
 Start

141980
 Prefix

Review52: 482–494.
snp15d—Confidence intervals for nonparametric statistics and their differences
Roger Newson, Guy’s, King’s and St Thomas’ School of Medicine, London, UK, roger.newson@kcl.ac.uk
Abstract:Rank order or socalled nonparametric methods are in fact based on population parameters, which are zero under the
null hypothesis. Two of these parameters are Kendall’saand
 Exact

Somers’D, the parameter tested by a Wilco
 Suffix

xon ranksum
test. Confidence limits for these parameters are more informative thanpvalues alone, for three reasons. Firstly, confidence
intervals show that a highpvalue does not prove a null hypothesis.
 (check this in PDF content)

 Start

142430
 Prefix

Secondly, for continuous data, Kendall’sacan often
be used to define robust confidence limits for Pearson’s correlation by Greiner’s relation. Thirdly, we can define confidence
limits for differences between two Kendall’sa’s or
 Exact

Somers’D’s, and these are informative, be
 Suffix

cause a larger Kendall’s
aor Somers’Dcannot be secondary to a smaller one. The programsomersdcalculates confidence intervals for Somers’
Dor Kendall’sa, using jackknife variances. There is a choice of transformations, including Fisher’sz, Daniels’ arcsine,
Greiner’s, and theztransform of Greiner’s.
 (check this in PDF content)

 Start

142469
 Prefix

Secondly, for continuous data, Kendall’sacan often
be used to define robust confidence limits for Pearson’s correlation by Greiner’s relation. Thirdly, we can define confidence
limits for differences between two Kendall’sa’s or Somerscause a larger Kendall’s
aor
 Exact

Somers’Dcannot be secondary to a smaller one.
 Suffix

The programsomersdcalculates confidence intervals for Somers’
Dor Kendall’sa, using jackknife variances. There is a choice of transformations, including Fisher’sz, Daniels’ arcsine,
Greiner’s, and theztransform of Greiner’s.
 (check this in PDF content)

 Start

142524
 Prefix

Secondly, for continuous data, Kendall’sacan often
be used to define robust confidence limits for Pearson’s correlation by Greiner’s relation. Thirdly, we can define confidence
limits for differences between two Kendall’sa’s or Somerscause a larger Kendall’s
aor Somers The programdcalculates confidence intervals for
 Exact

Somers’
Dor Kendall’sa, using jackknife varia
 Suffix

nces. There is a choice of transformations, including Fisher’sz, Daniels’ arcsine,
Greiner’s, and theztransform of Greiner’s.Aclusteroption is available. The estimation results are saved as for a
model fit, so that differences can be estimated usinglincom.
 (check this in PDF content)

 Start

142799
 Prefix

There is a choice of transformations, including Fisher’sz, Daniels’ arcsine,
Greiner’s, and theztransform of Greiner’s.Aclusteroption is available. The estimation results are saved as for a
model fit, so that differences can be estimated usinglincom.
Keywords:
 Exact

Somers’ D, Kendall’s tau, rank correlation, ra
 Suffix

nksum test, Wilcoxon test, confidence intervals, nonparametric methods.
Syntax
somersdvarlist
weight
ifexp
inrange
,cluster(varname)level(#)tauatdist
transf(transformationname)
wheretransformationnameis one of
idenjzjasinjrhojzrho
fweights,iweightsandpweights are allowed.
 (check this in PDF content)

 Start

143133
 Prefix

Syntax
dvarlist
weight
ifexp
inrange
,cluster(varname)level(#)tauatdist
transf(transformationname)
wheretransformationnameis one of
idenjzjasinjrhojzrho
fweights,iweightsandpweights are allowed.
Description
dcalculates the nonparametric statistics
 Exact

Somers’D(corresponding to ranksum tests)
 Suffix

and Kendall’sa, with
confidence limits. Somers’Dorais calculated for the first variable ofvarlistas a predictor of each of the other variables in
varlist, with estimates and jackknife variances and confidence intervals output and saved ine()as if for the parameters of a
model fit.
 (check this in PDF content)

 Start

143182
 Prefix

Syntax
dvarlist
weight
ifexp
inrange
,cluster(varname)level(#)tauatdist
transf(transformationname)
wheretransformationnameis one of
idenjzjasinjrhojzrho
fweights,iweightsandpweights are allowed.
Description
dcalculates the nonparametric statistics Somers and Kendall’sa, with
confidence limits.
 Exact

Somers’Dorais calculated for the first variab
 Suffix

le ofvarlistas a predictor of each of the other variables in
varlist, with estimates and jackknife variances and confidence intervals output and saved ine()as if for the parameters of a
model fit.
 (check this in PDF content)

 Start

143491
 Prefix

Somersle ofvarlistas a predictor of each of the other variables in
varlist, with estimates and jackknife variances and confidence intervals output and saved ine()as if for the parameters of a
model fit. It is possible to uselincomto output confidence limits for differences between the population
 Exact

Somers’Dor Kendall’s
avalues.
Options
cluster
 Suffix

(varname)specifies the variable which defines sampling clusters. Ifclusteris defined, then the betweencluster
Somers’Dorais calculated, and the variances are calculated assuming that the data are sampled from a population of
clusters, rather than a population of observations.
level(#)specifies the confidence level, in percent, for confidence intervals of the esti
 (check this in PDF content)

 Start

143618
 Prefix

It is possible to uselincomto output confidence limits for differences between the population Somers(varname)specifies the variable which defines sampling clusters. Ifclusteris defined, then the betweencluster
 Exact

Somers’Dorais calculated, and the varianc
 Suffix

es are calculated assuming that the data are sampled from a population of
clusters, rather than a population of observations.
level(#)specifies the confidence level, in percent, for confidence intervals of the estimates.
 (check this in PDF content)

 Start

143994
 Prefix

Ifclusteris defined, then the betweencluster
Somerses are calculated assuming that the data are sampled from a population of
clusters, rather than a population of observations.
level(#)specifies the confidence level, in percent, for confidence intervals of the estimates. The default islevel(95)or as
set bysetlevel.
tauacausesdto calculate Kendall’sa.Iftauais absent, then
 Exact

somersdcalculates Somers’D.
tdistspecifies tha
 Suffix

t the estimates are assumed to have atdistribution withn1 degrees of freedom, wherenis the number
of clusters ifclusteris specified, or the number of observations ifclusteris not specified.
transf(transformationname)specifies that the estimates are to be transformed, defining estimates for the transformed population
value.iden(identity or untransformed) is the default.zspecifies Fisher’sz(
 (check this in PDF content)

 Start

144994
 Prefix

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
 Exact

Somers’D(Somers 1962)
 Suffix

’D(Somers 1962) 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.
 (check this in PDF content)

 Start

145002
 Prefix

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
 Exact

(Somers 1962)
 Suffix

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.
 (check this in PDF content)

 Start

145506
 Prefix

DYXis the difference between the two corresponding conditional probabilities, given that the twoXvalues are not equal.
Kendall’sais the covariance betweensign(X1X2)andsign(Y1Y2), whereas
 Exact

Somers’Dis the regression coefficient
ofsig
 Suffix

n(Y1Y2)with respect tosign(X1X2). (The correlation coefficient betweensign(X1X2)andsign(Y1Y2)is
known as Kendall’sb, and is the geometric mean ofDYXandDXY.)
Given a sample of data points(Xi;Yi), we may estimate and test the population values of Kendall’saand Somers’Dby
the corresponding sample statisticsbXYandbDYX.
 (check this in PDF content)

 Start

145780
 Prefix

(The correlation coefficient betweensign(X1X2)andsign(Y1Y2)is
known as Kendall’sb, and is the geometric mean ofDYXandDXY.)
Given a sample of data points(Xi;Yi), we may estimate and test the population values of Kendall’saand
 Exact

Somers’Dby
the corresponding sample statistic
 Suffix

sbXYandbDYX. These are commonly known as nonparametric statistics, even thoughXY
andDYXare parameters. The two Wilcoxon ranksum tests (see [R]signrank) both test hypotheses predictingDYX=0. The
twosample ranksum test represents the case whereXis a binary variable indicating membership of one of two subpopulations.
 (check this in PDF content)

 Start

147189
 Prefix

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
 Exact

(Kendall 1949, Newson 1987)
 Suffix

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.
 (check this in PDF content)

 Start

147204
 Prefix

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.
 (check this in PDF content)

 Start

150091
 Prefix

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
 Exact

Hoeffding (1948)
 Suffix

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
 (check this in PDF content)

 Start

150325
 Prefix

(In the terminology of Hoeffding (1948), 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
 Exact

Somers’D, defined respectively by
XY=TXY=V;
 Suffix

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.
 (check this in PDF content)

 Start

150535
 Prefix

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 SomersDYX=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
 Exact

Arvesen (1969)
 Suffix

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
 (check this in PDF content)

 Start

150670
 Prefix

want to estimate
are Kendall’saand SomersDYX=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
 Exact

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)).
 (check this in PDF content)

 Start

151517
 Prefix

This is done using the jackknife influence matrix
, which hasnrows labeled by the cluster subscripts, andp+2 columns labeled (in Stata fashion) by the namesV,X,and
Y(i)for 1ip. It is defined by
[h;V]= (V)hbV;[h;X]= (XX)hbTXX;h
h;Y(i)
i
= (XY
(i))
hbTXY(i)(13)
The jackknife covariance matrix is then equal to
Cb=[n(n1)]10(14)
The estimates for Kendall’saand
 Exact

Somers’Dare defined by
bXY=bTXY=bV;
 Suffix

bDYX=bTXY=bTXX(15)
and the covariance matrices are defined using Taylor polynomials. In the case of Somers’D, we define thep(p+2)
matrix of estimated derivativesb(D), whose rows are labeled by the namesY(1);:::;Y(p), and whose columns are labeled by
V;X;Y(1);:::;Y(p).
 (check this in PDF content)

 Start

151636
 Prefix

It is defined by
[h;V]= (V)hbV;[h;X]= (XX)hbTXX;h
h;Y(i)
i
= (XY
(i))
hbTXY(i)(13)
The jackknife covariance matrix is then equal to
Cb=[n(n1)]10(14)
The estimates for Kendall’saand SomersbDYX=bTXY=bTXX(15)
and the covariance matrices are defined using Taylor polynomials. In the case of
 Exact

Somers’D, we define thep(p+2)
 Suffix

matrix of estimated derivativesb(D), whose rows are labeled by the namesY(1);:::;Y(p), and whose columns are labeled by
V;X;Y(1);:::;Y(p). This matrix is defined by
b(D)
h
Y(i);X
i
=
@bDYX
@bTXX
=TbXY
Tb2XX
b(D)
h
Y(i);Y(i)
i
=
@bDYX
@bTXY
=
1
TbXX
(16)
all other entries being zero.
 (check this in PDF content)

 Start

152371
 Prefix

This matrix is defined
by
b()[X;V]=@bXX
@bV
=TbXX
Vb2
b()[X;X]=@bXX
@bTXX
=
1
Vb
b()
h
Y(i);V
i
=
@bXY
@bV
=TbXY
Vb2
b()
h
Y(i);Y(i)
i
=
@bXY(i)
@bTXY(i)
=
1
Vb
(17)
all other entries again being zero. The estimated dispersion matrices of the
 Exact

Somers’Dandaestimates are thereforebC(D)
 Suffix

and
Cb(), respectively, defined by
Cb(D)=b(D)bCb(D)0;bC()=b()bCb()0(18)
Thetransfoption offers a choice of transformations. Since these are available both for Somers’Dand for Kendall’sa,we
will denote the original estimate as(which can stand forDor) and the transformed estimate as.
 (check this in PDF content)

 Start

152527
 Prefix

The estimated dispersion matrices of the Somersand
Cb(), respectively, defined by
Cb(D)=b(D)bCb(D)0;bC()=b()bCb()0(18)
Thetransfoption offers a choice of transformations. Since these are available both for
 Exact

Somers’Dand for Kendall’sa,we
will denote
 Suffix

the original estimate as(which can stand forDor) and the transformed estimate as. They are summarized
below, together with their derivativesd=dand their inverses().
transfTransform name()d=d()
idenUntransformed1zFisher’szarctanh()=(12)1tanh()=
12log[(1+)=(1)][exp(2)1]=[exp(2)+1]
asinDaniels’ arcsinearcsin()(12)1=2sin()
rhoGreiner’ssin(2)2cos(2)(2
 (check this in PDF content)

 Start

153542
 Prefix

()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
 Exact

Fisher (1921)
 Suffix

(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).
 (check this in PDF content)

 Start

153569
 Prefix

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)
 Suffix

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).
 (check this in PDF content)

 Start

153588
 Prefix

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
 (check this in PDF content)

 Start

153641
 Prefix

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 Edwardes (1995) recommended it specifically for
 Exact

Somers’Don the basis of simulation studie
 Suffix

s. 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 using the inverse function().
 (check this in PDF content)

 Start

155565
 Prefix

51950
Foreign22395.5825
+combined7427752775
unadjustedvariance7150.00
adjustmentforties1.06
adjustedvariance7148.94
Ho:weight(foreign==Domestic)=weight(foreign==Foreign)
z=5.080
Prob>z=0.0000
We note that American cars are typically heavier and travel fewer miles per gallon than foreign cars. For confidence
intervals, we use
 Exact

somersd:
.somersdforeignmpgweight
Somers'D
Tra
 Suffix

nsformation:Untransformed
Validobservations:74
Jackknife
foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+mpg.4571678.1351463.3830.001.1922866.7220491
weight.7508741.08324859.0200.000.9140383.58771

 (check this in PDF content)

 Start

156844
 Prefix

difference:
.lincomweightmpg
(1)mpgweight=0.0
foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+(1).2937063.08843973.3210.001.1203677.4670449
The difference between
 Exact

Somers’Dvalues is positive. This indicates t
 Suffix

hat, if there are two cars, one heavier and consuming
fewer gallons per mile, the other lighter and consuming more gallons per mile, then the second is more likely to be foreign.
So maybe 1970’s American cars were not as wasteful as some people think, and were, if anything, more fuelefficient for their
weight than nonAmerican cars at the time.
 (check this in PDF content)

 Start

157409
 Prefix

So maybe 1970’s American cars were not as wasteful as some people think, and were, if anything, more fuelefficient for their
weight than nonAmerican cars at the time. Figure 1 illustrates this graphically. Data points are domestic cars (“D”) and foreign
cars (“F”). A regression analysis could show the same thing, but
 Exact

Somers’Dshows it in stronger terms, witho
 Suffix

ut contentious
assumptions such as linearity. (On the other hand, a regression model is more informative if its assumptions are true, so the two
methods 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.
 (check this in PDF content)

 Start

157874
 Prefix

, a regression model is more informative if its assumptions are true, so the two
methods 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. Applyingdto theautodata.
The confidence intervals for such high values of
 Exact

Somers’Dwould probably be more reliable if we
 Suffix

used theztransform,
recommended by Edwardes (1995). The results of this are as follows:
.somersdforeignmpgweight,tran(z)
Somers'D
Transformation:Fisher'sz
Validobservations:74
Jackknife
foreignCoef.
 (check this in PDF content)

 Start

157915
 Prefix

if its assumptions are true, so the two
methods 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. Applyingdto theautodata.
The confidence intervals for such high values of Somersused theztransform,
recommended by
 Exact

Edwardes (1995)
 Suffix

The results of this are as follows:
.somersdforeignmpgweight,tran(z)
Somers'D
Transformation:Fisher'sz
Validobservations:74
Jackknife
foreignCoef.
 (check this in PDF content)

 Start

157996
 Prefix

(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. Applyingdto theautodata.
The confidence intervals for such high values of Somersused theztransform,
recommended by Edwardes (1995). The results of this are as follows:
.dforeignmpgweight,tran(z)
 Exact

Somers'D
Transformation:Fisher'sz
Validobserva
 Suffix

tions:74
Jackknife
foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+mpg.4937249.17085512.8900.004.1588551.8285947
weight.9749561.19085475.1080.0001.349024.6008878
95
 (check this in PDF content)

 Start

158424
 Prefix

Jackknife
foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+mpg.4937249.17085512.8900.004.1588551.8285947
weight.9749561.19085475.1080.0001.349024.6008878
95%CIforuntransformed
 Exact

Somers'D
Somers_DMinimumMaximum
mpg.45716783.1
 Suffix

'D
Somers_DMinimumMaximum
mpg.45716783.15753219.67972072
weight.75087413.87382282.53768098
.lincomweightmpg
(1)mpgweight=0.0
foreignCoef.
 (check this in PDF content)

 Start

158433
 Prefix

Jackknife
foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+mpg.4937249.17085512.8900.004.1588551.8285947
weight.9749561.19085475.1080.0001.349024.6008878
95%CIforuntransformedSomers'D
 Exact

Somers_DMinimumMaximum
mpg.45716783.15753219.6
 Suffix

_DMinimumMaximum
mpg.45716783.15753219.67972072
weight.75087413.87382282.53768098
.lincomweightmpg
(1)mpgweight=0.0
foreignCoef.
 (check this in PDF content)

 Start

158917
 Prefix

foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+(1).4812312.12354523.8950.000.2390871.7233753
Note thatdgives not only symmetric confidence limits for theztransformed
 Exact

Somers’Destimates, but also the
more inform
 Suffix

ative asymmetric confidence limits for the untransformed Somers’Destimates (corresponding to theeformoption).
The asymmetric confidence limits for the untransformed estimates are closer to zero than the symmetric confidence limits for the
untransformed estimates in the previous output, and are probably more realistic.
 (check this in PDF content)

 Start

158979
 Prefix

foreignCoef.Std.Err.zP>z[95%Conf.Interval]
+(1).4812312.12354523.8950.000.2390871.7233753
Note thatdgives not only symmetric confidence limits for theztransformed Somersative asymmetric confidence limits for the untransformed
 Exact

Somers’Destimates (corresponding to theeformop
 Suffix

tion).
The asymmetric confidence limits for the untransformed estimates are closer to zero than the symmetric confidence limits for the
untransformed estimates in the previous output, and are probably more realistic.
 (check this in PDF content)

 Start

159282
 Prefix

The asymmetric confidence limits for the untransformed estimates are closer to zero than the symmetric confidence limits for the
untransformed estimates in the previous output, and are probably more realistic. The output tolincomgives confidence limits
for the difference betweenztransformed
 Exact

Somers’Dvalues. This difference is expressed
 Suffix

inzunits, but must, of course, be in
the same direction as the difference between untransformed Somers’Dvalues. The conclusions are similar.
Example 2
In this example, we demonstrate Kendall’saby comparing weight (pounds) and displacement (cubic inches) as predictors
of fuel efficiency (miles per gallon).
 (check this in PDF content)

 Start

159384
 Prefix

asymmetric confidence limits for the untransformed estimates are closer to zero than the symmetric confidence limits for the
untransformed estimates in the previous output, and are probably more realistic. The output tolincomgives confidence limits
for the difference betweenztransformed Somersinzunits, but must, of course, be in
the same direction as the difference between untransformed
 Exact

Somers’Dvalues. The conclusions are similar.
 Suffix

Example 2
In this example, we demonstrate Kendall’saby comparing weight (pounds) and displacement (cubic inches) as predictors
of fuel efficiency (miles per gallon). We first usektauto carry out significance tests with no confidence limits:
.ktaumpgweight
Numberofobs=74
Kendall'staua=0.6857
Kendall'staub=0.7059
Kendall'sscore=1852
SEofscore=213.605(correctedforties)
TestofHo:mpgandweightind
 (check this in PDF content)

 Start

165251
 Prefix

method (Jackknife)e(wtype)weight type
e(transf)transformation specified bytransfe(tranlab)transformation label in output
Matrices
e(b)coefficient vectore(V)variance–covariance matrix of the estimators
Functions
e(sample)marks estimation sample
Note that (confusingly)e(depvar)is theXvariable, or predictor variable, in the conventional terminology for defining
 Exact

Somers’D.somersdis also different from most es
 Suffix

timation commands in that its results are not designed to be used bypredict.
 (check this in PDF content)