Tests of independence between two rankings

Description

Performs various independence tests based on rank correlations.

Usage

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

Arguments

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 and r_6 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} floor, m_{3,b}=lfloor m_{3,b}+0.5 floor, m_{4,b}=lfloor m_{4,a} floor.

The t-Student approximation 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^+, We conjecture that a similar result holds with Filliben's rank correlation r_6^+.

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{Fh(r_5)-Fh(r_5(1-0.6/(n+8)))}{√{(n-3)}}, r_6^*=frac{Fh(r_6)-Fh(r_6(1-0.6/(n+8)))}{√{(n-3)}}

where Fh(x)=0.5log[(1+x)/(1-x)]. 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). Fieller and Pearson (1961) obtains the Gaussian approximation to r_5. The generalization of the last result to r_6 is a conjecture, based on similarities of various kinds.

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:

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

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

	
# 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,"fil","wgh")$r
ranktes(r, length(Area), "fil", "ga",FALSE, "two", TRUE)
#####
#
## Not run: 
data(Atar);attach(Atar)
op<-par(mfrow=c(1,1))
plot(TBL,TFL,main="",xlab="Backward Linkage Index",ylab=
"Forward Linkage Index",pch=19, cex=0.9,col="magenta")
abline(h=mean(TFL),col="black",lty=2,lwd=1)
abline(v=mean(TBL),col="black",lty=2,lwd=1)
par(op)
r<-comprank(TBL,TFL,"fy","wgh")$r
ranktes(r, length(TBL), "fy", "vggfr",FALSE, "two", TRUE)
detach(Atar)

## End(Not run)
#####
#
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,"gini","wgh")$r
ranktes(r, length(AVR), "gini", "st",FALSE, "greater", TRUE)
detach(Sharpe)
#####
#
## Not run: 
# Sun,J.-G. and Jurisicova, A. and Casper, R.F. (1997). "Detection of
# Deoxyribonucleic Acid Fragmentation in Human Sperm: Correlation
# with Fertilization In Vitro".  Biology of Reproduction, 56, 602-607.
n<-c(222,298,143,143,291,148); r<-c(-0.18,-0.12,-0.16,-0.20,-0.06,-0.003)
App<-c("Ga","St","Vg")
N<-length(n);Ta<-matrix(NA,N,5)
for (i in 1:length(n)){Ta[i,1]<-r[i];Ta[i,2]<-n[i]
		for (j in 1:3){
				app<-App[j]
				a<-ranktes(r[i],n[i],"S",app,FALSE,"t",FALSE);Ta[i,2+j]<-a$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))

## End(Not run)
#####
#
## Not run: 
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,"ken","wgh")$r
a<-ranktes(r, length(Sm15F), "ken", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"\n")
r<-comprank(Sm15F,Sm15M,"g","wgh")$r
a<-ranktes(r, length(Sm15F), "g", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"\n")
r<-comprank(Sm15F,Sm15M,"fy","wgh")$r
a<-ranktes(r, length(Sm15F), "fy", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"\n")
r<-comprank(Sm15F,Sm15M,"fil","wgh")$r
a<-ranktes(r, length(Sm15F), "fil", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"\n")
detach(Starshi)

## End(Not run)
#####
#
## Not run: 
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(Gabbs[,i],Gabbs[,7],index, print=FALSE)$r
					B[,i]<-All.App(r,19,index,"less")}
print(round(B,5))
detach(Gabbs)

## End(Not run)
#####
#
## Not run: 
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)

## End(Not run)

Results