Introduction to Categorical Data Analysis

44 CONTINGENCY TABLES
when the data are very unbalanced, such as when some categories have many more
observations than other categories. Table 2.7 illustrates this. For the equally spaced
row scores (1, 2, 3, 4, 5), M 2 = 1.83, giving a much weaker conclusion (P = 0.18).
The magnitudes of r and M 2 do not change with transformations of the scores that
maintain the same relative spacings between the categories. For example, scores (1, 2,
3, 4, 5) yield the same correlation as scores (0, 1, 2, 3, 4) or (2, 4, 6, 8, 10) or (10, 20,
30, 40, 50).
An alternative approach assigns ranks to the subjects and uses them as the category
scores. For all subjects in a category, one assigns the average of the ranks that would
apply for a complete ranking of the sample from 1 to n. These are called midranks.
For example, in Table 2.7 the 17,114 subjects at level 0 for alcohol consumption
share ranks 1 through 17,114. We assign to each of them the average of these ranks,
which is the midrank (1 + 17,114)/2 = 8557.5. The 14,502 subjects at level