prediction: Prediction probabilities for Cox proportional hazard, Shared, Joint frailty models, Joint models for longitudinal data and a terminal event and Trivariate joint model for longitudinal data, recurrent events and a terminal event (linear and non-linear).

Description

For Cox proportional hazard model

A predictive probability of event between t and horizon time t+w, with w the window of prediction.

For Gamma Shared Frailty model for clustered (not recurrent) events

Two kinds of predictive probabilities can be calculated:

- a conditional predictive probability of event between t and horizon time t+w, i.e. given a specific group

- a marginal predictive probability of event between t and horizon time t+w, i.e. averaged over the population

For Gaussian Shared Frailty model for clustered (not recurrent) events

Two kinds of predictive probabilities can be calculated:

- a conditional predictive probability of event between t and horizon time t+w, i.e. given a specific group and given a specific Gaussian random effect \(\eta\)

- a marginal predictive probability of event between t and horizon time t+w, i.e. averaged over the population

For Gamma Shared Frailty model for recurrent events

Two kinds of predictive probabilities can be calculated:

- A marginal predictive probability of event between t and horizon time t+w, i.e. averaged over the population.

- a conditional predictive probability of event between t and horizon time t+w, i.e. given a specific individual.

This prediction method is the same as the conditional gamma prediction method applied for clustered events (see formula \(P\) before).

For Gaussian Shared Frailty model for recurrent events

Two kinds of predictive probabilities can be calculated:

- A marginal predictive probability of event between t and horizon time t+w, i.e. averaged over the population.

- a conditional predictive probability of event between t and horizon time t+w, i.e. given a specific individual.

This prediction method is the same as the conditional Gaussian prediction method applied for clustered events (see formula \(P\) before).

It is possible to compute all these predictions in two ways on a scale of times : - either you want a cumulative probability of developing the event between t and t+w (with t fixed, but with a varying window of prediction w); - either you want at a specific time the probability to develop the event in the next w (ie, for a varying prediction time t, but for a fixed window of prediction). See Details.

For Joint Frailty model

Prediction for two types of event can be calculated : for a terminal event or for a new recurrent event, knowing patient's characteristics.

- Prediction of death knowing patients' characteristics :

It is to predict the probability of death in a specific time window given the history of patient i before the time of prediction t. The history H, (l=1,2) is the information on covariates before time t, but also the number of recurrences and the time of occurences. Three types of marginal probabilities are computed:

- a prediction of death between t and t+w given that the patient had exactly J recurrences (H) before t

- a prediction of death between t and t+w given that the patient had at least J recurrences (H) before t

- a prediction of death between t and t+w considering the recurrence history only in the parameters estimation. It corresponds to the average probability of death between t and t+w for a patient with these given characteristics.

- Prediction of risk of a new recurrent event knowing patients' characteristics :

It is to predict the probability of a new recurrent event in a specific time window given the history of patient i before the time of prediction t. The history H is the information on covariates before time t, but also the number of recurrences and the time of occurences. The marginal probability computed is a prediction of a new recurrent event between t and t+w given that the patient had exactly J recurrences (\(H\)) before t:

It is possible to compute all these predictions in two ways : - either you want a cumulative probability of developing the event between t and t+w (with t fixed, but with a varying window of prediction w); - either you want at a specific time the probability to develop the event in the next w (ie, for a varying prediction time t, but for a fixed window of prediction). See Details.

With Gaussian frailties (\(\eta\)), the same expressions are used but with u replaced by \(exp\)(J\eqneta) and \(g(\eta)\) corresponds to the Gaussian distribution.

For Joint Nested Frailty models

Prediction of the probability of developing a terminal event between t and t+w for subject i who survived by time t based on the visiting and disease histories of their own and other family members observed by time t.

Let (Y(t)) be the history of subject i in family f, before time t, which includes all the recurrent events and covariate information. For disease history, let T(t) = min(T,t) be the observed time to an event before t ; \(\delta\)(t) the disease indicator by time t and X(t) the covariate information observed up to time t. We define the family history of subject i in family f by

which includes the visiting and disease history of all subjects except for subject i in family f as well as their covariate information by time t.

The prediction probability can be written as :

For Joint models for longitudinal data and a terminal event

The predicted probabilities are calculated in a specific time window given the history of biomarker measurements before the time of prediction t ((t)). The probabilities are conditional also on covariates before time t and that the subject was at risk at t. The marginal predicted probability of the terminal event is

These probabilities can be calculated in several time points with fixed time of prediction t and varying window w or with fixed window w and varying time of prediction t. See Details for an example of how to construct time windows.

For Trivariate joint models for longitudinal data, recurrent events and a terminal event

The predicted probabilities are calculated in a specific time window given the history of biomarker measurements (t) and recurrences H (complete history of recurrences with known J number of observed events) before the time of prediction t. The probabilities are conditional also on covariates before time t and that the subject was at risk at t. The marginal predicted probability of the terminal event is

The biomarker history can be represented using a linear (trivPenal) or non-linear mixed-effects model (trivPenalNL).

Usage

prediction(fit, data, data.Longi, t, window, event="Both", conditional =
FALSE, MC.sample=0, individual)

Arguments

fit

A frailtyPenal, jointPenal, longiPenal, trivPenal or trivPenalNL object.

data

Data frame for the prediction. See Details.

data.Longi

Data frame for the prediction used for joint models with longitudinal data. See Details.

Time or vector of times for prediction.

window

Window or vector of windows for prediction.

event

Only for joint and shared models. The type of event you want to predict : "Terminal" for a terminal event, "Recurrent" for a recurrent event or "Both". Default value is "Both". For joint nested model, only 'Terminal' is allowed. In a shared model, if you want to predict a new recurrent event then the argument "Recurrent" should be use. If you want to predict a new event from clustered data, do not use this option.

conditional

Only for prediction method applied on shared models. Provides distinction between the conditional and marginal prediction methods. Default is FALSE.

MC.sample

Number of samples used to calculate confidence bands with a Monte-Carlo method (with a maximum of 1000 samples). If MC.sample=0 (default value), no confidence intervals are calculated.

individual

Only for joint nested model. Vector of individuals (of the same family) you want to make prediction.

Value

The following components are included in a 'predFrailty' object obtained by using prediction function for Cox proportional hazard and shared frailty model.

npred

Number of individual predictions

x.time

A vector of prediction times of interest (used for plotting predictions): vector of prediction times t if fixed window. Otherwise vector of prediction times t+w

window

Prediction window or vector of prediction windows

pred

Predictions estimated for each profile

icproba

Logical value. Were confidence intervals estimated ?

predLow

Lower limit of Monte-Carlo confidence interval for each prediction

predHigh

Upper limit of Monte-Carlo confidence interval for each prediction

type

Type of prediction probability (marginal or conditional)

group

For conditional probability, the list of group on which you make predictions

The following components are included in a 'predJoint' object obtained by using prediction function for joint frailty model.

npred

Number of individual predictions

x.time

A vector of prediction times of interest (used for plotting predictions): vector of prediction times t if fixed window. Otherwise vector of prediction times t+w

window

Prediction window or vector of prediction windows

group

Id of each patient

pred1

Estimation of probability of type 1: exactly j recurrences

pred2

Estimation of probability of type 2: at least j recurrences

pred3

Estimation of probability of type 3

pred1_rec

Estimation of prediction of relapse

icproba

Logical value. Were confidence intervals estimated ?

predlow1

Lower limit of Monte-Carlo confidence interval for probability of type 1

predhigh1

Upper limit of Monte-Carlo confidence interval for probability of type 1

predlow2

Lower limit of Monte-Carlo confidence interval for probability of type 2

predhigh2

Upper limit of Monte-Carlo confidence interval for probability of type 2

predlow3

Lower limit of Monte-Carlo confidence interval for probability of type 3

predhigh3

Upper limit of Monte-Carlo confidence interval for probability of type 3

predhigh1_rec

Upper limit of Monte-Carlo confidence interval for prediction of relapse

predlow1_rec

Lower limit of Monte-Carlo confidence interval for prediction of relapse

The following components are included in a 'predLongi' object obtained by using prediction function for joint models with longitudinal data.

npred

Number of individual predictions

x.time

A vector of prediction times of interest (used for plotting predictions): vector of prediction times t if fixed window. Otherwise vector of prediction times t+w

window

Prediction window or vector of prediction windows

group

Id of each patient

pred

Estimation of probability

icproba

Logical value. Were confidence intervals estimated?

predLow

Lower limit of Monte-Carlo confidence intervals

predHigh

Upper limit of Monte-Carlo confidence intervals

trivariate

Logical value. Are the prediction calculated from the trivariate model?

Details

To compute predictions with a prediction time t fixed and a variable window:

prediction(fit, datapred, t=10, window=seq(1,10,by=1))

Otherwise, you can have a variable prediction time and a fixed window.

prediction(fit, datapred, t=seq(10,20,by=1), window=5)

Or fix both prediction time t and window.

prediction(fit, datapred,
t=10, window=5)

The data frame building is an important step. It will contain profiles of patient on which you want to do predictions. To make predictions on a Cox proportional hazard or a shared frailty model, only covariates need to be included. You have to distinguish between numerical and categorical variables (factors). If we fit a shared frailty model with two covariates sex (factor) and age (numeric), here is the associated data frame for three profiles of prediction.

 datapred <- data.frame(sex=0,age=0) datapred$sex <-
as.factor(datapred$sex) levels(datapred$sex)<- c(1,2) datapred[1,] <-
c(1,40) # man, 40 years old datapred[2,] <- c(2,45) # woman, 45 years old
datapred[3,] <- c(1,60) # man, 60 years old

Time-dependent covariates: In the context of time-dependent covariate, the last previous value of the covariate is used before the time t of prediction.

It should be noted, that in a data frame for both marginal and conditional prediction on a shared frailty model for clustered data, the group must be specified. In the case of marginal predictions this can be any number as it does not influence predictions. However, for conditional predictions, the group must be also included in the data set used for the model fitting. The conditional predictions apply the empirical Bayes estimate of the frailty from the specified cluster. Here, three individuals belong to group 5.

 datapred <- data.frame(group=0, sex=0,age=0) datapred$sex <-
as.factor(datapred$sex) levels(datapred$sex)<- c(1,2) datapred[1,] <-
c(5,1,40) # man, 40 years old (cluster 5) datapred[2,] <- c(5,2,45) # woman,
45 years old (cluster 5) datapred[3,] <- c(5,1,60) # man, 60 years old
(cluster 5)

To use the prediction function on joint frailty models and trivariate joint models, the construction will be a little bit different. In these cases, the prediction for the terminal event takes into account covariates but also history of recurrent event times for a patient. You have to create a data frame with the relapse times, the indicator of event, the cluster variable and the covariates. Relapses occurring after the prediction time may be included but will be ignored for the prediction. A joint model with calendar-timescale need to be fitted with Surv(start,stop,event), relapse times correspond to the "stop" variable and indicators of event correspond to the "event" variable (if event=0, the relapse will not be taken into account). For patients without relapses, all the values of "event" variable should be set to 0. Finally, the same cluster variable name needs to be in the joint model and in the data frame for predictions ("id" in the following example). For instance, we observe relapses of a disease and fit a joint model adjusted for two covariates sex (1:male 2:female) and chemo (treatment by chemotherapy 1:no 2:yes). We describe 3 different profiles of prediction all treated by chemotherapy: 1) a man with four relapses at 100, 200, 300 and 400 days, 2) a man with only one relapse at 1000 days, 3) a woman without relapse.

 datapred <- data.frame(time=0,event=0,id=0,sex=0,chemo=0)
datapred$sex <- as.factor(datapred$sex) levels(datapred$sex) <- c(1,2)
datapred$chemo <- as.factor(datapred$chemo) levels(datapred$chemo) <- c(1,2)
datapred[1,] <- c(100,1,1,1,2) # first relapse of the patient 1 datapred[2,]
<- c(200,1,1,1,2) # second relapse of the patient 1 datapred[3,] <-
c(300,1,1,1,2) # third relapse of the patient 1 datapred[4,] <-
c(400,1,1,1,2) # fourth relapse of the patient 1 datapred[5,] <-
c(1000,1,2,1,2) # one relapse at 1000 days for patient 2 datapred[6,] <-
c(100,0,3,2,2) # patient 3 did not relapse

The data can also be the dataset used to fit the joint model. In this case, you will obtain as many prediction rows as patients.

Finally, for the predictions using joint models for longitudinal data and a terminal event and trivariate joint models, a data frame with the history of the biomarker measurements must be provided. It must include data on measurements (values and time points), cluster variable and covariates. Measurements taken after the prediction time may be included but will be ignored for the prediction. The same cluster variable name must be in the data frame, in the data frame used for the joint model and in the data frame with the recurrent event and terminal event times. For instance, we observe two patients and each one had 5 tumor size measurements (patient 1 had an increasing tumor size and patient 2, decreasing). The joint model used for the predictions was adjusted on sex (1: male, 2: female), treatment (1: sequential arm, 2: combined arm), WHO baseline performance status (1: 0 status, 2: 1 status, 3: 2 status) and previous resection of the primate tumor (0: no, 1: yes). The data frame for the biomarker measurements can be:

 datapredj_longi <- data.frame(id = 0, year = 0, tumor.size =
0, treatment = 0, age = 0, who.PS = 0, prev.resection = 0)
datapredj_longi$treatment <- as.factor(datapredj_longi$treatment)
levels(datapredj_longi$treatment) <- 1:2 datapredj_longi$age <-
as.factor(datapredj_longi$age) levels(datapredj_longi$age) <- 1:3
datapredj_longi$who.PS <- as.factor(datapredj_longi$who.PS)
levels(datapredj_longi$who.PS) <- 1:3 datapredj_longi$prev.resection <-
as.factor (datapredj_longi$prev.resection)
levels(datapredj_longi$prev.resection) <- 1:2 # patient 1: increasing tumor
size datapredj_longi[1,] <- c(1, 0,1.2 ,2,1,1,1) datapredj_longi[2,] <-
c(1,0.3,1.4,2,1,1,1) datapredj_longi[3,] <- c(1,0.6,1.9,2,1,1,1)
datapredj_longi[4,] <- c(1,0.9,2.5,2,1,1,1) datapredj_longi[5,] <-
c(1,1.5,3.9,2,1,1,1)

# patient 2: decreasing tumor size datapredj_longi[6,] <- c(2, 0,1.2 ,2,1,1,1) datapredj_longi[7,] <- c(2,0.3,0.7,2,1,1,1) datapredj_longi[8,] <- c(2,0.5,0.3,2,1,1,1) datapredj_longi[9,] <- c(2,0.7,0.1,2,1,1,1) datapredj_longi[10,] <- c(2,0.9,0.1,2,1,1,1)

References

A. Krol, L. Ferrer, JP. Pignon, C. Proust-Lima, M. Ducreux, O. Bouche, S. Michiels, V. Rondeau (2016). Joint Model for Left-Censored Longitudinal Data, Recurrent Events and Terminal Event: Predictive Abilities of Tumor Burden for Cancer Evolution with Application to the FFCD 2000-05 Trial. Biometrics 72(3) 907-16.

A. Mauguen, B. Rachet, S. Mathoulin-Pelissier, G. MacGrogan, A. Laurent, V. Rondeau (2013). Dynamic prediction of risk of death using history of cancer recurrences in joint frailty models. Statistics in Medicine, 32(30), 5366-80.

V. Rondeau, A. Laurent, A. Mauguen, P. Joly, C. Helmer (2015). Dynamic prediction models for clustered and interval-censored outcomes: investigating the intra-couple correlation in the risk of dementia. Statistical Methods in Medical Research

Examples

Run this code

# NOT RUN {

# }
# NOT RUN {
#####################################################
#### prediction on a COX or SHARED frailty model ####
#####################################################

data(readmission)
#-- here is a generated cluster (31 clusters of 13 subjects)
readmission <- transform(readmission,group=id%%31+1)

#-- we compute predictions of death
#-- we extract last row of each subject for the time of death
readmission <- aggregate(readmission,by=list(readmission$id),
                         FUN=function(x){x[length(x)]})[,-1]

##-- predictions on a Cox proportional hazard model --##
cox <- frailtyPenal(Surv(t.stop,death)~sex+dukes,
n.knots=10,kappa=10000,data=readmission)

#-- construction of the data frame for predictions
datapred <- data.frame(sex=0,dukes=0)
datapred$sex <- as.factor(datapred$sex)
levels(datapred$sex)<- c(1,2)
datapred$dukes <- as.factor(datapred$dukes)
levels(datapred$dukes)<- c(1,2,3)
datapred[1,] <- c(1,2) # man, dukes 2
datapred[2,] <- c(2,3) # woman, dukes 3

#-- prediction of death for two patients between 100 and 100+w,
#-- with w in (50,100,...,1900)
pred.cox <- prediction(cox,datapred,t=100,window=seq(50,1900,50))
plot(pred.cox)

#-- prediction of death for two patients between t and t+400,
#-- with t in (100,150,...,1500)
pred.cox2 <- prediction(cox,datapred,t=seq(100,1500,50),window=400)
plot(pred.cox2)

##-- predictions on a shared frailty model for clustered data --##
sha <- frailtyPenal(Surv(t.stop,death)~cluster(group)+sex+dukes,
n.knots=10,kappa=10000,data=readmission)

#-- marginal prediction
# a group must be specified but it does not influence the results 
# in the marginal predictions setting
datapred$group[1:2] <- 1
pred.sha.marg <- prediction(sha,datapred,t=100,window=seq(50,1900,50))
plot(pred.sha.marg)

#-- conditional prediction, given a specific cluster (group=5)
datapred$group[1:2] <- 5
pred.sha.cond <- prediction(sha,datapred,t=100,window=seq(50,1900,50),
                            conditional = TRUE)
plot(pred.sha.cond)

##-- marginal prediction of a recurrent event, on a shared frailty model
data(readmission)

datapred <- data.frame(t.stop=0,event=0,id=0,sex=0,dukes=0)
datapred$sex <- as.factor(datapred$sex)
levels(datapred$sex)<- c(1,2)
datapred$dukes <- as.factor(datapred$dukes)
levels(datapred$dukes)<- c(1,2,3)

datapred[1,] <- c(100,1,1,1,2) #man, dukes 2, 3 recurrent events
datapred[2,] <- c(200,1,1,1,2) 
datapred[3,] <- c(300,1,1,1,2) 
datapred[4,] <- c(350,0,2,1,2) #man, dukes 2  0 recurrent event

#-- Shared frailty model with gamma distribution
sha <- frailtyPenal(Surv(t.stop,event)~cluster(id)+sex+dukes,n.knots=10,
kappa=10000,data=readmission)
pred.sha.rec.marg <- prediction(sha,datapred,t=200,window=seq(50,1900,50),
event='Recurrent',MC.sample=100)

plot(pred.sha.rec.marg,conf.bands=TRUE)

##-- conditional prediction of a recurrent event, on a shared frailty model
pred.sha.rec.cond <- prediction(sha,datapred,t=200,window=seq(50,1900,50),
event='Recurrent',conditional = TRUE,MC.sample=100)

plot(pred.sha.rec.cond,conf.bands=TRUE)
#####################################################
######## prediction on a JOINT frailty model ########
#####################################################

data(readmission)

##-- predictions of death on a joint model --##
joi <- frailtyPenal(Surv(t.start,t.stop,event)~cluster(id)
+sex+dukes+terminal(death),formula.terminalEvent=~sex
+dukes,data=readmission,n.knots=10,kappa=c(100,100),recurrentAG=TRUE)

#-- construction of the data frame for predictions
datapredj <- data.frame(t.stop=0,event=0,id=0,sex=0,dukes=0)
datapredj$sex <- as.factor(datapredj$sex)
levels(datapredj$sex) <- c(1,2)
datapredj$dukes <- as.factor(datapredj$dukes)
levels(datapredj$dukes) <- c(1,2,3)
datapredj[1,] <- c(100,1,1,1,2)
datapredj[2,] <- c(200,1,1,1,2)
datapredj[3,] <- c(300,1,1,1,2)
datapredj[4,] <- c(400,1,1,1,2)
datapredj[5,] <- c(380,1,2,1,2)

#-- prediction of death between 100 and 100+500 given relapses
pred.joint0 <- prediction(joi,datapredj,t=100,window=500,event = "Terminal")
print(pred.joint0)

#-- prediction of death between 100 and 100+w given relapses 
# (with confidence intervals)
pred.joint <- prediction(joi,datapredj,t=100,window=seq(50,1500,50),
event = "Terminal",MC.sample=100)
plot(pred.joint,conf.bands=TRUE)
# each y-value of the plot corresponds to the prediction between [100,x]

#-- prediction of death between t and t+500 given relapses
pred.joint2 <- prediction(joi,datapredj,t=seq(100,1000,50),
window=500,event = "Terminal")
plot(pred.joint2)
# each y-value of the plot corresponds to the prediction between [x,x+500], 
#or in the next 500

#-- prediction of relapse between 100 and 100+w given relapses 
# (with confidence intervals)
pred.joint <- prediction(joi,datapredj,t=100,window=seq(50,1500,50),
event = "Recurrent",MC.sample=100)
plot(pred.joint,conf.bands=TRUE)
# each y-value of the plot corresponds to the prediction between [100,x]

#-- prediction of relapse and death between 100 and 100+w given relapses 
# (with confidence intervals)
pred.joint <- prediction(joi,datapredj,t=100,window=seq(50,1500,50),
event = "Both",MC.sample=100)
plot(pred.joint,conf.bands=TRUE)
# each y-value of the plot corresponds to the prediction between [100,x]

#############################################################################
### prediction on a JOINT model for longitudinal data and a terminal event ####
#############################################################################


data(colorectal)
data(colorectalLongi)

# Survival data preparation - only terminal events 
colorectalSurv <- subset(colorectal, new.lesions == 0)

#-- construction of the data-frame for predictions
#-- biomarker observations
datapredj_longi <- data.frame(id = 0, year = 0, tumor.size = 0, treatment = 0,
 age = 0, who.PS = 0, prev.resection = 0)
datapredj_longi$treatment <- as.factor(datapredj_longi$treatment)
levels(datapredj_longi$treatment) <- 1:2
datapredj_longi$age <- as.factor(datapredj_longi$age)
levels(datapredj_longi$age) <- 1:3
datapredj_longi$who.PS <- as.factor(datapredj_longi$who.PS)
levels(datapredj_longi$who.PS) <- 1:3
datapredj_longi$prev.resection <- as.factor(datapredj_longi$prev.resection)
levels(datapredj_longi$prev.resection) <- 1:2

# patient 1: increasing tumor size
datapredj_longi[1,] <- c(1, 0,1.2 ,2,1,1,1)
datapredj_longi[2,] <- c(1,0.3,1.4,2,1,1,1)
datapredj_longi[3,] <- c(1,0.6,1.9,2,1,1,1)
datapredj_longi[4,] <- c(1,0.9,2.5,2,1,1,1)
datapredj_longi[5,] <- c(1,1.5,3.9,2,1,1,1)

# patient 2: decreasing tumor size
datapredj_longi[6,] <- c(2, 0,1.2 ,2,1,1,1)
datapredj_longi[7,] <- c(2,0.3,0.7,2,1,1,1)
datapredj_longi[8,] <- c(2,0.5,0.3,2,1,1,1)
datapredj_longi[9,] <- c(2,0.7,0.1,2,1,1,1)
datapredj_longi[10,] <- c(2,0.9,0.1,2,1,1,1)

#-- terminal event
datapredj <- data.frame(id = 0, treatment = 0, age = 0, who.PS = 0,
prev.resection = 0)
datapredj$treatment <- as.factor(datapredj$treatment)
levels(datapredj$treatment) <- 1:2
datapredj$age <- as.factor(datapredj$age)
levels(datapredj$age) <- 1:3
datapredj$who.PS <- as.factor(datapredj$who.PS)
datapredj$prev.resection <- as.factor(datapredj$prev.resection)
levels(datapredj$prev.resection) <- 1:2
levels(datapredj$who.PS) <- 1:3
datapredj[1,] <- c(1,2,1,1,1)
datapredj[2,] <- c(2,2,1,1,1)

model.spli.CL <- longiPenal(Surv(time1, state) ~ age + treatment + who.PS
+ prev.resection, tumor.size ~  year * treatment + age + who.PS , 
colorectalSurv, data.Longi = colorectalLongi, random = c("1", "year"),
id = "id", link = "Current-level", left.censoring = -3.33, n.knots = 6, 
kappa = 1)

#-- prediction of death between 1 year and 1+2 given history of the biomarker
pred.jointLongi0 <- prediction(model.spli.CL, datapredj, datapredj_longi,
t = 1, window = 2)
print(pred.jointLongi0)

#-- prediction of death between 1 year and 1+w given history of the biomarker
pred.jointLongi <- prediction(model.spli.CL, datapredj, datapredj_longi,
t = 1, window = seq(0.5, 2.5, 0.2), MC.sample = 100)
plot(pred.jointLongi, conf.bands = TRUE)
# each y-value of the plot corresponds to the prediction between [1,x]

#-- prediction of death between t and t+0.5 given history of the biomarker
pred.jointLongi2 <- prediction(model.spli.CL, datapredj, datapredj_longi,
t = seq(1, 2.5, 0.5), window = 0.5, MC.sample = 100)
plot(pred.jointLongi2, conf.bands = TRUE)
# each y-value of the plot corresponds to the prediction between [x,x+0.5], 
#or in the next 0.5

#############################################################################
##### marginal prediction on a JOINT NESTED model for a terminal event ######
#############################################################################
#*--Warning! You can compute this prediction method with ONLY ONE family 
#*--by dataset of prediction. 
#*--Please make sure your data frame contains a column for individuals AND a 
#*--column for the reference number of the family chosen.

data(readmission)
readmissionNested <- transform(readmission,group=id%%30+1)

#-- construction of the data frame for predictions : 
#-- family 5 was selected for the prediction

DataPred <- readmissionNested[which(readmissionNested$group==5),]

#-- Fitting the model
modJointNested_Splines <- 
frailtyPenal(formula = Surv(t.start, t.stop, event)~subcluster(id)+ 
cluster(group) + dukes + terminal(death),formula.terminalEvent
=~dukes, data = readmissionNested, recurrentAG = TRUE,n.knots = 8, 
kappa = c(9.55e+9, 1.41e+12), initialize = TRUE)

#-- Compute prediction over the individuals 274 and 4 of the family 5
predRead <- prediction(modJointNested_Splines, data=DataPred,t=500,
window=seq(100,1500,200), conditional=FALSE, individual = c(274, 4))


#########################################################################
##### prediction on TRIVARIATE JOINT model (linear and non-linear) ######
#########################################################################

data(colorectal)
data(colorectalLongi)

#-- construction of the data frame for predictions
#-- history of recurrences and terminal event
datapredj <- data.frame(time0 = 0, time1 = 0, new.lesions = 0, id = 0, 
treatment = 0, age = 0, who.PS = 0, prev.resection =0)
datapredj$treatment <- as.factor(datapredj$treatment)
levels(datapredj$treatment) <- 1:2
datapredj$age <- as.factor(datapredj$age)
levels(datapredj$age) <- 1:3
datapredj$who.PS <- as.factor(datapredj$who.PS)
levels(datapredj$who.PS) <- 1:3
datapredj$prev.resection <- as.factor(datapredj$prev.resection)
levels(datapredj$prev.resection) <- 1:2

datapredj[1,] <- c(0,0.4,1,1,2,1,1,1)
datapredj[2,] <- c(0.4,1.2,1,1,2,1,1,1)
datapredj[3,] <- c(0,0.5,1,2,2,1,1,1)

# Linear trivariate joint model
# (computation takes around 40 minutes)
model.trivPenal <-trivPenal(Surv(time0, time1, new.lesions) ~ cluster(id)
+ age + treatment + who.PS +  terminal(state),
formula.terminalEvent =~ age + treatment + who.PS + prev.resection, 
tumor.size ~ year * treatment + age + who.PS, data = colorectal, 
data.Longi = colorectalLongi, random = c("1", "year"), id = "id", 
link = "Random-effects", left.censoring = -3.33, recurrentAG = TRUE,
n.knots = 6, kappa=c(0.01, 2), method.GH="Pseudo-adaptive",
n.nodes=7, init.B = c(-0.07, -0.13, -0.16, -0.17, 0.42, #recurrent events covarates
-0.23, -0.1, -0.09, -0.12, 0.8, -0.23, #terminal event covariates
3.02, -0.30, 0.05, -0.63, -0.02, -0.29, 0.11, 0.74)) #biomarker covariates

#-- prediction of death between 1 year and 1+2
pred.jointTri0 <- prediction(model.trivPenal, datapredj, 
datapredj_longi, t = 1, window = 2)
print(pred.jointTri0)

#-- prediction of death between 1 year and 1+w
pred.jointTri <- prediction(model.trivPenal, datapredj, 
datapredj_longi, t = 1, window = seq(0.5, 2.5, 0.2), MC.sample = 100)
plot(pred.jointTri, conf.bands = TRUE)

#-- prediction of death between t and t+0.5
pred.jointTri2 <- prediction(model.trivPenal, datapredj, 
datapredj_longi, t = seq(1, 2.5, 0.5), window = 0.5, MC.sample = 100)
plot(pred.jointTri2, conf.bands = TRUE)


###############################

# No information on dose - creation of a dummy variable 
colorectalLongi$dose <- 1

# (computation can take around 40 minutes)
model.trivPenalNL <- trivPenalNL(Surv(time0, time1, new.lesions) ~ cluster(id) + age + treatment
 + terminal(state), formula.terminalEvent =~ age + treatment, biomarker = "tumor.size",
 formula.KG ~ 1, formula.KD ~ treatment, dose = "dose", time.biomarker = "year", 
 data = colorectal, data.Longi =colorectalLongi, random = c("y0", "KG"), id = "id", 
 init.B = c(-0.22, -0.16, -0.35, -0.19, 0.04, -0.41, 0.23), init.Alpha = 1.86,
 init.Eta = c(0.5, 0.57, 0.5, 2.34), init.Biomarker = c(1.24, 0.81, 1.07, -1.53),
 recurrentAG = TRUE, n.knots = 5, kappa = c(0.01, 2), method.GH = "Pseudo-adaptive")

#-- prediction of death between 1 year and 1+2
pred.jointTriNL0 <- prediction(model.trivPenalNL, datapredj, 
datapredj_longi, t = 1, window = 2)
print(pred.jointTriNL0)

#-- prediction of death between 1 year and 1+w 
pred.jointTriNL <- prediction(model.trivPenalNL, datapredj, 
datapredj_longi, t = 1, window = seq(0.5, 2.5, 0.2), MC.sample = 100)
plot(pred.jointTriNL, conf.bands = TRUE)

#-- prediction of death between t and t+0.5
pred.jointTriNL2 <- prediction(model.trivPenalNL, datapredj, 
datapredj_longi, t = seq(2, 3, 0.2), window = 0.5, MC.sample = 100)
plot(pred.jointTriNL2, conf.bands = TRUE)

# }
# NOT RUN {

# }

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