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145441
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Remarks
The population value of Kendall’sa(Kendall 1970) is defined as
XY=E[sign(X1X2)sign(Y1Y2)](1)
where(X1;Y1)and(X2;Y2)are bivariate random variables sampled independently from the same population, andE[]denotes
expectation. The population value of Somers’D
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(Somers 1962)
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is defined as
DYX=
XY
XX
(2)
Therefore,XYis the difference between two probabilities, namely the probability that the larger of the twoXvalues is
associated with the larger of the twoYvalues and the probability that the largerXvalue is associated with the smallerYvalue.
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147705
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Under this relation, Kendall’savalues of 0,13,12
and1correspond to Pearson’s correlations of 0,12,1p2and1, respectively. A similar correspondence is likely to
hold in a wider range of continuous bivariate distributions
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(Kendall 1949, Newson 1987)
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Newson 1987).
3. Kendall’sahas the desirable property that a largeracannot be secondary to a smallera, that is, if a positiveXYis
caused entirely by a monotonic positive relationship of both variables with a third variableW, thenWXandWYmust
both be greater thanXY.
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147720
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Under this relation, Kendall’savalues of 0,13,12
and1correspond to Pearson’s correlations of 0,12,1p2and1, respectively. A similar correspondence is likely to
hold in a wider range of continuous bivariate distributions (Kendall 1949,
 Exact

Newson 1987)
 Suffix

. Kendall’sahas the desirable property that a largeracannot be secondary to a smallera, that is, if a positiveXYis
caused entirely by a monotonic positive relationship of both variables with a third variableW, thenWXandWYmust
both be greater thanXY.
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150592
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notation to define (for instance)
vh:j:=
Xmh
i=1
Xmj
k=1
vhijk;t(XY)h:j:=
Xmh
i=1
Xmj
k=1
t(XY)hijk;vh:::=
Xn
j=1
vh:j:;t(XY)h:::=
Xn
j=1
t(XY)h:j:(8)
and any other sums over any other indices. Given that the clusters are sampled independently from a common population of
clusters, we can define
V=E[vh:j:];TXY=E
h
t(XY)h:j:
i
(9)
for allh6=j. (In the terminology of
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Hoeffding (1948)
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these quantities are regular functionals of the cluster population distribution,
and the expressions inside the square brackets are kernels of these regular functionals.) The quantities we really want to estimate
are Kendall’saand Somers’D, defined respectively by
XY=TXY=V;DYX=TXY=TXX=XY=XX(10)
(These are equal to the familiar formulas (1) and (2) if each cluster contains one observation with
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151095
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inside the square brackets are kernels of these regular functionals.) The quantities we really want to estimate
are Kendall’saand Somers’D, defined respectively by
XY=TXY=V;DYX=TXY=TXX=XY=XX(10)
(These are equal to the familiar formulas (1) and (2) if each cluster contains one observation with an importance weight of one.)
To estimate these, we use the jackknife method of
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Arvesen (1969)
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on the regular functionals (9) and use appropriate Taylor
polynomials. The functionalsVandTXYare estimated by the Hoeffding (1948)Ustatistics
Vb=v::::
n(n1)
;bTXY=
t(XY)::::
n(n1)
(11)
and the respective jackknife pseudovalues corresponding to thehth cluster are given by
(V)h=(n1)1v::::(n2)1[v::::2vh:::]
(XY)
h=(n1)
1t(XY)
::::(n2)
1
h
t(XY)::::2t
(XY)
h:::
i
(12)
somers
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151234
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Somers’D, defined respectively by
XY=TXY=V;DYX=TXY=TXX=XY=XX(10)
(These are equal to the familiar formulas (1) and (2) if each cluster contains one observation with an importance weight of one.)
To estimate these, we use the jackknife method of Arvesen (1969) on the regular functionals (9) and use appropriate Taylor
polynomials. The functionalsVandTXYare estimated by the
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Hoeffding (1948)
 Suffix

Ustatistics
Vb=v::::
n(n1)
;bTXY=
t(XY)::::
n(n1)
(11)
and the respective jackknife pseudovalues corresponding to thehth cluster are given by
(V)h=(n1)1v::::(n2)1[v::::2vh:::]
(XY)
h=(n1)
1t(XY)
::::(n2)
1
h
t(XY)::::2t
(XY)
h:::
i
(12)
somersdcalculates correlation measures for a single variableXwith a set ofYvariates(Y(1);:::;Y(p)).
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154273
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()is the covariance matrix for the untransformed estimates given by (18), andb()is the
diagonal matrix whose diagonal entries are thed=destimates specified in the table, then the transformed parameter and its
covariance matrix are
b=(b);bC()=b()bC()b()0(19)
Fisher’sztransform was originally recommended for the Pearson correlation coefficient by
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Fisher (1921)
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(see also Gayen
1951), but Edwardes (1995) recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine
was suggested as a normalizing transform in Daniels and Kendall (1947).
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154301
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for the untransformed estimates given by (18), andb()is the
diagonal matrix whose diagonal entries are thed=destimates specified in the table, then the transformed parameter and its
covariance matrix are
b=(b);bC()=b()bC()b()0(19)
Fisher’sztransform was originally recommended for the Pearson correlation coefficient by Fisher (1921) (see also
 Exact

Gayen
1951)
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but Edwardes (1995) recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine
was suggested as a normalizing transform in Daniels and Kendall (1947).
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154319
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estimates given by (18), andb()is the
diagonal matrix whose diagonal entries are thed=destimates specified in the table, then the transformed parameter and its
covariance matrix are
b=(b);bC()=b()bC()b()0(19)
Fisher’sztransform was originally recommended for the Pearson correlation coefficient by Fisher (1921) (see also Gayen
1951), but
 Exact

Edwardes (1995)
 Suffix

recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine
was suggested as a normalizing transform in Daniels and Kendall (1947). Iftransf(z)ortransf(asin)is specified, then
somersdprints asymmetric confidence intervals for the untransformedDoravalues, calculated from symmetric confidence
intervals for the transformed parameters
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154491
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parameter and its
covariance matrix are
b=(b);bC()=b()bC()b()0(19)
Fisher’sztransform was originally recommended for the Pearson correlation coefficient by Fisher (1921) (see also Gayen
1951), but Edwardes (1995) recommended it specifically for Somers’Don the basis of simulation studies. Daniels’ arcsine
was suggested as a normalizing transform in
 Exact

Daniels and Kendall (1947)
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Iftransf(z)ortransf(asin)is specified, then
somersdprints asymmetric confidence intervals for the untransformedDoravalues, calculated from symmetric confidence
intervals for the transformed parameters using the inverse function().
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158790
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are mutually complementary.)
Weight (lbs.)
1,0002,0003,0004,0005,000
0
5
10
15
20
25
30
35
40
45
F
D
D
FF
F
F
F
F
F
F
DD
F
F
D
F
DF
FF
F
D
DDD
F
F
DDD
FD
F
D
F
D
DD
DD
D
DDD
DDDDDD
F
DDD
D
D
D
DD
D
D
D
D
D
D
D
D
DD
D
D
DD
Figure 1. Applyingsomersdto theautodata.
The confidence intervals for such high values of Somers’Dwould probably be more reliable if we used theztransform,
recommended by
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Edwardes (1995)
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The results of this are as follows:
.somersdforeignmpgweight,tran(z)
Somers'D
Transformation:Fisher'sz
Validobservations:74
Jackknife
foreignCoef.
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