Introduction to Categorical Data Analysis

3.4 STATISTICAL INFERENCE AND MODEL CHECKING 87
residual is a standardized difference
Pearson residual = ei = yi −ˆμi
�
�Var(yi)
For example, for Poisson GLMs the Pearson residual for count i equals
ei = yi −ˆμi
�
ˆμi
(3.9)
(3.10)
It divides by the estimated Poisson standard deviation. The reason for calling ei a
Pearson residual is that � e2 �
i = i (yi −ˆμi) 2 / ˆμi. When the GLM is the model
corresponding to independence for cells in a two-way contingency table, this is the
Pearson chi-squared statistic X2 for testing independence [equation (2.8)]. Therefore,
X2 decomposes into terms describing the lack of fit for separate observations.
Components of the deviance, called deviance residuals, are alternative measures
of lack of fit.
Pearson residuals fluctuate around zero, following approximately a normal distribution
when μi is large. When the model holds, these residuals are less variable than
standard normal, however, because the numerator must use the fitted value ˆμi rather
than the true mean μi. Since the sample data determine the fitted value, (yi −ˆμi)
tends to be smaller than yi − μi.
The standardized residual takes (yi −ˆμi) and divides it by its estimated standard
error, that is
Standardized residual = yi −ˆμi
SE
It 3 does have an approximate standard normal distribution when μi is large. With
standardized residuals it is easier to tell when a deviation (yi −ˆμi) is “large.” Standardized
residuals larger than about 2 or 3 in absolute value are worthy of attention,
although some values of this size occur by chance alone when the number of observations
is large. Section 2.4.5 introduced standardized residuals that follow up tests
of independence in two-way contingency tables. We will use standardized residuals
with logistic regression in Chapter 5.
Other diagnostic tools from regression modeling are also helpful in assessing fits
of GLMs. For instance, to assess the influence of an observation on the overall fit,
one can refit the model with that observation deleted (Section 5.2.6).
3 SE =[�Var(yi)(1 − hi)] 1/2 , where hi is the leverage of observation i (Section 5.2.6). The greater the
value of hi, the more potential that observation has for influencing the model fit.