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

Description Usage Arguments Details Value Author(s) References Examples

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

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)

`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.

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.

`alpha`	a list of output for α 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 γ 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 β 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

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

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

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# 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))

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