Residuals

• A model fits well if the expected and observed counts approximately coincide.
• We normally use an overall measure of this match, such as $G^2$ or $X^2$.
• However, this is based on cell-wise comparisons of fitted and observed values, and we can look at the residuals.
• Simple residuals are defined as:
  
  $$R_{ij} = O_{ij} - E_{ij}$$
  
  
  That is, if the model underfits that cell there is a positive residual.

• The size of residuals will be related to the cell size, so they are usually scaled. A `Pearson Residual' is defined as
  
  $$e_{ij} = \frac{O_{ij} - E_{ij}}{\sqrt E_{ij}}$$
  
  
  Notice that $X^2$ is thus equal to $\Sigma e^2$.

• We can also partition the $G^2$ across the cells to give what are called deviance residuals.

• If the model fits, these residuals have a large-sample standard normal distribution so residuals with an absolute value much greater than 2 are worthy of attention.

• But remember that in a big table there are many residuals so we expect to see a few in the tails of the probability distribution.

• In practice we will use adjusted residuals. These are equivalent to Pearson residuals scaled by $\sqrt{1/(1-h_i)}$ where $h_i$ is the leverage of that case (cell):
  
  $$e_{ij}^{adj} = \frac{O_{ij} - E_{ij}}{\sqrt{ E_{ij} (1 - h_{ij})}}$$
  
  

• Examination of residuals can tell us a lot about how the model fails to fit. SPSS normally displays observed and expected counts, and raw, adjusted and deviance residuals in the GENLOG output. It can also save them as a variable for examination elsewhere:

  GENLOG
    fsex housing vote
    /MODEL=POISSON
    /DESIGN fsex housing vote
    /SAVE ADJRESID .
  

  This will create a variable ADJ_n where n is a number containing the adjusted residuals.

• Using a macro available in http://teaching.sociology.ul.ie/~brendan/CDA/macros/ these can be inspected in tables:

  temp.
  select if fsex=1.
  tabu housing vote adj_1.
  temp.
  select if fsex=2.
  tabu housing vote adj_1.
  

  Something similar can be achieved using the menu interface, via Statistics Custom Tables Basic Tables; the macro handles the case for two classifying variables.

  Men
  +----------------+---------------------------------------+
  |                |                 VOTE                  |
  |                +---------+---------+---------+---------+
  |                |Conservat| Labour  | Lib-dem |  Other  |
  |                |ive      |         |         |         |
  |                +---------+---------+---------+---------+
  |                |  Mean   |  Mean   |  Mean   |  Mean   |
  +----------------+---------+---------+---------+---------+
  |HOUSING         |         |         |         |         |
  |Owned/mortgage  |         |         |         |         |
  |  Adjusted      |         |         |         |         |
  |     Residual   |    8.65 |   -2.87 |   -2.32 |    -.22 |
  |LA etc rented   |         |         |         |         |
  |  Adjusted      |         |         |         |         |
  |     Residual   |  -10.05 |    4.81 |   -4.58 |     .51 |
  |Private rented  |         |         |         |         |
  |  Adjusted      |         |         |         |         |
  |     Residual   |    -.18 |     .79 |    1.01 |     .32 |
  +----------------+---------+---------+---------+---------+
  Women
  +----------------+---------+---------+---------+---------+
  |HOUSING         |         |         |         |         |
  |Owned/mortgage  |         |         |         |         |
  |  Adjusted      |         |         |         |         |
  |     Residual   |    3.62 |   -8.63 |    4.57 |   -2.09 |
  |LA etc rented   |         |         |         |         |
  |  Adjusted      |         |         |         |         |
  |     Residual   |   -7.51 |   12.35 |    -.85 |    2.28 |
  |Private rented  |         |         |         |         |
  |  Adjusted      |         |         |         |         |
  |     Residual   |   -2.92 |     .36 |     .80 |    1.13 |
  +----------------+---------+---------+---------+---------+
  

Loglinear Analysis Unit 4