[R] Cochran-Armitage-trend-test
Torsten Hothorn
Torsten.Hothorn at rzmail.uni-erlangen.de
Tue Nov 29 13:36:42 CET 2005
On Thu, 28 Jul 2005, Eric Lecoutre wrote:
> Hi there,
>
> I often do receive some mails about this piece of code regarding
> Cochran-Armitage or Mantel Chi square.
This is a very late reply but maybe still interesting. The
conditional version of the Cochran-Armitage test for trend
for proportions is part of the framework implemented by `coin'
and can be computed as follows:
R> library("coin")
R>
R> ### Cochran-Armitage trend test for proportions
R> ### Lung tumors in female mice exposed to 1,2-dichloroethane
R> ### Encyclopedia of Biostatistics (Armitage & Colton, 1998),
R> ### Chapter Trend Test for Counts and Proportions, page 4578, Table 2
R> lungtumor <- data.frame(dose = rep(c(0, 1, 2), c(40, 50, 48)),
+ tumor = c(rep(c(0, 1), c(38, 2)),
+ rep(c(0, 1), c(43, 7)),
+ rep(c(0, 1), c(33, 15))))
R> table(lungtumor$dose, lungtumor$tumor)
0 1
0 38 2
1 43 7
2 33 15
R>
R> ### Cochran-Armitage test (permutation equivalent to correlation
R> ### between dose and tumor), cf. Table 2 for results
R> independence_test(tumor ~ dose, data = lungtumor, teststat = "quad")
Asymptotic General Independence Test
data: tumor by dose
T = 10.6381, df = 1, p-value = 0.001108
R>
R> ### approximate distribution by Monte-Carlo
R> independence_test(tumor ~ dose, data = lungtumor, teststat = "quad",
+ distribution = approximate(B = 50000))
Approximative General Independence Test
data: tumor by dose
T = 10.6381, p-value = 0.00092
R>
Moreover, we can use a linear-by-linear association test with the
appropriate scores which, of course, could be changed as well
R> ### linear-by-linear association test with scores 0, 1, 2
R> ### is identical with Cochran-Armitage test
R> lungtumor$dose <- ordered(lungtumor$dose)
R> independence_test(tumor ~ dose, data = lungtumor, teststat = "quad",
+ scores = list(dose = c(0, 1, 2)))
Asymptotic General Independence Test
data: tumor by groups 0 < 1 < 2
T = 10.6381, df = 1, p-value = 0.001108
R>
Just for the record ...
Best,
Torsten
> The archived mail does unfortunately lack some pieces of code (function
> "scores").
> I copy there all my raw code that I did implement to mimic SAS PROC FREQ
> statistics regarding the analysis of contingency tables. Whoever is
> interested to take it and rework it a little bit (for example redefining
> outputs so that they suits a htest object) is welcome.
>
> Best wishes,
>
> Eric
>
>
> -----
>
>
>
> # R functions to provides statistics on contingency tables
> # Mimics SAS PROC FREQ outputs
> # Implementation is the one described in SAS PROC FREQ manual
>
> # Eric Lecoutre <ericlecoutre at gmail.com
>
> # Feel free to use / modify / document / add to a package
>
>
>
> #------------------------------------ UTILITARY FUNCTIONS
> ------------------------------------#
>
>
> print.ordtest=function(l,...)
> {
> tmp=matrix(c(l$estimate,l$ASE),nrow=1)
> dimnames(tmp)=list(l$name,c("Estimate","ASE"))
> print(round(tmp,4),...)
> }
>
>
> compADPQ=function(x)
> {
> nr=nrow(x)
> nc=ncol(x)
> Aij=matrix(0,nrow=nr,ncol=nc)
> Dij=matrix(0,nrow=nr,ncol=nc)
> for (i in 1:nr) {
> for (j in 1:nc) {
>
> Aij[i,j]=sum(x[1:i,1:j])+sum(x[i:nr,j:nc])-sum(x[i,])-sum(x[,j])
>
> Dij[i,j]=sum(x[i:nr,1:j])+sum(x[1:i,j:nc])-sum(x[i,])-sum(x[,j])
> }}
> P=sum(x*Aij)
> Q=sum(x*Dij)
> return(list(Aij=Aij,Dij=Dij,P=P,Q=Q))
> }
>
>
> scores=function(x,MARGIN=1,method="table",...)
> {
> # MARGIN
> # 1 - row
> # 2 - columns
>
> # Methods for ranks are
> #
> # x - default
> # rank
> # ridit
> # modridit
>
> if (method=="table")
> {
> if (is.null(dimnames(x))) return(1:(dim(x)[MARGIN]))
> else {
> options(warn=-1)
> if
> (sum(is.na(as.numeric(dimnames(x)[[MARGIN]])))>0)
> {
> out=(1:(dim(x)[MARGIN]))
> }
> else
> {
> out=(as.numeric(dimnames(x)[[MARGIN]]))
> }
> options(warn=0)
> }
> }
> else {
> ### method is a rank one
> Ndim=dim(x)[MARGIN]
> OTHERMARGIN=3-MARGIN
>
> ranks=c(0,(cumsum(apply(x,MARGIN,sum))))[1:Ndim]+(apply(x,MARGIN,sum)+1)
> /2
> if (method=="ranks") out=ranks
> if (method=="ridit") out=ranks/(sum(x))
> if (method=="modridit") out=ranks/(sum(x)+1)
> }
>
> return(out)
> }
>
>
> #------------------------------------ FUNCTIONS
> ------------------------------------#
>
> tablegamma=function(x)
> {
> # Statistic
> tmp=compADPQ(x)
> P=tmp$P
> Q=tmp$Q
> gamma=(P-Q)/(P+Q)
> # ASE
> Aij=tmp$Aij
> Dij=tmp$Dij
> tmp1=4/(P+Q)^2
> tmp2=sqrt(sum((Q*Aij - P*Dij)^2 * x))
> gamma.ASE=tmp1*tmp2
> # Test
> var0=(4/(P+Q)^2) * (sum(x*(Aij-Dij)^2) - ((P-Q)^2)/sum(x))
> tb=gamma/sqrt(var0)
> p.value=2*(1-pnorm(tb))
> # Output
>
> out=list(estimate=gamma,ASE=gamma.ASE,statistic=tb,p.value=p.value,name=
> "Gamma",bornes=c(-1,1))
> class(out)="ordtest"
> return(out)
> }
>
>
> tabletauc=function(x)
> {
> tmp=compADPQ(x)
> P=tmp$P
> Q=tmp$Q
> m=min(dim(x))
> n=sum(x)
> # statistic
>
> tauc=(m*(P-Q))/(n^2*(m-1))
> # ASE
> Aij=tmp$Aij
> Dij=tmp$Dij
> dij=Aij-Dij
> tmp1=2*m/((m-1)*n^2)
> tmp2= sum(x * dij^2) - (P-Q)^2/n
> ASE=tmp1*sqrt(tmp2)
>
> # Test
> tb=tauc/ASE
> p.value=2*(1-pnorm(tb))
> # Output
>
> out=list(estimate=tauc,ASE=ASE,statistic=tb,p.value=p.value,name="Kendal
> l's tau-c",bornes=c(-1,1))
> class(out)="ordtest"
> return(out)
> }
>
> tabletaub=function(x)
> {
> # Statistic
> tmp=compADPQ(x)
> P=tmp$P
> Q=tmp$Q
> n=sum(x)
> wr=n^2 - sum(apply(x,1,sum)^2)
> wc=n^2 - sum(apply(x,2,sum)^2)
> taub=(P-Q)/sqrt(wr*wc)
> # ASE
> Aij=tmp$Aij
> Dij=tmp$Dij
> w=sqrt(wr*wc)
> dij=Aij-Dij
> nidot=apply(x,1,sum)
> ndotj=apply(x,2,sum)
> n=sum(x)
> vij=outer(nidot,ndotj, FUN=function(a,b) return(a*wc+b*wr))
> tmp1=1/(w^2)
> tmp2= sum(x*(2*w*dij + taub*vij)^2)
> tmp3=n^3*taub^2*(wr+wc)^2
> tmp4=sqrt(tmp2-tmp3)
> taub.ASE=tmp1*tmp4
> # Test
> var0=4/(wr*wc) * (sum(x*(Aij-Dij)^2) - (P-Q)^2/n)
> tb=taub/sqrt(var0)
> p.value=2*(1-pnorm(tb))
> # Output
>
> out=list(estimate=taub,ASE=taub.ASE,statistic=tb,p.value=p.value,name="K
> endall's tau-b",bornes=c(-1,1))
> class(out="ordtest")
> return(out)
> }
>
> tablesomersD=function(x,dep=2)
> {
> # dep: which dimension stands for the dependant variable
> # 1 - ROWS
> # 2 - COLS
> # Statistic
> if (dep==1) x=t(x)
> tmp=compADPQ(x)
> P=tmp$P
> Q=tmp$Q
> n=sum(x)
> wr=n^2 - sum(apply(x,1,sum)^2)
> somers=(P-Q)/wr
> # ASE
> Aij=tmp$Aij
> Dij=tmp$Dij
> dij=Aij-Dij
> tmp1=2/wr^2
> tmp2=sum(x*(wr*dij - (P-Q)*(n-apply(x,1,sum)))^2)
> ASE=tmp1*sqrt(tmp2)
> # Test
> var0=4/(wr^2) * (sum(x*(Aij-Dij)^2) - (P-Q)^2/n)
> tb=somers/sqrt(var0)
> p.value=2*(1-pnorm(tb))
> # Output
> if (dep==1) dir="R|C" else dir= "C|R"
> name=paste("Somer's D",dir)
>
> out=list(estimate=somers,ASE=ASE,statistic=tb,p.value=p.value,name=name,
> bornes=c(-1,1))
> class(out)="ordtest"
> return(out)
>
>
> }
>
> #out=table.somersD(data)
>
>
>
> tablepearson=function(x,scores.type="table")
> {
>
> # Statistic
> sR=scores(x,1,scores.type)
> sC=scores(x,2,scores.type)
> n=sum(x)
> Rbar=sum(apply(x,1,sum)*sR)/n
> Cbar=sum(apply(x,2,sum)*sC)/n
> ssr=sum(x*(sR-Rbar)^2)
> ssc=sum(t(x)* (sC-Cbar)^2)
> tmpij=outer(sR,sC,FUN=function(a,b) return((a-Rbar)*(b-Cbar)))
> ssrc= sum(x*tmpij)
> v=ssrc
> w=sqrt(ssr*ssc)
> r=v/w
> # ASE
> bij=outer(sR,sC, FUN=function(a,b)return((a-Rbar)^2*ssc +
> (b-Cbar)^2*ssr))
> tmp1=1/w^2
> tmp2=x*(w*tmpij - (bij*v)/(2*w))^2
> tmp3=sum(tmp2)
> ASE=tmp1*sqrt(tmp3)
> # Test
> var0= (sum(x*tmpij) - (ssrc^2/n))/ (ssr*ssc)
> tb=r/sqrt(var0)
> p.value=2*(1-pnorm(tb))
> # Output
>
> out=list(estimate=r,ASE=ASE,statistic=tb,p.value=p.value,name="Pearson
> Correlation",bornes=c(-1,1))
> class(out)="ordtest"
> return(out)
> }
>
> # table.pearson(data)
>
>
> tablespearman=function(x)
> {
> # Details algorithme manuel SAS PROC FREQ page 540
> # Statistic
> n=sum(x)
> nr=nrow(x)
> nc=ncol(x)
> tmpd=cbind(expand.grid(1:nr,1:nc))
> ind=rep(1:(nr*nc),as.vector(x))
> tmp=tmpd[ind,]
> rhos=cor(apply(tmp,2,rank))[1,2]
> # ASE
> Ri=scores(x,1,"ranks")- n/2
> Ci=scores(x,2,"ranks")- n/2
> sr=apply(x,1,sum)
> sc=apply(x,2,sum)
> F=n^3 - sum(sr^3)
> G=n^3 - sum(sc^3)
> w=(1/12)*sqrt(F*G)
> vij=data
> for (i in 1:nrow(x))
> {
> qi=0
> if (i<nrow(x))
> {
> for (k in i:nrow(x)) qi=qi+sum(x[k,]*Ci)
> }
> }
> for (j in 1:ncol(x))
> {
> qj=0
> if (j<ncol(x))
> {
> for (k in j:ncol(x)) qj=qj+sum(x[,k]*Ri)
> }
> vij[i,j]=n*(Ri[i]*Ci[j] +
> 0.5*sum(x[i,]*Ci)+0.5*sum(data[,j]*Ri) +qi+qj)
> }
>
>
> v=sum(data*outer(Ri,Ci))
> wij=-n/(96*w)*outer(sr,sc,FUN=function(a,b) return(a^2*G+b^2*F))
> zij=w*vij-v*wij
> zbar=sum(data*zij)/n
> vard=(1/(n^2*w^4))*sum(x*(zij-zbar)^2)
> ASE=sqrt(vard)
> # Test
> vbar=sum(x*vij)/n
> p1=sum(x*(vij-vbar)^2)
> p2=n^2*w^2
> var0=p1/p2
> stat=rhos/sqrt(var0)
>
> # Output
> out=list(estimate=rhos,ASE=ASE,name="Spearman
> correlation",bornes=c(-1,1))
> class(out)="ordtest"
> return(out)
> }
>
> #tablespearman(data)
>
>
>
>
> tablelambdasym=function(x)
> {
> # Statistic
> ri = apply(x,1,max)
> r=max(apply(x,2,sum))
> n=sum(x)
> cj=apply(x,2,max)
> c=max(apply(x,1,sum))
> sri=sum(ri)
> w=2*n - r -c
> v=2*n - sri - sum(cj)
> lambda=(w-v)/w
> # ASE ...
>
> tmpSi=0
> l=min(which(apply(x,2,sum)==r))
> for (i in 1:length(ri))
> {
> li=min(which(x[i,]==ri[i]))
> if (li==l) tmpSi=tmpSi+x[i,li]
> }
>
> tmpSj=0
> k=min(which(apply(x,1,sum)==c))
> for (j in 1:length(cj))
> {
> kj=min(which(x[,j]==cj[j]))
> if (kj==k) tmpSj=tmpSj+x[kj,j]
> }
>
> rk=max(x[k,])
> cl=max(x[,l])
> tmpx=tmpSi+tmpSj+rk+cl
> y=8*n-w-v-2*tmpx
>
> nkl=x[k,l]
> tmpSij=0
> for (i in 1:nrow(x))
> {
> for (j in 1:ncol(x))
> {
> li=min(which(x[i,]==ri[i]))
> kj=min(which(x[,j]==cj[j]))
> tmpSij=tmpSij+x[kj,li]
> }
> }
>
>
> ASE=(1/(w^2))*sqrt(w*v*y-(2*(w^2)*(n-tmpSij))-2*(v^2)*(n-nkl))
> # Output
>
> out=list(estimate=lambda,ASE=ASE,name="Lambda
> Symetric",bornes=c(0,1))
> class(out)="ordtest"
> return(out)
>
> }
> #tablelambdasym(data)
>
> tablelambdaasym=function(x,transpose=FALSE)
> {
> # Statistic
> if (transpose==TRUE) x=t(x)
> ri = apply(x,1,max)
> r=max(apply(x,2,sum))
> sri=sum(ri)
> n=sum(x)
> lambda=(sum(ri)-r)/(n-r)
> # ASE
> l=min(which(apply(x,2,sum)==r))
> tmp=0
> for (i in 1:length(ri))
> {
> li=min(which(x[i,]==ri[i]))
> if (li==l) tmp=tmp+x[i,li]
> }
> ASE=sqrt(((n-sri)/(n-r)^3) *(sri+r-2*tmp))
> # Output
> if (transpose) dir="R|C" else dir= "C|R"
> name=paste("Lambda asymetric",dir)
> out=list(estimate=lambda,ASE=ASE,name=name,bornes=c(0,1))
> class(out)="ordtest"
> return(out)
> }
>
>
>
>
> tableUCA=function(x,transpose=TRUE)
> {
> if (transpose==TRUE) x=t(x)
> # Statistic
> n=sum(x)
> ni=apply(x,1,sum)
> nj=apply(x,2,sum)
> Hx=-sum((ni/n)*log(ni/n))
> Hy=-sum((nj/n)*log(nj/n))
> Hxy=-sum((x/n)*log(x/n))
> v=Hx+Hy- Hxy
> w=Hy
> U=v/w
> # ASE
> tmp1=1/((n)*(w^2))
> tmpij=0
> for (i in 1:nrow(x))
> {
> for (j in 1:ncol(x))
> {
> tmpij=tmpij+( x[i,j]* (
> Hy*log(x[i,j]/ni[i])+(Hx-Hxy)* log(nj[j]/n) )^2 )
> }
> }
> ASE=tmp1*sqrt(tmpij)
> # Output
> if (transpose) dir="R|C" else dir= "C|R"
> name=paste("Uncertainty Coefficient",dir)
> out=list(estimate=U,ASE=ASE,name=name,bornes=c(0,1))
> class(out)="ordtest"
> return(out)
> }
>
> #tableUCA(data)
>
> tableUCS=function(x)
> {
>
> # Statistic
> n=sum(x)
> ni=apply(x,1,sum)
> nj=apply(x,2,sum)
> Hx=-sum((ni/n)*log(ni/n))
> Hy=-sum((nj/n)*log(nj/n))
> Hxy=-sum((x/n)*log(x/n))
> U=(2*(Hx+Hy-Hxy))/(Hx+Hy)
> # ASE
> tmpij=0
> for (i in 1:nrow(x))
> {
> for (j in 1:ncol(x))
> {
> tmpij=tmpij+( x[i,j]*
> (Hxy*log(ni[i]*nj[j]/n^2) - (Hx+Hy)*log(x[i,j]/n))^2 /(n^2*(Hx+Hy)^4)
> )
> }
> }
> ASE=2*sqrt(tmpij)
> # Output
> name="Uncertainty Coefficient Symetric"
> out=list(estimate=U,ASE=ASE,name=name,bornes=c(0,1))
> class(out)="ordtest"
> return(out)
> }
>
>
>
>
>
> tablelinear=function(x,scores.type="table")
> {
> r=tablepearson(x,scores.type)$estimate
> n=sum(x)
> ll=r^2*(n-1)
> out=list(estimate=ll)
> return(out)
> }
>
> tablephi=function(x)
> {
> if (all.equal(dim(x),c(2,2))==TRUE)
> {
> rtot=apply(x,1,sum)
> ctot=apply(x,2,sum)
> phi= det(x)/sqrt(prod(rtot)*prod(ctot))
> }
> else {
> Qp=chisq.test(x)$statistic
> phi=sqrt(Qp/sum(x))
> }
> names(phi)="phi"
> return(phi=phi)
> }
>
>
> tableCramerV=function(x)
> {
> if (all.equal(dim(x),c(2,2))==TRUE)
> {
> cramerV=tablephi(x)
> }
> else
> {
> Qp=tableChisq(x)$estimate
> cramerV=sqrt((Qp/n)/min(dim(x)-1))
> }
> names(cramerV)="Cramer's V"
> return(cramerV)
> }
>
>
> tableChisq=function(x)
> {
> nidot=apply(x,1,sum)
> ndotj=apply(x,2,sum)
> n=sum(nidot)
> eij=outer(nidot,ndotj,"*")/n
> R=length(nidot)
> C=length(ndotj)
> dll=(R-1)*(C-1)
> Qp=sum((x-eij)^2/eij)
> p.value=1-pchisq(Qp,dll)
>
> out=list(estimate=Qp,dll=dll,p.value=p.value,dim=c(R,C),name="Pearson's
> Chi-square")
> return(out)
> }
>
>
> tableChisqLR=function(x)
> {
> # Likelihood ratio Chi-squared test
> nidot=apply(x,1,sum)
> ndotj=apply(x,2,sum)
> n=sum(nidot)
> eij=outer(nidot,ndotj,"*")/n
> R=length(nidot)
> C=length(ndotj)
> dll=(R-1)*(C-1)
> G2=2*sum(x*log(x/eij))
> p.value=1-pchisq(G2,dll)
>
> out=list(estimate=G2,dll=dll,p.value=p.value,dim=c(R,C),name="Likelihood
> ratio Chi-square")
> return(out)
> }
>
> tableChisqCA=function(x)
> {
> if (all.equal(dim(x),c(2,2))==TRUE)
> {
> nidot=apply(x,1,sum)
> ndotj=apply(x,2,sum)
> n=sum(nidot)
> eij=outer(nidot,ndotj,"*")/n
> R=length(nidot)
> C=length(ndotj)
> dll=(R-1)*(C-1)
> tmp=as.vector(abs(x-eij))
> tmp=pmax(tmp-0.5,0)
> tmp=matrix(tmp,byrow=TRUE,ncol=C)
> Qc=sum(tmp^2/eij)
> p.value=1-pchisq(Qc,dll)
>
> out=list(estimate=Qc,dll=dll,p.value=p.value,dim=c(R,C),name="Continuity
> adjusted Chi-square")
> return(out)
> }
> else
> { stop("Continuity-adjusted chi-square must be used with
> (2,2)-tables",call.=FALSE) }
> }
>
> tableChisqMH=function(x)
> {
> n=sum(x)
> G2=(n-1)*(tablepearson(x)$estimate^2)
> dll=1
> p.value=1-pchisq(G2,dll)
>
> out=list(estimate=G2,dll=dll,p.value=p.value,dim=dim(x),name="Mantel-Hae
> nszel Chi-square")
> return(out)
>
> }
>
> tableCC=function(x)
> {
> Qp=tableChisq(x)$estimate
> n=sum(x)
> P=sqrt(Qp/(Qp+n))
> m=min(dim(x))
>
> out=list(estimate=P,dim=dim(x),bornes=c(0,sqrt((m-1)/m)),name="Contingen
> cy coefficient")
> return(out)
>
> }
>
> tabletrend=function(x,transpose=FALSE)
> {
> if (any(dim(x)==2))
> {
> if (transpose==TRUE) {
> x=t(x)
> }
>
> if (dim(x)[2]!=2){stop("Cochran-Armitage test for trend must be
> used with a (R,2) table. Use transpose argument",call.=FALSE) }
>
> nidot=apply(x,1,sum)
> n=sum(nidot)
>
> Ri=scores(x,1,"table")
> Rbar=sum(nidot*Ri)/n
>
> s2=sum(nidot*(Ri-Rbar)^2)
> pdot1=sum(x[,1])/n
> T=sum(x[,1]*(Ri-Rbar))/sqrt(pdot1*(1-pdot1)*s2)
> p.value.uni=1-pnorm(abs(T))
> p.value.bi=2*p.value.uni
>
> out=list(estimate=T,dim=dim(x),p.value.uni=p.value.uni,p.value.bi=p.valu
> e.bi,name="Cochran-Armitage test for trend")
> return(out)
>
> }
> else {stop("Cochran-Armitage test for trend must be used with a
> (2,C) or a (R,2) table",call.=FALSE) }
> }
>
>
>
>
>
>
>
>
>
>
>
> Eric Lecoutre
> UCL / Institut de Statistique
> Voie du Roman Pays, 20
> 1348 Louvain-la-Neuve
> Belgium
>
> tel: (+32)(0)10473050
> lecoutre at stat.ucl.ac.be
> http://www.stat.ucl.ac.be/ISpersonnel/lecoutre
>
> If the statistics are boring, then you've got the wrong numbers. -Edward
> Tufte
>
>
> > -----Original Message-----
> > From: r-help-bounces at stat.math.ethz.ch
> > [mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Vito Ricci
> > Sent: jeudi 28 juillet 2005 16:30
> > To: r-help at stat.math.ethz.ch
> > Cc: amelie2000 at gmx.de
> > Subject: Re: [R] Cochran-Armitage-trend-test
> >
> >
> > Hi,
> > see:
> > http://finzi.psych.upenn.edu/R/Rhelp02a/archive/20396.html
> >
> > Regards,
> > Vito
> >
> >
> >
> > amelie2000 at gmx.de wrote:
> >
> > Hi!
> >
> > I am searching for the Cochran-Armitage-trend-test. Is
> > it included in an
> > R-package?
> >
> > Thank you!
> >
> >
> >
> > Diventare costruttori di soluzioni
> > Became solutions' constructors
> >
> > "The business of the statistician is to catalyze
> > the scientific learning process."
> > George E. P. Box
> >
> > "Statistical thinking will one day be as necessary for
> > efficient citizenship as the ability to read and write" H. G. Wells
> >
> > Top 10 reasons to become a Statistician
> >
> > 1. Deviation is considered normal
> > 2. We feel complete and sufficient
> > 3. We are 'mean' lovers
> > 4. Statisticians do it discretely and continuously
> > 5. We are right 95% of the time
> > 6. We can legally comment on someone's posterior distribution
> > 7. We may not be normal, but we are transformable
> > 8. We never have to say we are certain
> > 9. We are honestly significantly different
> > 10. No one wants our jobs
> >
> >
> > Visitate il portale http://www.modugno.it/
> > e in particolare la sezione su Palese
> > http://www.modugno.it/archivio/palesesanto_spirito/
> >
> > ______________________________________________
> > R-help at stat.math.ethz.ch mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide!
> > http://www.R-project.org/posting-guide.html
> >
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
>
>
More information about the R-help
mailing list