Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

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^maxފ½ 2 lnðL^F=L^maxފ

= [–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