| Title: | Kernel Hazard Estimation with Best One-Sided and Double One-Sided Cross-Validation |
|---|---|
| Description: | Local linear hazard estimator and its multiplicatively bias correction, including three bandwidth selection methods: best one-sided cross-validation, double one-sided cross-validation, and standard cross-validation. |
| Authors: | M.L. Gamiz, E. Mammen, M.D. Martinez-Miranda and J.P. Nielsen |
| Maintainer: | Maria Dolores Martinez-Miranda <[email protected]> |
| License: | GPL-2 |
| Version: | 1.1.0 |
| Built: | 2024-12-18 06:49:02 UTC |
| Source: | CRAN |
Local linear hazard estimator and its multiplicatively bias correction, including three bandwidth selection methods: best one-sided cross-validation, double one-sided cross-validation, and standard cross-validation.
| Package: | DOvalidation |
| Type: | Package |
| Version: | 1.1.0 |
| Date: | 2017-10-20 |
| License: | GPL-2 |
M.L. Gamiz, E. Mammen, M.D. Martinez-Miranda and J.P. Nielsen
Maintainer: Maria Dolores Martinez-Miranda <[email protected]>
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years ## Local linear hazard estimator ## and its multiplicatively bias corrected version (MBC) ## with best one-sided cross-validated bandwidths ## Note: use functions b.BO and b.BO.MBC to get these bandwidths ## (48.7 and 14.6, respectively) res.LL<-hazard.LL(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=14.6) res.MBC<-hazard.MBC(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=48.7) plot(ti,res.LL$hLL,main='Hazard estimates',xlab='age',ylab='', type='l',col=4,lwd=2) lines(ti,res.MBC$hMBC,col=2,lwd=2) legend("topleft",bt="n",c("Local linear", "MBC"),col=c(4,2),lwd=2)data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years ## Local linear hazard estimator ## and its multiplicatively bias corrected version (MBC) ## with best one-sided cross-validated bandwidths ## Note: use functions b.BO and b.BO.MBC to get these bandwidths ## (48.7 and 14.6, respectively) res.LL<-hazard.LL(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=14.6) res.MBC<-hazard.MBC(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=48.7) plot(ti,res.LL$hLL,main='Hazard estimates',xlab='age',ylab='', type='l',col=4,lwd=2) lines(ti,res.MBC$hMBC,col=2,lwd=2) legend("topleft",bt="n",c("Local linear", "MBC"),col=c(4,2),lwd=2)
Bandwidth selection for local linear hazard estimation using best one-sided cross-validation
b.BO(grid.b, nb , K = "sextic", type.bo = "Oi", xi, Oi, Ei, wei = "same")b.BO(grid.b, nb , K = "sextic", type.bo = "Oi", xi, Oi, Ei, wei = "same")
grid.b |
Optional. A vector of bandwidths to minimise the cross-validation score. If not specified it will be considered an equally-spaced grid of |
nb |
Optional. The number of bandwidths used to minimise the cross-validation score. If |
K |
Indicates the kernel function to be considered in the local linear hazard estimator. Choose between values |
type.bo |
Choose between |
xi |
Vector of time points where the count data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
wei |
Indicates the weights used in the cross-validation score. Choose between the value |
It is assumed that the data are given as count data i.e. number of occurrences and exposures.
The BO-validated bandwidth is calculated as the minimizer of a cross-validation score with a indirect kernel. If the score is strictly increasing or decreasing then a warning will be shown together with the selected bandwidth (in this case one of the extremes in grid.b, adjusted by the rescaling constant, which is 0.5371 for the Epanechnikov kernel, and 0.5874 for the sextic kernel).
The score is defined with two different weighting functions. This is controlled with the parameter wei. By default wei="exposure" that means that only areas where the exposure is significant contribute to the criterion. Specify wei="same" to allow all time points contribute the same to the criterion (see Gamiz et al. 2017).
bbo |
The onesided cross-validated bandwidth. |
ind.bo |
The position of the best one-sided cross-validated bandwidth into "grid.b". |
cvbo.values |
The values of the cross-validation score for each bandwidth in |
b.grid |
The grid of bandwidths where the score has been evaluated. |
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(20,40,length.out=30) res.bo<-b.BO(grid.b=my.bs,K="sextic",type.bo="Oi",xi=ti,Oi=Oi,Ei=Ei,wei="same") bbo<-res.bo$bbo cvs<-res.bo$cvbo.values plot(my.bs,cvs,main="BO-validation score",xlab="Bandwidth") print(paste("The best one-sided cross-validated bandwidth is:", bbo,sep=" "))data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(20,40,length.out=30) res.bo<-b.BO(grid.b=my.bs,K="sextic",type.bo="Oi",xi=ti,Oi=Oi,Ei=Ei,wei="same") bbo<-res.bo$bbo cvs<-res.bo$cvbo.values plot(my.bs,cvs,main="BO-validation score",xlab="Bandwidth") print(paste("The best one-sided cross-validated bandwidth is:", bbo,sep=" "))
Bandwidth selection for multiplicatively biased corrected local linear hazard estimation using best one-sided cross-validation
b.BO.MBC(grid.b, nb , K = "sextic", type.bo = "Oi", xi, Oi, Ei, wei = "same")b.BO.MBC(grid.b, nb , K = "sextic", type.bo = "Oi", xi, Oi, Ei, wei = "same")
grid.b |
Optional. A vector of bandwidths to minimise the cross-validation score. If not specified it will be considered an equally-spaced grid of |
nb |
Optional. The number of bandwidths used to minimise the cross-validation score. If |
K |
Indicates the kernel function to be considered in the hazard. Choose between values |
type.bo |
Choose between |
xi |
Vector of time points where the count data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
wei |
Indicates the weights used in the cross-validation score. Choose between the value |
The BO-validated bandwidth is calculated as the minimizer of a cross-validation score with a indirect kernel. If the score is strictly increasing or decreasing then a warning will be shown together with the selected bandwidth (in this case one of the extremes in grid.b, adjusted by the rescaling constant, which is 0.5948 for the Epanechnikov kernel, and 0.6501 for the sextic kernel).
The score is defined with two different weighting functions. This is controlled with the parameter wei. By default wei="exposure" that means that only areas where the exposure is significant contribute to the criterion. Specify wei="same" to allow all time points contribute the same to the criterion (see Gamiz et al. 2017).
bbo |
The onesided cross-validated bandwidth. |
ind.bo |
The position of the best one-sided cross-validated bandwidth into "grid.b". |
cvbo.values |
The values of the cross-validation score for each bandwidth in |
b.grid |
The grid of bandwidths where the score has been evaluated. |
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
hazard.MBC,b.OSCV.MBC,b.CV.MBC
data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(50,80,length.out=30) res.bo<-b.BO.MBC(grid.b=my.bs,K="sextic",type.bo = "Oi",xi=ti,Oi=Oi,Ei=Ei,wei="same") bbo<-res.bo$bbo cvs<-res.bo$cvbo.values plot(my.bs,cvs,main="BO-validation score",xlab="Bandwidth") print(paste("The best one-sided cross-validated bandwidth is:", bbo,sep=" "))data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(50,80,length.out=30) res.bo<-b.BO.MBC(grid.b=my.bs,K="sextic",type.bo = "Oi",xi=ti,Oi=Oi,Ei=Ei,wei="same") bbo<-res.bo$bbo cvs<-res.bo$cvbo.values plot(my.bs,cvs,main="BO-validation score",xlab="Bandwidth") print(paste("The best one-sided cross-validated bandwidth is:", bbo,sep=" "))
Bandwidth selection for local linear hazard estimation using least squares cross-validation
b.CV(grid.b, nb , K = "epa", xi, Oi, Ei, wei = "exposure")b.CV(grid.b, nb , K = "epa", xi, Oi, Ei, wei = "exposure")
grid.b |
Optional. A vector of bandwidths to minimise the cross-validation score. If not specified it will be considered an equally-spaced grid of |
nb |
Optional. The number of bandwidths used to minimise the cross-validation score. If |
K |
Indicates the kernel function to be considered in the local linear hazard estimator. Choose between values |
xi |
Vector of time points where the count data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
wei |
Indicates the weights used in the cross-validation score. Choose between the value |
It is assumed that the data are given as count data i.e. number of occurrences and exposures.
If the cross-validation score is strictly increasing or decreasing then a warning will be shown together with the cross-validated bandwidth (in this case one of the extremes in grid.b).
The cross-validation score is defined with two different weighting functions. This is controlled with the parameter wei. By default wei="exposure" that means that only areas where the exposure is significant contribute to the criterion. Specify wei="same" to allow all time points contribute the same to the criterion (see Gamiz et al. 2017).
bcv |
The cross-validated bandwidth. |
ind.cv |
The position of the cross-validated bandwidth into |
cv.values |
The values of the cross-validation score for each bandwidth in |
b.grid |
The grid of bandwidths where the cross-validation score has been evaluated. |
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics, 28, 675-698.
data(UK) Oi<-UK$D Ei<-UK$E ti<-40:110 # time is age and it goes from 40 to 110 years M<-length(ti) my.bs<-seq(1,5,length=50) res.cv<-b.CV(grid.b=my.bs,K="sextic",xi=ti,Oi=Oi,Ei=Ei) bcv<-res.cv$bcv cv.values<-res.cv$cv.values plot(my.bs,cv.values,main="Cross-validation score",xlab="Bandwidth") print(paste("The cross-validated bandwidth is:", bcv,sep=" "))data(UK) Oi<-UK$D Ei<-UK$E ti<-40:110 # time is age and it goes from 40 to 110 years M<-length(ti) my.bs<-seq(1,5,length=50) res.cv<-b.CV(grid.b=my.bs,K="sextic",xi=ti,Oi=Oi,Ei=Ei) bcv<-res.cv$bcv cv.values<-res.cv$cv.values plot(my.bs,cv.values,main="Cross-validation score",xlab="Bandwidth") print(paste("The cross-validated bandwidth is:", bcv,sep=" "))
Bandwidth selection for multiplicatively bias corrected local linear hazard estimation using least squares cross-validation
b.CV.MBC(grid.b, nb , K = "sextic", xi, Oi, Ei, wei = "same")b.CV.MBC(grid.b, nb , K = "sextic", xi, Oi, Ei, wei = "same")
grid.b |
Optional. A vector of bandwidths to minimise the cross-validation score. If not specified it will be considered an equally-spaced grid of |
nb |
Optional. The number of bandwidths used to minimise the cross-validation score. If "grid.b" is provided then the argument "nb" will be ignored (if specified). |
K |
Indicates the kernel function to be considered in the local linear hazard estimator. Choose between values "epa" (for the epanechnikov kernel) or "sextic" (see details of |
xi |
Vector of time points where the count data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point (xi). |
Ei |
Vector with the observed exposure at each time point (xi). |
wei |
Indicates the weights used in the cross-validation score. Choose between the value "exposure" or "same". See details below. |
It is assumed that the data are given as count data i.e. number of occurrences and exposures.
If the cross-validation score is strictly increasing or decreasing then a warning will be shown together with the cross-validated bandwidth (in this case one of the extremes in "grid.b").
The cross-validation score is defined with two different weighting functions. This is controlled with the parameter wei. By default wei="exposure" that means that only areas where the exposure is significant contribute to the criterion. Specify wei="same" to allow all time points contribute the same to the criterion (see Gamiz et al. 2017).
bcv |
The cross-validated bandwidth. |
ind.cv |
The position of the cross-validated bandwidth into "grid.b". |
cv.values |
The values of the cross-validation score for each bandwidth in "grid.b". |
b.grid |
The grid of bandwidths where the cross-validation score has been evaluated. |
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics, 28, 675-698.
data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(50,80,length.out=30) res.cv<-b.CV.MBC(grid.b=my.bs,K="sextic",xi=ti,Oi=Oi,Ei=Ei,wei="same") bcv<-res.cv$bcv cv.values<-res.cv$cv.values plot(my.bs,cv.values,main="Cross-validation score",xlab="Bandwidth") print(paste("The cross-validated bandwidth is:", bcv,sep=" "))data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(50,80,length.out=30) res.cv<-b.CV.MBC(grid.b=my.bs,K="sextic",xi=ti,Oi=Oi,Ei=Ei,wei="same") bcv<-res.cv$bcv cv.values<-res.cv$cv.values plot(my.bs,cv.values,main="Cross-validation score",xlab="Bandwidth") print(paste("The cross-validated bandwidth is:", bcv,sep=" "))
Bandwidth selection for local linear hazard estimation using DO-validation and one-sided (left or right) cross-validation
b.OSCV(grid.b, nb , K = "epa", Ktype = "left", xi, Oi, Ei, wei = "exposure")b.OSCV(grid.b, nb , K = "epa", Ktype = "left", xi, Oi, Ei, wei = "exposure")
grid.b |
Optional. A vector of bandwidths to minimise the cross-validation score. If not specified it will be considered an equally-spaced grid of |
nb |
Optional. The number of bandwidths used to minimise the cross-validation score. If |
K |
Indicates the kernel function to be considered in the local linear hazard estimator. Choose between values |
Ktype |
Choose between |
xi |
Vector of time points where the count data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
wei |
Indicates the weights used in the cross-validation score. Choose between the value |
It is assumed that the data are given as count data i.e. number of occurrences and exposures.
The DO-validated bandwidth is calculated as the average of left- and right- sided cross-validation (see example below).
If the one-sided cross-validation score is strictly increasing or decreasing then a warning will be shown together with the onesided cross-validated bandwidth (in this case one of the extremes in grid.b, adjusted by the rescaling constant, which is 0.5371 for the Epanechnikov kernel, and 0.5874 for the sextic kernel).
The score is defined with two different weighting functions. This is controlled with the parameter wei. By default wei="exposure" that means that only areas where the exposure is significant contribute to the criterion. Specify wei="same" to allow all time points contribute the same to the criterion (see Gamiz et al. 2017).
boscv |
The onesided cross-validated bandwidth. |
ind.oscv |
The position of the one-sided cross-validated bandwidth into |
oscv.values |
The values of the one-sided cross-validation score for each bandwidth in |
b.grid |
The grid of bandwidths where the one-sided cross-validation score has been evaluated. |
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
data(UK) Oi<-UK$D Ei<-UK$E ti<-40:110 # time is age and it goes from 40 to 110 years M<-length(ti) my.bs<-seq(3,10,length=50) ## The left one-sided cross-validated bandwidth res.left<-b.OSCV(grid.b=my.bs,K="sextic",Ktype="left",xi=ti,Oi=Oi,Ei=Ei) bleft<-res.left$boscv ## The right one-sided cross-validated bandwidth res.right<-b.OSCV(grid.b=my.bs,K="sextic",Ktype="right",xi=ti,Oi=Oi,Ei=Ei) bright<-res.right$boscv ## The DO-validated bandwidth bdo<-(bleft+bright)/2 print(paste("DO-validated bandwidth= ", bdo, sep=""))data(UK) Oi<-UK$D Ei<-UK$E ti<-40:110 # time is age and it goes from 40 to 110 years M<-length(ti) my.bs<-seq(3,10,length=50) ## The left one-sided cross-validated bandwidth res.left<-b.OSCV(grid.b=my.bs,K="sextic",Ktype="left",xi=ti,Oi=Oi,Ei=Ei) bleft<-res.left$boscv ## The right one-sided cross-validated bandwidth res.right<-b.OSCV(grid.b=my.bs,K="sextic",Ktype="right",xi=ti,Oi=Oi,Ei=Ei) bright<-res.right$boscv ## The DO-validated bandwidth bdo<-(bleft+bright)/2 print(paste("DO-validated bandwidth= ", bdo, sep=""))
Bandwidth selection for the multiplicatively bias corrected local linear hazard estimation using DO-validation and one-sided (left or right) cross-validation
b.OSCV.MBC(grid.b, nb , K = "sextic", Ktype = "left", xi, Oi, Ei, wei = "same")b.OSCV.MBC(grid.b, nb , K = "sextic", Ktype = "left", xi, Oi, Ei, wei = "same")
grid.b |
Optional. A vector of bandwidths to minimise the cross-validation score. If not specified it will be considered an equally-spaced grid of |
nb |
Optional. The number of bandwidths used to minimise the cross-validation score. If |
K |
Indicates the kernel function to be considered in the hazard estimator. Choose between values |
Ktype |
Choose between |
xi |
Vector of time points where the count data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
wei |
Indicates the weights used in the cross-validation score. Choose between the value |
It is assumed that the data are given as count data i.e. number of occurrences and exposures.
The DO-validated bandwidth is calculated as the average of left- and right- sided cross-validation (see example below).
If the one-sided cross-validation score is strictly increasing or decreasing then a warning will be shown together with the onesided cross-validated bandwidth (in this case one of the extremes in grid.b, adjusted by the rescaling constant, which is 0.5948 for Epanechnikov kernel, and 0.6501 for the sextic kernel).
The score is defined with two different weighting functions. This is controlled with the parameter wei. By default wei="exposure" that means that only areas where the exposure is significant contribute to the criterion. Specify wei="same" to allow all time points contribute the same to the criterion (see Gamiz et al. 2017).
boscv |
The one-sided cross-validated bandwidth. |
ind.oscv |
The position of the one-sided cross-validated bandwidth into |
oscv.values |
The values of the one-sided cross-validation score for each bandwidth in |
b.grid |
The grid of bandwidths where the one-sided cross-validation score has been evaluated. |
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(50,80,length.out=30) ## The left one-sided cross-validated bandwidth res.left<-b.OSCV.MBC(grid.b=my.bs,K="sextic",Ktype="left", xi=ti,Oi=Oi,Ei=Ei,wei="same") bleft<-res.left$boscv ## The right one-sided cross-validated bandwidth res.right<-b.OSCV.MBC(grid.b=my.bs,K="sextic",Ktype="right", xi=ti,Oi=Oi,Ei=Ei,wei="same") bright<-res.right$boscv ## The do-validated bandwidth bdo<-(bleft+bright)/2 print(paste("DO-validated bandwidth= ", bdo, sep=""))data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years my.bs<-seq(50,80,length.out=30) ## The left one-sided cross-validated bandwidth res.left<-b.OSCV.MBC(grid.b=my.bs,K="sextic",Ktype="left", xi=ti,Oi=Oi,Ei=Ei,wei="same") bleft<-res.left$boscv ## The right one-sided cross-validated bandwidth res.right<-b.OSCV.MBC(grid.b=my.bs,K="sextic",Ktype="right", xi=ti,Oi=Oi,Ei=Ei,wei="same") bright<-res.right$boscv ## The do-validated bandwidth bdo<-(bleft+bright)/2 print(paste("DO-validated bandwidth= ", bdo, sep=""))
Aggregate data in the form of occurrences and exposures from individual survival data (possibly right censored and/or left truncated).
discretise.data(Li, Zi, deltai, xi, M)discretise.data(Li, Zi, deltai, xi, M)
Li |
Vector of truncation levels: the datum is registered only if the life time is greater than the truncation level. |
Zi |
Vector of observed life times (Zi=min(Ti,Ci) with Ci censoring value and Ti the true life time). |
deltai |
Vector with non-censoring indicator values (0 if datum is censored, 1 otherwise). |
xi |
Optional. Vector with the grid of time points where the occurrences and exposures should be calculated. If not provide the grid is calculated automatically. |
M |
Optional. A positive scalar used as the grid size. If not provided it is chosen automatically. |
The hazard estimators and bandwidth selectors available in the DOvalidation package work from data aggregated in the form of occurrences and exposures. This function can be used to work with individual survival data in the form (Li,Zi,deltai) – left-truncation level (Li), observed time (Zi) and non-censoring indicator (deltai). If data are not truncated then Li can be chosen as 0.
xi |
Vector with the time grid points. |
Oi |
Vector with the calculated occurrences at the time grid points. |
Ei |
Vector with the calculated exposures at the time grid points. |
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
## A simple example with right censored data Zi<-c(3,6,7,7,8,10,11,11,11,12,13,13,14,16,20,20,22,32,34,36) n<-length(Zi) Li<-deltai<-rep(0,n) ind.cens<-c(1,3,4,8,9,13,14,15,16) deltai[-ind.cens]<-1 ## Obtain the occurrences and exposures for a grid of 6 time points res<-discretise.data(Li,Zi,deltai,M=6) ## Now calculate the local linear hazard estimator hazard.LL(res$xi,res$Oi,res$Ei,res$xi,b=10)## A simple example with right censored data Zi<-c(3,6,7,7,8,10,11,11,11,12,13,13,14,16,20,20,22,32,34,36) n<-length(Zi) Li<-deltai<-rep(0,n) ind.cens<-c(1,3,4,8,9,13,14,15,16) deltai[-ind.cens]<-1 ## Obtain the occurrences and exposures for a grid of 6 time points res<-discretise.data(Li,Zi,deltai,M=6) ## Now calculate the local linear hazard estimator hazard.LL(res$xi,res$Oi,res$Ei,res$xi,b=10)
Mortality data of women in the calendar year 2006 from Denmark. The data were obtained from the Humam Mortality Database. Only ages from 40 to 110 have been included.
data(DK)data(DK)
This data frame contains 71 rows and the following 2 columns.
DDeath counts for women of ages between 40 and 110 during the calendar year 2006.
E"Person-years" lived in the female population during the year 2006 for each age-group (from 40 to 110)
Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Spreeuw, J., Nielsen, J.P. and Jarner, S.F. (2013). A visual test of mixed hazard models, SORT, 37, 149-170.
data(DK)data(DK)
Local linear estimator of the unidimensional hazard (or hazard rate) with natural weighting introduced by Nielsen and Tanggaard (2001).
hazard.LL(xi, Oi, Ei, x, b, K="epa", Ktype="symmetric" , CI=FALSE)hazard.LL(xi, Oi, Ei, x, b, K="epa", Ktype="symmetric" , CI=FALSE)
xi |
Vector of time points where the counts data are given. |
Oi |
Vector with the number of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
x |
Vector (or scalar) with the (time) grid points where the hazard estimator will be evaluated. |
b |
A positive scalar used as the bandwidth. |
K |
Indicates the kernel function to be considered in the estimator. Choose between values |
Ktype |
Indicates the type of kernel to be used. Choose among |
CI |
Logical. If |
The estimator is calculated assuming that the data are given as count data i.e. number of occurrences and exposures.
The function allows to consider two different kernels using the argument K. These are: Epanechnikov, K(u)=.75*(1-u^2)*(abs(u)<1), and sextic K(u)=(3003/2048)*(1-(u)^2)^6)*(abs(u)<1). The argument Ktype will define the usual estimator with whole support kernel as it is defined by K or the one-sided versions using left-sided kernel, 2*K(u)*(u<0), or right-sided kernel 2*K(u)*(u>0). See more details in Gamiz et al. (2016).
x |
Vector (or scalar) with the (time) grid points where the hazard estimator has been evaluated. |
OLL |
Vector with the smoothed occurrences (using the local linear kernel). |
ELL |
Vector with the smoothed exposures (using the local linear kernel). |
hLL |
Vector (or scalar) with the resulting hazard estimates at grid points |
OLL.norm |
Vector with the normalized smoothed occurrences (the smoothing weights are defined as for |
ELL.norm |
Vector with the normalized smoothed exposures (the smoothing weights are defined as for |
CI.inf |
Vector with the lower limits for the 95% confidence intervals. If |
CI.sup |
Vector with the upper limits for the 95% confidence intervals. If |
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
## Calculation of the local linear hazard estimator with do-validated bandwidth. ## The hazard estimator is shown and decomposed into smoothed occurrences and exposures. ## This example is described in Gamiz et al. (2016). data(UK) Oi<-UK$D Ei<-UK$E ti<-40:110 # time is age and it goes from 40 to 110 years M<-length(ti) country<-'UK' bdo<-5.11 resLL.do<-hazard.LL(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=bdo,K="sextic",Ktype="symmetric",CI=TRUE) ## The local linear hazard estimate is hLL.do below hLL.do<-resLL.do$hLL ## The smoothed occurrences and exposures are: ELL.norm.do<-resLL.do$ELL.norm OLL.norm.do<-resLL.do$OLL.norm ## The 95% pointwise confidence intervals based on the asymptotics are hLL.do.inf<-resLL.do$CI.inf hLL.do.sup<-resLL.do$CI.sup # Now we plot the hazard estimator with confidence intervals old.par<-par(mar=c(3,1.5,1.5,1.5),oma=c(2,0.5,0.5,0.2), mgp=c(1.5,0.5,0),cex.axis=1,cex.main=1.5,mfrow=c(3,2)) #hazard estimate tit<-paste(country,"Hazard estimate",sep= ' - ' ) yy<-range(c(hLL.do.inf,hLL.do.sup),na.rm=TRUE) plot(ti,hLL.do,main=tit,xlab='age',ylab='',type='l',lwd=2,ylim=yy) # the confidence bands x1<-ti;x2<-ti[M:1] y1<-hLL.do.sup;y2<-hLL.do.inf[M:1] polygon(c(x1,x2,x1[1]),c(y1,y2,y1[1]),col=gray(0.7),border=FALSE) lines(ti,hLL.do,lty=1,lwd=2,col=1) ## Zooming at the old mortality ind.ages<- -c(1:60) ## only women with ages 100 or higher ti2<-ti[ind.ages];M2<-length(ti2) yy2<-range(c(hLL.do.inf[ind.ages],hLL.do.sup[ind.ages]),na.rm=TRUE) plot(ti2,hLL.do[ind.ages],main=tit,xlab='age',ylab='',type='l', lwd=2,ylim=yy2) # the confidence intervals x1<-ti2;x2<-ti2[M2:1] y1<-hLL.do.sup[ind.ages];hLL.do.inf2<-hLL.do.inf[ind.ages] y2<-hLL.do.inf2[M2:1] polygon(c(x1,x2,x1[1]),c(y1,y2,y1[1]),col=gray(0.7),border=FALSE) lines(ti2,hLL.do[ind.ages],lty=1,lwd=2,col=1) ## We decompose the estimator in the smooth occurrences and exposures # The occurrences with a zoom at old-age mortality yy<-range(OLL.norm.do,na.rm=TRUE) plot(ti,OLL.norm.do,main="Smoothed occurrences",xlab='age',ylab='',type='l', lwd=2,ylim=yy) yy2<-range(OLL.norm.do[ind.ages],na.rm=TRUE) plot(ti2,OLL.norm.do[ind.ages],main="Smoothed occurrences",xlab='age', ylab='',type='l',lwd=2,ylim=yy2) # The exposures with a zoom at old-age mortality yy<-range(ELL.norm.do,na.rm=TRUE) plot(ti,ELL.norm.do,main="Smoothed exposures",xlab='age',ylab='',type='l', lwd=2,ylim=yy) yy2<-range(ELL.norm.do[ind.ages],na.rm=TRUE) plot(ti2,ELL.norm.do[ind.ages],main="Smoothed exposures",xlab='age',ylab='', type='l',lwd=2,ylim=yy2) # Revert the changes made in the graphics options par(old.par)## Calculation of the local linear hazard estimator with do-validated bandwidth. ## The hazard estimator is shown and decomposed into smoothed occurrences and exposures. ## This example is described in Gamiz et al. (2016). data(UK) Oi<-UK$D Ei<-UK$E ti<-40:110 # time is age and it goes from 40 to 110 years M<-length(ti) country<-'UK' bdo<-5.11 resLL.do<-hazard.LL(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=bdo,K="sextic",Ktype="symmetric",CI=TRUE) ## The local linear hazard estimate is hLL.do below hLL.do<-resLL.do$hLL ## The smoothed occurrences and exposures are: ELL.norm.do<-resLL.do$ELL.norm OLL.norm.do<-resLL.do$OLL.norm ## The 95% pointwise confidence intervals based on the asymptotics are hLL.do.inf<-resLL.do$CI.inf hLL.do.sup<-resLL.do$CI.sup # Now we plot the hazard estimator with confidence intervals old.par<-par(mar=c(3,1.5,1.5,1.5),oma=c(2,0.5,0.5,0.2), mgp=c(1.5,0.5,0),cex.axis=1,cex.main=1.5,mfrow=c(3,2)) #hazard estimate tit<-paste(country,"Hazard estimate",sep= ' - ' ) yy<-range(c(hLL.do.inf,hLL.do.sup),na.rm=TRUE) plot(ti,hLL.do,main=tit,xlab='age',ylab='',type='l',lwd=2,ylim=yy) # the confidence bands x1<-ti;x2<-ti[M:1] y1<-hLL.do.sup;y2<-hLL.do.inf[M:1] polygon(c(x1,x2,x1[1]),c(y1,y2,y1[1]),col=gray(0.7),border=FALSE) lines(ti,hLL.do,lty=1,lwd=2,col=1) ## Zooming at the old mortality ind.ages<- -c(1:60) ## only women with ages 100 or higher ti2<-ti[ind.ages];M2<-length(ti2) yy2<-range(c(hLL.do.inf[ind.ages],hLL.do.sup[ind.ages]),na.rm=TRUE) plot(ti2,hLL.do[ind.ages],main=tit,xlab='age',ylab='',type='l', lwd=2,ylim=yy2) # the confidence intervals x1<-ti2;x2<-ti2[M2:1] y1<-hLL.do.sup[ind.ages];hLL.do.inf2<-hLL.do.inf[ind.ages] y2<-hLL.do.inf2[M2:1] polygon(c(x1,x2,x1[1]),c(y1,y2,y1[1]),col=gray(0.7),border=FALSE) lines(ti2,hLL.do[ind.ages],lty=1,lwd=2,col=1) ## We decompose the estimator in the smooth occurrences and exposures # The occurrences with a zoom at old-age mortality yy<-range(OLL.norm.do,na.rm=TRUE) plot(ti,OLL.norm.do,main="Smoothed occurrences",xlab='age',ylab='',type='l', lwd=2,ylim=yy) yy2<-range(OLL.norm.do[ind.ages],na.rm=TRUE) plot(ti2,OLL.norm.do[ind.ages],main="Smoothed occurrences",xlab='age', ylab='',type='l',lwd=2,ylim=yy2) # The exposures with a zoom at old-age mortality yy<-range(ELL.norm.do,na.rm=TRUE) plot(ti,ELL.norm.do,main="Smoothed exposures",xlab='age',ylab='',type='l', lwd=2,ylim=yy) yy2<-range(ELL.norm.do[ind.ages],na.rm=TRUE) plot(ti2,ELL.norm.do[ind.ages],main="Smoothed exposures",xlab='age',ylab='', type='l',lwd=2,ylim=yy2) # Revert the changes made in the graphics options par(old.par)
Local linear estimator of the unidimensional hazard (or hazard rate) with Ramlau-Hansen weighting as was defined by Nielsen and Tanggaard (2001).
hazard.LL.RH(xi , Oi , Ei , x , b , K="epa")hazard.LL.RH(xi , Oi , Ei , x , b , K="epa")
xi |
Vector of time points where the counts data are given. |
Oi |
Vector with the number (counts) of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
x |
Vector (or scalar) with the (time) grid points where the hazard estimator will be evaluated. |
b |
A positive scalar used as the bandwidth. |
K |
Indicates the kernel function to be considered in the estimator. Choose between values |
The estimator is calculated assuming that the data are given as count data i.e. number of occurrences and exposures.
The function allows to consider two different kernels using the argument K. These are: Epanechnikov, K(u)=.75*(1-u^2)*(abs(u)<1), and sextic K(u)=(3003/2048)*(1-(u)^2)^6)*(abs(u)<1).
x |
Vector (or scalar) with the (time) grid points where the hazard estimator has been evaluated. |
hLL |
Vector (or scalar) with the resulting hazard estimates at grid points x. |
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
## This example was described by Gamiz et al. (2016) to analyse the exposure robustness of ## local linear hazards with natural weigthing and Ramlau-Hansen weighting data(Iceland) Ei<-Iceland$E Oi<-Iceland$D xi<-40:110 n<-length(xi) x<-seq(xi[1],xi[n],length=100) ## Hazard estimates with the original data b0<-11.9899 alphaIC17<-hazard.LL.RH(xi,Oi,Ei,x,b=b0,K="sextic")$hLL alLL17<-hazard.LL(xi,Oi,Ei,x,b=b0,K="sextic",Ktype="symmetric")$hLL hi<-Oi/Ei;hi[Ei==0]<-0 print(round(hi[60:71],3)) ## Hazard estimates with the modified data (one change in the exposure) Ei2<-Ei; Ei2[67]<-2/365 alphaIC005<-hazard.LL.RH(xi,Oi,Ei2,x,b=b0,K="sextic")$hLL alLL005<-hazard.LL(xi,Oi,Ei2,x,b=b0,K="sextic",Ktype="symmetric")$hLL ## Figure: Exposure robustness old.par<-par(mfrow=c(2,2)) plot(x[73:100],alphaIC17[73:100],lwd=2,type='l',main='Exposure: 0.17', xlab='',ylab='Ramlau-Hansen weighting') plot(x[73:100],alphaIC005[73:100],lwd=2,type='l',main='Exposure: 0.005', xlab='',ylab='Ramlau-Hansen weighting') plot(x[73:100],alLL17[73:100],lwd=2,type='l',main='Exposure: 0.17', xlab='',ylab='Natural weighting') plot(x[73:100],alLL005[73:100],lwd=2,type='l',main='Exposure: 0.005', xlab='',ylab='Natural weighting') par(old.par)## This example was described by Gamiz et al. (2016) to analyse the exposure robustness of ## local linear hazards with natural weigthing and Ramlau-Hansen weighting data(Iceland) Ei<-Iceland$E Oi<-Iceland$D xi<-40:110 n<-length(xi) x<-seq(xi[1],xi[n],length=100) ## Hazard estimates with the original data b0<-11.9899 alphaIC17<-hazard.LL.RH(xi,Oi,Ei,x,b=b0,K="sextic")$hLL alLL17<-hazard.LL(xi,Oi,Ei,x,b=b0,K="sextic",Ktype="symmetric")$hLL hi<-Oi/Ei;hi[Ei==0]<-0 print(round(hi[60:71],3)) ## Hazard estimates with the modified data (one change in the exposure) Ei2<-Ei; Ei2[67]<-2/365 alphaIC005<-hazard.LL.RH(xi,Oi,Ei2,x,b=b0,K="sextic")$hLL alLL005<-hazard.LL(xi,Oi,Ei2,x,b=b0,K="sextic",Ktype="symmetric")$hLL ## Figure: Exposure robustness old.par<-par(mfrow=c(2,2)) plot(x[73:100],alphaIC17[73:100],lwd=2,type='l',main='Exposure: 0.17', xlab='',ylab='Ramlau-Hansen weighting') plot(x[73:100],alphaIC005[73:100],lwd=2,type='l',main='Exposure: 0.005', xlab='',ylab='Ramlau-Hansen weighting') plot(x[73:100],alLL17[73:100],lwd=2,type='l',main='Exposure: 0.17', xlab='',ylab='Natural weighting') plot(x[73:100],alLL005[73:100],lwd=2,type='l',main='Exposure: 0.005', xlab='',ylab='Natural weighting') par(old.par)
Multiplicatively bias corrected local linear estimator of the unidimensional hazard with natural weighting introduced by Nielsen and Tanggaard (2001).
hazard.MBC(xi, Oi, Ei, x, b, K="sextic", Ktype="symmetric")hazard.MBC(xi, Oi, Ei, x, b, K="sextic", Ktype="symmetric")
xi |
Vector of time points where the counts data are given. |
Oi |
Vector with the number of occurrences observed at each time point ( |
Ei |
Vector with the observed exposure at each time point ( |
x |
Vector (or scalar) with the (time) grid points where the hazard estimator will be evaluated. |
b |
A positive scalar used as the bandwidth. |
K |
Indicates the kernel function to be considered in the estimator. Choose between values |
Ktype |
Indicates the type of kernel to be used. Choose among |
The estimator is calculated assuming that the data are given as count data i.e. number of occurrences and exposures.
The function allows to consider two different kernels using the argument K. These are: Epanechnikov, K(u)=.75*(1-u^2)*(abs(u)<1), and sextic K(u)=(3003/2048)*(1-(u)^2)^6)*(abs(u)<1). The argument Ktype will define the usual estimator with whole support kernel as it is defined by K or the one-sided versions using left-sided kernel, 2*K(u)*(u<0), or right-sided kernel 2*K(u)*(u>0). See more details in Gamiz et al. (2017).
x |
Vector (or scalar) with the (time) grid points where the hazard estimator has been evaluated. |
hMBC |
Vector (or scalar) with the resulting hazard estimates at grid points |
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Martinez-Miranda, M.D. and Nielsen, J.P. (2017). Multiplicative local linear hazard estimation and best one-sided cross-validation. Available at http://arxiv.org/abs/1710.05575
Nielsen, J.P. and Tanggaard, C. (2001). Boundary and bias correction in kernel hazard estimation. Scandinavian Journal of Statistics,28, 675-698.
data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years res<-hazard.MBC(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=48.7) plot(ti,res$hMBC,main='Hazard estimate',xlab='age',ylab='',type='l',lwd=2)data(Iceland) Oi<-Iceland$D Ei<-Iceland$E ti<-40:110 # time is age and it goes from 40 to 110 years res<-hazard.MBC(xi=ti,Oi=Oi,Ei=Ei,x=ti,b=48.7) plot(ti,res$hMBC,main='Hazard estimate',xlab='age',ylab='',type='l',lwd=2)
Mortality data of women in the calendar year 2006 from Iceland. The data were obtained from the Humam Mortality Database. Only ages from 40 to 110 have been included.
data(Iceland)data(Iceland)
This data frame contains 71 rows and the following 2 columns.
DDeath counts for women of ages between 40 and 110 during the calendar year 2006
E"Person-years" lived in the female population during the year 2006 for each age-group (from 40 to 110).
Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Spreeuw, J., Nielsen, J. P. and Jarner, S. F. (2013). A visual test of mixed hazard models, SORT, 37, 149-??170.
data(Iceland)data(Iceland)
Evaluation of the Epanechnikov kernel function
K.epa(u)K.epa(u)
u |
A vector (or scalar) with the evaluation point(s). |
The value of the kernel function at u.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
curve(K.epa,-1.5,1.5,main="Epanechnikov kernel",ylab="K(u)",xlab="u") # The left onesided K.epa.left<-function(u) return(2*K.epa(u)*(u<0)) curve(K.epa.left,-1.5,1.5,main="Left onesided Epanechnikov kernel",ylab="K(u)",xlab="u")curve(K.epa,-1.5,1.5,main="Epanechnikov kernel",ylab="K(u)",xlab="u") # The left onesided K.epa.left<-function(u) return(2*K.epa(u)*(u<0)) curve(K.epa.left,-1.5,1.5,main="Left onesided Epanechnikov kernel",ylab="K(u)",xlab="u")
Evaluation of the Sextic kernel function
K.sextic(u)K.sextic(u)
u |
A vector (or scalar) with the evaluation point(s). |
The value of the kernel function at u.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
curve(K.sextic,-1.5,1.5,main="Sextic kernel",ylab="K(u)",xlab="u") # The left onesided K.sextic.left<-function(u) return(2*K.sextic(u)*(u<0)) curve(K.sextic.left,-1.5,1.5,main="Left onesided sextic kernel",ylab="K(u)",xlab="u")curve(K.sextic,-1.5,1.5,main="Sextic kernel",ylab="K(u)",xlab="u") # The left onesided K.sextic.left<-function(u) return(2*K.sextic(u)*(u<0)) curve(K.sextic.left,-1.5,1.5,main="Left onesided sextic kernel",ylab="K(u)",xlab="u")
Mortality data of women in the calendar year 2006 from United Kingdom. The data were obtained from the Humam Mortality Database. Only ages from 40 to 110 have been included.
data(UK)data(UK)
This data frame contains 71 rows and the following 2 columns.
DDeath counts for women of ages between 40 and 110 during the calendar year 2006.
E"Person-years" lived in the female population during the year 2006 for each age-group (from 40 to 110)
Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Spreeuw, J., Nielsen, J.P. and Jarner, S.F. (2013). A visual test of mixed hazard models, SORT, 37, 149-170.
data(UK)data(UK)
Mortality data of women in the calendar year 2006 from United States. The data were obtained from the Humam Mortality Database. Only ages from 40 to 110 have been included.
data(US)data(US)
This data frame contains 71 rows and the following 2 columns.
DDeath counts for women of ages between 40 and 110 during the calendar year 2006. Some of these numbers are estimates (of population size or numbers of deaths), not actual counts, and therefore may be expressed as non-integers.
E"Person-years" lived in the female population during the year 2006 for each age-group (from 40 to 110)
Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de
Gamiz, M.L., Mammen, E., Martinez-Miranda, M.D. and Nielsen, J.P.(2016). Double one-sided cross-validation of local linear hazards. Journal of the Royal Statistical Society B, 78, 755-779.
Spreeuw, J., Nielsen, J.P. and Jarner, S.F. (2013). A visual test of mixed hazard models, SORT, 37, 149-170.
data(US)data(US)