On Testing Significance of the Multivariate Rank Correlation Coefficient

Transcription

1 Folia Oeconomica Acta Universitatis Loziensis ISSN e-issn (335) 208 DOI: Grzegorz Kończak University of Economics in Katowice, Faculty of Management, Department of Statistics, Econometrics an Mathematics, On Testing Significance of the Multivariate Rank Correlation Coefficient Abstract: The Spearman s rho is a measure of the strength of the association between two variables. There are some extensions of this coefficient for the multivariate case. Measures of the multivariate association which are the generalisation of the bivariate Spearman s rho are consiere in the literature. These measures are base on copula functions. This article presents a proposal of the testing for the multivariate Spearman s rank correlation coefficient. The propose test is base on the permutation metho. The test statistic use in the permutation test is base on the empirical copula function. The properties of the propose metho have been escribe using computer simulations. Keywors: multivariate Spearman s rho, copula function, permutation tests, Monte Carlo stuy JEL: C2, C4, C5 [2]

2 22 Grzegorz Kończak. Introuction an basic notations The Spearman s rho ρ S is a well known measure for the strength of the association between two ranom variables X an Y. Let us consier n objects ranke from to n. Let R x an R y be the ranks of the variables X an Y. In this case, R x an R y are the permutations of the same set containing the numbers, 2,, n. The Spearman rank correlation coefficient for the sample of size n has the form (Wywiał, 2004: 97): R s n 2 6 ( Rx R ) i yi i=. 3 = Let us consier the hypothesis n n () H : 0 0 r S = with the alternative H: r 0 or H: r S > 0 or H : r S < 0. S The hypothesis H 0 coul be teste using the test statistic t = R s n 2, R where R S is the Spearman correlation coefficient base on the sample an n > 0 (see Sheskin, 2004). Uner the null hypothesis, the test statistic (2) has t istribution with n 2 egree of freeom (Zar, 972: ). Wywiał (2004: 97) pointe that for the sample of size n uner the null hypothesis the istribution of the test statistic 2 s (2) z = Rs n (3) coul be approximate by the stanar normal istribution. The above presente Spearman s rho measures the strength of the association only for two variables. There are some extensions of this measure to the imensional ( > 2) cases. The multivariate Spearman s rho extensions were consiere by Joe (990) an Schmi an Schmit (2006). Beő an Ong (205) use this measure for aggregating ranks. Multivariate extensions of Spearman s rho are base on copula functions. FOE 3(335) 208

3 On Testing Significance of the Multivariate Rank Correlation Coefficient On the measuring of multivariate epenences One of the statistical methos use to measure multivariate epenences are copulas. Copulas are very useful tools for escribing an unerstaning the epenence between two or more ranom variables. A copula is a function which joins a multivariate function to its marginal istribution functions. It is a multivariate istribution function efine on the unit cube [0, ], with a uniformly istribute marginal. Formally, the efinition of copulas coul be written as follows (Nelsen, 999: 8 9): A imensional copula is a function C with omain [0, ] such that. C(u) is zero for all u in [0, ] for which at least one coorinate is equal to 0 2. C(u) = u k if all coorinates of u are except the k th one 3. C is increasing in the sense that for every a b (a i b i for i =, 2,, ) in [0, ] the volume assigne by C to the box [a, b] = [a, b ] [a 2, b 2 ] [a, b ] is nonnegative. Let (X, X 2,, X ) an (Y, Y 2,, Y ) be two inepenent vectors with joint istributions C X (F(x)) an C Y (F(y)) where F(x) = (F (x ),, F (x )) an F(y) = (F (y ),, F (y )) are the marginal istributions an C X, C Y are the respective copulas. Then the concorance function (see Beő, Ong, 205: 2) is given by Q( CX, CY) = P ( X j Yj) > 0 P ( X j Yj) < 0 = 2 C ( ) ( ), [0,] X v CY u j= j= where u = F(x) an ν = F(y). There are methos of multivariate extensions for the Spearman s rho coefficient. Some of them are erive from multivariate epenence concepts (Nelsen, 996: 223). The three following multivariate ( 2) versions of Spearman s rho were analyse by Schmi an Schmit (2006: 760) r = h ( ) 2 C( ), u u (4) [0,] r2 = h( ) 2 ( ) C( ) Π, 2 u u [0,] (5) r3 = h C u v uv 2 (2) 2 (, ), 2 2 [0,] kl k< l (6) FOE 3(335) 208

4 24 Grzegorz Kończak + where h ( ) = =, M(u) is the upper Frėchet Hoeffing QM (, Π) 2 ( + ) boun given by M(u) = max{u + u u ( ), 0} an Π(u) is the inepenence copula given by Π ( u ) = u, u ϵ [0, ]. i i= The measures ρ, ρ 2 an ρ 3 are multivariate extensions of two imensional Spearman s rho, because for = 2 there is (Schmi, Schmit, 2006: 76) ρ = ρ 2 = ρ 3 = ρ s. For > 2, the values of ρ, ρ 2 an ρ 3 are ifferent in general. Empirical copula Let us consier a ranom sample X j = (X j, X 2j,, X j ) (j =, 2,, n) from a imensional ranom vector X with the joint istribution function F an the copula C which are unknown. The istribution function F coul be estimate as follows: n ˆ F ( x ) =, for i =, 2,, an x ϵ R. i { Xij x)} n j= The copula function C coul be estimate by (Schmi, Schmit, 2006): n ˆ C ( u ) ij i =, for 2 n { Uˆ u } n j= i= u = ( u, u,..., u ) [0,], where U ˆ ˆ( ij = Fi Xij ) for i =, 2,,, j =, 2,, n an Uˆ ˆ ˆ ˆ j = ( Uj, U2 j,..., Uj ). Empirical copulas will be use to estimate the multivariate ( > 2) Spearman s rho correlation coefficient. Let R, R 2,, R be the rankings of experts. Then the ranking R i (i =, 2,, ) is an n imensional vector. This vector is the permutation of the numbers, 2,, n. The normalise ranks (Beő, Ong, 205) are calculate as follows: Rij R ij = (i =, 2,, ). Using the empirical copula (Schmi, Schmit, 2007) n + expression in the Spearman s formula, we obtain an empirical expression of multivariate Spearman s correlation coefficient (Beő, Ong, 205: 2; Schmi, Schmit, 2007: 40) n 2 ˆ r = h ( ) ( Rij ), n j= i= (7) FOE 3(335) 208

5 On Testing Significance of the Multivariate Rank Correlation Coefficient 25 n 2 ˆ r2 = h ( ) Rij, n j= i= (8) 2 ˆ r3 = ( R kj )( R lj ). (9) n 2 k< l j= The formulas for ρ, ρ 2 an ρ 3 are ifferent in general. In this paper, formula (4) is consiere as well as the estimator given by (7). This formula will be use for testing the significance of the multivariate Spearman s rho. 3. The properties of two an multivariate Spearman s rho The Spearman s rho is a nonparametric measure of rank correlation. It assesses how well the relationship between two variables can be escribe using a monotonic function. This coefficient takes values from to. The Spearman s coefficient is equal to if the rankings are ientical, for example: Ranking :, 2,, n. Ranking 2:, 2,, n. This coefficient is equal to if they are in reverse orer, for example: Ranking :, 2,, n. Ranking 2: n, n,,. Typical histograms for R s for samples of the size n = 5, n = 0 an n = 20 for inepenent rankings (ρ s = 0) are presente in Figure. Figure. Empirical istributions of R S for inepenent rankings (n = 5, 0 an 20) FOE 3(335) 208

6 26 Grzegorz Kończak The exact istributions of Spearman s R S for inepenent rankings for the sample of size 5, 8 an 0 are presente in Figure 2. Figure 2. Theoretical istributions of R s (n = 5, 8 an 0) for inepenent rankings For n = 5, there are 5! = 20 permutations of the secon variable (the first variable is fixe). The Spearman s ρ S for the sample of the size n = 5 can take 2 following variants of the values:.0; 0.9; ; 0.; 0.0; 0.; ; 0.9 an.0. The values of the potential variants of Spearman s ρ S are presente in Table. Table. The number of variants of Spearman s ρ S values for the sample size n = 5, 6,, 0 Sample size n No. of variants Domański an Pruska (2000: 5) escribe ifficulties in constructing tables with critical values for the Spearman s rho ue to the number of possible permutations of variables. For n = 0, there are 0! = 3,628,800 permutations of the ranking. For the multivariate extension of the Spearman s coefficient ( > 2), calculations are much more complicate. The number of possible permutations grows raically for the imension > 2. The number of permutations of the 2, 3,, variable (the first ranking is fixe) is Np = n!. There are 26 variants of ifferent values of ρ in the 3 imensional case an,94 variants of ifferent values in 4 imensional case for the sample size of n = 5. The number of permutations of variables for the sample sizes n = 5, 6,, 0 are presente in Table 2. i= 2 FOE 3(335) 208

7 On Testing Significance of the Multivariate Rank Correlation Coefficient 27 Table 2. The number of possible permutations (the first variable is fixe) Sample size n = 2 = 3 = c.a * c.a *0 8 The empirical istributions of the multivariate Spearman s coefficient ρ for = 3 are presente in Figure 3 an for = 4 in Figure 4. Figure 3. The empirical istributions of the multivariate ( = 3) Spearman s coefficient ˆr for inepenent rankings (n = 5, 0 an 20) Figure 4. The empirical istributions of the multivariate ( = 4) Spearman s coefficient ˆr for inepenent rankings (n = 5, 0 an 20) FOE 3(335) 208

8 28 Grzegorz Kończak The istribution of the multivariate Spearman s coefficient for inepenent rankings for > 2 in general is not symmetric. The exact istributions of the multivariate Spearman s ˆr for the sample of the size n = 5 for = 3 an = 4 are presente in Figure 5. Figure 5. The exact istributions of the multivariate ( = 3 an = 4) Spearman s coefficient for inepenent rankings (n = 5) The value of Spearman s rank correlation coefficient varies between an. The maximum, minimum an estimate values of quantiles of the multivariate Spearman s ρ are presente in Table 3 (n = 5) an Table 4 (n = 0). These values were obtaine in series of computer simulations. In each case, there were generate 000 times ( = 2, 3,, 0) inepenent rankings an the value of the multivariate Spearman s coefficient ρ was calculate using formula (7). Table 3. The estimate quantiles of the multivariate Spearman s ˆr for the sample of the size n = 5 Dim Quantile Min Max* 2* * Exact values. Source: computer simulation FOE 3(335) 208

9 On Testing Significance of the Multivariate Rank Correlation Coefficient 29 Table 4. The estimate quantiles of the multivariate Spearman s ˆr for the sample of the size n = 0 Dim Quantile Min Max* 2* * Exact values. Source: computer simulation The examples of the complete agreement in the rankings for 4 imension an the highest iscrepancy for the sample of the size n = 5 are presente in Figure 6 an Figure 7. If the first ranking is fixe, then there exists the one an only combination of three other rankings which gives the maximum ˆr = (see Figure 6). In this case, there are 288 rankings with the minimum ˆr = One of these combinations is presente in Figure Figure 6. The complete agreement in 4 rankings Figure 7. One of the highest iscrepancy in 4 rankings FOE 3(335) 208

10 30 Grzegorz Kończak Figure 8. Intervals of variations for the imensional correlation coefficient for the samples of the size n = 5 an n = 0 The area of variations of the imensional ( = 2, 3,, 0) Spearman s r for the sample sizes of n = 5 an n = 0 is presente in Figure 8. The istribution of the multivariate Spearman s r for the imension greater than 2 is not symmetric. Due to the asymmetry of the istribution, the critical region for the H 0 shoul be asymmetric. To test the significance of the multivariate Spearman coefficient, the permutation test will be propose. 4. Testing multivariate epenences Zar (200: 773) presente tables of critical values of the Spearman s ranke correlation coefficient. These tables coul be use only for the two imensional version of Spearman s rank coefficient. For the case where > 2, the permutation test coul be use. Permutation tests were introuce by R.A. Fisher an E.J.G. Pitman in 930s (Berry, Johnston, Mielke, 204: 20). Lehmann (2009: 439) shows that permutation tests are generally asymptotically as goo as the best parametric ones. The concept of permutation tests is simpler than that of tests base on normal istribution. Efron an Tibshirani (993: 202) point out that the main application of these tests is a two sample problem. In permutation tests, the observe value of the test statistic (T 0 ) is compare with the empirical istribution of this statistic uner the null hypothesis. The following steps are taken in ealing with permutation tests (Goo, 2005: 8; Kończak; 206: 29): FOE 3(335) 208

11 On Testing Significance of the Multivariate Rank Correlation Coefficient 3. Assume the significance level a. 2. Ientify the null hypothesis an the alternative hypothesis. 3. Choose a form of the test statistic T. 4. Calculate the value T 0 of the test statistic for the sample ata. 5. Determine by a series of permutations the frequency istribution of the test statistic uner the null hypothesis (T, T 2,, T N, where N 000). 6. Make a ecision using this empirical istribution as a guie. The ASL (Achieve Significance Level) has the following form: ASL= P ( T T 0 ). () The ASL is unknown an coul be estimate by the following formula: car{: i Ti T0 ) ASL. (2) N This notation applies where the H 0 rejection area is right sie. In the case of the left sie rejection area in the above notation, inequality shoul be change. If the value of ASL is lower than the assume level of significance a, then H 0 shoul be rejecte. The significance of the escribe multivariate Spearman s rank coefficient will be teste. The sample multivariate Spearman s rank coefficient given by (7) as a test statistic will be use in Monte Carlo stuy. The empirical istribution of this coefficient will be obtaine in the proceure of permutation testing. The null hypothesis will be rejecte for ASL < a. 5. The test proceure Monte Carlo stuy Let us consier the null hypothesis that all rankings are inepenent. This hypothesis coul be written as follows: with the alternative H : r = 0 0 S H : r > 0. There were consiere hypotheses for three, four an five imensional rankings. Two following variants were consiere: ) H 0 is true there is no correlation between vectors R, R 2,, R. S FOE 3(335) 208

12 32 Grzegorz Kończak 2) H 0 is false two rankings are ientical, an the others were no correlate. The probabilities of rejection of H 0 were estimate by a sequence of 000 computer simulations of permutation tests. In each permutation test, there were 000 permutations consiere. In all the simulations, the significance level a = 0.05 was assume. The estimate probabilities of rejection of H 0 are presente in Table 5 (H 0 true the size of the test) an in Table 6 (H 0 false). Dimension Table 5. Estimate probabilities of H 0 rejection (H 0 true) Sample size n Dimension Table 6. Estimate probabilities of H 0 rejection (H 0 false) Sample size n The size of the test is close to the assume significance level a = 0.05 (see Table 5). For the greater size of the sample in the case of false H 0, there is a greater probability of H 0 rejection. For the smaller imension in the case of false H 0, there is a greater probability of H 0 rejection (see Table 6). 6. Conclusions This article presents a proposal of the testing for multivariate extensions of Spearman s rho. There are some variants of such extensions. In the paper, one of them given by formula (4) was consiere. The properties of these multivariate measures were escribe. These multivariate Spearman s correlations coul be use for measuring the rankings agreement. The test for the significance of the multivariate Spearman s rho was propose. The propose testing proceure is base on the permutation metho. FOE 3(335) 208

13 On Testing Significance of the Multivariate Rank Correlation Coefficient 33 References Beő J., Ong Ch.S. (205), Multivariate Spearman s rho for rank aggregation, arxiv.org [accesse: ]. Berry K.J., Johnston J.E., Mielke Jr. P.W. (204), A Chronicle of Permutation Statistical Methos, Springer International Publishing, New York. Domański Cz., Pruska K. (2000), Nieklasyczne metoy statystyczne, Polskie Wyawnictwo Ekonomiczne, Warszawa. Efron B., Tibshirani R. (993), An Introuction to the Bootstrap, Chapman & Hall, New York. Goo P. (2005), Permutation, Parametric an Bootstrap Tests of Hypotheses, Science Business Meia Inc., New York. Joe H. (990), Multivariate Concorance, Journal of Multivariate Analysis, no. 35, pp Kończak G. (206), Testy permutacyjne. Teoria i zastosowania, Uniwersytet Ekonomiczny w Katowicach, Katowice. Lehmann E.L. (2009), Parametric vs. nonparametric: Two alternative methoologies, Journal of Nonparametric Statistics, no. 2 pp Nelsen R.B. (996), Nonparametric Measures of Multivariate Association, IMS Lecture Notes Monograph Series, no. 28, pp Nelsen R.B. (999), An Introuction to Copulas, Springer Verlag, New York. Schmi F., Schmit R. (2006), Bootstraping Spearman s Multivariate Rho, Proceeings of COMP STAT 2006, pp Schmi F., Schmit R. (2007), Multivariate Extensions of Spearman s Rho an Relate Statistics, Statistics & Probability Letters, no. 77, pp Sheskin D.J. (2004), Hanbook of Parametric an Nonparametric Statistical Proceures, Chapman & Hall/CRC, Boca Raton. Wywiał J. (2004), Wprowazenie o wnioskowania statystycznego, Akaemia Ekonomiczna w Katowicach, Katowice. Zar J.H. (972), Significance Testing of the Spearman Rank Correlation Coefficient, Journal of the American Statistical Association, vol. 67, no. 339, pp Zar J.H. (200), Biostatistical Analysis, Pearson Prentice Hall, New Jersey. O testowaniu istotności wielowymiarowego współczynnika korelacji rang Streszczenie: Współczynnik korelacji rang Spearmana pozwala na baanie siły zależności mięzy wiema zmiennymi, la których okonano pomiaru na skali porząkowej. W literaturze są prezentowane rozszerzenia tego współczynnika na przypaek wielowymiarowy. W tych konstrukcjach wykorzystywane są zwykle funkcje łączące (kopule). W artykule przestawiono propozycję testowania istotności zależności wielowymiarowej la anych mierzonych na skali rangowej. Przestawiony test la istotności wielowymiarowego współczynnika korelacji rang wykorzystuje metoę permutacyjną. Własności proponowanego testu scharakteryzowano z wykorzystaniem symulacji komputerowych. Słowa kluczowe: wielowymiarowy współczynnik rang Spearmana, kopuła, test permutacyjny, symulacja Monte Carlo JEL: C2, C4, C5 FOE 3(335) 208

14 34 Grzegorz Kończak by the author, licensee Łóź University Łóź University Press, Łóź, Polan. This article is an open access article istribute uner the terms an conitions of the Creative Commons Attribution license CC BY (http: //creativecommons.org/licenses/by/3.0/) Receive: ; verifie: Accepte: FOE 3(335) 208