FMA.historical function

Functional mediation analysis under historical influence model

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

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

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.

Returns

• 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

References

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

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

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