Introductory Biostatistics

The number ofconcordancesis calculated by

C¼a 1 ðb 2 þþbkÞþa 2 ðb 3 þþbkÞþþak
 1 bk

(The termconcordance pairas used above corresponds to a less severe injury
for the seat belt user.) The number ofdiscordancesis

D¼b 1 ða 2 þþakÞþb 2 ða 3 þþakÞþþbk
 1 ak

Tomeasurethe degree of association, we use the indexC=Dand call it the
generalized odds; if there are only two levels of injury, this new index is reduced
to the familiar odds ratio. When data are properly arranged, by an a priori
hypothesis, the products in the number of concordance pairsC(e.g.,a 1 b 2 )go
from upper left to lower right, and the products in the number of discordance
pairsD(e.g.,b 1 a 2 ) go from lower left to upper right. In that a priori hypothe-
sis, column 1 is associated with row 1; In the example above, the use of seat
belt (yes, first row) is hypothesized to be associated with less severe injury
(none, first column). Under this hypothesis, the resulting generalized odds is
greater than 1.

Example 1.15 For the study above on the use of seat belts in automobiles, we
have from the data shown in Table 1.13,

C¼ð 75 Þð 175 þ 135 þ 25 Þþð 160 Þð 135 þ 25 Þþð 100 Þð 25 Þ

¼ 53 ; 225

D¼ð 65 Þð 160 þ 100 þ 15 Þþð 175 Þð 100 þ 15 Þþð 135 Þð 15 Þ

¼ 40 ; 025

leading to generalized odds of

y¼

C

D

¼

53 ; 225

40 ; 025

¼ 1 : 33

That is,given two people with di¤erent levels of injury, the (generalized) odds
that the more severely injured person did not wear a seat belt is 1.33. In other
words, the people with the more severe injuries would be more likely than the
people with less severe injuries to be those who did not use a seat belt.

The following example shows the use of generalized odds in case–control
studies with an ordinal risk factor.

24 DESCRIPTIVE METHODS FOR CATEGORICAL DATA