Fitting an "Overdispersed" Poisson Regression

McCullagh and Nelder fit a Poisson regression in which the usual assumption that the scale parameter equals 1.0 is relaxed; we will follow their example and fit an "overdispersed" Poisson regression.

  1. To run a Generalized Linear Models analysis, from the menus choose:

    Analyze > Generalized Linear Models > Generalized Linear Models...

    Figure 1. Generalized Linear Models Type of Model tab
    Generalized Linear Models Type of Model tab
  2. Select Poisson loglinear as the type of model. This specifies a Poisson distribution with a log link function.
  3. Click the Response tab.
    Figure 2. Generalized Linear Models Response tab
    Generalized Linear Models Response tab
  4. On the Response tab, select Number of damage incidents as the dependent variable.
  5. Click the Predictors tab.
    Figure 3. Predictors tab
    Predictors tab
  6. On the Predictors tab, select Ship type, Year of construction, and Period of operation as factors.
  7. Select Logarithm of aggregate months of service as the offset variable.
  8. Click Options.
    Figure 4. Generalized Linear Models Options dialog box
    Generalized Linear Models Options dialog box
  9. In the Options dialog, select Descending in the Category Order for Factors group. This indicates that the first category of each factor will be its reference category; the effect of this selection on the model is in the interpretation of parameter estimates.
  10. Click Continue, and then click the Model tab.
    Figure 5. Model tab
    Model tab
  11. On the Model tab, select type (Ship type), construction (Year of construction), and operation (Period of operation) as main effects in the model.
  12. Click the Estimation tab.
    Figure 6. Estimation tab
    Estimation tab
  13. On the Estimation tab, select Pearson chi-square from the Scale Parameter Method drop-down list in the Parameter Estimation group. The scale parameter is usually assumed to be 1 in a Poisson regression, but McCullagh and Nelder use the Pearson chi-square estimate to obtain more conservative variance estimates and significance levels.
  14. Click the EM Means tab.
    Figure 7. EM Means tab
    EM Means tab
  15. Select type (Ship type) and construction (Year of construction) as terms to display means for and select Pairwise as the contrast for each.
  16. Select Compute means for linear predictor as the scale.
  17. Select Sequential Sidak as the adjustment method.
  18. Click the Save tab.
    Figure 8. Save tab
    Save tab
  19. On the Save tab, select Predicted value of linear predictor and Standardized deviance residual. These values are saved to the active dataset and can help you diagnose any problems with the model fit.
  20. Click OK.

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