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

2.3.5 Inference and interpretation
It is sometimes useful to think of scientific data as measurements composed of
a message, or signal, that is distorted by noise. For instance, in the example
about birthweight the ‘signal’ is the usual growth rate ofbabies and the ‘noise’
comes from all the genetic and environmental factors that lead to individual
variation. A goal ofstatistical modelling is to extract as much information
as possible about the signal. In practice, this has to be balanced against
other criteria such as simplicity. The Oxford Dictionary describes the law of
parsimony (otherwise known as Occam’s Razor) as the principle that no
more causes should be assumed than will account for the effect. Accordingly
a simpler or more parsimonious model that describes the data adequately
is preferable to a more complicated one which leaves little of the variability
‘unexplained’. To determine a parsimonious model consistent with the data,
we test hypotheses about the parameters.
Hypothesis testing is performed in the context of model fitting by defining
a series ofnested models corresponding to different hypotheses. Then the
question about whether the data support a particular hypothesis can be formulated
in terms ofthe adequacy offit ofthe corresponding model relative
to other more complicated models. This logic is illustrated in the examples
earlier in this chapter. Chapter 5 provides a more detailed explanation of
the concepts and methods used, including the sampling distributions for the
statistics used to describe ‘goodness offit’.
While hypothesis testing is useful for identifying a good model, it is much
less useful for interpreting it. Wherever possible, the parameters in a model
should have some natural interpretation; for example, the rate of growth of
babies, the relative risk ofacquiring a disease or the mean difference in profit
from two marketing strategies. The estimated magnitude of the parameter and
the reliability ofthe estimate as indicated by its standard error or a confidence
interval are far more informative than significance levels or p-values. They
make it possible to answer questions such as: is the effect estimated with
sufficient precision to be useful, or is the effect large enough to be of practical,
social or biological significance?
2.3.6 Further reading
An excellent discussion ofthe principles ofstatistical modelling is in the introductory
part ofCox and Snell (1981). The importance ofadopting a systematic
approach is stressed by Kleinbaum et al. (1998). The various steps ofmodel
choice, criticism and validation are outlined by Krzanowski (1998). The use of
residuals is described in Neter et al. (1996), Draper and Smith (1998), Belsley
et al. (1980) and Cook and Weisberg (1999).
© 2002 by Chapman & Hall/CRC