“Somers' D”的意思、由来-开放百科全书

In statistics, Somers’ D, sometimes incorrectly referred to as Somer’s D, is a measure of ordinal association between two possibly dependent random variables {{mvar|X}} and {{mvar|Y}}. Somers’ D takes values between

when all pairs of the variables disagree and

when all pairs of the variables agree. Somers’ D is named after Robert H. Somers, who proposed it in 1962.^[1]

Somers’ D plays a central role in rank statistics and is the parameter behind many nonparametric methods.^[2] It is also used as a quality measure of binary choice or ordinal regression (e.g., logistic regressions) and credit scoring models.

Somers’ D for sample

We say that two pairs

and

are concordant if the ranks of both elements agree, or

and

or if

and

. We say that two pairs

and

are discordant, if the ranks of both elements disagree, or if

and

or if

and

. If

, the pair is neither concordant nor discordant.

Let

be a set of observations of two possibly dependent random vectors {{mvar|X}} and {{mvar|Y}}. Define Kendall tau rank correlation coefficient

where

is the number of concordant pairs and

is the number of discordant pairs. Somers’ D of {{mvar|Y}} with respect to {{mvar|X}} is defined as

.^[2] Note that Kendall's tau is symmetric in {{mvar|X}} and {{mvar|Y}}, whereas Somers’ D is asymmetric in {{mvar|X}} and {{mvar|Y}}.

quantifies the number of pairs with unequal {{mvar|X}} values, Somers’ D is the difference between the number of concordant and discordant pairs, divided by the number of pairs with {{mvar|X}} values in the pair being unequal.

Somers’ D for distribution

Let two independent bivariate random variables

and

have the same probability distribution

. Again, Somers’ D, which measures ordinal association of random variables {{mvar|X}} and {{mvar|Y}} in

, can be defined through Kendall's tau

or the difference between the probabilities of concordance and discordance. Somers’ D of {{mvar|Y}} with respect to {{mvar|X}} is defined as

. Thus,

is the difference between the two corresponding probabilities, conditional on the {{mvar|X}} values not being equal.

If {{mvar|X}} has a continuous probability distribution, then

and Kendall's tau and Somers’ D coincide. Somers’ D normalizes Kendall's tau for possible mass points of variable {{mvar|X}}.

If {{mvar|X}} and {{mvar|Y}} are both binary with values 0 and 1, then Somers’ D is the difference between two probabilities:

Somers' D for binary dependent variables

In practice, Somers' D is most often used when the dependent variable Y is a binary variable,^[2] i.e. for binary classification or prediction of binary outcomes including binary choice models in econometrics. Methods for fitting such models include logistic and probit regression.

Several statistics can be used to quantify the quality of such models: area under the receiver operating characteristic (ROC) curve, Goodman and Kruskal's gamma, Kendall's tau (Tau-a), Somers’ D, etc. Somers’ D is probably the most widely used of the available ordinal association statistics.^[3] If there are no ties on independent variable, Somers’ D is related to the area under the receiver operating characteristic curve (AUC),

In the case where the independent (predictor) variable {{mvar|X}} is {{em|discrete}} and the dependent (outcome) variable {{mvar|Y}} is binary, Somers’ D equals

where

is the number of neither concordant nor discordant pairs that are tied on variable {{mvar|Y}} and not on variable {{mvar|X}}.

Example

Suppose that the independent (predictor) variable {{mvar|X}} takes three values, {{val|0.25}}, {{val|0.5}}, or {{val|0.75}}, and dependent (outcome) variable {{mvar|Y}} takes two values, {{val|0}} or {{val|1}}. The table below contains observed combinations of {{mvar|X}} and {{mvar|Y}}:

References

1. ^{{cite journal|last1=Somers|first1=R. H.|title=A new asymmetric measure of association for ordinal variables|journal=American Sociological Review|date=1962|volume=27|issue=6|jstor=2090408|doi=10.2307/2090408 }}
2. ^¹²{{cite journal|last1=Newson|first1=Roger|title=Parameters behind "nonparametric" statistics: Kendall's tau, Somers' D and median differences|journal=Stata Journal|date=2002|volume=2|issue=1|pages=45–64| url=http://www.stata-journal.com/article.html?article=st0007}}
3. ^{{cite book |last1=O'Connell |first1=A. A.|year=2006|title=Logistic Regression Models for Ordinal Response Variables|publisher=SAGE Publications}}