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LAMBDA AND THE PRE PRINCIPLE

Table 21.5 is a hypothetical 2 X 2 table showing the breakdown, by gender, of adult monolingual Indians and adult bilingual Indian/Spanish speakers in a Mexican village.

Table 21.5 Monolingual and Bilingual Speakers, by Gender, in a Mexican Village, 1962

Men

Women

Row totals

Bilingual

  • 61
  • (82%)
  • 24
  • (36%)

85

Monolingual

  • 13
  • (18%)
  • 42
  • (64%)

55

Column totals

74

66

140

old error = 55

new error = 13 + 24 = 37

' - ^ - 33

I’ve included the marginals and the ns in the cells to make it easier to do the calculation here.

Reading across table 21.5, we see that 82% of the men were bilingual, compared to 36% of the women. Clearly, gender is related to whether someone is a bilingual Indian/ Spanish speaker or whether he or she is monolingual in the Indian language only.

Suppose that for the 140 persons in table 21.5 you were asked to guess whether they were bilingual or monolingual, but you didn’t know their gender. The mode for the dependent variable in this table is ‘‘bilingual’’ (85 bilinguals compared to 55 monolin- guals), so you should guess that everybody is bilingual. If you did that, you’d make 55 mistakes out of the 140 choices, for an error rate of 55/140, or 39%. We’ll call this the old error.

Suppose, though, that you have all the data in table 21.5—you know the mode for gender as well as for bilingual status. Your best guess now would be that every man is bilingual and every woman is monolingual. You’d still make some mistakes, but fewer than if you just guessed that everyone is bilingual.

How many fewer? When you guess that every male is bilingual, you make exactly 13 mistakes, and when you guess that every female is monolingual, you make 24 mistakes, for a total of 37 out of 140 or 37/140 = 26%. This is the new error. The difference between the old error (39%) and the new error (26%), divided by the old error is the proportionate reduction of error, or PRE. Thus,

This PRE measure of association for nominal variables is called lambda, written either or L. Like all PRE measures of association, lambda has the nice quality of being intuitively and directly interpretable. A of .33 means that if you know the scores on an independent variable, you can guess the scores on the dependent variable 33% more of the time than if you didn’t know anything about the independent variable.

 
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