Functional mediation analysis under concurrent regression model

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

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 coefficientthe 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

References

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

Author

Yi Zhao, Johns Hopkins University, zhaoyi1026@gmail.com;

Xi Luo, Brown University xi.rossi.luo@gmail.com;

Martin Lindquist, Johns Hopkins University, mal2053@gmail.com;

Brian Caffo, Johns Hopkins University, bcaffo@gmail.com

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)

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