Introduction to Measures of Association, with Emphasis on Lambda

Consider an election with 4 candidates. Suppose you want to know to what extent a voter's candidate preference can be predicted from the voter's occupational category: professional, skilled, or unskilled. Lambda measures this quality. Let the occupational categories be A, B, C, and let the candidates be I, II, III, IV. Suppose a sample of 315 people breaks down as shown in the following table:

Preferences for 4 Candidates
Occ.	I	II	III	IV
A	36	11	15	21
B	14	28	51	32
C	18	45	32	12
Total	68	84	98	65

To compute lambda, first determine how many times you could guess candidate preference correctly without knowing the voter's occupational category. In the sample of 315 voters the most popular candidate is III. You would have 98 correct guesses by guessing candidate III for every voter. That's the best you can do without knowing voters' occupational categories.

Then determine the number of correct guesses you could achieve by using occupational category. The table shows that among voters in category A, the most popular candidate is I. Therefore if you knew a voter was in category A, you best guess would be I, and you would make 36 correct guesses. Similarly, your optimum strategy is to select the column with the largest number in each row. In the table below, the largest frequency in each row is starred. The starred values add up to 132, so you could make 132 correct guesses altogether if you knew each voter's occupational category. That's a gain of (132 - 98) or 34 over the best you can do without knowing occupation.

Favored Candidates in Each Row
Occ.	I	II	III	IV
A	36*	11	15	21
B	14	28	51*	32
C	18	45*	32	12
Total	68	84	98	65

But a hypothetical perfect judge could make 315 correct guesses (the total sample size), a gain of (315 - 98) or 217 over the same guesses. Since 34/217 = .157, we can say that knowing occupational category produces 15.7% of the increase in correct guesses that could be achieved by guessing all cases correctly. That value is lambda; lambda equals the increase in number of correct guesses of column membership produced by knowing row membership, expressed as a proportion of the largest possible increase. In more mechanical terms,

lambda = (sum of row maxima - largest column total)/(N - largest column total)

Lambda is a good measure of association for a very specific problem, but there are many instances where it would be inappropriate:

If you want to give more weight to correct "long shot" guesses than other guesses.
If you want to allow the guesser to distribute a guess across columns, saying for instance, "The chance is .4 that this voter will choose candidate I, .2 for candidate II", and so on.
If you want a measure of the degree to which the row variable affects the column variable. To see the inappropriateness of lambda for this purpose, see the later discussion of the "unique zero" property.
If you want a measure of association that treats the row and column variables symmetrically, so a table would yield the same measure of association if the table were "flipped", with each row becoming a column and vice versa.
If the row categories are necessarily the same as the column categories, as when a table relates husband's religion to wife's religion. Then the number of "correct predictions" couldn't be the highest entry in each row, as it is with lambda. Rather it should be the number in the diagonal cell, as when husband and wife have the same religion.
If the categories of both rows and columns are naturally ordered from low to high. In the voting example one might argue that occupation is so ordered, but candidates are certainly not.

Thus a great many conditions can suggest the need for alternatives to lambda. In the outline in the main section of this paper, these various criteria are applied in reverse order. For instance, measures of association for ordered categories appear first in the outline, and lamba is one of the last measures shown.

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