Introductory Biostatistics

10.3.3 Overdispersion

The Poisson is a very special distribution; its meanmand variances^2 are equal.
If we use the variance/mean ratio as a dispersion parameter, it is 1 in a stan-
dard Poisson model, less than 1 in an underdispersed model, and greater than 1
in an overdispersed model. Overdispersion is a common phenomenon in prac-
tice and it causes concerns because the implication is serious; analysis that
assumes the Poisson model often underestimates standard error(s) and, thus
wrongly inflates the level of significance.

Measuring and Monitoring Dispersion After a Poisson regression model is
fitted, dispersion is measured by the scaled deviance or scaled Peason chi-
square; it is the deviance or Pearson chi-square divided by the degrees of free-
dom (the number of observations minus the number of parameters). Thedevi-
anceis defined as twice the di¤erence between the maximum achievable log
likelihood and the log likelihood at the maximum likelihood estimates of the
regression parameters. The contribution to the Pearson chi-square from theith
observation is

ðyi
mm^iÞ^2

mm^i

Example 10.14 Refer to the data set on emergency service of Example 10.5
(Table 10.2). With all four covariates, we have the results shown in Table
10.10. Both indices are greater than 1, indicating an overdispersion. In this
example we have a sample size of 44 but 5 df lost, due to the estimation of the
five regression parameters, including the intercept.

Fitting an Overdispersed Poisson Model PROC GENMOD allows the specifi-
cation of a scale parameter to fit overdispersed Poisson regression models. The
GENMOD procedure does not use the Poisson density function; it fits gener-

TABLE 10.9

Source df LRw^2 pValue

Residency 1 3.199 0.0741

Gender 1 0.84 0.3599

Revenue 1 0.71 0.3997

Hours 1 4.18 0.0409

TABLE 10.10

Criterion df Value

Scaled

Value

Deviance 39 54.518 1.398

Pearson chi-square 39 54.417 1.370

368 METHODS FOR COUNT DATA