Properties of deviance
similar to
properties ofw^2 statistic
Common to test for GOF
by comparing deviance
withw^2 nk 1 value:
ðquestionably legitimateÞ
GOFH 0 : model fits
HA: model does not fit
Deviance “significant”)poor fit
Devðb^Þ¼ 2 lnðL^c=L^maxÞ
¼ 2 lnL^cð 2 lnL^maxÞ
Recall
LR¼ 2 lnL^Rð 2 lnL^FÞw^2
R¼reduced model
F¼full model
DevRð^bÞDevFð^bÞ
¼½ 2 lnðL^R=L^maxÞ½ 2 lnðL^F=L^maxÞ
= [–2 ln LˆR – (–2 ln Lˆˆˆmax)] – [–2 ln LF – (– 2 ln Lmax)]
¼ 2 lnL^Rð 2 lnL^FÞLR
Nevertheless for thelogistic model:
w^2 approximation of
deviance statistic is
questionable(see below)
These properties of the deviance, i.e., its values
range from zero to larger and larger positive
numbers, correspond to the properties of a chi-
square statistic as used in a likelihood ratio test.
In fact, when using the deviance to test for
GOF, it is common, though not strictly legiti-
mate, to compare the deviance to chi-square
values withnk 1 degrees of freedom when
the current model containskþ 1 parameters.
The GOF null hypothesis is that the “model
fits,” and the alternative hypothesis is that the
“model does not fit.” Thus, if the deviance is
“significantly” large, the model is considered to
have poor fit.
Note that thedeviance statistic is, by definition, a
likelihood ratio (LR) statistic for comparing one’s
current model to the saturated model. Thus, the
use of the chi-square distribution to test for the
significance of the deviance appears justified
because the LR statistic has an approximate
chi-square distribution underH 0 when compar-
ing full vs. reduced (nonsaturated) models that
are fitted using ML estimation.
In particular, it can be shown from simple
algebra that theLR test for comparing two hier-
archical nonsaturated regression models is
equivalent to the difference in deviances
between the two models. This follows, as
shown at the left, because the maximized like-
lihood for the saturated model drops out of the
difference in deviance scores.
Nevertheless, as we shall soon explain, when
using the deviance statistic to assess GOF for
a single (nonsaturated) logistic model, the
chi-square approximation for the LR test is
questionable.
Presentation: III. The Deviance Statistic 313