an introduction to generalized linear models - GDM@FUDAN ...

2.3.4 Residuals and model checking
Firstly, consider residuals for a model involving the Normal distribution. Suppose
that the response variable Yi is modelled by
E(Yi) =µi; Yi ∼ N(µi,σ 2 ).
The fitted values are the estimates �µ i. Residuals can be defined as yi − �µ i and
the approximate standardized residuals as
ri =(yi − �µ i)/�σ,
where �σ is an estimate ofthe unknown parameter σ. These standardized residuals
are slightly correlated because they all depend on the estimates �µ i and
�σ that were calculated from the observations. Also they are not exactly Normally
distributed because σ has been estimated by �σ. Nevertheless, they are
approximately Normally distributed and the adequacy ofthe approximation
can be checked using appropriate graphical methods (see below).
The parameters µi are functions of the explanatory variables. If the model
is a good description ofthe relationship between the response and the explanatory
variables, this should be well ‘captured’ or ‘explained’ by the �µ i’s.
Therefore there should be little remaining information in the residuals yi − �µ i.
This too can be checked graphically (see below). Additionally, the sum of
squared residuals � (yi − �µ i) 2 provides an overall statistic for assessing the
adequacy ofthe model; in fact, it is the component ofthe log-likelihood function
or least squares expression which is optimized in the estimation process.
Secondly, consider residuals from a Poisson model. Recall the model for
chronic medical conditions
E(Yi) =θi; Yi ∼ P oisson(θi).
In this case approximate standardized residuals are ofthe form
ri = yi − �θi � .
�θi These can be regarded as signed square roots ofcontributions to the Pearson
goodness-of-fit statistic
� (oi − ei) 2
,
i
where oi is the observed value yi and ei is the fitted value � θi ‘expected’ from
the model.
For other distributions a variety ofdefinitions ofstandardized residuals
are used. Some ofthese are transformations ofthe terms (yi − �µ i) designed
to improve their Normality or independence (for example, see Chapter 9 of
Neter et al., 1996). Others are based on signed square roots ofcontributions
to statistics, such as the log-likelihood function or the sum of squares, which
are used as overall measures ofthe adequacy ofthe model (for example, see
© 2002 by Chapman & Hall/CRC
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