pvrank (version 1.0)

ranktes: Tests of independence between two rankings

Description

Performs various independence tests based on rank correlations.

Usage

ranktes(r, n, index = "spearman", 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". 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".

type

type of alternative hypothesis. The options are "two-sided" (default), "greater" or "less". Only enough of the string to be unique is required; "greater" corresponds to positive association, "less"

print

FALSE suppresses some of the output.

Value

• A list containing the following components:
• n
  number of ranks
• Statistic
  coefficient of monotone association
• r
  observed value
• 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

Details

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 $\alpha$. 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 $\alpha$.

If approx="exact" then rankes uses precomputed exact null distributions for $n\le 26$ (Spearman or $r_1$; see Gustafson, 2009). $n\le 24$ (Gini or $r_3$; see Girone et al. 2010). $n\le60$ (Kendall or $r_2$; here we apply the recursive formula given by Panneton and Robillard, 1972). $n\le 15$ ($r_4$); see Tarsitano and Amerise (2015).

In the case approx$\ne$ "exact", this routine computes critical values and significance levels by applying the density required in approx. Now, of course, conservative and liberal p-values coincide. The statistics involved in the t-Student approximation are

$$r_h^+=r_h\sqrt{\frac{m_h}{1-r_h^2}} \sim t_{m_h} \ h=1,\cdots,4$$.

where $m_1=n-2$, $m_2=[9n(n-1)/(4n+10)-1]$, $m_3=[3(n-1)(n^2-k_n)/2(n^2+2+ k_n)]$, $m_4=(n-2.01524)/1.00762$ with $k_n=n\ mod \ 2$.

The statistics involved in the Gaussian distribution are $$r_1^*=r_1\sqrt{n-1}, \ r_2^*=r_2\sqrt{\frac{9n(n-1)}{4n+10}}, \ r_3^*=r_3\sqrt{1.5n}, \ r_4^*=r_4\sqrt{1.00762(n-1)}.$$

The t-Student approximation of Spearman's $r_1$ and the Gaussian approximation of Kendall's $r_2$ are well known.

Vittadini (1996) proposes the t-Student approximation for Kendall's tau. Landenna et. al. (1989) suggest the symmetrical t-Student for the Gini cograduation coefficient $r_3$.

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). The same authors propose a t-Student approximation.

If approx="vggfr" the Vianelli generalized Gaussian distribution with finite range is fitted by using the method of moments. The approx="vggfr"approximation to the null distribution of $r_4$ seems to be very poor. Presumably this is an effect the reduced number of ranks for which the exact frequencies of $r_4$ are fully known, which prevent us from having better possibilities for the fitting. The combination approx="vggfr" and index="r4" is not implemented at the moment.

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). Spearman rho null distribution. Available at http://www.luke-g.com/math/spearman/index.html

Iman, L. and Conover, W. J. (1978). Approximations of the critical region for Spearman's rho with and without ties present. 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. (2013). "Approximation of the null distribution of rank correlations". Submitted.

Tarsitano, A. and Amerise, I. L. (2015). "On a measure of rank order association". Submitted.

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

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="olivedrab")
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,"spearman","wgh")
ranktes(r, length(TBL), "spearman", "st",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="violetred",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")
ranktes(r, length(AVR), "gini", "st",FALSE, "two", 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$Cpv
		}}
Df<-matrix(Ta,6,5)
rownames(Df)<-c("Conc. sperm/mL","Motility", "Fertilization rate", "Cleavage rate",  
"Male age","Abstinence days")
colnames(Df)<-c("Spearman's rho","n of samples","Appr. Gaussian","Appr. t-Student", 
"Appr. GGFR")
Df<-as.data.frame(Df)
print(round(Df,5))
###
data(Starshi);attach(Starshi)
op<-par(mfrow=c(1,1))
plot(Sm15F,Sm15M, type = "p",pch=19,cex=0.9,col="violetred",main="Smokers ")
text(Sm15F,Sm15M,labels = rownames(Starshi),cex=0.6,pos=c(1,rep(2,10),3,2))
abline(h=mean(Sm15M),col="black",lty=2,lwd=1)
abline(v=mean(Sm15F),col="black",lty=2,lwd=1)
par(op)
r<-comprank(Sm15F,Sm15M,"r4","wgh")
a<-ranktes(r, length(Sm15F), "r4", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"")
r<-comprank(Sm15F,Sm15M,"s","wgh")
a<-ranktes(r, length(Sm15F), "s", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"")
r<-comprank(Sm15F,Sm15M,"k","wgh")
a<-ranktes(r, length(Sm15F), "k", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"")
r<-comprank(Sm15F,Sm15M,"g","wgh")
a<-ranktes(r, length(Sm15F), "g", "ex",TRUE, "two", FALSE)
cat(a$Value,a$Cpv,a$Lpv,"")
detach(Starshi)