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#'@title Approximate Variance for the Potential Impact Fraction using the
#' approximate method
#'
#'@description Function that calculates approximate variance to the potential
#' impact fraction.
#'
#'@param X Mean value of exposure levels from a cross-sectional random
#' sample.
#'
#'@param Xvar Variance of exposure levels.
#'
#'@param thetahat Estimator (vector or matrix) of \code{theta} for the Relative
#' Risk function.
#'
#'@param thetavar Estimator of variance of \code{thetahat}
#'
#'@param rr Function for Relative Risk which uses parameter \code{theta}.
#' The order of the parameters shound be \code{rr(X, theta)}.
#'
#'
#' **Optional**
#'
#'@param cft Function \code{cft(X)} for counterfactual. Leave empty for
#' the Population Attributable Fraction \code{\link{paf}} where counterfactual
#' is 0 exposure.
#'
#'@param nsim Number of simulations for estimation of variance
#'
#'@param check_thetas Checks that theta parameters are correctly inputed
#'
#'@param check_cft Check if counterfactual function \code{cft} reduces
#' exposure.
#'
#'@param check_xvar Check if it is covariance matrix.
#'
#'@param check_exposure Check that exposure \code{X} is positive and numeric
#'
#'@param check_rr Check that Relative Risk function \code{rr} equals
#' \code{1} when evaluated at \code{0}
#'
#'@param deriv.method.args \code{method.args} for
#' \code{\link[numDeriv]{hessian}}.
#'
#'@param deriv.method \code{method} for \code{\link[numDeriv]{hessian}}.
#' Don't change this unless you know what you are doing.
#'
#'@param check_integrals Check that counterfactual and relative risk's expected
#' values are well defined for this scenario.
#'
#'@param is_paf Force evaluation as paf
#'
#' @author Rodrigo Zepeda-Tello \email{rzepeda17@gmail.com}
#' @author Dalia Camacho-GarcĂa-FormentĂ \email{daliaf172@gmail.com}
#'
#'@seealso \code{\link{pif.variance.linear}} for \code{\link{pif}} variance and
#' \code{\link{pif.confidence}} for confidence intervals of \code{\link{pif}}
#'
#' @examples
#' \dontrun{
#' #Example 1: Exponential Relative risk
#' #--------------------------------------------
#' set.seed(46987)
#' rr <- function(X,theta){exp(X*theta)}
#' cft <- function(X){0.5*X}
#' X <- rbeta(100, 2, 3)
#' Xmean <- data.frame(mean(X))
#' Xvar <- var(X)
#' theta <- 1.2
#' thetavar <- 0.15
#' pif.variance.approximate.linear(Xmean, theta, rr, thetavar, Xvar, cft)
#'
#' #Example 2: Multivariate example
#' #--------------------------------------------
#' X1 <- rnorm(1000,3,.5)
#' X2 <- rnorm(1000,4,1)
#' X <- data.frame(cbind(X1,X2))
#' Xmean <- matrix(colMeans(X), ncol = 2)
#' Xvar <- cov(X)
#' theta <- c(0.12, 0.17)
#' thetavar <- matrix(c(0.001, 0.00001, 0.00001, 0.004), byrow = TRUE, nrow = 2)
#' rr <- function(X, theta){exp(theta[1]*X[,1] + theta[2]*X[,2])}
#' pif.variance.approximate.linear(Xmean, theta, rr, thetavar, Xvar,
#' cft = function(X){cbind(0.5*X[,1],0.4*X[,2])}, check_integrals = FALSE)
#' }
#'@importFrom MASS mvrnorm
#'@importFrom stats weighted.mean
#'@importFrom numDeriv grad
#'@keywords internal
#'@export
pif.variance.approximate.linear <- function(X, thetahat, rr, thetavar, Xvar,
cft = NA,
check_thetas = TRUE, check_cft = TRUE, check_xvar = TRUE, check_rr = TRUE,
check_integrals = TRUE, check_exposure = TRUE, deriv.method.args = list(),
deriv.method = c("Richardson", "complex"), nsim = 1000,
is_paf = FALSE){
#Function for checking that thetas are correctly inputed
if(check_thetas){ check.thetas(thetavar, thetahat, NA, NA, "approximate") }
#Check counterfactual function
if(!is.function(cft)){ is_paf <- TRUE }
#Set X as matrix
if(check_xvar) {Xvar <- check.xvar(Xvar)}
.Xvar <- Xvar
.X <- as.matrix(X)
#Set a minimum for nsim
.nsim <- max(nsim,10)
#Get the expected pif
.pifexp <- function(theta){
pif.approximate(X = .X, Xvar = .Xvar, thetahat = theta, rr = rr, cft = cft,
deriv.method.args = deriv.method.args, deriv.method = deriv.method,
check_exposure = check_exposure, check_rr = check_rr,
check_integrals = check_integrals, is_paf = is_paf)
}
#Get the variance of pif
.pifvar <- function(theta){
#Rewrite functions as functions of X
rr.fun.x <- function(X){
rr(X,theta)
}
rr.fun.cft <- function(X){
cftX <- cft(X)
rr(cftX, theta)
}
#Estimate relative risks
dR0 <- as.matrix(grad(rr.fun.x, .X, method = as.vector(deriv.method)[1],
method.args = deriv.method.args))
R0 <- rr.fun.x(.X)
#Estimate cft
if (is_paf){
dR1 <- 0
R1 <- 1
} else {
dR1 <- as.matrix(grad(rr.fun.cft, .X, method = as.vector(deriv.method)[1],
method.args = deriv.method.args))
R1 <- rr.fun.cft(.X)
}
#Calculate Taylor this way
aux <- ((dR1*R0 - dR0*R1)/(R0^2))
vr <- t(aux)%*%.Xvar%*%(aux)
return(vr)
}
#Get expected value and variance of that
.meanvec <- rep(NA, .nsim)
.varvec <- rep(NA, .nsim)
.thetasim <- mvrnorm(.nsim, thetahat, thetavar, empirical = TRUE)
for (i in 1:.nsim){
.meanvec[i] <- .pifexp(.thetasim[i,])
.varvec[i] <- .pifvar(.thetasim[i,])
}
#Get variance of that
.varpif <- var(.meanvec) + mean(.varvec)
#Return variance
return(.varpif)
}
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