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In [[probability theory]] and [[statistics]], '''Goodman & Kruskal's lambda''' ('''<math>\lambda</math>''') is a measure of proportional reduction in error in [[cross tabulation]] analysis. For any sample with a [[level of measurement#Nominal_measurement|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. <math>\lambda</math> can be calculated with the equation
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:<math>\lambda = \frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_1}.</math>
 
where
 
:<math>\varepsilon_1</math> is the overall non-modal frequency, and
:<math>\varepsilon_2</math> 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 variable|independent]] and dependent variables) to one ([[perfect association]]).
 
==Weaknesses==
{{Disputed-section|date=December 2011}}
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 (statistics)|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.
 
{| class="wikitable" style="text-align:center; width:50%"
|+ '''Relationship Status and Blood Pressure (fictitious)'''
|-
!colspan="2" rowspan="2"| !!colspan="2"|Relationship Status !!rowspan="2"|Total
|-
! Unmarried !! Married
|-
! rowspan="2"|Blood Pressure!! Normal
| 80%<br>(120) || 51%<br>(102) || 63.4%<br>(222)
|-
! High
| 20%<br>(30) || 49%<br>(98) || 36.6%<br>(128)
|-
!colspan="2"|Total
| 42.9%<br>(150) || 57.1%<br>(200) || 100%<br>(350)
|}
 
For this sample,
 
:<math>\lambda = \frac{128 - (30 + 98)}{128} = 0</math>
 
even though the data demonstrate a pronounced relationship between the independent and dependent variables.
 
==See also==
*[[Proportional reduction in loss]]
 
{{primary sources|date=July 2012}}
==References==
*Goodman, L.A., Kruskal, W.H. (1954) [http://www.nssl.noaa.gov/users/brooks/public_html/feda/papers/goodmankruskal1.pdf "Measures of association for cross classifications"]. Part I. ''[[Journal of the American Statistical Association]]'', 49, 732&ndash;764. {{jstor|281536}}
*Goodman, L.A., Kruskal, W.H. (1959) [https://bitbucket.org/mhunter/readinglists/src/d8e8010f0b0d/ReadingList_NotreDame/GoodmanKruskal1959CrossClassificationMeasures2.pdf "Measures of Association for Cross Classifications. II: Further Discussion and References"]. ''Journal of the American Statistical Association'', 52, 123&ndash;163. {{jstor|2282143}}
*Goodman, L.A., Kruskal, W.H. (1963) "Measures of Association for Cross Classifications III: Approximate Sampling Theory", ''Journal of the American Statistical Association'', 58, 310&ndash;364. {{jstor|2283271}} {{DOI|10.1080/01621459.1963.10500850}}
 
[[Category:Categorical data]]
[[Category:Statistical dependence]]
[[Category:Statistical ratios]]
[[Category:Summary statistics for contingency tables]]

Revision as of 18:12, 19 February 2014

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