Parameter   Estimate         SE    Z-value      Lower      Upper
  
          1     6.6073      .0367     179.80       6.54       6.68
          2     1.9677      .0392      50.15       1.89       2.04
          3      .5192      .0464      11.19        .43        .61
          4      .0000      .            .          .          .
  

• This generates the following results:

  Housing = 1: log freq = 6.6073 + 1.9677 = 8.575 => freq = 5297
  Housing = 2: log freq = 6.6073 +  .5192 = 7.127 => freq = 1244
  Housing = 3: log freq = 6.6073 +  .0000 = 6.607 => freq =  740
  

  which correspond to the observed values (with a little bit of rounding error).