Package 'cfma' reference manual

Title:	Causal Functional Mediation Analysis
Description:	Performs causal functional mediation analysis (CFMA) for functional treatment, functional mediator, and functional outcome. This package includes two functional mediation model types: (1) a concurrent mediation model and (2) a historical influence mediation model. See Zhao et al. (2018), Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data, <arXiv:1805.06923> for details.
Authors:	Yi Zhao <[email protected]>, Xi Luo <[email protected]>, Martin Lindquist <[email protected]>, Brian Caffo <[email protected]>
Maintainer:	Yi Zhao <[email protected]>
License:	GPL (>=2)
Version:	1.0.1
Built:	2024-11-27 05:55:13 UTC
Source:	https://github.com/zhaoyi1026/cfma

Causal Functional Mediation Analysis

Description

cfma package performs causal functional mediation analysis (CFMA) for functional treatment, functional mediator, and functional outcome. This package includes two functional mediation model type: (1) a concurrent mediation model and (2) a historical influence mediation model.

Details

Package:	cfma
Type:	Package
Version:	1.0
Date:	2018-05-16
License:	GPL (>=2)

Author(s)

Yi Zhao <[email protected]> and Xi Luo <[email protected]> and Martin Lindquist <[email protected]> and Brian Caffo <[email protected]>

Maintainer: Yi Zhao <[email protected]>

Simulated data from the concurrent mediation model

Description

"env.concurrent" is an R environment containing the data generated from a concurrent mediation model.

Usage

data("env.concurrent")data("env.concurrent")

Format

An R environment

Z: a $n\times T$ data matrix, treatment trajectory of $n$ subjects for $T$ time points.
M: a $n\times T$ data matrix, mediator trajectory of $n$ subjects for $T$ time points.
Y: a $n\times T$ data matrix, outcome trajectory of $n$ subjects for $T$ time points.
alpha: a length $T$ vector model coefficient.
beta: a length $T$ vector model coefficient.
gamma: a length $T$ vector model coefficient.

Details

The data was generated from the concurrent mediation model

$M(t)=Z(t)\alpha(t)+\epsilon_{1}(t),$

$R(t)=Z(t)\gamma(t)+M(t)\beta(t)+\epsilon_{2}(t).$

$Z(t)$ is the convolution of hemodynamic response function (HRF) and event onsets.

Examples

data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)

Simulated data from the historical influence mediation model

Description

"env.historical" is an R environment containing the data generated from a historical influence mediation model.

Usage

data("env.historical")data("env.historical")

Format

An R environment

Z: a $n\times T$ data matrix, treatment trajectory of $n$ subjects for $T$ time points.
M: a $n\times T$ data matrix, mediator trajectory of $n$ subjects for $T$ time points.
Y: a $n\times T$ data matrix, outcome trajectory of $n$ subjects for $T$ time points.
alpha: a $T\times T$ matrix model coefficient.
beta: a $T\times T$ matrix model coefficient.
gamma: a $T\times T$ matrix model coefficient.

Details

The data was generated from the historical influence mediation model

$M(t)=\int_{\Omega_{t}^{1}}Z(s)\alpha(s,t)ds+\epsilon_{1}(t),$

$Y(t)=\int_{\Omega_{t}^{2}}Z(s)\gamma(s,t)ds+\int_{\Omega_{t}^{3}}M(s)\beta(s,t)ds+\epsilon_{2}(t),$

where $\alpha(s,t)$ , $\beta(s,t)$ , $\gamma(s,t)$ are coefficient curves; $\Omega_{t}^{j}=[(t-\delta_{j})\vee 0,t]$ for $j=1,2,3$ . $Z(t)$ is the convolution of hemodynamic response function (HRF) and event onsets.

Examples

data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)

Functional mediation analysis under concurrent regression model

Description

This function performs functional mediation regression under the concurrent model with given tuning parameter.

Usage

FMA.concurrent(Z, M, Y, intercept = TRUE, basis = NULL, Ld2.basis = NULL, 
    basis.type = c("fourier"), nbasis = 3, timeinv = c(0, 1), timegrids = NULL, 
    lambda.m = 0.01, lambda.y = 0.01)
FMA.concurrent(Z, M, Y, intercept = TRUE, basis = NULL, Ld2.basis = NULL, 
    basis.type = c("fourier"), nbasis = 3, timeinv = c(0, 1), timegrids = NULL, 
    lambda.m = 0.01, lambda.y = 0.01)

Arguments

`Z`	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`M`	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`Y`	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`intercept`	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
`basis`	a data matrix. Basis function used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis`	a data matrix. The second derivative of the basis function. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis.type`	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
`nbasis`	an integer, the number of basis function included. If `basis` is provided, this argument will be ignored.
`timeinv`	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
`timegrids`	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
`lambda.m`	a numeric value of the tuning parameter in the mediator model.
`lambda.y`	a numeric value of the tuning parameter in the outcome model.

Details

The concurrent mediation model is

$M(t)=Z(t)\alpha(t)+\epsilon_{1}(t),$

$Y(t)=Z(t)\gamma(t)+M(t)\beta(t)+\epsilon_{2}(t),$

where $\alpha(t)$ , $\beta(t)$ , $\gamma(t)$ are coefficient curves. The model coefficient curves are estimated by minimizing the penalized $L_{2}$ -loss.

Value

`basis`	the basis functions used in the analysis.
`M`	a list of output for the mediator model `coefficient`the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `M` `lambda`: $\lambda$ value
`Y`	a list of output for the outcome model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `Y` `lambda`: $\lambda$ value
`IE`	a list of output for the indirect effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `coefficients`: the coefficient with respect to the basis function `curve`: the estimated causal curve
`DE`	a list of output for the direct effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `coefficients`: the coefficient with respect to the basis function `curve`: the estimated causal curve

Author(s)

Yi Zhao, Johns Hopkins University, [email protected];

Xi Luo, Brown University [email protected];

Martin Lindquist, Johns Hopkins University, [email protected];

Brian Caffo, Johns Hopkins University, [email protected]

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Examples


##################################################
# Concurrent functional mediation model
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)

# consider Fourier basis
fit<-FMA.concurrent(Z,M,Y,intercept=FALSE,timeinv=c(0,300))

# estimate of alpha
plot(fit$M$curve[1,],type="l",lwd=5)
lines(get("alpha",env.concurrent),lty=2,lwd=2,col=2)

# estimate of gamma
plot(fit$Y$curve[1,],type="l",lwd=5)
lines(get("gamma",env.concurrent),lty=2,lwd=2,col=2)

# estimate of beta
plot(fit$Y$curve[2,],type="l",lwd=5)
lines(get("beta",env.concurrent),lty=2,lwd=2,col=2)

# estimate of causal curves
plot(fit$IE$curve,type="l",lwd=5)
plot(fit$DE$curve,type="l",lwd=5)
##################################################
##################################################
# Concurrent functional mediation model
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)

# consider Fourier basis
fit<-FMA.concurrent(Z,M,Y,intercept=FALSE,timeinv=c(0,300))

# estimate of alpha
plot(fit$M$curve[1,],type="l",lwd=5)
lines(get("alpha",env.concurrent),lty=2,lwd=2,col=2)

# estimate of gamma
plot(fit$Y$curve[1,],type="l",lwd=5)
lines(get("gamma",env.concurrent),lty=2,lwd=2,col=2)

# estimate of beta
plot(fit$Y$curve[2,],type="l",lwd=5)
lines(get("beta",env.concurrent),lty=2,lwd=2,col=2)

# estimate of causal curves
plot(fit$IE$curve,type="l",lwd=5)
plot(fit$DE$curve,type="l",lwd=5)
##################################################

Functional mediation analysis under concurrent regression model with point-wise bootstrap confidence interval

Description

This function performs functional mediation regression under the concurrent model with given tuning parameter. Point-wise confidence bands are obtained from bootstrap.

Usage

FMA.concurrent.boot(Z, M, Y, intercept = TRUE, basis = NULL, Ld2.basis = NULL, 
    basis.type = c("fourier"), nbasis = 3, timeinv = c(0, 1), timegrids = NULL, 
    lambda.m = 0.01, lambda.y = 0.01, sims = 1000, boot = TRUE, 
    boot.ci.type = c("bca", "perc"), conf.level = 0.95, verbose = TRUE)
FMA.concurrent.boot(Z, M, Y, intercept = TRUE, basis = NULL, Ld2.basis = NULL, 
    basis.type = c("fourier"), nbasis = 3, timeinv = c(0, 1), timegrids = NULL, 
    lambda.m = 0.01, lambda.y = 0.01, sims = 1000, boot = TRUE, 
    boot.ci.type = c("bca", "perc"), conf.level = 0.95, verbose = TRUE)

Arguments

`Z`	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`M`	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`Y`	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`intercept`	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
`basis`	a data matrix. Basis function used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis`	a data matrix. The second derivative of the basis function. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis.type`	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
`nbasis`	an integer, the number of basis function included. If `basis` is provided, this argument will be ignored.
`timeinv`	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
`timegrids`	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
`lambda.m`	a numeric value of the tuning parameter in the mediator model.
`lambda.y`	a numeric value of the tuning parameter in the outcome model.
`sims`	an integer indicating the number of simulations for inference.
`boot`	a logical value, indicating whether or not bootstrap should be used. Default is `TRUE`.
`boot.ci.type`	a character of confidence interval method. `boot.ci.type = "bca"` bias corrected confidence interval; `boot.ci.type = "perc"` percentile confidence interval.
`conf.level`	a number of significance level. Default is 0.95.
`verbose`	a logical value, indicating whether print out bootstrap replications.

Details

The concurrent mediation model is

$M(t)=Z(t)\alpha(t)+\epsilon_{1}(t),$

$Y(t)=Z(t)\gamma(t)+M(t)\beta(t)+\epsilon_{2}(t),$

where $\alpha(t)$ , $\beta(t)$ , $\gamma(t)$ are coefficient curves. The model coefficient curves are estimated by minimizing the penalized $L_{2}$ -loss.

Value

`alpha`	a list of output for $\alpha$ estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`gamma`	: a list of output for $\gamma$ estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`beta`	a list of output for $\beta$ estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`IE`	a list of output for indirect effect estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`DE`	a list of output for direct effect estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve

Author(s)

Yi Zhao, Johns Hopkins University, [email protected];

Xi Luo, Brown University [email protected];

Martin Lindquist, Johns Hopkins University, [email protected];

Brian Caffo, Johns Hopkins University, [email protected]

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Examples


##################################################
# Concurrent functional mediation model
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)


# consider Fourier basis
fit.boot<-FMA.concurrent.boot(Z,M,Y,intercept=FALSE,timeinv=c(0,300))

##################################################
##################################################
# Concurrent functional mediation model
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)


# consider Fourier basis
fit.boot<-FMA.concurrent.boot(Z,M,Y,intercept=FALSE,timeinv=c(0,300))

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

Functional mediation analysis under concurrent regression model

Description

This function performs functional mediation regression under the concurrent model. Tuning parameter is chosen based on cross-validation.

Usage

FMA.concurrent.CV(Z, M, Y, intercept = TRUE, basis = NULL, Ld2.basis = NULL, 
    basis.type = c("fourier"), nbasis = 3, timeinv = c(0, 1), timegrids = NULL, 
    lambda = NULL, nfolds = 5)
FMA.concurrent.CV(Z, M, Y, intercept = TRUE, basis = NULL, Ld2.basis = NULL, 
    basis.type = c("fourier"), nbasis = 3, timeinv = c(0, 1), timegrids = NULL, 
    lambda = NULL, nfolds = 5)

Arguments

`Z`	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`M`	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`Y`	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`intercept`	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
`basis`	a data matrix. Basis function used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis`	a data matrix. The second derivative of the basis function. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis.type`	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
`nbasis`	an integer, the number of basis function included. If `basis` is provided, this argument will be ignored.
`timeinv`	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
`timegrids`	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
`lambda`	a numeric vector of tuning parameter values.
`nfolds`	a number gives the number of folds in cross-validation.

Details

The concurrent mediation model is

$M(t)=Z(t)\alpha(t)+\epsilon_{1}(t),$

$Y(t)=Z(t)\gamma(t)+M(t)\beta(t)+\epsilon_{2}(t),$

where $\alpha(t)$ , $\beta(t)$ , $\gamma(t)$ are coefficient curves. The model coefficient curves are estimated by minimizing the penalized $L_{2}$ -loss. Tuning parameter $\lambda$ controls the smoothness of the estimated curves, and is chosen by cross-validation.

Value

`basis`	the basis functions used in the analysis.
`M`	a list of output for the mediator model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `M` `lambda`: the chosen $\lambda$ value
`Y`	a list of output for the outcome model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `Y` `lambda`: the chosen $\lambda$ value
`IE`	a list of output for the indirect effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `coefficients`: the coefficient with respect to the basis function `curve`: the estimated causal curve
`DE`	a list of output for the direct effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `coefficients`: the coefficient with respect to the basis function `curve`: the estimated causal curve

Author(s)

Yi Zhao, Johns Hopkins University, [email protected];

Xi Luo, Brown University [email protected];

Martin Lindquist, Johns Hopkins University, [email protected];

Brian Caffo, Johns Hopkins University, [email protected]

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Examples


##################################################
# Concurrent functional mediation model
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)

## Not run: 
# consider Fourier basis
fit<-FMA.concurrent.CV(Z,M,Y,intercept=FALSE,timeinv=c(0,300))

# estimate of alpha
plot(fit$M$curve[1,],type="l",lwd=5)
lines(get("alpha",env.concurrent),lty=2,lwd=2,col=2)

# estimate of gamma
plot(fit$Y$curve[1,],type="l",lwd=5)
lines(get("gamma",env.concurrent),lty=2,lwd=2,col=2)

# estimate of beta
plot(fit$Y$curve[2,],type="l",lwd=5)
lines(get("beta",env.concurrent),lty=2,lwd=2,col=2)

# estimate of causal curves
plot(fit$IE$curve,type="l",lwd=5)
plot(fit$DE$curve,type="l",lwd=5)

## End(Not run)
##################################################
##################################################
# Concurrent functional mediation model
data(env.concurrent)
Z<-get("Z",env.concurrent)
M<-get("M",env.concurrent)
Y<-get("Y",env.concurrent)

## Not run: 
# consider Fourier basis
fit<-FMA.concurrent.CV(Z,M,Y,intercept=FALSE,timeinv=c(0,300))

# estimate of alpha
plot(fit$M$curve[1,],type="l",lwd=5)
lines(get("alpha",env.concurrent),lty=2,lwd=2,col=2)

# estimate of gamma
plot(fit$Y$curve[1,],type="l",lwd=5)
lines(get("gamma",env.concurrent),lty=2,lwd=2,col=2)

# estimate of beta
plot(fit$Y$curve[2,],type="l",lwd=5)
lines(get("beta",env.concurrent),lty=2,lwd=2,col=2)

# estimate of causal curves
plot(fit$IE$curve,type="l",lwd=5)
plot(fit$DE$curve,type="l",lwd=5)

## End(Not run)
##################################################

Functional mediation analysis under historical influence model

Description

This function performs functional mediation regression under the historical influence model with given tuning parameter.

Usage

FMA.historical(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, 
    lambda1.m = 0.01, lambda2.m = 0.01, lambda1.y = 0.01, lambda2.y = 0.01)
FMA.historical(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, 
    lambda1.m = 0.01, lambda2.m = 0.01, lambda1.y = 0.01, lambda2.y = 0.01)

Arguments

`Z`	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`M`	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`Y`	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`delta.grid1`	a number indicates the width of treatment-mediator time interval in the mediator model.
`delta.grid2`	a number indicates the width of treatment-outcome time interval in the outcome model.
`delta.grid3`	a number indicates the width of mediator-outcome time interval in the outcome model.
`intercept`	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
`basis1`	a data matrix. Basis function on the $s$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis1`	a data matrix. The second derivative of the basis function on the $s$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis2`	a data matrix. Basis function on the $t$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis2`	a data matrix. The second derivative of the basis function on the $t$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis.type`	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
`nbasis1`	an integer, the number of basis function on the $s$ domain included. If `basis1` is provided, this argument will be ignored.
`nbasis2`	an integer, the number of basis function on the $t$ domain included. If `basis2` is provided, this argument will be ignored.
`timeinv`	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
`timegrids`	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
`lambda1.m`	a numeric vector of tuning parameter values on the $s$ domain in the mediator model.
`lambda2.m`	a numeric vector of tuning parameter values on the $t$ domain in the mediator model.
`lambda1.y`	a numeric vector of tuning parameter values on the $s$ domain in the outcome model.
`lambda2.y`	a numeric vector of tuning parameter values on the $t$ domain in the outcome model.

Details

The historical influence mediation model is

$M(t)=\int_{\Omega_{t}^{1}}Z(s)\alpha(s,t)ds+\epsilon_{1}(t),$

$Y(t)=\int_{\Omega_{t}^{2}}Z(s)\gamma(s,t)ds+\int_{\Omega_{t}^{3}}M(s)\beta(s,t)ds+\epsilon_{2}(t),$

Value

`basis1`	the basis functions on the $s$ domain used in the analysis.
`basis2`	the basis functions on the $t$ domain used in the analysis.
`M`	a list of output for the mediator model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `M` `lambda1`: the $\lambda$ value on the $s$ domain `lambda2`: the $\lambda$ value on the $t$ domain
`Y`	a list of output for the outcome model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `Y` `lambda1`: the $\lambda$ value on the $s$ domain `lambda2`: the $\lambda$ value on the $t$ domain
`IE`	a list of output for the indirect effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `curve`: the estimated causal curve
`DE`	a list of output for the direct effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `curve`: the estimated causal curve

Author(s)

Yi Zhao, Johns Hopkins University, [email protected];

Xi Luo, Brown University [email protected];

Martin Lindquist, Johns Hopkins University, [email protected];

Brian Caffo, Johns Hopkins University, [email protected]

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Examples


##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)

# consider Fourier basis
fit<-FMA.historical(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))

# estimate of causal curves
plot(fit$IE$curve,type="l",lwd=5)
plot(fit$DE$curve,type="l",lwd=5)
##################################################
##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)

# consider Fourier basis
fit<-FMA.historical(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))

# estimate of causal curves
plot(fit$IE$curve,type="l",lwd=5)
plot(fit$DE$curve,type="l",lwd=5)
##################################################

Functional mediation analysis under historical influence regression model with point-wise bootstrap confidence interval

Description

This function performs functional mediation regression under the historical influence model with given tuning parameter. Point-wise confidence bands are obtained from bootstrap.

Usage

FMA.historical.boot(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, 
    lambda1.m = 0.01, lambda2.m = 0.01, lambda1.y = 0.01, lambda2.y = 0.01, 
    sims = 1000, boot = TRUE, boot.ci.type = c("bca", "perc"), 
    conf.level = 0.95, verbose = TRUE)
FMA.historical.boot(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, 
    lambda1.m = 0.01, lambda2.m = 0.01, lambda1.y = 0.01, lambda2.y = 0.01, 
    sims = 1000, boot = TRUE, boot.ci.type = c("bca", "perc"), 
    conf.level = 0.95, verbose = TRUE)

Arguments

`Z`	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`M`	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`Y`	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`delta.grid1`	a number indicates the width of treatment-mediator time interval in the mediator model.
`delta.grid2`	a number indicates the width of treatment-outcome time interval in the outcome model.
`delta.grid3`	a number indicates the width of mediator-outcome time interval in the outcome model.
`intercept`	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
`basis1`	a data matrix. Basis function on the $s$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis1`	a data matrix. The second derivative of the basis function on the $s$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis2`	a data matrix. Basis function on the $t$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis2`	a data matrix. The second derivative of the basis function on the $t$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis.type`	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
`nbasis1`	an integer, the number of basis function on the $s$ domain included. If `basis1` is provided, this argument will be ignored.
`nbasis2`	an integer, the number of basis function on the $t$ domain included. If `basis2` is provided, this argument will be ignored.
`timeinv`	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
`timegrids`	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
`lambda1.m`	a numeric vector of tuning parameter values on the $s$ domain in the mediator model.
`lambda2.m`	a numeric vector of tuning parameter values on the $t$ domain in the mediator model.
`lambda1.y`	a numeric vector of tuning parameter values on the $s$ domain in the outcome model.
`lambda2.y`	a numeric vector of tuning parameter values on the $t$ domain in the outcome model.
`sims`	an integer indicating the number of simulations for inference.
`boot`	a logical value, indicating whether or not bootstrap should be used. Default is `TRUE`.
`boot.ci.type`	a character of confidence interval method. `boot.ci.type = "bca"` bias corrected confidence interval; `boot.ci.type = "perc"` percentile confidence interval.
`conf.level`	a number of significance level. Default is 0.95.
`verbose`	a logical value, indicating whether print out bootstrap replications.

Details

The historical influence mediation model is

$M(t)=\int_{\Omega_{t}^{1}}Z(s)\alpha(s,t)ds+\epsilon_{1}(t),$

$Y(t)=\int_{\Omega_{t}^{2}}Z(s)\gamma(s,t)ds+\int_{\Omega_{t}^{3}}M(s)\beta(s,t)ds+\epsilon_{2}(t),$

Value

`alpha`	a list of output for $\alpha$ estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`gamma`	a list of output for $\gamma$ estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`beta`	a list of output for $\beta$ estimate `coefficients`: the result of the coefficient estimates corresponding to the basis function `curve`: the point-wise estimate of the coefficient curve
`IE`	a list of output for indirect effect estimate `curve`: the point-wise estimate of the coefficient curve
`DE`	a list of output for direct effect estimate `curve`: the point-wise estimate of the coefficient curve

Author(s)

Yi Zhao, Johns Hopkins University, [email protected];

Xi Luo, Brown University [email protected];

Martin Lindquist, Johns Hopkins University, [email protected];

Brian Caffo, Johns Hopkins University, [email protected]

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Examples


##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)


# consider Fourier basis
fit.boot<-FMA.historical.boot(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))

##################################################
##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)


# consider Fourier basis
fit.boot<-FMA.historical.boot(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))

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

Functional mediation analysis under historical influence model

Description

This function performs functional mediation regression under the historical influence model. Tuning parameter is chosen based on cross-validation.

Usage

FMA.historical.CV(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, lambda1 = NULL, lambda2 = NULL, nfolds = 5)
FMA.historical.CV(Z, M, Y, delta.grid1 = 1, delta.grid2 = 1, delta.grid3 = 1, 
    intercept = TRUE, basis1 = NULL, Ld2.basis1 = NULL, basis2 = NULL, Ld2.basis2 = NULL, 
    basis.type = c("fourier"), nbasis1 = 3, nbasis2 = 3, 
    timeinv = c(0, 1), timegrids = NULL, lambda1 = NULL, lambda2 = NULL, nfolds = 5)

Arguments

`Z`	a data matrix. `Z` is the treatment trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`M`	a data matrix. `M` is the mediator trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`Y`	a data matrix. `Y` is the outcome trajectory in the mediation analysis. The number of rows is the number of subjects, and the number of columns is the number of measured time points.
`delta.grid1`	a number indicates the width of treatment-mediator time interval in the mediator model.
`delta.grid2`	a number indicates the width of treatment-outcome time interval in the outcome model.
`delta.grid3`	a number indicates the width of mediator-outcome time interval in the outcome model.
`intercept`	a logic variable. Default is `TRUE`, an intercept term is included in the regression model.
`basis1`	a data matrix. Basis function on the $s$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis1`	a data matrix. The second derivative of the basis function on the $s$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis2`	a data matrix. Basis function on the $t$ domain used in the functional data analysis. The number of columns is the number of basis function considered. If `basis = NULL`, Fourier basis functions will be generated.
`Ld2.basis2`	a data matrix. The second derivative of the basis function on the $t$ domain. The number of columns is the number of basis function considered. If `Ld2.basis = NULL`, the second derivative of Fourier basis functions will be generated.
`basis.type`	a character of basis function type. Default is Fourier basis (`basis.type = "fourier"`).
`nbasis1`	an integer, the number of basis function on the $s$ domain included. If `basis1` is provided, this argument will be ignored.
`nbasis2`	an integer, the number of basis function on the $t$ domain included. If `basis2` is provided, this argument will be ignored.
`timeinv`	a numeric vector of length two, the time interval considered in the analysis. Default is (0,1).
`timegrids`	a numeric vector of time grids of measurement. If `timegrids = NULL`, it is assumed the between measurement time interval is constant.
`lambda1`	a numeric vector of tuning parameter values on the $s$ domain.
`lambda2`	a numeric vector of tuning parameter values on the $t$ domain.
`nfolds`	a number gives the number of folds in cross-validation.

Details

The historical influence mediation model is

$M(t)=\int_{\Omega_{t}^{1}}Z(s)\alpha(s,t)ds+\epsilon_{1}(t),$

$Y(t)=\int_{\Omega_{t}^{2}}Z(s)\gamma(s,t)ds+\int_{\Omega_{t}^{3}}M(s)\beta(s,t)ds+\epsilon_{2}(t),$

where $\alpha(s,t)$ , $\beta(s,t)$ , $\gamma(s,t)$ are coefficient curves; $\Omega_{t}^{j}=[(t-\delta_{j})\vee 0,t]$ for $j=1,2,3$ . The model coefficient curves are estimated by minimizing the penalized $L_{2}$ -loss. Tuning parameter $\lambda$ controls the smoothness of the estimated curves, and is chosen by cross-validation.

Value

`basis1`	the basis functions on the $s$ domain used in the analysis.
`basis2`	the basis functions on the $t$ domain used in the analysis.
`M`	a list of output for the mediator model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `M` `lambda1`: the chosen $\lambda$ value on the $s$ domain `lambda2`: the chosen $\lambda$ value on the $t$ domain
`Y`	a list of output for the outcome model `coefficient`: the estimated coefficient with respect to the basis function `curve`: the estimated coefficient curve `fitted`: the fitted value of `Y` `lambda1`: the chosen $\lambda$ value on the $s$ domain `lambda2`: the chosen $\lambda$ value on the $t$ domain
`IE`	a list of output for the indirect effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `curve`: the estimated causal curve
`DE`	a list of output for the direct effect comparing $Z_{1}(t)=1$ versus $Z_{0}(t)=0$ `curve`: the estimated causal curve

Author(s)

Yi Zhao, Johns Hopkins University, [email protected];

Xi Luo, Brown University [email protected];

Martin Lindquist, Johns Hopkins University, [email protected];

Brian Caffo, Johns Hopkins University, [email protected]

References

Zhao et al. (2017). Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data. arXiv preprint arXiv:1805.06923.

Examples


##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)

## Not run: 
# consider Fourier basis
fit<-FMA.historical.CV(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))

## End(Not run)
##################################################
##################################################
# Historical influence functional mediation model
data(env.historical)
Z<-get("Z",env.historical)
M<-get("M",env.historical)
Y<-get("Y",env.historical)

## Not run: 
# consider Fourier basis
fit<-FMA.historical.CV(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3,
    intercept=FALSE,timeinv=c(0,300))

## End(Not run)
##################################################

Package 'cfma'

Help Index

Causal Functional Mediation Analysis

Description

Details

Author(s)

Simulated data from the concurrent mediation model

Description

Usage

Format

Details

Examples

Simulated data from the historical influence mediation model

Description

Usage

Format

Details

Examples

Functional mediation analysis under concurrent regression model

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Functional mediation analysis under concurrent regression model with point-wise bootstrap confidence interval

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Functional mediation analysis under concurrent regression model

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Functional mediation analysis under historical influence model

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Functional mediation analysis under historical influence regression model with point-wise bootstrap confidence interval

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Functional mediation analysis under historical influence model

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples