Introductory Biostatistics

alized linear models of which Poisson model is a special case. Instead of

VarðYÞ¼m

it allows the variance function to have a multiplicative overdispersion factorf:

VarðYÞ¼fm

The models are fit in the usual way, and the point estimates of regression
coe‰cients are not a¤ected. The covariance matrix, however, is multiplied by
f. There are two options available for fitting overdispersed models; the users
can control either the scaled deviance (by specifying DSCALE in the model
statement) or the scaled Pearson chi-square (by specifying PSCALE in the
model statement). The value of the controlled index becomes 1; the value of the
other index is close to, but may not be equal, 1.

Example 10.15 Refer to the data set on emergency service of Example 10.5
(Table 10.2) with all four covariates. By fitting an overdispersed model con-
trolling the scaled deviance, we have the results shown in Table 10.11. As
compared to the results in Example 10.8, the point estimates remain the same
but the standard errors are larger; the e¤ect of workload (hours) is no longer
significant at the 5% level.
Note: An SAS program would include the instruction

MODEL CASES = GENDER RESIDENCY REVENUE HOURS/
DIST = POISSON LINK = LOG OFFSET = LN DSCALE;

and the measures of dispersion become those shown in Table 10.12. We would
obtain similar results by controlling the scaled Pearson chi-square.

TABLE 10.12

Criterion df Value

Scaled

Value

Deviance 39 39.000 1.000

Pearson chi-square 39 38.223 0.980

TABLE 10.11

Variable Coe‰cient Standard Error zStatistic pValue

Intercept
8.1338 1.0901
7.462 <0.0001
No residency 0.2090 0.2378 0.879 0.3795
Female
0.1954 0.2579
0.758 0.4486
Revenue 0.0016 0.0033 0.485 0.6375
Hours 0.0007 0.0004 1.694 0.0903

POISSON REGRESSION MODEL 369