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

Using the notation just described, the formulae

for these discrimination measures are shown

at the left, with the first of these formulae (for

the AUC) provided in the previous section.

The calculation of the AUC for the fitted model

is shown at the left. The value forwin this

formula is 13,635(.718), or 9,789.91 and the

value forzis (13,635)(.053), or 722.655.

Based on the AUC result of 0.745 for these data,

there is evidence of fair (Grade C) discrimina-

tion using the fitted model.

A plot of the ROC curve for these data can also

be obtained and is shown here. Notice that the

points on the plot that represent the coordi-

nates of Se by 1Sp at different cut-pts have

not been connected by the program. Neverthe-

less, it is possible to fit a cubic regression to the

plotted points of sensitivity by 1specificity

(not shown, but see Computer Appendix).

where
w¼no. of case/noncase pairs for
which
^PðXcaseÞ>P^ðXnoncaseÞ

d¼no. of case/noncase pairs for
which
^PðXnoncaseÞ>P^ðXcaseÞ

z¼no. of case/noncase pairs for
which
^PðXcaseÞ¼P^ðXnoncaseÞ

Formulae for discrimination measures:

c =w^ + 0.5z= AUC

np

Somer’s D =wn – d

p

Gamma =w – d

w + d

Tau-a = w – d

0.5ΣYi(ΣYi – 1)

i i

EXAMPLE

c¼

wþ 0 : 5 z

np

¼

13,635ð: 718 Þþ 0 : 5 ð13,635Þð: 053 Þ

13,635

¼ 0 : 745

AUC¼ 0 : 745 )Fair discrimination

ðgrade CÞ

1.0

ROC plot

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.0 0.1 0.2 0.3 0.4 0.5

1 – Specificity

Sensitivity

AUC = 0.745

0.6 0.7 0.8 0.9 1.0

368 10. Assessing Discriminatory Performance of a Binary Logistic Model