# FMA.historical.CV: Functional mediation analysis under historical influence... In cfma: Causal Functional Mediation Analysis

## Description

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

## Usage

 1 2 3 4 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_{Ω_{t}^{1}}Z(s)α(s,t)ds+ε_{1}(t),

Y(t)=\int_{Ω_{t}^{2}}Z(s)γ(s,t)ds+\int_{Ω_{t}^{3}}M(s)β(s,t)ds+ε_{2}(t),

where α(s,t), β(s,t), γ(s,t) are coefficient curves; Ω_{t}^{j}=[(t-δ_{j})\vee 0,t] for j=1,2,3. The model coefficient curves are estimated by minimizing the penalized L_{2}-loss. Tuning parameter λ 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 λ value on the s domain lambda2: the chosen λ 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 λ value on the s domain lambda2: the chosen λ 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, 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

## References

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

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 ################################################## # 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.CV(Z,M,Y,delta.grid1=3,delta.grid2=3,delta.grid3=3, intercept=FALSE,timeinv=c(0,300)) ################################################## 

cfma documentation built on May 2, 2019, 2:07 a.m.