Type: Package
Title: Optimal Treatment Regimes in Survival Contexts with Kaplan-Meier-Like Estimators
Version: 0.2.1
Description: Provide methods for estimating optimal treatment regimes in survival contexts with Kaplan-Meier-like estimators when no unmeasured confounding assumption is satisfied (Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) <doi:10.1111/rssb.12201>) and when no unmeasured confounding assumption fails to hold and a binary instrument is available (Xia, J., Zhan, Z., Zhang, J. (2022) <doi:10.48550/arXiv.2210.05538>).
License: MIT + file LICENSE
Encoding: UTF-8
LazyData: true
Imports: rgenoud, survival
Depends: R (≥ 3.10)
RoxygenNote: 7.2.2
NeedsCompilation: no
Packaged: 2023-07-04 13:35:10 UTC; xiajunwen
Author: Junwen Xia [aut, cre], Zishu Zhan [aut], Jingxiao Zhang [aut]
Maintainer: Junwen Xia <xiajunwen@ruc.edu.cn>
Repository: CRAN
Date/Publication: 2023-07-06 15:10:02 UTC

The (S)AIWKME estimator.

Description

Given a predetermined t0 and eta, calculate t0-year potential survival probability based on the (S)AIWKME estimator.

Usage

AIWKME(eta, datalist, ps, prep, t0, smooth = TRUE)

Arguments

eta

The parameters of the regime.

datalist

A list used to calculate the (S)AIWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving treatment given baseline covariates named fal. Fps.AIWKME can produce ps by positing logistic model.

prep

A list including the augmented terms in the numerator with treatment all to 1 named gamma.num.1 and all to 0 named gamma.num.0 and in the denominator with treatment all to 1 named gamma.den.1 and all to 0 named gamma.den.0; gamma.num.1 and the others are matrix with ordered observed time as rows and patients as columns. Fprep.AIWKME can produce prep by positing Cox proportional hazards model.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed estimator should be used.

Details

More details can be found in references.

Value

Estimated potential survival probability given eta and t0.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0 and eta
t0=5
eta=c(1,2,3)

# calculate ps and prep
ps=Fps.AIWKME(datalist)
prep=Fprep.AIWKME(datalist, t0)

AIWKME(eta, datalist, ps, prep, t0, smooth=TRUE)

The (S)DRKMEIV estimator.

Description

Given a predetermined t0 and eta, calculate t0-year potential survival probability based on the (S)DRKMEIV estimator.

Usage

DRKMEIV(eta, datalist, ps, prep, t0, smooth = TRUE)

Arguments

eta

The parameters of the regime.

datalist

A list used to calculate the (S)DRKMEIV estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving instrument given baseline covariates named fzl, the probability of receiving treatment given baseline covariates and instrument equaling 0 named fal0, the probability of receiving treatment given baseline covariates and instrument equaling 1 named fal1, and the difference between fal1 and fal0 named deltal. Fps.DRKMEIV can produce ps by positing logistic model.

prep

A list including estimates \hat{\gamma}_1(\boldsymbol{L};s) with treatment all to 1 named gamma.num.1 and all to 0 named gamma.num.0, \hat{\gamma}_1'(\boldsymbol{L};s) with treatment all to 1 named gammaa.num.1 and all to 0 named gammaa.num.0, \hat{\gamma}_2(\boldsymbol{L};s) with treatment all to 1 named gamma.den.1 and all to 0 named gamma.den.0, and \hat{\gamma}_2'(\boldsymbol{L};s) with treatment all to 1 named gammaa.den.1 and all to 0 named gammaa.den.0; gamma.num.1 and the others are matrix with ordered observed time as rows and patients as columns. There are also estimates for the last term of the (S)DRKMEIV estimator. More details can be found in references. Fprep.DRKMEIV can produce prep by positing Cox proportional hazards model.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed version should be used.

Details

More details can be found in references.

Value

Estimated potential survival probability given eta and t0.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0 and eta
t0=5
eta=c(1,2,3)

# calculate ps and prep
ps=Fps.DRKMEIV(datalist,t0)
prep=Fprep.DRKMEIV(datalist, ps, t0)

DRKMEIV(eta, datalist, ps, prep, t0, smooth=TRUE)

Cox proportional hazards model for eta-free terms in the (S)AIWKME estimator.

Description

Cox proportional hazards model for eta-free terms in the (S)AIWKME estimator.

Usage

Fprep.AIWKME(datalist, t0)

Arguments

datalist

A list used to calculate the (S)AIWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

t0

A predetermined t.

Details

More details can be found in references, AIWKME, and Genetic.optim.AIWKME.

Value

A list including the augmented terms in the numerator with treatment all to 1 named gamma.num.1 and all to 0 named gamma.num.0 and in the denominator with treatment all to 1 named gamma.den.1 and all to 0 named gamma.den.0; gamma.num.1 and the others are matrix with ordered observed time as rows and patients as columns. More details can be found in references.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Cox proportional hazards model for eta-free terms in the (S)DRKMEIV estimator.

Description

Cox proportional hazards model for eta-free terms in the (S)DRKMEIV estimator.

Usage

Fprep.DRKMEIV(datalist, ps, t0)

Arguments

datalist

A list used to calculate the (S)DRKMEIV estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving instrument given baseline covariates named fzl, the probability of receiving treatment given baseline covariates and instrument equaling 0 named fal0, the probability of receiving treatment given baseline covariates and instrument equaling 1 named fal1, and the difference between fal1 and fal0 named deltal. Fps.DRKMEIV can produce ps by positing logistic model.

t0

A predetermined t.

Details

More details can be found in references, DRKMEIV, and Genetic.optim.DRKMEIV.

Value

A list including estimates \hat{\gamma}_1(\boldsymbol{L};s) with treatment all to 1 named gamma.num.1 and all to 0 named gamma.num.0, \hat{\gamma}_1'(\boldsymbol{L};s) with treatment all to 1 named gammaa.num.1 and all to 0 named gammaa.num.0, \hat{\gamma}_2(\boldsymbol{L};s) with treatment all to 1 named gamma.den.1 and all to 0 named gamma.den.0, and \hat{\gamma}_2'(\boldsymbol{L};s) with treatment all to 1 named gammaa.den.1 and all to 0 named gammaa.den.0; gamma.num.1 and the others are matrix with ordered observed time as rows and patients as columns. There are also estimates for the last term of the (S)DRIWKMEIV estimator. More details can be found in references.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

Logistic regression for observed treatment used for the (S)AIWKME estimator.

Description

Logistic regression for observed treatment used for the (S)AIWKME estimator.

Usage

Fps.AIWKME(datalist)

Arguments

datalist

A list used to calculate the (S)AIWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

Details

More details can be found in references, AIWKME, and Genetic.optim.AIWKME.

Value

A list including the probability of receiving treatment given baseline covariates named fal.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Logistic regression for observed treatment and instrument used for the (S)DRKMEIV estimator.

Description

Logistic regression for observed treatment and instrument used for the (S)DRKMEIV estimator.

Usage

Fps.DRKMEIV(datalist, t0)

Arguments

datalist

A list used to calculate the (S)DRKMEIV estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

t0

A predetermined t.

Details

More details can be found in references, DRKMEIV, and Genetic.optim.DRKMEIV.

Value

A list including the probability of receiving instrument given baseline covariates named fzl, the probability of receiving treatment given baseline covariates and instrument equaling 0 named fal0, the probability of receiving treatment given baseline covariates and instrument equaling 1 named fal1, the difference between fal1 and fal0 named deltal, and the censoring survival function given baseline covariates and treatment 1 or 0 named surv.C.1 or surv.C.0.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

Logistic regression for observed treatment used for the (S)IWKME estimator.

Description

Logistic regression for observed treatment used for the (S)IWKME estimator.

Usage

Fps.IWKME(datalist)

Arguments

datalist

A list used to calculate the (S)IWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

Details

More details can be found in references, IWKME, and Genetic.optim.IWKME.

Value

A list including the probability of receiving treatment given baseline covariates named fal.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Logistic regression for observed treatment and instrument used for the (S)IWKMEIV estimator.

Description

Logistic regression for observed treatment and instrument used for the (S)IWKMEIV estimator.

Usage

Fps.IWKMEIV(datalist, t0)

Arguments

datalist

A list used to calculate the (S)IWKMEIV estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

t0

A predetermined t.

Details

More details can be found in references, IWKMEIV, and Genetic.optim.IWKMEIV.

Value

A list including the probability of receiving instrument given baseline covariates named fzl, the difference between fal1 and fal0 named deltal, where fal0 denotes the probability of receiving treatment given baseline covariates and instrument equaling 0, and fal1 denotes the probability of receiving treatment given baseline covariates and instrument equaling 1, and the censoring survival function given baseline covariates and treatment 1 or 0 named surv.C.1 or surv.C.0.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

The optimal treatment regime based on the (S)AIWKME estimator.

Description

Given a predetermined t0, estimate the optimal treatment regime by maximizing t0-year survival probability based on the (S)AIWKME estimator.

Usage

Genetic.optim.AIWKME(datalist, ps, prep, t0, smooth = TRUE)

Arguments

datalist

A list used to calculate the (S)AIWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving treatment given baseline covariates named fal. Fps.AIWKME can produce ps by positing logistic model.

prep

A list including the augmented terms in the numerator with treatment all to 1 named gamma.num.1 and all to 0 named gamma.num.0 and in the denominator with treatment all to 1 named gamma.den.1 and all to 0 named gamma.den.0; gamma.num.1 and the others are matrix with ordered observed time as rows and patients as columns. Fprep.AIWKME can produce prep by positing Cox proportional hazards model.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed version should be used.

Details

More details can be found in references.

Value

A numeric vector in which the last number is the estimated optimal t0-year survival probability and others are the estimated parameters of the optimal treatment regime.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0
t0=5

# calculate ps and prep
ps=Fps.AIWKME(datalist)
prep=Fprep.AIWKME(datalist, t0)

Genetic.optim.AIWKME(datalist, ps, prep, t0, smooth=TRUE)

The optimal treatment regime based on the (S)DRKMEIV estimator.

Description

Given a predetermined t0, estimate the optimal treatment regime by maximizing t0-year survival probability based on the (S)DRKMEIV estimator.

Usage

Genetic.optim.DRKMEIV(datalist, ps, prep, t0, smooth = TRUE)

Arguments

datalist

A list used to calculate the (S)DRKMEIV estimator including instrument named z, treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving instrument given baseline covariates named fzl, the probability of receiving treatment given baseline covariates and instrument equaling 0 named fal0, the probability of receiving treatment given baseline covariates and instrument equaling 1 named fal1, and the difference between fal1 and fal0 named deltal. Fps.DRKMEIV can produce ps by positing logistic model.

prep

A list including estimates \hat{\gamma}_1(\boldsymbol{L};s) with treatment all to 1 named gamma.num.1 and all to 0 named gamma.num.0, \hat{\gamma}_1'(\boldsymbol{L};s) with treatment all to 1 named gammaa.num.1 and all to 0 named gammaa.num.0, \hat{\gamma}_2(\boldsymbol{L};s) with treatment all to 1 named gamma.den.1 and all to 0 named gamma.den.0, and \hat{\gamma}_2'(\boldsymbol{L};s) with treatment all to 1 named gammaa.den.1 and all to 0 named gammaa.den.0; gamma.num.1 and the others are matrix with ordered observed time as rows and patients as columns. There are also estimates for the last term of the (S)DRKMEIV estimator. More details can be found in references. Fprep.DRKMEIV can produce prep by positing Cox proportional hazards model.

t0

A predetermined time to point out that t0-year survival probability is our estimate

smooth

A logic variable indicating wether a smoothed version should be used.

Details

More details can be found in references.

Value

A numeric vector in which the last number is the estimated optimal t0-year survival probability and the others are the estimated parameter of the optimal treatment regime.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

Examples

# load data
data(simulation)

# order the data by observed time and select the first 200 patients to speed up 
simulation=simulation[order(simulation$Survival)[1:200],]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0
t0=1

# calculate ps and prep
ps=Fps.DRKMEIV(datalist, t0)
prep=Fprep.DRKMEIV(datalist, ps, t0)


Genetic.optim.DRKMEIV(datalist, ps, prep, t0, smooth=TRUE)

The optimal treatment regime based on the (S)IWKME estimator.

Description

Given a predetermined t0, estimate the optimal treatment regime by maximizing t0-year survival probability based on the (S)IWKME estimator.

Usage

Genetic.optim.IWKME(datalist, ps, t0, smooth = TRUE)

Arguments

datalist

A list used to calculate the (S)IWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving treatment given baseline covariates named fal. Fps.IWKME can produce ps by positing logistic model.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed version should be used.

Details

More details can be found in references.

Value

A numeric vector in which the last number is the estimated optimal t0-year survival probability and the others are the estimated parameter of the optimal treatment regime.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0
t0=5

# calculate ps
ps=Fps.IWKME(datalist)

Genetic.optim.IWKME(datalist, ps, t0, smooth=TRUE)

The optimal treatment regime based on the (S)IWKMEIV estimator.

Description

Given a predetermined t0, estimate the optimal treatment regime by maximizing t0-year survival probability based on the (S)IWKMEIV estimator.

Usage

Genetic.optim.IWKMEIV(datalist, ps, t0, smooth = TRUE)

Arguments

datalist

A list used to calculate the (S)IWKMEIV estimator including instrument named z, treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving instrument given baseline covariates named fzl and the difference between fal1 and fal0 named deltal, where fal0 denotes the probability of receiving treatment given baseline covariates and instrument equaling 0, and fal1 denotes the probability of receiving treatment given baseline covariates and instrument equaling 1. Fps.IWKMEIV can produce ps by positing logistic model.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed version should be used.

Details

More details can be found in references.

Value

A numeric vector in which the last number is the estimated optimal t0-year survival probability and others are the estimated parameter of the optimal treatment regime.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0
t0=1

# calculate ps and prep
ps=Fps.IWKMEIV(datalist, t0)

Genetic.optim.IWKMEIV(datalist, ps, t0, smooth=TRUE)

The (S)IWKME estimator.

Description

Given a predetermined t0 and eta, calculate t0-year potential survival probability based on the (S)IWKME estimator.

Usage

IWKME(eta, datalist, ps, t0, smooth = TRUE)

Arguments

eta

The parameters of the regime.

datalist

A list used to calculate the (S)IWKME estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Fps.IWKME can produce ps by positing logistic model. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving treatment given baseline covariates named fal.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed estimator should be used.

Details

More details can be found in references.

Value

Estimated potential survival probability given eta and t0.

References

Jiang, R., Lu, W., Song, R., and Davidian, M. (2017) On estimation of optimal treatment regimes for maximizing t‐year survival probability. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79: 1165-1185. DOI:10.1111/rssb.12201

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# calculate ps
ps=Fps.IWKME(datalist)

# predetermined t0 and eta
t0=5
eta=c(1,2,3)

IWKME(eta, datalist, ps, t0, smooth=TRUE)

The (S)IWKMEIV estimator.

Description

Given a predetermined t0 and eta, calculate t0-year potential survival probability based on the (S)IWKMEIV estimator.

Usage

IWKMEIV(eta, datalist, ps, t0, smooth = TRUE)

Arguments

eta

The parameters of the regime.

datalist

A list used to calculate the (S)IWKMEIV estimator including treatment named a, observed time named obs.t, censoring indicator (0, censored) named delta, and baseline covariates used to assign treatment named l. Notice that all the data in the datalist should be ordered by observed time.

ps

A list including the probability of receiving instrument given baseline covariates named fzl and the difference between fal1 and fal0 named deltal, where fal0 denotes the probability of receiving treatment given baseline covariates and instrument equaling 0, and fal1 denotes the probability of receiving treatment given baseline covariates and instrument equaling 1. Fps.IWKMEIV can produce ps by positing logistic model.

t0

A predetermined time.

smooth

A logic variable indicating wether a smoothed version should be used.

Details

More details can be found in references.

Value

Estimated potential survival probability given eta and t0.

References

Xia, J., Zhan, Z., Zhang, J. (2022) Estimating optimal treatment regime in survival contexts using an instrumental variable. Under Review.

Examples

# load data
data(simulation)
simulation=simulation[order(simulation$Survival),]

# convert the data into a datalist
datalist=list(z=simulation$Instrument,a=simulation$Treatment,
              obs.t=simulation$Survival,delta=simulation$Status,
              l=cbind(simulation$Covariate1,simulation$Covariate2))

# predetermined t0 and eta
t0=5
eta=c(1,2,3)

# calculate ps and prep
ps=Fps.IWKMEIV(datalist, t0)

IWKMEIV(eta, datalist, ps, t0, smooth=TRUE)

clip function.

Description

Limit the number not to be too large or too small.

Usage

clipp(x)

Arguments

x

A vector or matrix.

Value

A vector or matrix same as the input.

Simulation

Description

A simulation data to help understand and implement the functions in the package. 'Instrument' denotes the binary instrumental variable. 'Treatment' denotes the binary treatment. 'Survival' denotes the observed survival time. 'Status' denotes whether the the data is censoring where '0' means censoring. 'Covariate' denotes the baseline characteristics.

Usage

simulation

Format

An object of class data.frame with 500 rows and 6 columns.

Examples

  head(simulation)