FMA.historical {cfma}

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)

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),

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.

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, [e-mail];

Xi Luo, Brown University [e-mail];

Martin Lindquist, Johns Hopkins University, [e-mail];

Brian Caffo, Johns Hopkins University, [e-mail]

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)
##################################################