AUC

Description

Compute area under the ROC curve

Usage

auc(ss) 

Arguments

Details

Area under the ROC curve.

Value

A scalar with the AUC.

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods & Applications, in press

See Also

roc, butstrap, maxauc

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

## an important marker 

ro=roc(S2,Ti,delta,u,tt,s,vtimes) 
auc(ro)

## an unrelated marker 

ro=roc(S1,Ti,delta,u,tt,s,vtimes) 
auc(ro)

Bootstrapping AUC

Description

Boostrap the AUC for significance testing and confidence interval calculation

Usage

butstrap(X,etime,status,u=NULL,tt,s,vtimes,auc1,B=50,fc=NULL)

Arguments

Details

This function can be used to resample the AUC. The resulting p-value is obtained after assumption that the resampled AUC is Gaussian. Non-parametric confidence interval is obtained as the 2.5 and 97.5 confidence interval is simply given by a Gaussian approximation.

Value

A list with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

roc, auc, maxauc

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## an unimportant marker 

ro=roc(S1,Ti,delta,u,tt,s,vtimes) 
but=butstrap(S1,Ti,delta,u,tt,s,vtimes,ro)

Bootstrapping NRI

Description

Boostrap the AUC for significance testing and confidence interval calculation

Usage

butstrap.nri(risk1,risk2,etime,status,u,tt,nri1,wh,B=1000)

Arguments

Details

This function can be used to resample the NRI. The resulting p-value is obtained after assumption that the resampled NRI is Gaussian. Non-parametric confidence interval is obtained as the 2.5 and 97.5 confidence interval is simply given by a Gaussian approximation.

Value

A list with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

nri

Examples

# parameters
n=25
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=1000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

risk1=apply(S1[,1:s],1,sum)
risk1=(risk1-min(risk1))/(max(risk1)-min(risk1))
risk2=apply(S2[,1:s],1,sum)
risk2=(risk2-min(risk2))/(max(risk2)-min(risk2))
butstrap.nri(risk1,risk2,Ti,delta,u,tt,nri(risk1,risk2,Ti,delta,u,tt)$nri,wh=1,B=500)

Bootstrapping AUC

Description

Boostrap the AUC for significance testing and confidence interval calculation

Usage

butstrap.s(X,etime,status,u=NULL,tt,s,vtimes,auc1,B=50,fc=NULL)

Arguments

Details

This function can be used to resample the AUC. The resulting p-value is obtained after assumption that the resampled AUC is Gaussian. Non-parametric confidence interval is obtained as the 2.5 and 97.5 confidence interval is simply given by a Gaussian approximation.

Value

A list with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

roc, auc, maxauc

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=sample(3,n,replace=TRUE)
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## an unimportant marker 

ro=roc.s(S1,Ti,delta,u,tt,s,vtimes) 
but=butstrap.s(S1,Ti,delta,u,tt,s,vtimes,ro)

Optimal Score

Description

Compute optimal score for AUC

Usage

maxauc(X,etime,status,u=NULL,tt,s,vtimes,fc=NULL)

Arguments

Details

This function can be used to find an optimal linear combination of p scores/biomarkers repeatedly measured over time. The resulting score is optimal as it maximizes the AUC among all possible linear combinations. The first biomarker in array X plays a special role, as by default its coefficient is unitary.

Value

A list with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

auc, butstrap, maxauc

Examples

# parameters
n=25
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=500
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

X=array(NA,c(2,nrow(S1),ncol(S1)))
X[1,,]=round(S2) #fewer different values, quicker computation 
X[2,,]=S1

sc=maxauc(X,Ti,delta,u,tt,s,vtimes)

# beta coefficients

sc$beta

# final score (X[1,,]+X[2,,]*sc$beta[1]+...+X[p,,]*sc$beta[p-1])

sc$score

Optimal Score

Description

Compute optimal score for AUC

Usage

maxauc.s(X,etime,status,u=NULL,tt,s,vtimes,fc=NULL)

Arguments

Details

This function can be used to find an optimal linear combination of p scores/biomarkers repeatedly measured over time. The resulting score is optimal as it maximizes the AUC among all possible linear combinations. The first biomarker in array X plays a special role, as by default its coefficient is unitary.

Value

A list with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

auc, butstrap, maxauc

Examples

# parameters
n=20
tt=3
Tmax=10
u=1.5
s=sample(3,n,replace=TRUE)
vtimes=c(0,1,2,5)

# generate data 

ngrid=500
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

X=array(NA,c(2,nrow(S1),ncol(S1)))
X[1,,]=round(S2) #fewer different values, quicker computation 
X[2,,]=S1

sc=maxauc.s(X,Ti,delta,u,tt,s,vtimes)

# beta coefficients

sc$beta

# final score (X[1,,]+X[2,,]*sc$beta[1]+...+X[p,,]*sc$beta[p-1])

sc$score

NRI

Description

Compute NRI

Usage

nri(risk1, risk2, etime,status,u,tt)

Arguments

Details

This function gives the continuous NRI to compare two risk measures.

Value

A list with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

butstrap.nri

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

risk1=apply(S1[,1:s],1,sum)
risk1=(risk1-min(risk1))/(max(risk1)-min(risk1))
risk2=apply(S2[,1:s],1,sum)
risk2=(risk2-min(risk2))/(max(risk2)-min(risk2))
nri(risk1,risk2,Ti,delta,u,tt)

AUC as a function of time

Description

Compute area under the ROC curve for several values of time horizon

Usage

plotAUC(X,etime,status,u=NULL,tt,s,vtimes,fc=NULL,plot=TRUE)

Arguments

Details

Area under the ROC curve is computed for each value of the vector tt. The resulting vector is returned. If plot=TRUE (which is the default) also a plot of tt vs AUC is displayed.

Value

A vector with AUCs

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods & Applications, in press

See Also

roc, butstrap, auc

Examples

# parameters
n=25
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=1000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

## an important marker 

aucs=plotAUC(S2,Ti,delta,u,seq(2,5,length=5),s,vtimes) 

AUC as a function of time

Description

Compute area under the ROC curve for several values of the time horizon

Usage

plotAUC.s(X,etime,status,u=NULL,tt,s,vtimes,fc=NULL,plot=TRUE)

Arguments

Details

Area under the ROC curve is computed for each value of the vector tt. The resulting vector is returned. If plot=TRUE (which is the default) also a plot of tt vs AUC is displayed.

Value

A vector with AUCs

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods & Applications, in press

See Also

roc.s, butstrap.s, auc

Examples

# parameters
n=25
tt=3
Tmax=10
u=1.5
s=sample(3,n,replace=TRUE)
vtimes=c(0,1,2,5)

# generate data 

ngrid=1000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

## an important marker 

aucs=plotAUC.s(S2,Ti,delta,u,seq(2,5,length=5),s,vtimes) 

Plot ROC

Description

Plot the ROC curve

Usage

plotROC(ro, add=FALSE, col=NULL)

Arguments

Details

Plots the area under the ROC curve.

Value

A plot or a new line in an open plot.

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods & Applications, in press

See Also

roc, roc.s, auc

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

## an important marker 

ro=roc(S2,Ti,delta,u,tt,s,vtimes) 
plotROC(ro)

## an unrelated marker 

ro=roc(S1,Ti,delta,u,tt,s,vtimes) 
plotROC(ro)

ROC curve

Description

Compute ROC curve

Usage

roc(X,etime,status,u=NULL,tt,s,vtimes,fc=NULL)

Arguments

Details

ROC curve is defined as the curve given by (1-specificities, sensitivities). Here these are obtained for a time-dependent multiply-measured marker are defined as

Se(t,c,s,u) = Pr(f_c(X(t_1),X(t_2),...,X(t_s_i))| u <= T <= t),

and

Sp(t,c,s,u) = 1-Pr(f_c(X(t_1),X(t_2),...,X(t_s_i)) | T > t)

for some fixed f_c, where c is a cutoff. The default for f_c is that a positive diagnosis is given as soon as any measurement among the s considered is above the threshold.

Value

A matrix with the following columns:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

auc, butstrap, maxauc

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=2
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

## an important marker 

ro=roc(S2,Ti,delta,u,tt,s,vtimes) 
plot(ro,type="l",col="red")
abline(a=0,b=1)

## an unrelated marker 

ro=roc(S1,Ti,delta,u,tt,s,vtimes) 
plot(ro,type="l",col="red")
abline(a=0,b=1)

ROC curve

Description

Compute ROC curve

Usage

roc.s(X,etime,status,u=NULL,tt,s,vtimes,fc=NULL)

Arguments

Details

ROC curve is defined as the curve given by (1-specificities, sensitivities). Here these are obtained for a time-dependent multiply-measured marker are defined as

Se(t,c,s,u) = Pr(f_c(X(t_1),X(t_2),...,X(t_s_i))| u <= T <= t),

and

Sp(t,c,s,u) = 1-Pr(f_c(X(t_1),X(t_2),...,X(t_s_i)) | T > t)

for some fixed f_c, where c is a cutoff. The default for f_c is that a positive diagnosis is given as soon as any measurement among the s considered is above the threshold.

Value

A matrix with the following columns:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

auc, butstrap, maxauc

Examples

# parameters
n=100
tt=3
Tmax=10
u=1.5
s=sample(3,n,replace=TRUE)
vtimes=c(0,1,2,5)

# generate data 

ngrid=5000
ts=seq(0,Tmax,length=ngrid)
X2=matrix(rnorm(n*ngrid,0,0.1),n,ngrid)
for(i in 1:n) {
sa=sample(ngrid/6,1)
vals=sample(3,1)-1
X2[i,1:sa[1]]=vals[1]+X2[i,1:sa[1]]
X2[i,(sa[1]+1):ngrid]=vals[1]+sample(c(-2,2),1)+X2[i,(sa[1]+1):ngrid]
}

S1=matrix(sample(4,n,replace=TRUE),n,length(vtimes))
S2=matrix(NA,n,length(vtimes))

S2[,1]=X2[,1]

for(j in 2:length(vtimes)) {
tm=which.min(abs(ts-vtimes[j]))
S2[,j]=X2[,tm]}

cens=runif(n)
ripart=1-exp(-0.01*apply(exp(X2),1,cumsum)*ts/1:ngrid)

Ti=rep(NA,n)
for(i in 1:n) {
Ti[i]=ts[which.min(abs(ripart[,i]-cens[i]))]
}

cens=runif(n,0,Tmax*2)
delta=ifelse(cens>Ti,1,0)
Ti[cens<Ti]=cens[cens<Ti]

## 

## an important marker 

ro=roc.s(S2,Ti,delta,u,tt,s,vtimes) 
plot(ro,type="l",col="red")
abline(a=0,b=1)

## an unrelated marker 

ro=roc.s(S1,Ti,delta,u,tt,s,vtimes) 
plot(ro,type="l",col="red")
abline(a=0,b=1)

Sensitivity and Specificity

Description

Compute sensitivity and specificity

Usage

sensspec(X,etime,status,u=NULL,tt,s,vtimes,cutoff=0,fc=NULL)

Arguments

Details

Sensitivity and specificities for a time-dependent multiply-measured marker are defined as

Se(t,c,s,u) = Pr(f_c(X(t_1),X(t_2),...,X(t_s_i))| u <= T <= t),

and

Sp(t,c,s,u) = 1-Pr(f_c(X(t_1),X(t_2),...,X(t_s_i)) | T > t)

for some fixed f_c, where c is a cutoff. The default for f_c is that a positive diagnosis is given as soon as any measurement among the s considered is above the threshold.

Value

A vector with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

roc, auc, butstrap, maxauc

Sensitivity and Specificity

Description

Compute sensitivity and specificity

Usage

sensspec.s(X,etime,status,u=NULL,tt,s,vtimes,cutoff=0,fc=NULL)

Arguments

Details

Sensitivity and specificities for a time-dependent multiply-measured marker are defined as

Se(t,c,s,u) = Pr(f_c(X(t_1),X(t_2),...,X(t_s_i))| u <= T <= t),

and

Sp(t,c,s,u) = 1-Pr(f_c(X(t_1),X(t_2),...,X(t_s_i)) | T > t)

for some fixed f_c, where c is a cutoff. The default for f_c is that a positive diagnosis is given as soon as any measurement among the s considered is above the threshold.

Value

A vector with the following elements:

Author(s)

Alessio Farcomeni alessio.farcomeni@uniroma1.it

References

Barbati, G. and Farcomeni, A. (2017) Prognostic assessment of repeatedly measured time-dependent biomarkers, with application to dilated cardiomuopathy, Statistical Methods \& Applications, in press

See Also

roc, auc, butstrap, maxauc