Goodman and Kruskal's lambda
Encyclopedia
In probability theory
and statistics
, Goodman & Kruskal's 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 independent variable differ from the overall modal category and frequency, i.e. for all values of the independent variable together.
can be calculated with the equation
where
is the overall non-modal frequency, and
is the sum of the non-modal frequencies for each value of the independent variable.
Values for lambda range from zero (no association between independent and dependent variables) to one (perfect association).
For this sample,
even though the data demonstrate a pronounced relationship between the independent and dependent variables.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, Goodman & Kruskal's lambda (
) is a measure of proportional reduction in error in cross tabulationCross tabulation
Cross tabulation is the process of creating a contingency table from the multivariate frequency distribution of statistical variables. Heavily used in survey research, cross tabulations can be produced by a range of statistical packages, including some that are specialised for the task. Survey...
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 independent variable differ from the overall modal category and frequency, i.e. for all values of the independent variable together.
can be calculated with the equationwhere
is the overall non-modal frequency, and is the sum of the non-modal frequencies for each value of the independent variable.Values for lambda range from zero (no association between independent and dependent variables) 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.| |Total | ||||
|---|---|---|---|---|
| Unmarried | Married | |||
| Blood Pressure | Normal | 80% (120) |
51% (102) |
63.4% (222) |
| High | 20% (30) |
49% (98) |
36.6% (128) |
|
| Total | 42.9% (150) |
57.1% (200) |
100% (350) |
|
For this sample,
even though the data demonstrate a pronounced relationship between the independent and dependent variables.