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

Answers to
Practice
Exercises

1. a. X¼(CAT¼1, AGE¼50, CHL¼200, ECG¼0,
   SMK¼0, HPT¼0)
   P^ðXÞ¼ 1 =f 1 þexp½logitP^ðX*ފg,

where

logitP^ðX*Þ¼ 4 : 0497 þð 12 : 6894 Þð 1 Þþ 0 : 0350 ð 50 Þ

þð 0 : 00545 Þð 200 Þþ: 3671 ð 0 Þþ 0 : 7732 ð 0 Þ

þ 1 : 0466 ð 0 Þþ 0 : 0692 ð 1 Þð 200 Þ

þð 2 : 3318 Þð 1 Þð 0 Þ

b. logitP^ðX*Þ¼ 2 : 2391

P^ðX*Þ¼ 1 =f 1 þexp½logit^PðX*ފg

¼ 1 =f 1 þexp½ 2 : 2391 Šg¼ 0 : 096

c. Cut-point¼0.200

SinceP^ðX*Þ¼ 0 : 096 < 0 : 200 , we would predict sub-

jectX* to be a noncase.

d. If the cut-point is 0 and there is at least one true

case, than every case in the dataset will have

P^ðX*Þ> 0 , i.e., all 71 true cases will exceed the

cut-point and therefore be predicted to be cases.

Thus, the sensitivity percent is 100(71/71)¼100.

e. It is not possible that an increase in the cut-point

could result in a decrease in the specificity.

f. The denominator for computing 1 minus the speci-

ficity is the number of true noncases (538), whereas

the denominator for the false positive percentage in

the SAS output is the number of persons classified

as positive (74). Thus, we obtain different results as

follows:

Percentage specificity¼(100)(1Sp)¼(100)43/

538 ¼8%, whereas

Percentage false positive¼(100)43/74¼58.1%.

2. a. AUC¼c¼0.789. Grade C, i.e., fair discrimination.
   b. 38,198¼ 71 538, where 71 is the number of true
   cases and 538 is the number of true noncases.
   Thus, 38,198 is the number of distinct case/non-
   case pairs in the dataset.
   c. Percent Concordant¼ 100 w/np, where w is the
   number of case/noncase pairs for which the case
   has a higher predicted probability than the noncase
   andnpis the total number of case/noncase pairs
   (38,198).
   Percent Tied¼ 100 z/np, wherezis the number of
   case/noncase pairs for which the case has the same
   predicted probability as the noncase.

386 10. Assessing Discriminatory Performance of a Binary Logistic Model