Introductory Biostatistics

The distinction between concordance and association is that for two
responses to be associated perfectly, we require only that we can predict the
category on one response from the category of the other response, while for
two responses to have a perfect concordance, they must fall into the iden-
tical category. However, the proportions of concordance, overall or category-
specific, do not measure agreement. Among other reasons, they are a¤ected by
the marginal totals. One possibility is to compare the overall concordance,

y 1 ¼

X

i

pii

wherep’s are the proportions in the second 22 table above, with thechance
concordance,

y 2 ¼

X

i

piþpþi

which occurs if the row variable is independent of the column variable, because
if two events are independent, the probability of their joint occurrence is the
product of their individual or marginal probabilities (the multiplication rule).
This leads to a measure of agreement,

k¼

y 1 
y 2

1 
y 2

called thekappa statistic,0aka1, which can be expressed as

k¼

2 ðn 11 n 22 
n 12 n 21 Þ

n 1 þnþ 2 þnþ 1 n 2 þ

and the following are guidelines for the evaluation of kappa in clinical research:

k> 0 : 75 : excellent reproducibility

0 : 40 aka 0 : 75 : good reproducibility

0 ak< 0 : 40 : marginal=poor reproducibility

In general, reproducibility that is not good indicates the need for multiple
assessment.

Example 3.1 Two nurses perform ear examinations, focusing on the color of
the eardrum (tympanic membrane); each independently assigns each of 100
ears to one of two categories: (a) normal or gray, or (b) not normal (white,
pink, orange, or red). The data are shown in Table 3.6. The result,

PROBABILITY 119