# ranktes: Tests of independence between two rankings In pvrank: Rank Correlations

## Description

Performs various independence tests based on rank correlations.

## Usage

 1 2 ranktes(r, n, index = "", approx = "exact", CC = FALSE, type = "two-sided", print = TRUE)

## Arguments

 r the value of the test statistic. n the number of ranks. index a character string that specifies the rank correlation that is used in the test statistic. Acceptable values are: "spearman","kendall","gini","r4", "fy1" (FY-means), "fy2" (FY-medians) and "sbz" (Symmetrical Borroni-Zenga). Only enough of the string to be unique is required. approx a character string that specifies the type of approximation to the null distribution of the statistic required in index: "vggfr", "exact","gaussian","student". Only enough of the string to be unique is required. CC if true, a continuity correction is applied. Ignored if approx= "exact" or if index="r4" or if index="fy1" or if index="fy2" or if index="sbz". type type of alternative hypothesis. The options are "two-sided" (independence), "greater" (concordance) or "less" (discordance). Only enough of the string to be unique is required. print FALSE suppresses some of the output.

## Details

Upon computing r_h, it is common practice to determine whether the value is large enough, in absolute terms, to lead to the conclusion that the rank correlation coefficient that would be obtained from the entire set of permutations, say it ρ_h is different from zero. To this end, we consider the test H_0:ρ_h=0 against:

H_1:ρ_h>0 (concordance). Only a large positive r_h can be considered in line with this alternative hypothesis. To be considered significant, r_h must be greater and its value must be equal to or larger than the critical values corresponding to the prespecified α level of significance.

H_1:ρ_h<0 (discordance). Only a large negative r_h value will provide support for this alternative hypothesis. More specifically, r_h must be less and |r_h| must be equal to or larger than the critical values corresponding to the prespecified α level of significance.

H_1:ρ_h\ne 0 (independence). Either a large negative or a large positive value of r_h are conform to this alternative hypothesis. A significant value of r_h is obtained if the prespecified α level of significance is equal to or greater than 2Prob(-|r_h|) where Prob(.) is the probability density used to approximate the null distribution of r_h.

It is important to note that, as correctly observe Iman and Conover (1978), the discreteness of rank correlations often leads into situations where no critical region has exactly the size α. If approx="exact" the routine provides the next smaller exact size called conservative p-value or the next larger exact size called liberal p-value. A test can be considered conclusive if both the conservative and the liberal significance levels lie on the same side with respect of α. Compared to an exact p-value, a conservative p-value tends to understate the evidence against H_0, whereas a liberal p-value tends to overstate it.

If approx="exact" then ranktes uses precomputed exact null distributions. In particular, n≤ 26 for r_1 (see Gustafson, 2009); n≤ 24 for r_3 (see Girone et al. 2010). We have obtained the null distribution of r_4, r_5, r_6 and r_7 for n≤ 15 by calculating rank correlations for all the n! permutations of the integers 1,2, \cdots, n. Kendall's r_2 benefits from a recurrence relationship (Panneton and Robillard, 1972) that can handle r_2 for n up 60. In this regard, it is very useful the package Mpfr.

In the case approx \ne "exact", this routine computes critical values and significance levels by applying the density function indicated in approx. Now, of course, conservative and liberal p-values coincide.

The statistics involved in t-Student approximations are:

r_h^+=r_h√{\frac{m_{h,a}}{1-r_h^2}} \sim t_{m_{h,b}}, h=1,\cdots,4

where m_{1,a}=n-2, m_{2,a}=[9n(n-1)/(4n+10)-1], m_{3,a}=[3(n-1)(n^2-k_n)/2(n^2+2+ k_n)] with k_n=n\ mod \ 2 and m_{4,a}=2(n-2.01524)/1.00762. Furthermore, m_{1,b}=n-2, m_{2,b}=\lfloor m_{2,a} \rfloor, m_{3,b}=\lfloor m_{3,b}+0.5 \rfloor, m_{4,b}=\lfloor m_{4,a} \rfloor.

The t-Student approximations of r_1 and the Gaussian approximation of Kendall's r_2 are well known. Vittadini (1996) proposes the t-Student approximation to Kendall's r_2. Landenna et. al. (1989) suggest the t-Student for the Gini cograduation coefficient r_3. Tarsitano and Amerise (2015) developed a t-Student approximation to the null distribution of r_4. Terry (1952) has pointed out that the t-distribution with n-2 degrees of freedom provides a good approximation to the null distribution of r_5^+. The t-student approximation of r_6 and r_7 have not yet been developed.

The statistics involved in the Gaussian distribution are:

r_1^*=\frac{r_1}{√{n-1}}, \ r_2^*=r_2√{\frac{4n+10}{9n(n-1)}}, \ r_3^*=\frac{r_3}{√{1.5n}}, \ r_4^*=\frac{r_4}{√{1.00762(n-1)}}

r_5^*=\frac{r_5}{√{n-1}}, \ r_6^*=\frac{r_6}{√{n-1}}, \ r_7^*=\frac{r_7}{√{1.806452n}}

Cifarelli and Regazzini (1977) give the Gaussian approximation to the null distribution of Gini coefficient. See also Genest et al. (2010). The Gaussian approximation to the null distribution of r_4 is obtained in Tarsitano and Amerise (2015). Borroni (2013) derived the Gaussian approximation for r_7. The Gaussian approximations to r_5, r_6 and r_7 are asymptotic results.

If approx="vggfr" the Vianelli generalized Gaussian distribution with finite range is fitted by using the method of moments.

## Value

A list containing the following components:

 n number of ranks statistic type of rank correlation r observed value of the coefficient approx type of approximation tails type of alternative hypothesis Cpv conservative p-value Lpv liberal p-value Lambda if approx="vggfr" returns the vector containing the two shape parameters of the VGGFR density that best fits the null distribution of the required coefficients. It is NULL otherwise

## Note

In the case of a two-sided test, the p-value is defined as two times the one-sided p-value, bounded above by 1.

## Author(s)

Agostino Tarsitano, Ilaria Lucrezia Amerise, Marco Marozzi

## References

Borroni, G. C. (2013). "A new rank correlation measure". Statistical Papers, 54, 255–270.

Cifarelli, D. M. and Regazzini, E. (1977). "On a distribution-free test of independence based on Gini's rank association coefficient". Recent Developments in Statistics (Proceedings of the European Meeting of Statisticians, Grenoble, 1976), Amsterdam, North-Holland, 375–385.

Genest, N. B. and Neslehova, J. and Ben Ghorbal, N. (2010). "'Spearman's footrule and Gini's gamma: a review with complements" Journal of Nonparametric Statistics, 22, 937–954.

Girone, G. et al. (2010). "La distribuzione campionaria dell'indice di cograduazione di Gini per dimensioni campionarie fino a 24". Annali del Dipartimento di Scienze Statistiche "Carlo Cecchi" - Universita' di Bari, 24, 246–271.

Gustafson, L. (2009). rho null distribution. Available at http://www.luke-g.com/

Iman, L. and Conover, W. J. (1978). "Approximations of the critical region for 's rho with and without ties presen"t. Communication in Statistics - Simulation and Computation, 7, 269–282.

Landenna, G. and Scagni, A. and Boldrini, M. (1989). "An approximated distribution of the Gini's rank association coefficient". Communications in Statistics. Theory and Methods, 18, 2017–2026.

Tarsitano, A. and Amerise, I. L. (2015). "On a measure of rank-order association". Journal of Statistical and Econometric Methods, 4, 83–105.

Tarsitano, A. and Amerise, I. L. (2016). "Modelling the null distribution of rank correlations". Submitted.

Terry, M. E. (1952). "Some rank order tests which are most powerful against specific parameteric alternatives". Annals of Mathematical Statistics, 23, 346–366.

Vittadini, G. (1996). "Una famiglia di distribuzione per i test di associazione". In Atti della XXXVIII riunione scientifica della S.I.S., Rimini 9-13 Aprile 1996, 2, 521–528.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 # G. P. Watkins (1933). An Ordinal Index of Correlation, Journal of the # American Statistical Association, 28:182, 139-151. # 20-item series for area and density have been made up to cover the # original 13 states and the four others earliest admitted to the Union. State<-c("Georgia","North_Carolina","New_York","Luisiana","Pennsylvania", "Virginia","Tennessee","Ohio","Kentucky","Maine","South_Carolina", "West_Virginia","Maryland","Vermont","New_Hampshire","Massachusetts", "New_Jersey","Connecticut","Delaware","Rhode_Island") Area<-c(1,2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20) Density<-c(17,13,5,18,6,14,12,7,11,20,15,10,8,19,16,2,3,4,9,1) op<-par(mfrow=c(1,1)) plot(Area,Density,main="",xlab="Area",ylab="Density",pch=19,cex=0.9, col="darkgreen" ) abline(h=mean(Area),col="black",lty=2,lwd=1) abline(v=mean(Density),col="darkblue",lty=2,lwd=1) par(op) r<-comprank(Area,Density,"fy2","wgh")$r ranktes(r, length(Area), "fy2", "ga",FALSE, "two", TRUE) ##### # data(Atar);attach(Atar) op<-par(mfrow=c(1,1)) plot(TBL,TFL,main="",xlab="Backward Linkage Index",ylab= "Forward Linkage Index",pch=19, cex=0.9,col="magenta") abline(h=mean(TFL),col="black",lty=2,lwd=1) abline(v=mean(TBL),col="black",lty=2,lwd=1) par(op) r<-comprank(TBL,TFL,"fy1","wgh")$r ranktes(r, length(TBL), "fy1", "vggfr",FALSE, "two", TRUE) detach(Atar) ##### data(Sharpe);attach(Sharpe) op<-par(mfrow=c(1,1)) plot(AVR,VAR, type = "p",pch=19,cex=1.1,col="tomato",main="Mutual fund performance") text(AVR,VAR, labels = rownames(Sharpe), cex=0.5, pos=3) abline(h=mean(AVR),col="black",lty=2,lwd=1) abline(v=mean(VAR),col="black",lty=2,lwd=1) par(op) r<-comprank(AVR,VAR,"sbz","wgh")$r ranktes(r, length(AVR), "sbz", "st",FALSE, "greater", TRUE) detach(Sharpe) ##### # # Sun,J.-G. and Jurisicova, A. and Casper, R.F. (1997). "Detection of # Deoxyribonucleic Acid Fragmentation in Human Sperm: Correlation # with Fertilization In Vitro". Biology of Reproduction, 56, 602-607. n<-c(222,298,143,143,291,148) r<-c(-0.18,-0.12,-0.16,-0.20,-0.06,-0.003) App<-c("Ga","St","Vg") N<-length(n);Ta<-matrix(NA,N,5) for (i in 1:length(n)){Ta[i,1]<-r[i];Ta[i,2]<-n[i] for (j in 1:3){ app<-App[j] a<-ranktes(r[i],n[i],"S",app,FALSE,"t",FALSE);Ta[i,2+j]<-a$Cpv }} Df<-matrix(Ta,6,5) rownames(Df)<-c("Conc. sperm/mL","Motility", "Fertilization rate", "Cleavage rate", "Male age","Abstinence days") colnames(Df)<-c("Spearman","n of samples","Appr. Gaussian", "Appr. t-Student", "Appr. GGFR") Df<-as.data.frame(Df) print(round(Df,5)) ##### # data(Starshi);attach(Starshi) op<-par(mfrow=c(1,1)) plot(Sm15F,Sm15M, type = "p",pch=19,cex=0.9,col="darkorange", main="Smokers ") text(Sm15F,Sm15M,labels = rownames(Starshi),cex=0.6,pos= c(1,rep(2,10),3,2)) abline(h=mean(Sm15M),col="black",lty=2,lwd=1) abline(v=mean(Sm15F),col="black",lty=2,lwd=1) par(op) r<-comprank(Sm15F,Sm15M,"r4","wgh")$r a<-ranktes(r, length(Sm15F), "r4", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") r<-comprank(Sm15F,Sm15M,"sp","wgh")$r a<-ranktes(r, length(Sm15F), "sp", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") r<-comprank(Sm15F,Sm15M,"ke","wgh")$r a<-ranktes(r, length(Sm15F), "ke", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") r<-comprank(Sm15F,Sm15M,"gi","wgh")$r a<-ranktes(r, length(Sm15F), "gi", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") r<-comprank(Sm15F,Sm15M,"fy1","wgh")$r a<-ranktes(r, length(Sm15F), "fy1", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") r<-comprank(Sm15F,Sm15M,"fy2","wgh")$r a<-ranktes(r, length(Sm15F), "fy2", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") r<-comprank(Sm15F,Sm15M,"sbz","wgh")$r a<-ranktes(r, length(Sm15F), "sbz", "ex",TRUE, "two", FALSE) cat(a$Value,a$Cpv,a$Lpv,"\n") detach(Starshi) ##### # All.App<-function(r,n,index,type){ # Computes p-values of an observed rank correlation statistic A<-rep(r,9) names(A)<-encodeString(c(index,"t-Student, CC=F","Gaussian, CC=F", "VGGFR, CC=F","t-Student, CC=T","Gaussian, CC=T", "VGGFR, CC=T", "Exact p Conservative","Exact p Liberal"),justify="right") a<-ranktes(r,n,index,"St",FALSE,type,FALSE);A[2]<-a$Cpv a<-ranktes(r,n,index,"Ga",FALSE,type,FALSE);A[3]<-a$Cpv a<-ranktes(r,n,index,"Vg",FALSE,type,FALSE);A[4]<-a$Cpv a<-ranktes(r,n,index,"St",TRUE,type,FALSE);A[5]<-a$Cpv a<-ranktes(r,n,index,"Ga",TRUE,type,FALSE);A[6]<-a$Cpv a<-ranktes(r,n,index,"Vg",TRUE,type,FALSE);A[7]<-a$Cpv a<-ranktes(r,n,index,"Ex",FALSE,type,FALSE);A[8]<-a$Cpv;A[9]<-a$Lpv A<-as.matrix(A) return(A)} data(Gabbs);attach(Gabbs) B<-matrix(0,9,6) colnames(B)<-colnames(Gabbs[1:6]) rownames(B)<-encodeString(c("index","t-Student, CC=F","Gaussian, CC=F", "VGGFR, CC=F","t-Student, CC=T", "Gaussian, CC=T", "VGGFR, CC=T", "Exact p Conservative","Exact p Liberal"), justify="right") index<-"spearman" for (i in 1:6){r<-comprank(,i],,7],index, print=FALSE)$r B[,i]<-All.App(r,19,index,"less")} print(round(B,5)) detach(Gabbs) ##### # data(Dalyww);attach(Dalyww) op<-par(mfrow=c(1,1)) plot(ACLS,ASHR,main="The paradox of high rates of suicide in happy places", xlab="Adjusted Life Satisfaction", ylab="Adjusted Suicide Risk",pch=19, cex=0.8,col="steelblue") text(ACLS,ASHR,labels=rownames(Dalyww),cex=0.7,pos=2) abline(h=mean(ASHR),col="black",lty=2,lwd=1) abline(v=mean(ACLS),col="black",lty=2,lwd=1) par(op) r<-comprank(ACLS,ASHR,"spearman")$r;n<-length(ASHR) out<-ranktes(r,n,"s","ga",FALSE,"greater",FALSE) cat(round(out$Value,3),round(out$Cpv,5),round(out$Lpv,5),"\n") out<-ranktes(r,n,"s","st",FALSE,"greater",FALSE) cat(round(out$Value,3),round(out$Cpv,5),round(out$Lpv,5),"\n") out<-ranktes(r,n,"s","vg",FALSE,"greater",FALSE) cat(round(out$Value,3),round(out$Cpv,5),round(out$Lpv,5),"\n") # r<-comprank(ACLS,ASHR,"kendall")$r out<-ranktes(r,n,"kendall","st",FALSE,"greater",FALSE) cat(round(out$Value,3),round(out$Cpv,5),round(out$Lpv,5),"\n") # r<-comprank(ACLS,ASHR,"r4")$r out<-ranktes(r,n,"r4","st",FALSE,"greater",FALSE) cat(round(out$Value,3),round(out$Cpv,5),round(out\$Lpv,5),"\n") detach(Dalyww)

pvrank documentation built on May 17, 2018, 9:03 a.m.