- 1
- Arvesen, J. N. 1969. Jackknifing U-statistics.Annals of Mathematical Statistics40: 2076–2100.

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- population distribution, and the expressions inside the square brackets are kernels of these regular functionals.) The quantities we really want to estimate are Kendall’s-aand 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)U-statistics Vb=v:::: n(n-1) ;bTXY= t(XY):::: n(n-1) (11) and the respective jackknife pseudovalues corresponding to thehth cluster are given by (V)h=(n-1)-1v::::-(n-2)-1[v::::-2vh:::] (XY) h=(n-1) -1t(XY) ::::-(n-2) -1 h t(XY)::::-2t (XY) h::: i (12) somers

- In-text reference with the coordinate start=150535
- 3
- Edwardes M. D. deB. 1995. A confidence interval for Pr(X<Y)-Pr(X>Y) estimated from simple cluster samples.Biometrics51: 571–578.

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- estimates given by (18), andb-(-)is the diagonal matrix whose diagonal entries are thed-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform 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 untransformedDor-avalues, calculated from symmetric confidence intervals for the transformed parameters

- In-text reference with the coordinate start=157915
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- 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 thez-transform, 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 foreign|Coef.

- In-text reference with the coordinate start=153588
- 4
- Fisher, R. A. 1921. On the “probable error” of a coefficient of correlation deduced from a small sample.Metron1(4): 3–32.

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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-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform was originally recommended for the Pearson correlation coefficient by
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- 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).

- In-text reference with the coordinate start=153542
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- Gayen, A. K. 1951. The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes.Biometrika38: 219–247.

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- for the untransformed estimates given by (18), andb-(-)is the diagonal matrix whose diagonal entries are thed-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform 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).

- In-text reference with the coordinate start=153569
- 6
- Hoeffding, W. 1948. A class of statistics with asymptotically normal distribution.Annals of Mathematical Statistics19: 293–325.

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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
- 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’s-aand 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

- In-text reference with the coordinate start=150670
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- want to estimate are Kendall’s-aand 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
- U-statistics Vb=v:::: n(n-1) ;bTXY= t(XY):::: n(n-1) (11) and the respective jackknife pseudovalues corresponding to thehth cluster are given by (V)h=(n-1)-1v::::-(n-2)-1[v::::-2vh:::] (XY) h=(n-1) -1t(XY) ::::-(n-2) -1 h t(XY)::::-2t (XY) h::: i (12) somersdcalculates correlation measures for a single variableXwith a set ofY-variates(Y(1);:::;Y(p)).

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- Kendall, M. G. 1949. Rank and product-moment correlation.Biometrika36: 177–193. ——. 1970.Rank Correlation Methods. 4th ed. London: Griffin.

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- Under this relation, Kendall’s-a-values of 0,-13,-12 and-1correspond to Pearson’s correlations of 0,-12,-1p2and-1, 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’s-ahas the desirable property that a larger-acannot be secondary to a smaller-a, that is, if a positive-XYis caused entirely by a monotonic positive relationship of both variables with a third variableW, then-WXand-WYmust both be greater than-XY.

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- Newson, R. B. 1987. An analysis of cinematographic cell division data using U-statistics [Dphil dissertation]. Brighton, UK: Sussex University, 301–310.

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- Under this relation, Kendall’s-a-values of 0,-13,-12 and-1correspond to Pearson’s correlations of 0,-12,-1p2and-1, respectively. A similar correspondence is likely to hold in a wider range of continuous bivariate distributions (Kendall 1949,
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- Newson 1987)
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- . Kendall’s-ahas the desirable property that a larger-acannot be secondary to a smaller-a, that is, if a positive-XYis caused entirely by a monotonic positive relationship of both variables with a third variableW, then-WXand-WYmust both be greater than-XY.

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- Somers, R. H. theSTBare presently categorized as follows: General Categories: anannouncementsipinstruction on programming cccommunications & lettersosoperating system, hardware, & dmdata managementinterprogram communication dtdatasetsqsquestions and suggestions grgraphicstttea

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- 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 so-called nonparametric methods are in fact based on population parameters, which are zero under the null hypothesis. Two of these parameters are Kendall’s-aand
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- Somers’D, the parameter tested by a Wilco
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- xon rank-sum test. Confidence limits for these parameters are more informative thanp-values alone, for three reasons. Firstly, confidence intervals show that a highp-value does not prove a null hypothesis.

- In-text reference with the coordinate start=142430
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- Secondly, for continuous data, Kendall’s-acan 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’s-a’s or
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- Somers’D’s, and these are informative, be
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- cause a larger Kendall’s -aor Somers’Dcannot be secondary to a smaller one. The programsomersdcalculates confidence intervals for Somers’ Dor Kendall’s-a, using jackknife variances. There is a choice of transformations, including Fisher’sz, Daniels’ arcsine, Greiner’s-, and thez-transform of Greiner’s-.

- In-text reference with the coordinate start=142469
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- Secondly, for continuous data, Kendall’s-acan 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’s-a’s or Somerscause a larger Kendall’s -aor
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- Somers’Dcannot be secondary to a smaller one.
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- The programsomersdcalculates confidence intervals for Somers’ Dor Kendall’s-a, using jackknife variances. There is a choice of transformations, including Fisher’sz, Daniels’ arcsine, Greiner’s-, and thez-transform of Greiner’s-.

- In-text reference with the coordinate start=142524
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- Secondly, for continuous data, Kendall’s-acan 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’s-a’s or Somerscause a larger Kendall’s -aor Somers The programdcalculates confidence intervals for
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- Somers’ Dor Kendall’s-a, using jackknife varia
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- nces. There is a choice of transformations, including Fisher’sz, Daniels’ arcsine, Greiner’s-, and thez-transform of Greiner’s-.Aclusteroption is available. The estimation results are saved as for a model fit, so that differences can be estimated usinglincom.

- In-text reference with the coordinate start=142799
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- There is a choice of transformations, including Fisher’sz, Daniels’ arcsine, Greiner’s-, and thez-transform of Greiner’s-.Aclusteroption is available. The estimation results are saved as for a model fit, so that differences can be estimated usinglincom. Keywords:
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- Somers’ D, Kendall’s tau, rank correlation, ra
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- nk-sum 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.

- In-text reference with the coordinate start=143133
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- Syntax dvarlist weight -ifexp -inrange -,cluster(varname)level(#)tauatdist transf(transformationname) wheretransformationnameis one of idenjzjasinjrhojzrho fweights,iweightsandpweights are allowed. Description dcalculates the nonparametric statistics
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- Somers’D(corresponding to rank-sum tests)
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- and Kendall’s-a, with confidence limits. Somers’Dor-ais 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.

- In-text reference with the coordinate start=143182
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- 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’s-a, with confidence limits.
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- Somers’Dor-ais calculated for the first variab
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- 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.

- In-text reference with the coordinate start=143491
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- 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
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- Somers’Dor Kendall’s -avalues. Options cluster
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- (varname)specifies the variable which defines sampling clusters. Ifclusteris defined, then the between-cluster Somers’Dor-ais 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

- In-text reference with the coordinate start=143618
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- 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 between-cluster
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- Somers’Dor-ais calculated, and the varianc
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- 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.

- In-text reference with the coordinate start=143994
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- Ifclusteris defined, then the between-cluster 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’s-a.Iftauais absent, then
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- somersdcalculates Somers’D. tdistspecifies tha
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- t the estimates are assumed to have at-distribution withn-1 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(

- In-text reference with the coordinate start=144994
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- Remarks The population value of Kendall’s-a(Kendall 1970) is defined as -XY=E[sign(X1-X2)sign(Y1-Y2)](1) where(X1;Y1)and(X2;Y2)are bivariate random variables sampled independently from the same population, andE[-]denotes expectation. The population value of
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- Somers’D(Somers 1962)
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- ’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 twoX-values is associated with the larger of the twoY-values and the probability that the largerX-value is associated with the smallerY-value.

- In-text reference with the coordinate start=145002
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- Remarks The population value of Kendall’s-a(Kendall 1970) is defined as -XY=E[sign(X1-X2)sign(Y1-Y2)](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 twoX-values is associated with the larger of the twoY-values and the probability that the largerX-value is associated with the smallerY-value.

- In-text reference with the coordinate start=145506
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- DYXis the difference between the two corresponding conditional probabilities, given that the twoX-values are not equal. Kendall’s-ais the covariance betweensign(X1-X2)andsign(Y1-Y2), whereas
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- Somers’Dis the regression coefficient ofsig
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- n(Y1-Y2)with respect tosign(X1-X2). (The correlation coefficient betweensign(X1-X2)andsign(Y1-Y2)is known as Kendall’s-b, and is the geometric mean ofDYXandDXY.) Given a sample of data points(Xi;Yi), we may estimate and test the population values of Kendall’s-aand Somers’Dby the corresponding sample statisticsb-XYandbDYX.

- In-text reference with the coordinate start=145780
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- (The correlation coefficient betweensign(X1-X2)andsign(Y1-Y2)is known as Kendall’s-b, and is the geometric mean ofDYXandDXY.) Given a sample of data points(Xi;Yi), we may estimate and test the population values of Kendall’s-aand
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- Somers’Dby the corresponding sample statistic
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- sb-XYandbDYX. These are commonly known as nonparametric statistics, even though-XY andDYXare parameters. The two Wilcoxon rank-sum tests (see [R]signrank) both test hypotheses predictingDYX=0. The two-sample rank-sum test represents the case whereXis a binary variable indicating membership of one of two subpopulations.

- In-text reference with the coordinate start=150325
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- (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’s-aand
- 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.

- In-text reference with the coordinate start=151517
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- 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 1-i-p. It is defined by -[h;V]= (V)h-bV;-[h;X]= (XX)h-bTXX;h h;Y(i) i = (XY (i)) h-bTXY(i)(13) The jackknife covariance matrix is then equal to Cb=[n(n-1)]-1-0-(14) The estimates for Kendall’s-aand
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- Somers’Dare defined by b-XY=bTXY=bV;
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- 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).

- In-text reference with the coordinate start=151636
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- It is defined by -[h;V]= (V)h-bV;-[h;X]= (XX)h-bTXX;h h;Y(i) i = (XY (i)) h-bTXY(i)(13) The jackknife covariance matrix is then equal to Cb=[n(n-1)]-1-0-(14) The estimates for Kendall’s-aand SomersbDYX=bTXY=bTXX(15) and the covariance matrices are defined using Taylor polynomials. In the case of
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- Somers’D, we define thep-(p+2)
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- 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.

- In-text reference with the coordinate start=152371
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- This matrix is defined by b-(-)[X;V]=@b-XX @bV =TbXX Vb2 b-(-)[X;X]=@b-XX @bTXX = 1 Vb -b(-) h Y(i);V i = @b-XY @bV =TbXY Vb2 b-(-) h Y(i);Y(i) i = @b-XY(i) @bTXY(i) = 1 Vb (17) all other entries again being zero. The estimated dispersion matrices of the
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- Somers’Dand-aestimates are thereforebC(D)
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- 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’s-a,we will denote the original estimate as-(which can stand forDor-) and the transformed estimate as-.

- In-text reference with the coordinate start=152527
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- 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
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- Somers’Dand for Kendall’s-a,we will denote
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- the original estimate as-(which can stand forDor-) and the transformed estimate as-. They are summarized below, together with their derivativesd-=d-and their inverses-(-). transfTransform name-(-)d-=d--(-) idenUntransformed-1zFisher’szarctanh(-)=(1--2)-1tanh(-)= 12log[(1+-)=(1--)][exp(2-)-1]=[exp(2-)+1] asinDaniels’ arcsinearcsin(-)(1--2)-1=2sin(-) rhoGreiner’s-sin(-2-)-2cos(-2-)(2

- In-text reference with the coordinate start=153641
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- diagonal matrix whose diagonal entries are thed-=d-estimates specified in the table, then the transformed parameter and its covariance matrix are -b=-(b-);bC(-)=b-(-)bC(-)b-(-)0(19) Fisher’sz-transform 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
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- 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 untransformedDor-avalues, calculated from symmetric confidence intervals for the transformed parameters using the inverse function-(-).

- In-text reference with the coordinate start=155565
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- 51950 Foreign|22395.5825 ---------+--------------------------------combined|7427752775 unadjustedvariance7150.00 adjustmentforties-1.06 ---------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
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- somersd: .somersdforeignmpgweight Somers'D Tra
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- nsformation:Untransformed Validobservations:74 -----------------------------------------------------------------------------|Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4571678.1351463.3830.001.1922866.7220491 weight|-.7508741.0832485-9.0200.000-.9140383-.58771 --------------------------------------------

- In-text reference with the coordinate start=156844
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- difference: .lincom-weight-mpg (1)-mpg-weight=0.0 -----------------------------------------------------------------------------foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------(1)|.2937063.08843973.3210.001.1203677.4670449 -----------------------------------------------------------------------------The difference between
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- Somers’D-values is positive. This indicates t
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- 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 fuel-efficient for their weight than non-American cars at the time.

- In-text reference with the coordinate start=157409
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- So maybe 1970’s American cars were not as wasteful as some people think, and were, if anything, more fuel-efficient for their weight than non-American 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
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- Somers’Dshows it in stronger terms, witho
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- 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.

- In-text reference with the coordinate start=157874
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- , 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
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- Somers’Dwould probably be more reliable if we
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- used thez-transform, recommended by Edwardes (1995). The results of this are as follows: .somersdforeignmpgweight,tran(z) Somers'D Transformation:Fisher'sz Validobservations:74 -----------------------------------------------------------------------------|Jackknife foreign|Coef.

- In-text reference with the coordinate start=157996
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- (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 thez-transform, recommended by Edwardes (1995). The results of this are as follows: .dforeignmpgweight,tran(z)
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- Somers'D Transformation:Fisher'sz Validobserva
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- tions:74 -----------------------------------------------------------------------------|Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4937249.17085512.8900.004.1588551.8285947 weight|-.9749561.1908547-5.1080.000-1.349024-.6008878 -----------------------------------------------------------------------------95

- In-text reference with the coordinate start=158424
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- Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4937249.17085512.8900.004.1588551.8285947 weight|-.9749561.1908547-5.1080.000-1.349024-.6008878 -----------------------------------------------------------------------------95%CIforuntransformed
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- Somers'D Somers_DMinimumMaximum mpg.45716783.1
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- 'D Somers_DMinimumMaximum mpg.45716783.15753219.67972072 weight-.75087413-.87382282-.53768098 .lincom-weight-mpg (1)-mpg-weight=0.0 -----------------------------------------------------------------------------foreign|Coef.

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- Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4937249.17085512.8900.004.1588551.8285947 weight|-.9749561.1908547-5.1080.000-1.349024-.6008878 -----------------------------------------------------------------------------95%CIforuntransformed
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- Somers'D Somers_DMinimumMaximum mpg.45716783.1
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- 'D Somers_DMinimumMaximum mpg.45716783.15753219.67972072 weight-.75087413-.87382282-.53768098 .lincom-weight-mpg (1)-mpg-weight=0.0 -----------------------------------------------------------------------------foreign|Coef.

- In-text reference with the coordinate start=158433
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- Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4937249.17085512.8900.004.1588551.8285947 weight|-.9749561.1908547-5.1080.000-1.349024-.6008878 -----------------------------------------------------------------------------95%CIforuntransformedSomers'D
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- Somers_DMinimumMaximum mpg.45716783.15753219.6
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- _DMinimumMaximum mpg.45716783.15753219.67972072 weight-.75087413-.87382282-.53768098 .lincom-weight-mpg (1)-mpg-weight=0.0 -----------------------------------------------------------------------------foreign|Coef.

- In-text reference with the coordinate start=158433
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- Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4937249.17085512.8900.004.1588551.8285947 weight|-.9749561.1908547-5.1080.000-1.349024-.6008878 -----------------------------------------------------------------------------95%CIforuntransformedSomers'D
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- Somers_DMinimumMaximum mpg.45716783.15753219.6
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- _DMinimumMaximum mpg.45716783.15753219.67972072 weight-.75087413-.87382282-.53768098 .lincom-weight-mpg (1)-mpg-weight=0.0 -----------------------------------------------------------------------------foreign|Coef.

- In-text reference with the coordinate start=158433
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- Jackknife foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------mpg|.4937249.17085512.8900.004.1588551.8285947 weight|-.9749561.1908547-5.1080.000-1.349024-.6008878 -----------------------------------------------------------------------------95%CIforuntransformedSomers'D
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- Somers_DMinimumMaximum mpg.45716783.15753219.6
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- _DMinimumMaximum mpg.45716783.15753219.67972072 weight-.75087413-.87382282-.53768098 .lincom-weight-mpg (1)-mpg-weight=0.0 -----------------------------------------------------------------------------foreign|Coef.

- In-text reference with the coordinate start=158433
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- Somers_DMinimumMaximum mpg.45716783.15753219.6
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- In-text reference with the coordinate start=158917
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- foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------(1)|.4812312.12354523.8950.000.2390871.7233753 -----------------------------------------------------------------------------Note thatdgives not only symmetric confidence limits for thez-transformed
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- Somers’Destimates, but also the more inform
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- 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.

- In-text reference with the coordinate start=158979
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- foreign|Coef.Std.Err.zP>|z|[95%Conf.Interval] ---------+-------------------------------------------------------------------(1)|.4812312.12354523.8950.000.2390871.7233753 -----------------------------------------------------------------------------Note thatdgives not only symmetric confidence limits for thez-transformed Somersative asymmetric confidence limits for the untransformed
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- Somers’Destimates (corresponding to theeformop
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- 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.

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- 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 betweenz-transformed
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- Somers’Dvalues. This difference is expressed
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- inz-units, 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’s-aby comparing weight (pounds) and displacement (cubic inches) as predictors of fuel efficiency (miles per gallon).

- In-text reference with the coordinate start=159384
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- 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 betweenz-transformed Somersinz-units, but must, of course, be in the same direction as the difference between untransformed
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- Somers’Dvalues. The conclusions are similar.
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- Example 2 In this example, we demonstrate Kendall’s-aby 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'stau-a=-0.6857 Kendall'stau-b=-0.7059 Kendall'sscore=-1852 SEofscore=213.605(correctedforties) TestofHo:mpgandweightind

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- 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 theX-variable, or predictor variable, in the conventional terminology for defining
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- Somers’D.somersdis also different from most es
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- timation commands in that its results are not designed to be used bypredict.

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