[R] Cochran-Armitage-trend-test
Eric Lecoutre
lecoutre at stat.ucl.ac.be
Thu Jul 28 17:11:09 CEST 2005
Hi there,
I often do receive some mails about this piece of code regarding
Cochran-Armitage or Mantel Chi square.
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
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>
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>
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>
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>
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