Definitions

# Goodman and Kruskal's lambda

In probability theory and statistics, Goodman & Kruskal's lambda ($lambda$) is a measure of proportional reduction in error in cross tabulation analysis. For any sample with a nominal independent variable and dependent variable (or ones that can be treated nominally), it indicates the extent to which the modal categories and frequencies for each value of the IV differ from the overall modal category and frequency, i.e. for all values of the IV together. $lambda$ can be calculated with the equation

$lambda = frac\left\{varepsilon_1 - varepsilon_2\right\}\left\{varepsilon_1\right\}.$

where

$varepsilon_1$ is the overall non-modal frequency, and
$varepsilon_2$ is the sum of the non-modal frequencies for each value of the IV.

Values for lambda range from zero (no association between IV and DV) to one (perfect association).

## Weaknesses

Although Goodman and Kruskal's lambda is used to calculate association between variables, it yields a value of 0 (no association) whenever two variables are in accord—that is, when the modal category is the same for all values of the independent variable, even if the modal frequencies or percentages vary. Consider the table below, which describes a fictitious sample of 350 individuals, categorized by relationship status and blood pressure.

Relationship Status and Blood Pressure
Relationship Status Total
Unmarried Married
Blood Pressure Normal 80%
(120)
51%
(102)
63.4%
(222)
High 20%
(30)
49%
(98)
36.6%
(128)
Total 100%
(150)
100%
(200)
100%
(350)

For this sample,

$lambda = frac\left\{128 - \left(30 + 98\right)\right\}\left\{128\right\} = 0$

even though the data demonstrate a pronounced relationship between the independent and dependent variables.

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