In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics, Goodman & Kruskal's lambda (
) is a measure of
proportional reduction in error
Proportional reduction in loss (PRL) is a general framework for developing and evaluating measures of the reliability of particular ways of making observations which are possibly subject to errors of all types. Such measures quantify how much havin ...
in
cross tabulation
In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business ...
analysis. For any sample with a
nominal
Nominal may refer to:
Linguistics and grammar
* Nominal (linguistics), one of the parts of speech
* Nominal, the adjectival form of "noun", as in "nominal agreement" (= "noun agreement")
* Nominal sentence, a sentence without a finite verb
* Nou ...
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.
is defined by 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
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
and dependent variables) to one (
perfect association
Perfect commonly refers to:
* Perfection, completeness, excellence
* Perfect (grammar), a grammatical category in some languages
Perfect may also refer to:
Film
* ''Perfect'' (1985 film), a romantic drama
* ''Perfect'' (2018 film), a science ...
).
Weaknesses
Although Goodman and Kruskal's lambda is a simple way to assess the 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. As an example, consider the table below, which describes a fictitious sample of 350 individuals, categorized by relationship status and blood pressure. Assume that the relationship status is the independent variable, the blood pressure is the dependent variable, i.e., the question asked is "can the blood pressure be predicted better if the relationship status is known?"
For this sample,
:
The reason is that the predicted nominal blood pressure is actually "Normal" in both columns (both upper numbers are higher than the corresponding lower number). Thus, considering the relationship status will not change the prediction that people have a normal blood pressure, even though the data indicate that being married increases the probability of high blood pressure.
If the question is changed, e.g. by asking "What is the predicted relationship status based on blood pressure?,"
will have a non-zero value.
That is:
See also
*
Proportional reduction in loss
Proportional reduction in loss (PRL) is a general framework for developing and evaluating measures of the reliability of particular ways of making observations which are possibly subject to errors of all types. Such measures quantify how much havin ...
References
*Goodman, L.A., Kruskal, W.H. (1954
"Measures of association for cross classifications" Part I. ''
Journal of the American Statistical Association
The ''Journal of the American Statistical Association (JASA)'' is the primary journal published by the American Statistical Association, the main professional body for statisticians in the United States. It is published four times a year in Mar ...
'', 49, 732–764.
*Goodman, L.A., Kruskal, W.H. (1959
"Measures of Association for Cross Classifications. II: Further Discussion and References" ''Journal of the American Statistical Association'', 52, 123–163.
*Goodman, L.A., Kruskal, W.H. (1963) "Measures of Association for Cross Classifications III: Approximate Sampling Theory", ''Journal of the American Statistical Association'', 58, 310–364. {{doi, 10.1080/01621459.1963.10500850
Statistical ratios
Summary statistics for contingency tables