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R/medsens.R
In mediation: Causal Mediation Analysis
Defines functions
medsens
summary.medsens
print.summary.medsens
plot.medsens
Documented in
medsens
plot.medsens
print.summary.medsens
summary.medsens
#' Sensitivity Analysis for Causal Mediation Effects
#'
#' 'medsens' is used to perform sensitivity analysis on the average causal
#' mediation effects and direct effects for violations of the sequential
#' ignorability assumption. The function takes output from '\code{mediate}' and
#' calculates the true average causal mediation effects and direct effects for
#' different values of the sensitivity parameter representing the degree of the
#' sequential ignorability violation.
#'
#' @details This is the workhorse function for sensitivity analyses for average
#' causal mediation effects. The sensitivity analysis can be used to assess
#' the robustness of the findings from \code{mediate} to the violation of
#' sequential ignorability, the crucial identification assumption necessary
#' for the estimates to be valid. The analysis proceeds by quantifying the
#' degree of sequential ignorability violation as the correlation between the
#' error terms of the mediator and outcome models, and then calculating the
#' true values of the average causal mediation effect for given values of this
#' sensitivity parameter, rho. The original findings are deemed sensitive if
#' the true effects are found to vary widely as function of rho.
#'
#' The sensitivity analysis is only implemented for the following three model
#' combinations: linear mediator and outcome models (both of class 'lm'),
#' binary probit mediator (fitted via 'glm' with family "binomial" and link
#' "probit") and linear outcome models, and linear mediator and binary probit
#' outcome models. In addition, the binary outcome model cannot include a
#' treatment-mediator interaction term. An error is returned if the 'mediate'
#' object in 'x' is based on other model combinations. As of version 3.0, the
#' sensitivity analysis can also be conducted with respect to the average
#' direct effect by setting 'effect.type' to "direct" (or "both" if results
#' for the average causal mediation effect are also desired).
#'
#' Users should note that computation can take significant time for
#' \code{medsens}. Setting 'rho.by' to a larger number significantly decreases
#' computational time, as does decreasing 'eps' (for the linear-linear case)
#' or the number of simulations 'sims' (for the binary-linear and
#' linear-binary cases).
#'
#' @param x an object of class 'mediate', typically an output from the
#' \code{mediate} function.
#' @param rho.by a numeric value between 0 and 1 indicating the increment for
#' the sensitivity parameter, rho.
#' @param sims the number of Monte Carlo draws for the calculation of confidence
#' intervals. Only used in cases where either the mediator or outcome variable
#' is binary.
#' @param eps convergence tolerance parameter for the iterative FGLS. Only used
#' when both the mediator and outcome models are linear.
#' @param effect.type a character string indicating which effect(s) to be
#' analyzed. Default is "indirect".
#'
#' @return \code{medsens} returns an object of class "\code{medsens}", a list
#' containing the following elements. Some of these elements are not available
#' depending on the 'effect.type' argument specified by the user. The output
#' can then be passed to the \code{\link{summary}} (i.e.,
#' \code{\link{summary.medsens}}) and \code{\link{plot}} (i.e.,
#' \code{\link{plot.medsens}}) functions to produce tabular and graphical
#' summaries of the results.
#'
#' \item{d0, d1}{vectors of point estimates for average causal mediation
#' effects under the control and treatment conditions for each value of
#' sensitivity parameter rho.}
#' \item{upper.d0, lower.d0, upper.d1, lower.d1}{vectors of upper and lower
#' confidence limits for average causal mediation effect under the control
#' and treatment conditions for each value of rho.}
#' \item{z0, z1}{vectors of point estimates for average direct effect under
#' the control and treatment conditions for each value of sensitivity
#' parameter rho.}
#' \item{upper.z0, lower.z0, upper.z1, lower.z1}{vectors of upper and lower
#' confidence limits for average direct effect under the control and
#' treatment conditions for each value of rho.}
#' \item{tau}{a vector of point estimates for total effect for each value of
#' rho. Only present when the outcome model is binary.}
#' \item{upper.tau, lower.tau}{vectors of upper and lower confidence limits
#' for total effect. Only present when the outcome model is binary.}
#' \item{nu}{a vector of point estimates for the proportion mediated for each
#' value of rho. Only present when the outcome model is binary.}
#' \item{upper.nu, lower.nu}{vectors of upper and lower confidence limits
#' for the proportion mediated. Only present when the outcome model is binary.}
#' \item{rho}{a numeric vector containing the values of sensitivity
#' parameter rho used.}
#' \item{rho.by}{a numeric value indicating the increment of rho used.}
#' \item{sims}{a numeric value indicating the number of Monte Carlo draws
#' used.}
#' \item{err.cr.d, err.cr.z}{the values of rho with which the average causal
#' mediation and direct effects are zero. Vectors of length two if 'INT' is
#' 'TRUE'; numeric values otherwise.}
#' \item{ind.d0, ind.d1, ind.z0, ind.z1}{vectors of 0s/1s, indicating whether
#' the confidence intervals of d0, d1, z0 and z1 do not cover zero for each
#' value of rho.}
#' \item{R2star.prod}{a numeric vector containing the values of the products
#' of the two "R square stars", representing the proportions of residual
#' variance in the mediator and outcome explained by the hypothesized
#' unobserved confounder. The values correspond to those of rho. See
#' \code{\link{plot.medsens}} for details.}
#' \item{R2tilde.prod}{a numeric vector containing the values of the products
#' of the two "R square tildes", representing the proportions of total
#' variance in the mediator and outcome explained by the hypothesized
#' unobserved confounder. The values correspond to those of rho. See
#' \code{\link{plot.medsens}} for details.}
#' \item{R2star.d.thresh, R2star.z.thresh}{the values of the product of "R
#' square stars" for which the average causal mediation and direct effects are
#' zero, respectively.}
#' \item{R2tilde.d.thresh, R2tilde.z.thresh}{the values of the product of "R
#' square tildes" for which the average causal mediation and direct effects
#' are zero, respectively.}
#' \item{r.square.y, r.square.m}{the usual R square statistics for the
#' outcome and mediator models.}
#' \item{INT}{a logical value indicating whether interaction between the
#' treatment and mediator is allowed in the original mediate object.}
#' \item{conf.level}{the confidence level used.}
#' \item{effect.type}{the 'effect.type' argument used.}
#' \item{type}{a character string indicating the type of the mediator and
#' outcome models used. Currently either "ct" (linear mediator and outcome
#' models), 'bm' (binary mediator and linear outcome models) or 'bo' (linear
#' mediator and binary outcome models).}
#' \item{robustSE}{`TRUE' or `FALSE'.}
#' \item{cluster}{the clusters used.}
#' @author Dustin Tingley, Harvard University,
#' \email{dtingley@@gov.harvard.edu}; Teppei Yamamoto, Massachusetts Institute
#' of Technology, \email{teppei@@mit.edu}; Jaquilyn Waddell-Boie, Princeton
#' University, \email{jwaddell@@princeton.edu}; Kentaro Hirose, Princeton
#' University, \email{hirose@@princeton.edu}; Luke Keele, Penn State
#' University, \email{ljk20@@psu.edu}; Kosuke Imai, Princeton University,
#' \email{kimai@@princeton.edu}.
#'
#' @seealso \code{\link{mediate}}, \code{\link{summary.medsens}},
#' \code{\link{plot.medsens}}.
#'
#' @references Tingley, D., Yamamoto, T., Hirose, K., Imai, K. and Keele, L.
#' (2014). "mediation: R package for Causal Mediation Analysis", Journal of
#' Statistical Software, Vol. 59, No. 5, pp. 1-38.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2011). Unpacking the
#' Black Box of Causality: Learning about Causal Mechanisms from Experimental
#' and Observational Studies, American Political Science Review, Vol. 105, No.
#' 4 (November), pp. 765-789.
#'
#' Imai, K., Keele, L. and Tingley, D. (2010) A General Approach to Causal
#' Mediation Analysis, Psychological Methods, Vol. 15, No. 4 (December), pp.
#' 309-334.
#'
#' Imai, K., Keele, L. and Yamamoto, T. (2010) Identification, Inference, and
#' Sensitivity Analysis for Causal Mediation Effects, Statistical Science,
#' Vol. 25, No. 1 (February), pp. 51-71.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2009) "Causal Mediation
#' Analysis Using R" in Advances in Social Science Research Using R, ed. H. D.
#' Vinod New York: Springer.
#'
#' @export
#' @examples
#' # Examples with JOBS II Field Experiment
#'
#' # **For illustration purposes a small number of simulations are used**
#'
#' data(jobs)
#'
#' ####################################################
#' # Example 1: Binary treatment
#' ####################################################
#' # Fit parametric models
#' b <- lm(job_seek ~ treat + econ_hard + sex + age, data=jobs)
#' c <- lm(depress2 ~ treat + job_seek + econ_hard + sex + age, data=jobs)
#'
#' # Pass model objects through mediate function
#' med.cont <- mediate(b, c, treat="treat", mediator="job_seek", sims=50)
#' # med.cont <- mediate(b, c, treat="treat", mediator="job_seek", sims=50, robustSE = T)
#' # jobs$cluster <- rep(1:30, each = 30)[-1]
#' # med.cont <- mediate(b, c, treat="treat", mediator="job_seek", sims=50, cluster = jobs$cluster)
#'
#' # Pass mediate output through medsens function
#' sens.cont <- medsens(med.cont, rho.by=.1, eps=.01, effect.type="both")
#'
#' # Use summary function to display results
#' summary(sens.cont)
#'
#' # Plot true ACMEs and ADEs as functions of rho
#' par.orig <- par(mfrow = c(2,2))
#' plot(sens.cont, main="JOBS", ylim=c(-.2,.2))
#'
#' # Plot true ACMEs and ADEs as functions of "R square tildes"
#' plot(sens.cont, sens.par="R2", r.type="total", sign.prod="positive")
#' par(par.orig)
#'
#' ####################################################
#' # Example 2: Categorical treatment
#' ####################################################
#' \dontrun{
#'
#' # Purely for illustration, think of educ as a ``treatment''
#'
#' b <- lm(job_seek ~ educ + sex, data=jobs)
#' c <- lm(depress2 ~ educ + job_seek + sex, data=jobs)
#'
#' # compare two categories of educ --- gradwk and somcol
#' med.cont <- mediate(b, c, treat="educ", mediator="job_seek", sims=50,
#' control.value = "gradwk", treat.value = "somcol")
#' sens.cont <- medsens(med.cont, rho.by=.1, eps=.01, effect.type="both")
#' summary(sens.cont)
#' }
#'
medsens <- function(x, rho.by = 0.1, sims = 1000, eps = sqrt(.Machine$double.eps),
effect.type = c("indirect", "direct", "both"))
{
effect.type <- match.arg(effect.type)
if(effect.type == "both"){
etype.vec <- c("indirect", "direct")
} else {
etype.vec <- effect.type
}
model.y <- x$model.y
model.m <- x$model.m
class.y <- class(model.y)[1]
class.m <- class(model.m)[1]
treat <- x$treat
mediator <- x$mediator
INT <- x$INT
low <- (1 - x$conf.level)/2
high <- 1 - low
robustSE <- x$robustSE
cluster <- x$cluster
control.value <- x$control.value
treat.value <- x$treat.value
# Setting Variable labels
## Uppercase letters (e.g. T) = labels in the input matrix
## Uppercase letters + ".out" (e.g. T.out) = labels in the regression output
y.data <- model.frame(model.y)
Y <- colnames(y.data)[1]
if(is.factor(y.data[,treat])){
t.levels <- levels(y.data[,treat])
n.t.levels <- length(t.levels)
## When treatment = more than two categories
## control.value => 1st factor level (baseline category)
## treat.value => 2nd factor level
## run regression again with this new data set
if(n.t.levels != 2){
number <- 1:n.t.levels
t.levels.mat <- data.frame(number, t.levels)
c.num <- t.levels.mat$number[t.levels.mat$t.levels == control.value]
t.num <- t.levels.mat$number[t.levels.mat$t.levels == treat.value]
number.new <- c(c.num, t.num, number[-c(c.num, t.num)])
y.data[, treat] <- factor(y.data[, treat], levels(y.data[, treat])[number.new])
model.y <- update(model.y, data = y.data)
}
t.levels <- levels(y.data[,treat])
cat.c <- t.levels[1]
cat.t <- t.levels[2]
} else {
cat.c <- NULL
cat.t <- NULL
}
T.out <- paste(treat, cat.t, sep="")
m.data <- model.frame(model.m)
if(is.factor(m.data[,treat])){
t.levels <- levels(m.data[,treat])
n.t.levels <- length(t.levels)
## When treatment = more than two categories
## control.value => 1st factor level (baseline category)
## treat.value => 2nd factor level
## run regression again with this new data set
if(n.t.levels != 2){
number <- 1:n.t.levels
t.levels.mat <- data.frame(number, t.levels)
c.num <- t.levels.mat$number[t.levels.mat$t.levels == control.value]
t.num <- t.levels.mat$number[t.levels.mat$t.levels == treat.value]
number.new <- c(c.num, t.num, number[-c(c.num, t.num)])
m.data[, treat] <- factor(m.data[, treat], levels(m.data[, treat])[number.new])
model.m <- update(model.m, data = m.data)
}
}
if(is.factor(y.data[,mediator])){
m.levels <- levels(y.data[,mediator])
if(length(m.levels) != 2){
stop("mediator with more than two categories currently not supported")
}
cat.m0 <- m.levels[1]
cat.m1 <- m.levels[2]
} else {
cat.m0 <- NULL
cat.m1 <- NULL
}
M.out <- paste(mediator, cat.m1, sep="")
if(INT){
if(paste(treat,mediator,sep=":") %in% attr(model.y$terms,"term.labels")){ # T:M
TM.out <- paste(T.out, M.out, sep=":")
TM <- paste(treat, mediator, sep=":")
} else { # M:T
TM.out <- paste(M.out, T.out, sep=":")
TM <- paste(mediator, treat, sep=":")
}
}
## Setting Up Sensitivity Parameters
rho <- seq(-1 + rho.by, 1 - rho.by, rho.by)
R2star.prod <- rho^2
## Extracting weights
weights <- model.weights(y.data)
if(is.null(weights)){
weights <- rep(1, nrow(y.data))
}
## Initializing Containers
d0 <- d1 <- z0 <- z1 <- nu0 <- nu1 <- tau <-
upper.d0 <- upper.d1 <- upper.z0 <- upper.z1 <- upper.nu0 <- upper.nu1 <- upper.tau <-
lower.d0 <- lower.d1 <- lower.z0 <- lower.z1 <- lower.nu0 <- lower.nu1 <- lower.tau <-
ind.d0 <- ind.d1 <- ind.z0 <- ind.z1 <- err.cr.d <- err.cr.z <-
R2star.d.thresh <- R2tilde.d.thresh <- R2star.z.thresh <- R2tilde.z.thresh <- NULL
getvcov <- function(dat, fm, cluster){
## Compute cluster robust standard errors
## fm is the model object
cluster <- factor(cluster) # remove missing levels and NA
M <- nlevels(cluster)
N <- sum(!is.na(cluster))
K <- fm$rank
dfc <- (M/(M-1))*((N-1)/(N-K))
uj <- apply(estfun(fm),2, function(x) tapply(x, cluster, sum));
dfc*sandwich(fm, meat. = crossprod(uj)/N)
}
#########################################################
## CASE 1: Continuous Outcome + Continuous Mediator
#########################################################
if(class.y=="lm" & class.m=="lm") {
type <- "ct"
d0 <- d1 <- z0 <- z1 <- matrix(NA, length(rho), 1)
d0.var <- d1.var <- z0.var <- z1.var <- matrix(NA, length(rho), 1)
for(i in 1:length(rho)){
e.cor <- rho[i]
b.dif <- 1
# Stacked Equations
m.mat <- model.matrix(model.m) * weights^(1/2)
y.mat <- model.matrix(model.y) * weights^(1/2)
m.k <- ncol(m.mat)
m.n <- nrow(m.mat)
y.k <- ncol(y.mat)
y.n <- nrow(y.mat)
n <- y.n
m.zero <- matrix(0, m.n, y.k)
y.zero <- matrix(0, y.n, m.k)
X.1 <- cbind(m.mat, m.zero)
X.2 <- cbind(y.zero, y.mat)
X <- rbind(X.1, X.2)
Y.c <- as.matrix(c(model.response(model.frame(model.m)),
model.response(model.frame(model.y)))) * weights^(1/2)
# Estimates of OLS Start Values
inxx <- solve(crossprod(X))
b.ols <- inxx %*% crossprod(X,Y.c)
b.tmp <- b.ols
while(abs(b.dif) > eps){
e.hat <- as.matrix(Y.c - (X %*% b.tmp))
e.1 <- e.hat[1:n]
e.2 <- e.hat[(n+1):(2*n)]
sd.1 <- sd(e.1)
sd.2 <- sd(e.2)
omega <- matrix(NA, 2,2)
omega[1,1] <- crossprod(e.1)/(n-1)
omega[2,2] <- crossprod(e.2)/(n-1)
omega[2,1] <- e.cor*sd.1*sd.2
omega[1,2] <- e.cor*sd.1*sd.2
I <- Diagonal(n)
omega.i <- solve(omega)
v.i <- kronecker(omega.i, I)
Xv.i <- t(X) %*% v.i
X.sur <- Xv.i %*% X
b.sur <- solve(X.sur) %*% Xv.i %*% Y.c
# Variance-Covariance Matrix of SUR
if(robustSE){
meat <- 0
for (ii in 1:n){
Xi.med <- X.1[ii, ]
Xi.out <- X.2[ii, ]
Xi <- rbind(Xi.med, Xi.out)
ui <- as.matrix(c(e.1[ii], e.2[ii]))
meat <- t(Xi) %*% solve(omega) %*% ui %*% t(ui) %*% solve(omega) %*% Xi + meat
}
v.cov <- solve(X.sur) %*% meat %*% solve(X.sur)
} else if(!is.null(cluster)){
meat <- 0
for (mm in unique(cluster)) {
X.med.m <- matrix(X.1[cluster == mm, ],, ncol(X.1))
X.out.m <- matrix(X.2[cluster == mm, ],, ncol(X.2))
e.1.m <- e.1[cluster == mm]
e.2.m <- e.2[cluster == mm]
half.meat <- 0
for (ii in 1:nrow(X.med.m)){
Xm <- rbind(X.med.m[ii, ], X.out.m[ii, ])
half.meat <- t(Xm) %*% solve(omega) %*% as.matrix(c(e.1.m[ii], e.2.m[ii])) + half.meat
}
meat <- half.meat %*% t(half.meat) + meat
}
v.cov <- solve(X.sur) %*% meat %*% solve(X.sur)
} else {
v.cov <- solve(X.sur)
}
b.old <- b.tmp
b.dif <- sum((b.sur - b.old)^2)
b.tmp <- b.sur
}
# Name Elements - Extract Quantities
m.names <- names(model.m$coef)
y.names <- names(model.y$coef)
b.names <- c(m.names, y.names)
row.names(b.sur) <- b.names
m.coefs <- as.matrix(b.sur[1:m.k])
y.coefs <- as.matrix(b.sur[(m.k+1):(m.k+y.k)])
row.names(m.coefs) <- m.names
row.names(y.coefs) <- y.names
rownames(v.cov) <- b.names
colnames(v.cov) <- b.names
v.m <- v.cov[1:m.k,1:m.k]
v.y <- v.cov[(m.k+1):(m.k+y.k),(m.k+1):(m.k+y.k)]
m.coef.treat <- m.coefs[T.out, ]
y.coef.treat <- y.coefs[T.out, ]
y.coef.med <- y.coefs[M.out, ]
v.m.treat <- v.m[T.out, T.out]
v.y.treat <- v.y[T.out, T.out]
v.y.med <- v.y[M.out, M.out]
if(INT){
y.coef.TM <- y.coefs[TM.out, ]
v.y.TM <- v.y[TM.out, TM.out]
}
if(INT && ("direct" %in% etype.vec)){
m.bar <- apply(model.matrix(model.m), 2, weighted.mean, w=weights)
m.covts.ind <- -(match(c("(Intercept)", paste(T.out)), names(model.m$coefficients), NULL))
m.coef.covts <- m.coefs[m.covts.ind, ]
m.bar.covts <- m.bar[m.covts.ind]
m.intercept <- m.coefs["(Intercept)", ]
v.m.covts <- v.m[m.covts.ind, m.covts.ind]
v.m.intercept <- v.m["(Intercept)", "(Intercept)"]
v.m.int.treat <- v.m["(Intercept)", T.out]
if(length(m.coef.covts) != 0){
model.m.bar.z0 <- m.intercept + sum(crossprod(m.coef.covts, m.bar.covts))
v.model.m.bar.z0 <- v.m.intercept + sum(crossprod(m.bar.covts, v.m.covts) %*% m.bar.covts)
model.m.bar.z1 <- m.intercept + m.coef.treat + sum(m.coef.covts * m.bar.covts)
v.m.treat.covts <- v.m[T.out, m.covts.ind]
v.m.int.covts <- v.m["(Intercept)", m.covts.ind]
v.model.m.bar.z1 <- ( v.m.intercept + v.m.treat +
sum(crossprod(m.bar.covts, v.m.covts) %*% m.bar.covts) +
2 * ( v.m.int.treat +
sum(crossprod(m.bar.covts, v.m.treat.covts)) +
sum(crossprod(m.bar.covts, v.m.int.covts)) ) )
} else {
model.m.bar.z0 <- m.intercept
v.model.m.bar.z0 <- v.m.intercept
model.m.bar.z1 <- m.intercept + m.coef.treat
v.model.m.bar.z1 <- v.m.intercept + v.m.treat + 2 * v.m.int.treat
}
}
# Save Estimates
if(INT){
if("indirect" %in% etype.vec){
d0[i,] <- m.coef.treat * y.coef.med
d1[i,] <- m.coef.treat * (y.coef.med + y.coef.TM)
}
if("direct" %in% etype.vec){ # direct effects with interaction
if(length(m.coef.covts) != 0){ # with covariates
z0[i,] <- ( y.coef.treat +
y.coef.TM * (m.intercept + sum(crossprod(m.coef.covts, m.bar.covts))) )
z1[i,] <- ( y.coef.treat +
y.coef.TM * (m.intercept + m.coef.treat +
sum(crossprod(m.coef.covts, m.bar.covts))) )
} else { # no covariates
z0[i,] <- y.coef.treat + y.coef.TM * m.intercept
z1[i,] <- y.coef.treat + y.coef.TM * (m.intercept + m.coef.treat)
}
}
} else {
if("indirect" %in% etype.vec){
d0[i,] <- d1[i,] <- m.coef.treat * y.coef.med
}
if("direct" %in% etype.vec) {
z0[i,] <- z1[i,] <- y.coef.treat
}
}
# Save Variance Estimates
if(INT){
if("indirect" %in% etype.vec){
d0.var[i,] <- y.coef.med^2 * v.m.treat + m.coef.treat^2 * v.y.med
d1.var[i,] <- ( (y.coef.med + y.coef.TM)^2 * v.m.treat +
m.coef.treat^2 * (v.y.med + v.y.TM + 2 * v.y[M.out, TM.out]) )
}
if("direct" %in% etype.vec){
z0.var[i,] <- ( v.y.treat + model.m.bar.z0^2 * v.y.TM +
y.coef.TM^2 * v.model.m.bar.z0 +
v.y.TM * v.model.m.bar.z0 +
2 * model.m.bar.z0 * v.y[T.out, TM.out] )
z1.var[i,] <- ( v.y.treat + model.m.bar.z1^2 * v.y.TM +
y.coef.TM^2 * v.model.m.bar.z1 +
v.y.TM * v.model.m.bar.z1 +
2 * model.m.bar.z1 * v.y[T.out,TM.out] )
}
} else {
if("indirect" %in% etype.vec){
d0.var[i,] <- d1.var[i,] <- (m.coef.treat^2 * v.y.med) + (y.coef.med^2 * v.m.treat)
}
if("direct" %in% etype.vec){
z0.var[i,] <- z1.var[i,] <- v.y.treat
}
}
rm(b.sur, m.coefs, y.coefs, v.cov, v.m, v.y)
} # END of Rho Loop
if("indirect" %in% etype.vec){
upper.d0 <- d0 + qnorm(high) * sqrt(d0.var)
lower.d0 <- d0 + qnorm(low) * sqrt(d0.var)
upper.d1 <- d1 + qnorm(high) * sqrt(d1.var)
lower.d1 <- d1 + qnorm(low) * sqrt(d1.var)
ind.d0 <- as.numeric(lower.d0 < 0 & upper.d0 > 0)
ind.d1 <- as.numeric(lower.d1 < 0 & upper.d1 > 0)
}
if("direct" %in% etype.vec){
upper.z0 <- z0 + qnorm(high) * sqrt(z0.var)
lower.z0 <- z0 + qnorm(low) * sqrt(z0.var)
upper.z1 <- z1 + qnorm(high) * sqrt(z1.var)
lower.z1 <- z1 + qnorm(low) * sqrt(z1.var)
ind.z0 <- as.numeric(lower.z0 < 0 & upper.z0 > 0)
ind.z1 <- as.numeric(lower.z1 < 0 & upper.z1 > 0)
}
# Save R2 tilde values
r.sq.m <- summary(model.m)$r.squared
r.sq.y <- summary(model.y)$r.squared
R2tilde.prod <- rho^2 * (1 - r.sq.m) * (1 - r.sq.y)
## END OF CASE 1: Continuous Outcome + Continuous Mediator
} else
#########################################################
## CASE 2: Continuous Outcome + Binary Mediator
#########################################################
if (class.y == "lm" & class.m == "glm") {
type <- "bm"
if(model.m$family$link == "logit"){
stop("medsens is only valid for the probit link")
}
## Variable values (LABEL.value)
Y.value <- y.data[,1]
y.k <- length(model.y$coef)
# Step 1: Pre-loop computations
# Step 1-1: Sample M model parameters
Mmodel.coef <- model.m$coef
if(robustSE){
Mmodel.var.cov <- vcovHC(model.m)
} else if(!is.null(cluster)) {
dta <- merge(m.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.m, data=dta)
MModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Mmodel.var.cov <- vcov(model.m)
}
Mmodel.coef.sim <- rmvnorm(sims, mean=Mmodel.coef, sigma=Mmodel.var.cov)
# simulate M-model parameters
# Step 1-2: Sample lambda_0 and lambda_1; lambdas are (n x sims) matrix
m.mat <- m.mat.1 <- m.mat.0 <- model.matrix(model.m)
m.mat.1[,T.out] <- 1 # M-model matrix with t=1
m.mat.0[,T.out] <- 0 # M-model matrix with t=0
mu.sim <- tcrossprod(m.mat, Mmodel.coef.sim) # E(M|T,X)
mu.1.sim <- tcrossprod(m.mat.1, Mmodel.coef.sim) # E(M|T=1,X)
mu.0.sim <- tcrossprod(m.mat.0, Mmodel.coef.sim) # E(M|T=0,X)
# Step 1-3: Define lambda function (inverse Mill's ratio)
lambda <- function(mmodel, mcoef) {
mu <- model.matrix(mmodel) %*% mcoef
m <- mmodel$y # this is M
return((m*dnorm(-mu)-(1-m)*dnorm(-mu))/(m*pnorm(mu)+(1-m)*pnorm(-mu)))
}
# Step 2: Rho loop
## Step 2-0: Initialize containers
if("indirect" %in% etype.vec){
d0 <- d1 <- upper.d0 <- upper.d1 <-
lower.d0 <- lower.d1 <- ind.d0 <- ind.d1 <- matrix(NA, length(rho), 1)
}
if("direct" %in% etype.vec){
z0 <- z1 <- upper.z0 <- upper.z1 <-
lower.z0 <- lower.z1 <- ind.z0 <- ind.z1 <- matrix(NA, length(rho), 1)
}
Ymodel.coef.sim <- matrix(NA, sims, y.k)
colnames(Ymodel.coef.sim) <- names(model.y$coef)
sigma.3.sim <- rep(NA, sims)
d0.sim <- d1.sim <- z0.sim <- z1.sim <- rep(NA, sims)
## START OF RHO LOOP
for(i in 1:length(rho)){
## START OF SIMULATION LOOP
for(k in 1:sims){
## Step 2-1: Obtain the initial Y model with the correction term
adj <- lambda(model.m, Mmodel.coef.sim[k,]) * rho[i] # the adjustment term
w <- 1 - rho[i]^2*lambda(model.m, Mmodel.coef.sim[k,]) *
(lambda(model.m, Mmodel.coef.sim[k,]) + mu.sim[,k])
y.data.adj <- data.frame(y.data, w, adj, weights)
model.y.adj <- update(model.y, as.formula(paste(". ~ . + adj")),
weights = w*weights, data = y.data.adj)
sigma.3 <- summary(model.y.adj)$sigma
## Step 2-2: Update the Y model via Iterative FGLS
sigma.dif <- 1
while(abs(sigma.dif) > eps){
Y.star <- Y.value - sigma.3 * adj
y.data.star <- data.frame(Y.star, y.data.adj)
model.y.update <- update(model.y, as.formula(paste("Y.star ~ .")),
weights = w*weights, data = y.data.star)
sigma.3.temp <- summary(model.y.update)$sigma
sigma.dif <- sigma.3.temp - sigma.3
sigma.3 <- sigma.3.temp
}
## Step 2-3: Simulate Y model parameters
Ymodel.coef <- model.y.update$coef
if(robustSE){
Ymodel.var.cov <- vcovHC(model.y.update)
} else if(!is.null(cluster)) {
dta <- merge(y.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.y.update, data=dta)
YModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Ymodel.var.cov <- vcov(model.y.update)
}
Ymodel.coef.sim[k,] <- rmvnorm(1, mean=Ymodel.coef, sigma=Ymodel.var.cov)
# draw one simulation sample of Y-model parameters for each k
## Step 2-4: Simulate ACMEs; means are over observations
if(INT){
if("indirect" %in% etype.vec){
d1.sim[k] <- weighted.mean( (Ymodel.coef.sim[k,M.out] + Ymodel.coef.sim[k,TM.out]) *
(pnorm(mu.1.sim[,k]) - pnorm(mu.0.sim[,k])), w = weights )
d0.sim[k] <- weighted.mean( (Ymodel.coef.sim[k,M.out]) * (pnorm(mu.1.sim[,k]) -
pnorm(mu.0.sim[,k])), w = weights )
}
if("direct" %in% etype.vec){
z1.sim[k] <- weighted.mean( sum(Ymodel.coef.sim[k,c(T.out,TM.out)]) *
pnorm(mu.1.sim[,k]) + Ymodel.coef.sim[k,T.out] *
pnorm(-mu.1.sim[k]), w = weights )
z0.sim[k] <- weighted.mean( sum(Ymodel.coef.sim[k,c(T.out,TM.out)]) *
pnorm(mu.0.sim[,k]) + Ymodel.coef.sim[k,T.out] *
pnorm(-mu.0.sim[k]), w = weights )
}
} else {
if("indirect" %in% etype.vec){
d0.sim[k] <- d1.sim[k] <- weighted.mean( Ymodel.coef.sim[k,M.out] *
(pnorm(mu.1.sim[,k]) - pnorm(mu.0.sim[,k])), w = weights )
}
if("direct" %in% etype.vec){
z0.sim[k] <- z1.sim[k] <- Ymodel.coef.sim[k,T.out]
}
}
## END OF SIMULATION LOOP
}
## Step 2-5: Compute Outputs
if("indirect" %in% etype.vec){
d0[i] <- mean(d0.sim)
d1[i] <- mean(d1.sim)
upper.d0[i] <- quantile(d0.sim, high)
upper.d1[i] <- quantile(d1.sim, high)
lower.d0[i] <- quantile(d0.sim, low)
lower.d1[i] <- quantile(d1.sim, low)
ind.d0[i] <- as.numeric(lower.d0[i] < 0 & upper.d0[i] > 0)
ind.d1[i] <- as.numeric(lower.d1[i] < 0 & upper.d1[i] > 0)
}
if("direct" %in% etype.vec){
z0[i] <- mean(z0.sim)
z1[i] <- mean(z1.sim)
upper.z0[i] <- quantile(z0.sim, high)
upper.z1[i] <- quantile(z1.sim, high)
lower.z0[i] <- quantile(z0.sim, low)
lower.z1[i] <- quantile(z1.sim, low)
ind.z0[i] <- as.numeric(lower.z0[i] < 0 & upper.z0[i] > 0)
ind.z1[i] <- as.numeric(lower.z1[i] < 0 & upper.z1[i] > 0)
}
## END OF RHO LOOP
}
# Save R2 tilde values
fitted <- m.mat %*% Mmodel.coef
var.mstar <- var(fitted)
r.sq.m <- var.mstar/(1+var.mstar)
r.sq.y <- summary(model.y)$r.squared
R2tilde.prod <- rho^2 * (1 - r.sq.m) * (1 - r.sq.y)
## END OF CASE 2: Continuous Outcome + Binary Mediator
} else
#########################################################
## CASE 3: Binary Outcome + Continuous Mediator
#########################################################
if(class.y=="glm" & class.m=="lm") {
if(INT){
stop("sensitivity analysis is not available for binary outcome with interactions")
}
if(model.y$family$link == "logit"){
stop("medsens is only valid for the probit link")
}
type <- "bo"
# Step 1: Obtain Model Parameters
## Step 1-1: Simulate M model parameters
Mmodel.coef <- model.m$coef
if(robustSE){
Mmodel.var.cov <- vcovHC(model.m)
} else if(!is.null(cluster)) {
dta <- merge(m.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.m, data=dta)
MModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Mmodel.var.cov <- vcov(model.m)
}
m.k <- length(Mmodel.coef)
Mmodel.coef.sim <- rmvnorm(sims, mean = Mmodel.coef, sigma = Mmodel.var.cov)
beta2.sim <- Mmodel.coef.sim[, T.out]
sigma.2 <- summary(model.m)$sigma
sig2.shape <- model.m$df/2
sig2.invscale <- (model.m$df/2) * sigma.2^2
sigma.2.sim <- sqrt(1 / rgamma(sims, shape = sig2.shape, scale = 1/sig2.invscale))
## Step 1-2: Simulate Y model parameters
Ymodel.coef <- model.y$coef
if(robustSE){
Ymodel.var.cov <- vcovHC(model.y)
} else if(!is.null(cluster)) {
dta <- merge(y.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.y, data=dta)
YModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Ymodel.var.cov <- vcov(model.y)
}
y.k <- length(Ymodel.coef)
Ymodel.coef.sim <- rmvnorm(sims, mean = Ymodel.coef, sigma = Ymodel.var.cov)
colnames(Ymodel.coef.sim) <- names(Ymodel.coef)
gamma.tilde <- Ymodel.coef.sim[, M.out]
# Step 2: Compute ACME via the procedure in IKT
## Step 2-1: Estimate Error Correlation from inconsistent estimate of Y on M
rho12.sim <- (sigma.2.sim * gamma.tilde) / (1 + sqrt(sigma.2.sim^2 * gamma.tilde^2))
## Step 2-2: Calculate alpha_1, beta_1 and xi_1
YTmodel.coef.sim <- ( Ymodel.coef.sim[, !colnames(Ymodel.coef.sim) %in% M.out] *
sqrt(1-rho12.sim^2) %x% t(rep(1,y.k-1)) +
Mmodel.coef.sim * (rho12.sim/sigma.2.sim) %x% t(rep(1, y.k-1)) )
## Step 2-3: Calculate Gamma
## Data matrices for the Y model less M
y.mat.1 <- y.mat.0 <- model.matrix(model.y)[, !colnames(model.matrix(model.y)) %in% M.out]
y.mat.1[,T.out] <- 1
y.mat.0[,T.out] <- 0
## Initialize objects before the rho loop
d0 <- d1 <- z0 <- z1 <-
upper.d0 <- upper.d1 <- lower.d0 <- lower.d1 <-
upper.z0 <- upper.z1 <- lower.z0 <- lower.z1 <-
ind.d0 <- ind.d1 <- ind.z0 <- ind.z1 <-
tau <- nu0 <- nu1 <-
upper.tau <- upper.nu0 <- upper.nu1 <-
lower.tau <- lower.nu0 <- lower.nu1 <- rep(NA, length(rho))
d0.sim <- d1.sim <- z0.sim <- z1.sim <-
sigma.1star.sim <- beta3.sim <-
tau.sim <- nu0.sim <- nu1.sim <- rep(NA, sims)
## START OF RHO LOOP
for(i in 1:length(rho)){
gamma.sim <- (-rho[i] + rho12.sim * sqrt((1 - rho[i]^2)/(1 - rho12.sim^2)))/sigma.2.sim
for(k in 1:sims){
sigma.1star.sim[k] <- sqrt(gamma.sim[k]^2 *
sigma.2.sim[k]^2 + 2 * gamma.sim[k] * rho[i] * sigma.2.sim[k] + 1)
YTcoef.k <- YTmodel.coef.sim[k,]
if("indirect" %in% etype.vec){
d0.sim[k] <- weighted.mean( pnorm(y.mat.0 %*% YTcoef.k +
gamma.sim[k] * beta2.sim[k] / sigma.1star.sim[k]) -
pnorm(y.mat.0 %*% YTcoef.k), w = weights )
d1.sim[k] <- weighted.mean( pnorm(y.mat.1 %*% YTcoef.k) -
pnorm(y.mat.1 %*% YTcoef.k -
gamma.sim[k] * beta2.sim[k] / sigma.1star.sim[k]),
w = weights )
tau.sim[k] <- weighted.mean( pnorm(y.mat.1 %*% YTcoef.k) -
pnorm(y.mat.0 %*% YTcoef.k), w = weights )
nu0.sim[k] <- d0.sim[k]/tau.sim[k]
nu1.sim[k] <- d1.sim[k]/tau.sim[k]
}
if("direct" %in% etype.vec){
beta3.sim[k] <- sigma.1star.sim[k] * YTcoef.k[T.out] -
beta2.sim[k] * gamma.sim[k]
z0.sim[k] <- weighted.mean( pnorm(y.mat.0 %*% YTcoef.k +
beta3.sim[k] / sigma.1star.sim[k]) -
pnorm(y.mat.0 %*% YTcoef.k), w = weights )
z1.sim[k] <- weighted.mean( pnorm(y.mat.1 %*% YTcoef.k) -
pnorm(y.mat.1 %*% YTcoef.k -
beta3.sim[k] / sigma.1star.sim[k]), w = weights )
}
}
## Step 2-4: Compute Outputs
if("indirect" %in% etype.vec){
d0[i] <- mean(d0.sim) # ACME(t=0)
d1[i] <- mean(d1.sim) # ACME(t=1)
upper.d0[i] <- quantile(d0.sim, high)
upper.d1[i] <- quantile(d1.sim, high)
lower.d0[i] <- quantile(d0.sim, low)
lower.d1[i] <- quantile(d1.sim, low)
tau[i] <- mean(tau.sim) # ATE
nu0[i] <- mean(nu0.sim) # Proportion Mediated (t=0)
nu1[i] <- mean(nu1.sim) # Proportion Mediated (t=0)
upper.tau[i] <- quantile(tau.sim, high)
upper.nu0[i] <- quantile(nu0.sim, high)
upper.nu1[i] <- quantile(nu1.sim, high)
lower.tau[i] <- quantile(tau.sim, low)
lower.nu0[i] <- quantile(nu0.sim, low)
lower.nu1[i] <- quantile(nu1.sim, low)
ind.d0[i] <- as.numeric(lower.d0[i] < 0 & upper.d0[i] > 0)
ind.d1[i] <- as.numeric(lower.d1[i] < 0 & upper.d1[i] > 0)
}
if("direct" %in% etype.vec){
z0[i] <- mean(z0.sim) # ACME(t=0)
z1[i] <- mean(z1.sim) # ACME(t=1)
upper.z0[i] <- quantile(z0.sim, high)
upper.z1[i] <- quantile(z1.sim, high)
lower.z0[i] <- quantile(z0.sim, low)
lower.z1[i] <- quantile(z1.sim, low)
ind.z0[i] <- as.numeric(lower.z0[i] < 0 & upper.z0[i] > 0)
ind.z1[i] <- as.numeric(lower.z1[i] < 0 & upper.z1[i] > 0)
}
## END OF RHO LOOP
}
# Save R2 tilde values
r.sq.m <- summary(model.m)$r.squared
y.mat <- model.matrix(model.y)
fitted <- y.mat %*% Ymodel.coef
var.ystar <- var(fitted)
r.sq.y <- var.ystar/(1 + var.ystar)
R2tilde.prod <- rho^2 * (1 - r.sq.m) * (1 - r.sq.y)
## END OF CASE 3: Binary Outcome + Continuous Mediator
} else {
stop("mediate object fitted with non-supported model combinations")
}
#########################################################
## Compute Output
#########################################################
# Calculate rho at which ACME=0
if(INT || type=="bo"){
if("indirect" %in% etype.vec){
ii <- which(abs(d0 - 0) == min(abs(d0 - 0)))
kk <- which(abs(d1 - 0) == min(abs(d1 - 0)))
err.cr.1.d <- rho[ii]
err.cr.2.d <- rho[kk]
err.cr.d <- c(err.cr.1.d, err.cr.2.d)
}
if("direct" %in% etype.vec){
ii.z <- which(abs(z0 - 0) == min(abs(z0 - 0)))
kk.z <- which(abs(z1 - 0) == min(abs(z1 - 0)))
err.cr.1.z <- rho[ii.z]
err.cr.2.z <- rho[kk.z]
err.cr.z <- c(err.cr.1.z, err.cr.2.z)
}
} else {
if("indirect" %in% etype.vec){
ii <- which(abs(d0 - 0) == min(abs(d0 - 0)))
err.cr.d <- rho[ii]
}
if("direct" %in% etype.vec){
ii.z <- which(abs(z0 - 0) == min(abs(z0 - 0)))
err.cr.z <- rho[ii.z]
}
}
if("indirect" %in% etype.vec){
R2star.d.thresh <- err.cr.d^2
R2tilde.d.thresh <- err.cr.d^2 * (1 - r.sq.m) * (1 - r.sq.y)
}
if("direct" %in% etype.vec){
R2star.z.thresh <- err.cr.z^2
R2tilde.z.thresh <- err.cr.z^2 * (1 - r.sq.m) * (1 - r.sq.y)
}
out <- list(
d0=d0, d1=d1,
upper.d0=upper.d0, lower.d0=lower.d0,
upper.d1=upper.d1, lower.d1=lower.d1,
z0=z0, z1=z1,
upper.z0=upper.z0, lower.z0=lower.z0,
upper.z1=upper.z1, lower.z1=lower.z1,
tau=tau, upper.tau=upper.tau, lower.tau=lower.tau,
nu0=nu0, nu1=nu1, upper.nu0=upper.nu0, upper.nu1=upper.nu1,
lower.nu0=lower.nu0, lower.nu1=lower.nu1,
rho = rho, rho.by=rho.by, sims=sims,
err.cr.d=err.cr.d, err.cr.z=err.cr.z,
ind.d0=ind.d0, ind.d1=ind.d1, ind.z0=ind.z0, ind.z1=ind.z1,
R2star.prod=R2star.prod, R2tilde.prod=R2tilde.prod,
R2star.d.thresh=R2star.d.thresh, R2tilde.d.thresh=R2tilde.d.thresh,
R2star.z.thresh=R2star.z.thresh, R2tilde.z.thresh=R2tilde.z.thresh,
r.square.y=r.sq.y, r.square.m=r.sq.m,
INT=INT, conf.level = x$conf.level, effect.type=effect.type, type=type,
robustSE = robustSE, cluster = cluster)
class(out) <- "medsens"
out
}
#' Summarizing Results from Sensitivity Analysis for Causal Mediation Effects
#'
#' Functions to report results from the sensitivity analysis for causal
#' mediation effects via \code{\link{medsens}} in a tabular form.
#'
#' @aliases summary.medsens print.summary.medsens
#'
#' @param object output from medsens function.
#' @param x output from summary.medsens function.
#' @param ... additional arguments affecting the summary produced.
#'
#' @author Dustin Tingley, Harvard University,
#' \email{dtingley@@gov.harvard.edu}; Teppei Yamamoto, Massachusetts Institute
#' of Technology, \email{teppei@@mit.edu}; Jaquilyn Waddell-Boie, Princeton
#' University, \email{jwaddell@@princeton.edu}; Luke Keele, Penn State
#' University, \email{ljk20@@psu.edu}; Kosuke Imai, Princeton University,
#' \email{kimai@@princeton.edu}.
#'
#' @seealso \code{\link{medsens}}, \code{\link{summary}}.
#'
#' @references Tingley, D., Yamamoto, T., Hirose, K., Imai, K. and Keele, L.
#' (2014). "mediation: R package for Causal Mediation Analysis", Journal of
#' Statistical Software, Vol. 59, No. 5, pp. 1-38.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2011). Unpacking the
#' Black Box of Causality: Learning about Causal Mechanisms from Experimental
#' and Observational Studies, American Political Science Review, Vol. 105, No.
#' 4 (November), pp. 765-789.
#'
#' Imai, K., Keele, L. and Tingley, D. (2010) A General Approach to Causal
#' Mediation Analysis, Psychological Methods, Vol. 15, No. 4 (December), pp.
#' 309-334.
#'
#' Imai, K., Keele, L. and Yamamoto, T. (2010) Identification, Inference, and
#' Sensitivity Analysis for Causal Mediation Effects, Statistical Science,
#' Vol. 25, No. 1 (February), pp. 51-71.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2009) "Causal Mediation
#' Analysis Using R" in Advances in Social Science Research Using R, ed. H. D.
#' Vinod New York: Springer.
#'
#' @export
summary.medsens <- function(object, ...)
structure(object, class = c("summary.medsens", class(object)))
#' @rdname summary.medsens
#' @export
print.summary.medsens <- function(x, ...){
etype.vec <- x$effect.type
if(etype.vec == "both"){
etype.vec <- c("indirect","direct")
}
if(x$INT || x$type=="bo"){
if("indirect" %in% etype.vec){
tab.d0 <- cbind(round(x$rho,4), round(x$d0,4), round(x$lower.d0,4),
round(x$upper.d0, 4), x$ind.d0, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.d0)==1){
tab.d0 <- as.matrix(tab.d0[x$ind.d0==1, -5])
tab.d0 <- t(tab.d0)
} else {
tab.d0 <- tab.d0[x$ind.d0==1, -5]
}
colnames(tab.d0) <- c("Rho", "ACME(control)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.d0) <- NULL
tab.d1 <- cbind(round(x$rho,4), round(x$d1,4), round(x$lower.d1,4),
round(x$upper.d1, 4), x$ind.d1, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.d1)==1){
tab.d1 <- as.matrix(tab.d1[x$ind.d1==1, -5])
tab.d1 <- t(tab.d1)
} else {
tab.d1 <- tab.d1[x$ind.d1==1, -5]
}
colnames(tab.d1) <- c("Rho", "ACME(treated)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.d1) <- NULL
cat("\nMediation Sensitivity Analysis: Average Mediation Effect\n")
if(sum(x$ind.d0) > 0){
cat("\nSensitivity Region: ACME for Control Group\n\n")
print(tab.d0)
}
cat("\nRho at which ACME for Control Group = 0:", round(x$err.cr.d[1], 4))
cat("\nR^2_M*R^2_Y* at which ACME for Control Group = 0:", round(x$R2star.d.thresh[1], 4))
cat("\nR^2_M~R^2_Y~ at which ACME for Control Group = 0:", round(x$R2tilde.d.thresh[1], 4), "\n\n")
if(sum(x$ind.d1) > 0){
cat("\nSensitivity Region: ACME for Treatment Group\n\n")
print(tab.d1)
}
cat("\nRho at which ACME for Treatment Group = 0:", round(x$err.cr.d[2], 4))
cat("\nR^2_M*R^2_Y* at which ACME for Treatment Group = 0:", round(x$R2star.d.thresh[2], 4))
cat("\nR^2_M~R^2_Y~ at which ACME for Treatment Group = 0:", round(x$R2tilde.d.thresh[2], 4), "\n\n")
invisible(x)
}
if("direct" %in% etype.vec){
tab.z0 <- cbind(round(x$rho,4), round(x$z0,4), round(x$lower.z0,4),
round(x$upper.z0, 4), x$ind.z0, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.z0)==1){
tab.z0 <- as.matrix(tab.z0[x$ind.z0==1, -5])
tab.z0 <- t(tab.z0)
} else {
tab.z0 <- tab.z0[x$ind.z0==1, -5]
}
colnames(tab.z0) <- c("Rho", "ADE(control)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.z0) <- NULL
tab.z1 <- cbind(round(x$rho,4), round(x$z1,4), round(x$lower.z1,4),
round(x$upper.z1, 4), x$ind.z1, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.z1)==1){
tab.z1 <- as.matrix(tab.z1[x$ind.z1==1, -5])
tab.z1 <- t(tab.z1)
} else {
tab.z1 <- tab.z1[x$ind.z1==1, -5]
}
colnames(tab.z1) <- c("Rho", "ADE(treated)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.z1) <- NULL
cat("\nMediation Sensitivity Analysis: Average Direct Effect\n")
if(sum(x$ind.z0) > 0){
cat("\nSensitivity Region: ADE for Control Group\n\n")
print(tab.z0)
}
cat("\nRho at which ADE for Control Group = 0:", round(x$err.cr.z[1], 4))
cat("\nR^2_M*R^2_Y* at which ADE for Control Group = 0:", round(x$R2star.z.thresh[1], 4))
cat("\nR^2_M~R^2_Y~ at which ADE for Control Group = 0:", round(x$R2tilde.z.thresh[1], 4), "\n\n")
if(sum(x$ind.z1) > 0){
cat("\nSensitivity Region: ADE for Treatment Group\n\n")
print(tab.z1)
}
cat("\nRho at which ADE for Treatment Group = 0:", round(x$err.cr.z[2], 4))
cat("\nR^2_M*R^2_Y* at which ADE for Treatment Group = 0:", round(x$R2star.z.thresh[2], 4))
cat("\nR^2_M~R^2_Y~ at which ADE for Treatment Group = 0:", round(x$R2tilde.z.thresh[2], 4), "\n\n")
invisible(x)
}
} else {
if("indirect" %in% etype.vec){
tab <- cbind(round(x$rho,4), round(x$d0, 4), round(x$lower.d0, 4),
round(x$upper.d0, 4), x$ind.d0, round(x$R2star.prod,4), round(x$R2tilde.prod,4))
if(sum(x$ind.d0)==1){
tab <- as.matrix(tab[x$ind.d0==1, -5])
tab <- t(tab)
} else {
tab <- tab[x$ind.d0==1, -5]
}
colnames(tab) <- c("Rho", "ACME", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab) <- NULL
cat("\nMediation Sensitivity Analysis for Average Causal Mediation Effect\n")
if(sum(x$ind.d0) > 0){
cat("\nSensitivity Region\n\n")
print(tab)
}
cat("\nRho at which ACME = 0:", round(x$err.cr.d, 4))
cat("\nR^2_M*R^2_Y* at which ACME = 0:", round(x$R2star.d.thresh, 4))
cat("\nR^2_M~R^2_Y~ at which ACME = 0:", round(x$R2tilde.d.thresh, 4), "\n\n")
invisible(x)
}
if("direct" %in% etype.vec){
tab.z <- cbind(round(x$rho,4), round(x$z0, 4), round(x$lower.z0, 4),
round(x$upper.z0, 4), x$ind.z0, round(x$R2star.prod,4), round(x$R2tilde.prod,4))
if(sum(x$ind.z0)==1){
tab.z <- as.matrix(tab.z[x$ind.z0==1, -5])
tab.z <- t(tab.z)
} else {
tab.z <- tab.z[x$ind.z0==1, -5]
}
colnames(tab.z) <- c("Rho", "ADE", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.z) <- NULL
cat("\nMediation Sensitivity Analysis for Average Direct Effect\n")
if(sum(x$ind.z0) > 0){
cat("\nSensitivity Region\n\n")
print(tab.z)
}
cat("\nRho at which ADE = 0:", round(x$err.cr.z, 4))
cat("\nR^2_M*R^2_Y* at which ADE = 0:", round(x$R2star.z.thresh, 4))
cat("\nR^2_M~R^2_Y~ at which ADE = 0:", round(x$R2tilde.z.thresh, 4), "\n\n")
invisible(x)
}
}
}
#' Plotting Results from Sensitivity Analysis for Causal Mediation Effects
#'
#' This function is used to plot results from the 'medsens' function. Causal
#' average mediation effects (as well as average direct effects and proportions
#' mediated for selected models) can be plotted against two alternative
#' sensitivity parameters.
#'
#' @details The sensitivity analysis for causal mediation effects can be
#' conducted in terms of two alternative sensitivity parameters, which both
#' quantify the degree of violation of the sequential ignorability assumption.
#' The "rho" parameter represents the correlation between the two error terms
#' of the (latent) linear models for the mediator and outcome variables. A
#' large value of rho indicates the existence of important common unobserved
#' predictors for both the mediator and outcome and therefore a high degree of
#' sequential ignorability violation, while a value close to zero implies
#' there is no such confounders.
#'
#' The resulting "rho" figures plot the estimated true values of ACME (or ADE,
#' proportion mediated) against rho, along with the confidence intervals. When
#' rho is zero, sequantial ignorability holds, so the estimated value at that
#' point will be equal to the estimate returned by the \code{\link{mediate}}.
#' The confidence level is determined by the 'conf.level' value of the
#' original \code{\link{mediate}} object.
#'
#' The "R2" parameters represent the proportions of the mediator and outcome
#' variances that are explained by an unobserved pre-treatment confounder,
#' thereby indicating the importance of such a confounder in each model. When
#' 'r.type' is "residual", the R2 parameters represent the proportions of the
#' residual variances of the mediator and outcome models that become explained
#' by the inclusion of the hypothetical pre-treatment confounder. These are
#' denoted as "R square stars" in Imai, Keele and Yamamoto (2010) and can also
#' be specified as "star" or using a numeric value 1 in \code{medsens.plot}.
#' When 'r.type' is "total", the R2s represent the total mediator and outcome
#' variances the unobserved confounder would explain. This option can also be
#' specified using "tilde" or a numeric value 2.
#'
#' For both types of the "R2" parameters, 'sign.prod' indicates the
#' hypothesized direction in which the unobserved confounder affects the
#' mediator and outcome. (The name derives from the fact that this direction
#' is mathematically represented by the sign of the product of two regression
#' coefficients.) If "positive" (or a numeric value 1) is given, the
#' confounder is assumed to affect the mediator and outcome in the same
#' direction. If "negative" (or a numeric value -1), the effect is assumed to
#' be in opposite directions.
#'
#' The resulting contours in the "R2" plots represent the values of the ACME
#' (or ADE) for different combinations of the mediator R2 and outcome R2
#' values. When both values are zero (the lower-left corner of the plot), the
#' unobserved pre-treatment confounder has no effect on either mediator or
#' outcome and therefore sequantial ignorability is satisfied.
#'
#' @param x 'medsens' object, typically output from \code{medsens}.
#' @param sens.par a character string indicating the sensitivity parameter to be
#' used. Default plots effects as functions of "rho". See Details.
#' @param r.type type of the R square parameter to be used in "R2" plots. If
#' "residual", effects are plotted against the proportions of the residual
#' variances that are explained by the unobserved confounder. If "total", the
#' proportions of the total variances are used as sensitivity parameters. Only
#' relevant if 'sens.par' is "R2".
#' @param sign.prod a value indicating the direction of hypothesized confounding
#' in the sensitivity analysis. If "positive", the confounder is assumed to
#' affect the mediator and outcome variable in the same direction; if
#' "negative" the effects are assumed to be in opposite directions. Only
#' relevant if sens.par is set to "R2".
#' @param pr.plot a logical value. If 'TRUE', the "proportions mediated" will be
#' plotted instead of the average causal mediation effects or direct effects.
#' Currently only available if the object 'medsens' is based on the linear
#' mediator and binary probit outcome models. Default is 'FALSE'.
#' @param smooth.effect a logical value indicating whether the estimated
#' mediation effects are smoothed via \code{\link{lowess}} before being
#' plotted. Default is 'FALSE'.
#' @param smooth.ci a logical value indicating whether the confidence bands are
#' smoothed via \code{\link{lowess}} before being plotted. Default is 'FALSE'.
#' @param ask a logical value. If 'TRUE', the user is asked for input before a
#' new figure is plotted. Default is to ask only if the number of plots on
#' current screen is fewer than necessary.
#' @param levels vector of levels at which to draw contour lines. Only relevant
#' if 'sens.par' is set to "R2". If 'NULL', default values in
#' \code{\link{contour.default}} are used.
#' @param xlab label for the x axis. Default labels are used if 'NULL'.
#' @param ylab label for the y axis. Default labels are used if 'NULL'.
#' @param xlim limits of the x axis. If 'NULL' default values are used.
#' @param ylim limits of the y axis. If 'NULL' default values are used.
#' @param main main title for the plot. If 'NULL', default titles are used.
#' @param lwd width of the lines used in graphs.
#' @param ... additional arguments to be passed to plotting functions.
#'
#' @section Warning: The 'smooth.effect' and 'smooth.ci' options should be used
#' with caution since the smoothing could affect substantive implications of
#' the graphical analysis in a significant way.
#'
#' @author Dustin Tingley, Harvard University,
#' \email{dtingley@@gov.harvard.edu}; Teppei Yamamoto, Massachusetts Institute
#' of Technology, \email{teppei@@mit.edu}; Jaquilyn Waddell-Boie, Princeton
#' University, \email{jwaddell@@princeton.edu}; Luke Keele, Penn State
#' University, \email{ljk20@@psu.edu}; Kosuke Imai, Princeton University,
#' \email{kimai@@princeton.edu}.
#'
#' @seealso \code{\link{medsens}}, \code{\link{plot}}, \code{\link{contour}}.
#'
#' @references Tingley, D., Yamamoto, T., Hirose, K., Imai, K. and Keele, L.
#' (2014). "mediation: R package for Causal Mediation Analysis", Journal of
#' Statistical Software, Vol. 59, No. 5, pp. 1-38.
#'
#' Imai, K., Keele, L. and Tingley, D. (2010) A General Approach to Causal
#' Mediation Analysis, Psychological Methods, Vol. 15, No. 4 (December), pp.
#' 309-334.
#'
#' Imai, K., Keele, L. and Yamamoto, T. (2010) Identification, Inference, and
#' Sensitivity Analysis for Causal Mediation Effects, Statistical Science,
#' Vol. 25, No. 1 (February), pp. 51-71.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2009) "Causal Mediation
#' Analysis Using R" in Advances in Social Science Research Using R, ed. H. D.
#' Vinod New York: Springer.
#' @export
plot.medsens <- function(x, sens.par = c("rho", "R2"),
r.type = c("residual", "total"), sign.prod = c("positive", "negative"),
pr.plot = FALSE, smooth.effect = FALSE, smooth.ci = FALSE,
ask = prod(par("mfcol")) < nplots, levels = NULL,
xlab = NULL, ylab = NULL, xlim = NULL, ylim = NULL,
main = NULL, lwd = par("lwd"), ...){
# Compatibility with previous version
if(r.type[1L] == 1 | r.type[1L] == "star") r.type <- "residual"
if(r.type[1L] == 2 | r.type[1L] == "tilde") r.type <- "total"
if(sign.prod[1L] == 1) sign.prod <- "positive"
if(sign.prod[1L] == -1) sign.prod <- "negative"
# Match arguments
sens.par <- match.arg(sens.par)
r.type <- match.arg(r.type)
sign.prod <- match.arg(sign.prod)
# Effect types
etype.vec <- x$effect.type
if(x$effect.type == "both"){
etype.vec <- c("indirect","direct")
}
# Determine number of plots necessary
nplots <- ifelse(pr.plot, 2,
ifelse(x$INT || x$type == "bo", 2, 1) * ifelse(x$effect.type == "both", 2, 1))
if (ask) {
oask <- devAskNewPage(TRUE)
on.exit(devAskNewPage(oask))
}
##################################################
## A. Plots for proportions mediated
##################################################
if(pr.plot){
if(sens.par != "rho"){
stop("proportion mediated can only be plotted against rho")
}
if(x$type != "bo"){
stop("proportion mediated is only implemented for binary outcome")
}
out <- as.list(rep(NA, 2))
out[[1]] <- list(nu = x$nu0, lo = x$lower.nu0, up = x$upper.nu0)
out[[2]] <- list(nu = x$nu1, lo = x$lower.nu1, up = x$upper.nu1)
r <- x$rho
for(i in 1:2){
o <- out[[i]]
if(is.null(ylim)){
yl <- c(min(o$nu), max(o$nu))
} else yl <- ylim
if(is.null(main))
ma <- bquote(paste("Proportion Mediated"[.(i-1)], (rho)))
else ma <- main
if(smooth.effect)
nu <- lowess(r, o$nu)$y
else nu <- o$nu
if(smooth.ci){
lower <- lowess(r, o$lo)$y
upper <- lowess(r, o$up)$y
} else {
lower <- o$lo
upper <- o$up
}
plot.default(r, nu, type = "n", xlab = "", ylab = "",
main = ma, xlim = xlim, ylim = yl, ...)
polygon(x = c(r, rev(r)), y = c(lower, rev(upper)),
border = FALSE, col = 8, lty = 2, lwd = lwd)
lines(r, nu, lty = 1, lwd = lwd)
abline(h = 0)
abline(v = 0)
if(is.null(xlab))
title(xlab = expression(paste("Sensitivity Parameter: ", rho)),
line = 2.5, cex.lab = .9)
else title(xlab = xlab)
if(is.null(ylab))
title(ylab = bquote(paste("Proportion Mediated: ", bar(nu), (.(i-1)))),
line = 2.5, cex.lab = .9)
else title(ylab = ylab)
}
##################################################
## B. Plots for ACME/ADE wrt rho
##################################################
} else if(sens.par == "rho"){
out <- as.list(NA, 4)
out[[1]] <- list(eff = x$d0, lo = x$lower.d0, up = x$upper.d0,
labs = c("ACME", "Average Mediation Effect", 0))
out[[2]] <- list(eff = x$d1, lo = x$lower.d1, up = x$upper.d1,
labs = c("ACME", "Average Mediation Effect", 1))
out[[3]] <- list(eff = x$z0, lo = x$lower.z0, up = x$upper.z0,
labs = c("ADE", "Average Direct Effect", 0))
out[[4]] <- list(eff = x$z1, lo = x$lower.z1, up = x$upper.z1,
labs = c("ADE", "Average Direct Effect", 1))
r <- x$rho
rby <- x$rho.by
wh <- x$INT || x$type == "bo"
out[[1]]$show <- "indirect" %in% etype.vec
out[[2]]$show <- wh && "indirect" %in% etype.vec
out[[3]]$show <- "direct" %in% etype.vec
out[[4]]$show <- wh && "direct" %in% etype.vec
for(i in 1:length(out)){
o <- out[[i]]
if(!o$show) next
if(is.null(ylim)){
yl <- c(min(o$eff), max(o$eff))
} else yl <- ylim
if(is.null(main)){
if(wh){
ma <- bquote(paste(.(o$labs[1])[.(o$labs[3])], "(", rho, ")"))
} else {
ma <- bquote(paste(.(o$labs[1]), "(", rho, ")"))
}
} else ma <- main
if(smooth.effect)
eff <- lowess(r, o$eff)$y
else eff <- o$eff
if(smooth.ci){
lower <- lowess(r, o$lo)$y
upper <- lowess(r, o$up)$y
} else {
lower <- o$lo
upper <- o$up
}
plot.default(r, eff, type = "n", xlab = "", ylab = "",
main = ma, xlim = xlim, ylim = yl, ...)
polygon(x = c(r, rev(r)), y = c(lower, rev(upper)),
border = FALSE, col = 8, lty = 2, lwd = lwd)
lines(r, eff, lty = 1, lwd = lwd)
abline(h = 0)
abline(v = 0)
abline(h = weighted.mean(c(eff[floor(1/rby)], eff[ceiling(1/rby)]),
c(1 - 1/rby + floor(1/rby), 1/rby - floor(1/rby))),
lty = 2, lwd = lwd)
if(is.null(xlab))
title(xlab = expression(paste("Sensitivity Parameter: ", rho)),
line = 2.5, cex.lab = .9)
else title(xlab = xlab)
if(is.null(ylab)){
if(wh){
title(ylab = bquote(.(o$labs[2])[.(o$labs[3])]),
line = 2.5, cex.lab = .9)
} else {
title(ylab = o$labs[2], line = 2.5, cex.lab = .9)
}
} else title(ylab = ylab)
}
##################################################
## C. Plots for ACME/ADE wrt R2
##################################################
} else {
R2Mstar <- seq(0, 1 - x$rho.by, 0.01)
R2Ystar <- seq(0, 1 - x$rho.by, 0.01)
R2Mtilde <- (1 - x$r.square.m) * seq(0, 1 - x$rho.by, 0.01)
R2Ytilde <- (1 - x$r.square.y) * seq(0, 1 - x$rho.by, 0.01)
if(r.type == "residual"){
R2M <- R2Mstar; R2Y <- R2Ystar
} else {
R2M <- R2Mtilde; R2Y <- R2Ytilde
}
dlength <- length(seq(0, (1 - x$rho.by)^2, 0.0001))
R2prod.mat <- outer(R2Mstar, R2Ystar)
Rprod.mat <- round(sqrt(R2prod.mat), digits=4)
if(is.null(xlim))
xlim <- c(0,1)
if(is.null(ylim))
ylim <- c(0,1)
out <- as.list(NA, 4)
out[[1]] <- list(eff = x$d0, labs = c("ACME", "Average Mediation Effect", 0))
out[[2]] <- list(eff = x$d1, labs = c("ACME", "Average Mediation Effect", 1))
out[[3]] <- list(eff = x$z0, labs = c("ADE", "Average Direct Effect", 0))
out[[4]] <- list(eff = x$z1, labs = c("ADE", "Average Direct Effect", 1))
wh <- x$INT || x$type == "bo"
out[[1]]$show <- "indirect" %in% etype.vec
out[[2]]$show <- wh && "indirect" %in% etype.vec
out[[3]]$show <- "direct" %in% etype.vec
out[[4]]$show <- wh && "direct" %in% etype.vec
for(i in 1:length(out)){
o <- out[[i]]
if(!o$show) next
if(is.null(levels)){
levels <- pretty(quantile(o$eff, probs=c(0.1,0.9)), 10)
}
if(sign.prod == "positive"){
eff <- approx(o$eff[((length(o$eff)+1)/2):length(o$eff)], n=dlength)$y
sgnlab <- "1"
} else {
eff <- rev(approx(o$eff[1:((length(o$eff)+1)/2)], n=dlength)$y)
sgnlab <- "-1"
}
effmat <- matrix(eff[Rprod.mat/0.0001+1], nrow=length(R2M))
ma <- switch(wh + 1, bquote(paste(.(o$labs[1]))), # wh == F
bquote(paste(.(o$labs[1])[.(o$labs[3])]))) # wh == T
if(is.null(main) & r.type == "residual")
ma <- bquote(paste(.(ma), "(", R[M]^{2},"*,", R[Y]^2,"*), sgn",
(lambda[2]*lambda[3])==.(sgnlab)))
else if(is.null(main) & r.type == "total")
ma <- bquote(paste(.(ma), "(", tilde(R)[M]^{2}, "," , tilde(R)[Y]^2, "), sgn",
(lambda[2]*lambda[3])==.(sgnlab)))
else ma <- main
contour(R2M, R2Y, effmat, levels = levels, main = ma,
xlab = xlab, ylab = ylab, ylim = ylim, xlim = xlim, lwd = lwd, ...)
contour(R2M, R2Y, effmat, levels = 0, lwd = lwd + 1, add = TRUE, drawlabels = FALSE)
if(is.null(xlab) & r.type == "residual")
title(xlab=expression(paste(R[M]^{2},"*")), line=2.5, cex.lab=.9)
else if(is.null(xlab) & r.type == "total")
title(xlab=expression(paste(tilde(R)[M]^{2})), line=2.5, cex.lab=.9)
if(is.null(ylab) & r.type == "residual")
title(ylab=expression(paste(R[Y]^2,"*")), line=2.5, cex.lab=.9)
else if(is.null(ylab) & r.type == "total")
title(ylab=expression(paste(tilde(R)[Y]^2)), line=2.5, cex.lab=.9)
axis(2, at = seq(0, 1, by = .1))
axis(1, at = seq(0, 1, by = .1))
}
}
}
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Defines functions medsens summary.medsens print.summary.medsens plot.medsens
Documented in medsens plot.medsens print.summary.medsens summary.medsens
#' Sensitivity Analysis for Causal Mediation Effects
#'
#' 'medsens' is used to perform sensitivity analysis on the average causal
#' mediation effects and direct effects for violations of the sequential
#' ignorability assumption. The function takes output from '\code{mediate}' and
#' calculates the true average causal mediation effects and direct effects for
#' different values of the sensitivity parameter representing the degree of the
#' sequential ignorability violation.
#'
#' @details This is the workhorse function for sensitivity analyses for average
#' causal mediation effects. The sensitivity analysis can be used to assess
#' the robustness of the findings from \code{mediate} to the violation of
#' sequential ignorability, the crucial identification assumption necessary
#' for the estimates to be valid. The analysis proceeds by quantifying the
#' degree of sequential ignorability violation as the correlation between the
#' error terms of the mediator and outcome models, and then calculating the
#' true values of the average causal mediation effect for given values of this
#' sensitivity parameter, rho. The original findings are deemed sensitive if
#' the true effects are found to vary widely as function of rho.
#'
#' The sensitivity analysis is only implemented for the following three model
#' combinations: linear mediator and outcome models (both of class 'lm'),
#' binary probit mediator (fitted via 'glm' with family "binomial" and link
#' "probit") and linear outcome models, and linear mediator and binary probit
#' outcome models. In addition, the binary outcome model cannot include a
#' treatment-mediator interaction term. An error is returned if the 'mediate'
#' object in 'x' is based on other model combinations. As of version 3.0, the
#' sensitivity analysis can also be conducted with respect to the average
#' direct effect by setting 'effect.type' to "direct" (or "both" if results
#' for the average causal mediation effect are also desired).
#'
#' Users should note that computation can take significant time for
#' \code{medsens}. Setting 'rho.by' to a larger number significantly decreases
#' computational time, as does decreasing 'eps' (for the linear-linear case)
#' or the number of simulations 'sims' (for the binary-linear and
#' linear-binary cases).
#'
#' @param x an object of class 'mediate', typically an output from the
#' \code{mediate} function.
#' @param rho.by a numeric value between 0 and 1 indicating the increment for
#' the sensitivity parameter, rho.
#' @param sims the number of Monte Carlo draws for the calculation of confidence
#' intervals. Only used in cases where either the mediator or outcome variable
#' is binary.
#' @param eps convergence tolerance parameter for the iterative FGLS. Only used
#' when both the mediator and outcome models are linear.
#' @param effect.type a character string indicating which effect(s) to be
#' analyzed. Default is "indirect".
#'
#' @return \code{medsens} returns an object of class "\code{medsens}", a list
#' containing the following elements. Some of these elements are not available
#' depending on the 'effect.type' argument specified by the user. The output
#' can then be passed to the \code{\link{summary}} (i.e.,
#' \code{\link{summary.medsens}}) and \code{\link{plot}} (i.e.,
#' \code{\link{plot.medsens}}) functions to produce tabular and graphical
#' summaries of the results.
#'
#' \item{d0, d1}{vectors of point estimates for average causal mediation
#' effects under the control and treatment conditions for each value of
#' sensitivity parameter rho.}
#' \item{upper.d0, lower.d0, upper.d1, lower.d1}{vectors of upper and lower
#' confidence limits for average causal mediation effect under the control
#' and treatment conditions for each value of rho.}
#' \item{z0, z1}{vectors of point estimates for average direct effect under
#' the control and treatment conditions for each value of sensitivity
#' parameter rho.}
#' \item{upper.z0, lower.z0, upper.z1, lower.z1}{vectors of upper and lower
#' confidence limits for average direct effect under the control and
#' treatment conditions for each value of rho.}
#' \item{tau}{a vector of point estimates for total effect for each value of
#' rho. Only present when the outcome model is binary.}
#' \item{upper.tau, lower.tau}{vectors of upper and lower confidence limits
#' for total effect. Only present when the outcome model is binary.}
#' \item{nu}{a vector of point estimates for the proportion mediated for each
#' value of rho. Only present when the outcome model is binary.}
#' \item{upper.nu, lower.nu}{vectors of upper and lower confidence limits
#' for the proportion mediated. Only present when the outcome model is binary.}
#' \item{rho}{a numeric vector containing the values of sensitivity
#' parameter rho used.}
#' \item{rho.by}{a numeric value indicating the increment of rho used.}
#' \item{sims}{a numeric value indicating the number of Monte Carlo draws
#' used.}
#' \item{err.cr.d, err.cr.z}{the values of rho with which the average causal
#' mediation and direct effects are zero. Vectors of length two if 'INT' is
#' 'TRUE'; numeric values otherwise.}
#' \item{ind.d0, ind.d1, ind.z0, ind.z1}{vectors of 0s/1s, indicating whether
#' the confidence intervals of d0, d1, z0 and z1 do not cover zero for each
#' value of rho.}
#' \item{R2star.prod}{a numeric vector containing the values of the products
#' of the two "R square stars", representing the proportions of residual
#' variance in the mediator and outcome explained by the hypothesized
#' unobserved confounder. The values correspond to those of rho. See
#' \code{\link{plot.medsens}} for details.}
#' \item{R2tilde.prod}{a numeric vector containing the values of the products
#' of the two "R square tildes", representing the proportions of total
#' variance in the mediator and outcome explained by the hypothesized
#' unobserved confounder. The values correspond to those of rho. See
#' \code{\link{plot.medsens}} for details.}
#' \item{R2star.d.thresh, R2star.z.thresh}{the values of the product of "R
#' square stars" for which the average causal mediation and direct effects are
#' zero, respectively.}
#' \item{R2tilde.d.thresh, R2tilde.z.thresh}{the values of the product of "R
#' square tildes" for which the average causal mediation and direct effects
#' are zero, respectively.}
#' \item{r.square.y, r.square.m}{the usual R square statistics for the
#' outcome and mediator models.}
#' \item{INT}{a logical value indicating whether interaction between the
#' treatment and mediator is allowed in the original mediate object.}
#' \item{conf.level}{the confidence level used.}
#' \item{effect.type}{the 'effect.type' argument used.}
#' \item{type}{a character string indicating the type of the mediator and
#' outcome models used. Currently either "ct" (linear mediator and outcome
#' models), 'bm' (binary mediator and linear outcome models) or 'bo' (linear
#' mediator and binary outcome models).}
#' \item{robustSE}{`TRUE' or `FALSE'.}
#' \item{cluster}{the clusters used.}
#' @author Dustin Tingley, Harvard University,
#' \email{dtingley@@gov.harvard.edu}; Teppei Yamamoto, Massachusetts Institute
#' of Technology, \email{teppei@@mit.edu}; Jaquilyn Waddell-Boie, Princeton
#' University, \email{jwaddell@@princeton.edu}; Kentaro Hirose, Princeton
#' University, \email{hirose@@princeton.edu}; Luke Keele, Penn State
#' University, \email{ljk20@@psu.edu}; Kosuke Imai, Princeton University,
#' \email{kimai@@princeton.edu}.
#'
#' @seealso \code{\link{mediate}}, \code{\link{summary.medsens}},
#' \code{\link{plot.medsens}}.
#'
#' @references Tingley, D., Yamamoto, T., Hirose, K., Imai, K. and Keele, L.
#' (2014). "mediation: R package for Causal Mediation Analysis", Journal of
#' Statistical Software, Vol. 59, No. 5, pp. 1-38.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2011). Unpacking the
#' Black Box of Causality: Learning about Causal Mechanisms from Experimental
#' and Observational Studies, American Political Science Review, Vol. 105, No.
#' 4 (November), pp. 765-789.
#'
#' Imai, K., Keele, L. and Tingley, D. (2010) A General Approach to Causal
#' Mediation Analysis, Psychological Methods, Vol. 15, No. 4 (December), pp.
#' 309-334.
#'
#' Imai, K., Keele, L. and Yamamoto, T. (2010) Identification, Inference, and
#' Sensitivity Analysis for Causal Mediation Effects, Statistical Science,
#' Vol. 25, No. 1 (February), pp. 51-71.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2009) "Causal Mediation
#' Analysis Using R" in Advances in Social Science Research Using R, ed. H. D.
#' Vinod New York: Springer.
#'
#' @export
#' @examples
#' # Examples with JOBS II Field Experiment
#'
#' # **For illustration purposes a small number of simulations are used**
#'
#' data(jobs)
#'
#' ####################################################
#' # Example 1: Binary treatment
#' ####################################################
#' # Fit parametric models
#' b <- lm(job_seek ~ treat + econ_hard + sex + age, data=jobs)
#' c <- lm(depress2 ~ treat + job_seek + econ_hard + sex + age, data=jobs)
#'
#' # Pass model objects through mediate function
#' med.cont <- mediate(b, c, treat="treat", mediator="job_seek", sims=50)
#' # med.cont <- mediate(b, c, treat="treat", mediator="job_seek", sims=50, robustSE = T)
#' # jobs$cluster <- rep(1:30, each = 30)[-1]
#' # med.cont <- mediate(b, c, treat="treat", mediator="job_seek", sims=50, cluster = jobs$cluster)
#'
#' # Pass mediate output through medsens function
#' sens.cont <- medsens(med.cont, rho.by=.1, eps=.01, effect.type="both")
#'
#' # Use summary function to display results
#' summary(sens.cont)
#'
#' # Plot true ACMEs and ADEs as functions of rho
#' par.orig <- par(mfrow = c(2,2))
#' plot(sens.cont, main="JOBS", ylim=c(-.2,.2))
#'
#' # Plot true ACMEs and ADEs as functions of "R square tildes"
#' plot(sens.cont, sens.par="R2", r.type="total", sign.prod="positive")
#' par(par.orig)
#'
#' ####################################################
#' # Example 2: Categorical treatment
#' ####################################################
#' \dontrun{
#'
#' # Purely for illustration, think of educ as a ``treatment''
#'
#' b <- lm(job_seek ~ educ + sex, data=jobs)
#' c <- lm(depress2 ~ educ + job_seek + sex, data=jobs)
#'
#' # compare two categories of educ --- gradwk and somcol
#' med.cont <- mediate(b, c, treat="educ", mediator="job_seek", sims=50,
#' control.value = "gradwk", treat.value = "somcol")
#' sens.cont <- medsens(med.cont, rho.by=.1, eps=.01, effect.type="both")
#' summary(sens.cont)
#' }
#'
medsens <- function(x, rho.by = 0.1, sims = 1000, eps = sqrt(.Machine$double.eps),
effect.type = c("indirect", "direct", "both"))
{
effect.type <- match.arg(effect.type)
if(effect.type == "both"){
etype.vec <- c("indirect", "direct")
} else {
etype.vec <- effect.type
}
model.y <- x$model.y
model.m <- x$model.m
class.y <- class(model.y)[1]
class.m <- class(model.m)[1]
treat <- x$treat
mediator <- x$mediator
INT <- x$INT
low <- (1 - x$conf.level)/2
high <- 1 - low
robustSE <- x$robustSE
cluster <- x$cluster
control.value <- x$control.value
treat.value <- x$treat.value
# Setting Variable labels
## Uppercase letters (e.g. T) = labels in the input matrix
## Uppercase letters + ".out" (e.g. T.out) = labels in the regression output
y.data <- model.frame(model.y)
Y <- colnames(y.data)[1]
if(is.factor(y.data[,treat])){
t.levels <- levels(y.data[,treat])
n.t.levels <- length(t.levels)
## When treatment = more than two categories
## control.value => 1st factor level (baseline category)
## treat.value => 2nd factor level
## run regression again with this new data set
if(n.t.levels != 2){
number <- 1:n.t.levels
t.levels.mat <- data.frame(number, t.levels)
c.num <- t.levels.mat$number[t.levels.mat$t.levels == control.value]
t.num <- t.levels.mat$number[t.levels.mat$t.levels == treat.value]
number.new <- c(c.num, t.num, number[-c(c.num, t.num)])
y.data[, treat] <- factor(y.data[, treat], levels(y.data[, treat])[number.new])
model.y <- update(model.y, data = y.data)
}
t.levels <- levels(y.data[,treat])
cat.c <- t.levels[1]
cat.t <- t.levels[2]
} else {
cat.c <- NULL
cat.t <- NULL
}
T.out <- paste(treat, cat.t, sep="")
m.data <- model.frame(model.m)
if(is.factor(m.data[,treat])){
t.levels <- levels(m.data[,treat])
n.t.levels <- length(t.levels)
## When treatment = more than two categories
## control.value => 1st factor level (baseline category)
## treat.value => 2nd factor level
## run regression again with this new data set
if(n.t.levels != 2){
number <- 1:n.t.levels
t.levels.mat <- data.frame(number, t.levels)
c.num <- t.levels.mat$number[t.levels.mat$t.levels == control.value]
t.num <- t.levels.mat$number[t.levels.mat$t.levels == treat.value]
number.new <- c(c.num, t.num, number[-c(c.num, t.num)])
m.data[, treat] <- factor(m.data[, treat], levels(m.data[, treat])[number.new])
model.m <- update(model.m, data = m.data)
}
}
if(is.factor(y.data[,mediator])){
m.levels <- levels(y.data[,mediator])
if(length(m.levels) != 2){
stop("mediator with more than two categories currently not supported")
}
cat.m0 <- m.levels[1]
cat.m1 <- m.levels[2]
} else {
cat.m0 <- NULL
cat.m1 <- NULL
}
M.out <- paste(mediator, cat.m1, sep="")
if(INT){
if(paste(treat,mediator,sep=":") %in% attr(model.y$terms,"term.labels")){ # T:M
TM.out <- paste(T.out, M.out, sep=":")
TM <- paste(treat, mediator, sep=":")
} else { # M:T
TM.out <- paste(M.out, T.out, sep=":")
TM <- paste(mediator, treat, sep=":")
}
}
## Setting Up Sensitivity Parameters
rho <- seq(-1 + rho.by, 1 - rho.by, rho.by)
R2star.prod <- rho^2
## Extracting weights
weights <- model.weights(y.data)
if(is.null(weights)){
weights <- rep(1, nrow(y.data))
}
## Initializing Containers
d0 <- d1 <- z0 <- z1 <- nu0 <- nu1 <- tau <-
upper.d0 <- upper.d1 <- upper.z0 <- upper.z1 <- upper.nu0 <- upper.nu1 <- upper.tau <-
lower.d0 <- lower.d1 <- lower.z0 <- lower.z1 <- lower.nu0 <- lower.nu1 <- lower.tau <-
ind.d0 <- ind.d1 <- ind.z0 <- ind.z1 <- err.cr.d <- err.cr.z <-
R2star.d.thresh <- R2tilde.d.thresh <- R2star.z.thresh <- R2tilde.z.thresh <- NULL
getvcov <- function(dat, fm, cluster){
## Compute cluster robust standard errors
## fm is the model object
cluster <- factor(cluster) # remove missing levels and NA
M <- nlevels(cluster)
N <- sum(!is.na(cluster))
K <- fm$rank
dfc <- (M/(M-1))*((N-1)/(N-K))
uj <- apply(estfun(fm),2, function(x) tapply(x, cluster, sum));
dfc*sandwich(fm, meat. = crossprod(uj)/N)
}
#########################################################
## CASE 1: Continuous Outcome + Continuous Mediator
#########################################################
if(class.y=="lm" & class.m=="lm") {
type <- "ct"
d0 <- d1 <- z0 <- z1 <- matrix(NA, length(rho), 1)
d0.var <- d1.var <- z0.var <- z1.var <- matrix(NA, length(rho), 1)
for(i in 1:length(rho)){
e.cor <- rho[i]
b.dif <- 1
# Stacked Equations
m.mat <- model.matrix(model.m) * weights^(1/2)
y.mat <- model.matrix(model.y) * weights^(1/2)
m.k <- ncol(m.mat)
m.n <- nrow(m.mat)
y.k <- ncol(y.mat)
y.n <- nrow(y.mat)
n <- y.n
m.zero <- matrix(0, m.n, y.k)
y.zero <- matrix(0, y.n, m.k)
X.1 <- cbind(m.mat, m.zero)
X.2 <- cbind(y.zero, y.mat)
X <- rbind(X.1, X.2)
Y.c <- as.matrix(c(model.response(model.frame(model.m)),
model.response(model.frame(model.y)))) * weights^(1/2)
# Estimates of OLS Start Values
inxx <- solve(crossprod(X))
b.ols <- inxx %*% crossprod(X,Y.c)
b.tmp <- b.ols
while(abs(b.dif) > eps){
e.hat <- as.matrix(Y.c - (X %*% b.tmp))
e.1 <- e.hat[1:n]
e.2 <- e.hat[(n+1):(2*n)]
sd.1 <- sd(e.1)
sd.2 <- sd(e.2)
omega <- matrix(NA, 2,2)
omega[1,1] <- crossprod(e.1)/(n-1)
omega[2,2] <- crossprod(e.2)/(n-1)
omega[2,1] <- e.cor*sd.1*sd.2
omega[1,2] <- e.cor*sd.1*sd.2
I <- Diagonal(n)
omega.i <- solve(omega)
v.i <- kronecker(omega.i, I)
Xv.i <- t(X) %*% v.i
X.sur <- Xv.i %*% X
b.sur <- solve(X.sur) %*% Xv.i %*% Y.c
# Variance-Covariance Matrix of SUR
if(robustSE){
meat <- 0
for (ii in 1:n){
Xi.med <- X.1[ii, ]
Xi.out <- X.2[ii, ]
Xi <- rbind(Xi.med, Xi.out)
ui <- as.matrix(c(e.1[ii], e.2[ii]))
meat <- t(Xi) %*% solve(omega) %*% ui %*% t(ui) %*% solve(omega) %*% Xi + meat
}
v.cov <- solve(X.sur) %*% meat %*% solve(X.sur)
} else if(!is.null(cluster)){
meat <- 0
for (mm in unique(cluster)) {
X.med.m <- matrix(X.1[cluster == mm, ],, ncol(X.1))
X.out.m <- matrix(X.2[cluster == mm, ],, ncol(X.2))
e.1.m <- e.1[cluster == mm]
e.2.m <- e.2[cluster == mm]
half.meat <- 0
for (ii in 1:nrow(X.med.m)){
Xm <- rbind(X.med.m[ii, ], X.out.m[ii, ])
half.meat <- t(Xm) %*% solve(omega) %*% as.matrix(c(e.1.m[ii], e.2.m[ii])) + half.meat
}
meat <- half.meat %*% t(half.meat) + meat
}
v.cov <- solve(X.sur) %*% meat %*% solve(X.sur)
} else {
v.cov <- solve(X.sur)
}
b.old <- b.tmp
b.dif <- sum((b.sur - b.old)^2)
b.tmp <- b.sur
}
# Name Elements - Extract Quantities
m.names <- names(model.m$coef)
y.names <- names(model.y$coef)
b.names <- c(m.names, y.names)
row.names(b.sur) <- b.names
m.coefs <- as.matrix(b.sur[1:m.k])
y.coefs <- as.matrix(b.sur[(m.k+1):(m.k+y.k)])
row.names(m.coefs) <- m.names
row.names(y.coefs) <- y.names
rownames(v.cov) <- b.names
colnames(v.cov) <- b.names
v.m <- v.cov[1:m.k,1:m.k]
v.y <- v.cov[(m.k+1):(m.k+y.k),(m.k+1):(m.k+y.k)]
m.coef.treat <- m.coefs[T.out, ]
y.coef.treat <- y.coefs[T.out, ]
y.coef.med <- y.coefs[M.out, ]
v.m.treat <- v.m[T.out, T.out]
v.y.treat <- v.y[T.out, T.out]
v.y.med <- v.y[M.out, M.out]
if(INT){
y.coef.TM <- y.coefs[TM.out, ]
v.y.TM <- v.y[TM.out, TM.out]
}
if(INT && ("direct" %in% etype.vec)){
m.bar <- apply(model.matrix(model.m), 2, weighted.mean, w=weights)
m.covts.ind <- -(match(c("(Intercept)", paste(T.out)), names(model.m$coefficients), NULL))
m.coef.covts <- m.coefs[m.covts.ind, ]
m.bar.covts <- m.bar[m.covts.ind]
m.intercept <- m.coefs["(Intercept)", ]
v.m.covts <- v.m[m.covts.ind, m.covts.ind]
v.m.intercept <- v.m["(Intercept)", "(Intercept)"]
v.m.int.treat <- v.m["(Intercept)", T.out]
if(length(m.coef.covts) != 0){
model.m.bar.z0 <- m.intercept + sum(crossprod(m.coef.covts, m.bar.covts))
v.model.m.bar.z0 <- v.m.intercept + sum(crossprod(m.bar.covts, v.m.covts) %*% m.bar.covts)
model.m.bar.z1 <- m.intercept + m.coef.treat + sum(m.coef.covts * m.bar.covts)
v.m.treat.covts <- v.m[T.out, m.covts.ind]
v.m.int.covts <- v.m["(Intercept)", m.covts.ind]
v.model.m.bar.z1 <- ( v.m.intercept + v.m.treat +
sum(crossprod(m.bar.covts, v.m.covts) %*% m.bar.covts) +
2 * ( v.m.int.treat +
sum(crossprod(m.bar.covts, v.m.treat.covts)) +
sum(crossprod(m.bar.covts, v.m.int.covts)) ) )
} else {
model.m.bar.z0 <- m.intercept
v.model.m.bar.z0 <- v.m.intercept
model.m.bar.z1 <- m.intercept + m.coef.treat
v.model.m.bar.z1 <- v.m.intercept + v.m.treat + 2 * v.m.int.treat
}
}
# Save Estimates
if(INT){
if("indirect" %in% etype.vec){
d0[i,] <- m.coef.treat * y.coef.med
d1[i,] <- m.coef.treat * (y.coef.med + y.coef.TM)
}
if("direct" %in% etype.vec){ # direct effects with interaction
if(length(m.coef.covts) != 0){ # with covariates
z0[i,] <- ( y.coef.treat +
y.coef.TM * (m.intercept + sum(crossprod(m.coef.covts, m.bar.covts))) )
z1[i,] <- ( y.coef.treat +
y.coef.TM * (m.intercept + m.coef.treat +
sum(crossprod(m.coef.covts, m.bar.covts))) )
} else { # no covariates
z0[i,] <- y.coef.treat + y.coef.TM * m.intercept
z1[i,] <- y.coef.treat + y.coef.TM * (m.intercept + m.coef.treat)
}
}
} else {
if("indirect" %in% etype.vec){
d0[i,] <- d1[i,] <- m.coef.treat * y.coef.med
}
if("direct" %in% etype.vec) {
z0[i,] <- z1[i,] <- y.coef.treat
}
}
# Save Variance Estimates
if(INT){
if("indirect" %in% etype.vec){
d0.var[i,] <- y.coef.med^2 * v.m.treat + m.coef.treat^2 * v.y.med
d1.var[i,] <- ( (y.coef.med + y.coef.TM)^2 * v.m.treat +
m.coef.treat^2 * (v.y.med + v.y.TM + 2 * v.y[M.out, TM.out]) )
}
if("direct" %in% etype.vec){
z0.var[i,] <- ( v.y.treat + model.m.bar.z0^2 * v.y.TM +
y.coef.TM^2 * v.model.m.bar.z0 +
v.y.TM * v.model.m.bar.z0 +
2 * model.m.bar.z0 * v.y[T.out, TM.out] )
z1.var[i,] <- ( v.y.treat + model.m.bar.z1^2 * v.y.TM +
y.coef.TM^2 * v.model.m.bar.z1 +
v.y.TM * v.model.m.bar.z1 +
2 * model.m.bar.z1 * v.y[T.out,TM.out] )
}
} else {
if("indirect" %in% etype.vec){
d0.var[i,] <- d1.var[i,] <- (m.coef.treat^2 * v.y.med) + (y.coef.med^2 * v.m.treat)
}
if("direct" %in% etype.vec){
z0.var[i,] <- z1.var[i,] <- v.y.treat
}
}
rm(b.sur, m.coefs, y.coefs, v.cov, v.m, v.y)
} # END of Rho Loop
if("indirect" %in% etype.vec){
upper.d0 <- d0 + qnorm(high) * sqrt(d0.var)
lower.d0 <- d0 + qnorm(low) * sqrt(d0.var)
upper.d1 <- d1 + qnorm(high) * sqrt(d1.var)
lower.d1 <- d1 + qnorm(low) * sqrt(d1.var)
ind.d0 <- as.numeric(lower.d0 < 0 & upper.d0 > 0)
ind.d1 <- as.numeric(lower.d1 < 0 & upper.d1 > 0)
}
if("direct" %in% etype.vec){
upper.z0 <- z0 + qnorm(high) * sqrt(z0.var)
lower.z0 <- z0 + qnorm(low) * sqrt(z0.var)
upper.z1 <- z1 + qnorm(high) * sqrt(z1.var)
lower.z1 <- z1 + qnorm(low) * sqrt(z1.var)
ind.z0 <- as.numeric(lower.z0 < 0 & upper.z0 > 0)
ind.z1 <- as.numeric(lower.z1 < 0 & upper.z1 > 0)
}
# Save R2 tilde values
r.sq.m <- summary(model.m)$r.squared
r.sq.y <- summary(model.y)$r.squared
R2tilde.prod <- rho^2 * (1 - r.sq.m) * (1 - r.sq.y)
## END OF CASE 1: Continuous Outcome + Continuous Mediator
} else
#########################################################
## CASE 2: Continuous Outcome + Binary Mediator
#########################################################
if (class.y == "lm" & class.m == "glm") {
type <- "bm"
if(model.m$family$link == "logit"){
stop("medsens is only valid for the probit link")
}
## Variable values (LABEL.value)
Y.value <- y.data[,1]
y.k <- length(model.y$coef)
# Step 1: Pre-loop computations
# Step 1-1: Sample M model parameters
Mmodel.coef <- model.m$coef
if(robustSE){
Mmodel.var.cov <- vcovHC(model.m)
} else if(!is.null(cluster)) {
dta <- merge(m.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.m, data=dta)
MModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Mmodel.var.cov <- vcov(model.m)
}
Mmodel.coef.sim <- rmvnorm(sims, mean=Mmodel.coef, sigma=Mmodel.var.cov)
# simulate M-model parameters
# Step 1-2: Sample lambda_0 and lambda_1; lambdas are (n x sims) matrix
m.mat <- m.mat.1 <- m.mat.0 <- model.matrix(model.m)
m.mat.1[,T.out] <- 1 # M-model matrix with t=1
m.mat.0[,T.out] <- 0 # M-model matrix with t=0
mu.sim <- tcrossprod(m.mat, Mmodel.coef.sim) # E(M|T,X)
mu.1.sim <- tcrossprod(m.mat.1, Mmodel.coef.sim) # E(M|T=1,X)
mu.0.sim <- tcrossprod(m.mat.0, Mmodel.coef.sim) # E(M|T=0,X)
# Step 1-3: Define lambda function (inverse Mill's ratio)
lambda <- function(mmodel, mcoef) {
mu <- model.matrix(mmodel) %*% mcoef
m <- mmodel$y # this is M
return((m*dnorm(-mu)-(1-m)*dnorm(-mu))/(m*pnorm(mu)+(1-m)*pnorm(-mu)))
}
# Step 2: Rho loop
## Step 2-0: Initialize containers
if("indirect" %in% etype.vec){
d0 <- d1 <- upper.d0 <- upper.d1 <-
lower.d0 <- lower.d1 <- ind.d0 <- ind.d1 <- matrix(NA, length(rho), 1)
}
if("direct" %in% etype.vec){
z0 <- z1 <- upper.z0 <- upper.z1 <-
lower.z0 <- lower.z1 <- ind.z0 <- ind.z1 <- matrix(NA, length(rho), 1)
}
Ymodel.coef.sim <- matrix(NA, sims, y.k)
colnames(Ymodel.coef.sim) <- names(model.y$coef)
sigma.3.sim <- rep(NA, sims)
d0.sim <- d1.sim <- z0.sim <- z1.sim <- rep(NA, sims)
## START OF RHO LOOP
for(i in 1:length(rho)){
## START OF SIMULATION LOOP
for(k in 1:sims){
## Step 2-1: Obtain the initial Y model with the correction term
adj <- lambda(model.m, Mmodel.coef.sim[k,]) * rho[i] # the adjustment term
w <- 1 - rho[i]^2*lambda(model.m, Mmodel.coef.sim[k,]) *
(lambda(model.m, Mmodel.coef.sim[k,]) + mu.sim[,k])
y.data.adj <- data.frame(y.data, w, adj, weights)
model.y.adj <- update(model.y, as.formula(paste(". ~ . + adj")),
weights = w*weights, data = y.data.adj)
sigma.3 <- summary(model.y.adj)$sigma
## Step 2-2: Update the Y model via Iterative FGLS
sigma.dif <- 1
while(abs(sigma.dif) > eps){
Y.star <- Y.value - sigma.3 * adj
y.data.star <- data.frame(Y.star, y.data.adj)
model.y.update <- update(model.y, as.formula(paste("Y.star ~ .")),
weights = w*weights, data = y.data.star)
sigma.3.temp <- summary(model.y.update)$sigma
sigma.dif <- sigma.3.temp - sigma.3
sigma.3 <- sigma.3.temp
}
## Step 2-3: Simulate Y model parameters
Ymodel.coef <- model.y.update$coef
if(robustSE){
Ymodel.var.cov <- vcovHC(model.y.update)
} else if(!is.null(cluster)) {
dta <- merge(y.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.y.update, data=dta)
YModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Ymodel.var.cov <- vcov(model.y.update)
}
Ymodel.coef.sim[k,] <- rmvnorm(1, mean=Ymodel.coef, sigma=Ymodel.var.cov)
# draw one simulation sample of Y-model parameters for each k
## Step 2-4: Simulate ACMEs; means are over observations
if(INT){
if("indirect" %in% etype.vec){
d1.sim[k] <- weighted.mean( (Ymodel.coef.sim[k,M.out] + Ymodel.coef.sim[k,TM.out]) *
(pnorm(mu.1.sim[,k]) - pnorm(mu.0.sim[,k])), w = weights )
d0.sim[k] <- weighted.mean( (Ymodel.coef.sim[k,M.out]) * (pnorm(mu.1.sim[,k]) -
pnorm(mu.0.sim[,k])), w = weights )
}
if("direct" %in% etype.vec){
z1.sim[k] <- weighted.mean( sum(Ymodel.coef.sim[k,c(T.out,TM.out)]) *
pnorm(mu.1.sim[,k]) + Ymodel.coef.sim[k,T.out] *
pnorm(-mu.1.sim[k]), w = weights )
z0.sim[k] <- weighted.mean( sum(Ymodel.coef.sim[k,c(T.out,TM.out)]) *
pnorm(mu.0.sim[,k]) + Ymodel.coef.sim[k,T.out] *
pnorm(-mu.0.sim[k]), w = weights )
}
} else {
if("indirect" %in% etype.vec){
d0.sim[k] <- d1.sim[k] <- weighted.mean( Ymodel.coef.sim[k,M.out] *
(pnorm(mu.1.sim[,k]) - pnorm(mu.0.sim[,k])), w = weights )
}
if("direct" %in% etype.vec){
z0.sim[k] <- z1.sim[k] <- Ymodel.coef.sim[k,T.out]
}
}
## END OF SIMULATION LOOP
}
## Step 2-5: Compute Outputs
if("indirect" %in% etype.vec){
d0[i] <- mean(d0.sim)
d1[i] <- mean(d1.sim)
upper.d0[i] <- quantile(d0.sim, high)
upper.d1[i] <- quantile(d1.sim, high)
lower.d0[i] <- quantile(d0.sim, low)
lower.d1[i] <- quantile(d1.sim, low)
ind.d0[i] <- as.numeric(lower.d0[i] < 0 & upper.d0[i] > 0)
ind.d1[i] <- as.numeric(lower.d1[i] < 0 & upper.d1[i] > 0)
}
if("direct" %in% etype.vec){
z0[i] <- mean(z0.sim)
z1[i] <- mean(z1.sim)
upper.z0[i] <- quantile(z0.sim, high)
upper.z1[i] <- quantile(z1.sim, high)
lower.z0[i] <- quantile(z0.sim, low)
lower.z1[i] <- quantile(z1.sim, low)
ind.z0[i] <- as.numeric(lower.z0[i] < 0 & upper.z0[i] > 0)
ind.z1[i] <- as.numeric(lower.z1[i] < 0 & upper.z1[i] > 0)
}
## END OF RHO LOOP
}
# Save R2 tilde values
fitted <- m.mat %*% Mmodel.coef
var.mstar <- var(fitted)
r.sq.m <- var.mstar/(1+var.mstar)
r.sq.y <- summary(model.y)$r.squared
R2tilde.prod <- rho^2 * (1 - r.sq.m) * (1 - r.sq.y)
## END OF CASE 2: Continuous Outcome + Binary Mediator
} else
#########################################################
## CASE 3: Binary Outcome + Continuous Mediator
#########################################################
if(class.y=="glm" & class.m=="lm") {
if(INT){
stop("sensitivity analysis is not available for binary outcome with interactions")
}
if(model.y$family$link == "logit"){
stop("medsens is only valid for the probit link")
}
type <- "bo"
# Step 1: Obtain Model Parameters
## Step 1-1: Simulate M model parameters
Mmodel.coef <- model.m$coef
if(robustSE){
Mmodel.var.cov <- vcovHC(model.m)
} else if(!is.null(cluster)) {
dta <- merge(m.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.m, data=dta)
MModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Mmodel.var.cov <- vcov(model.m)
}
m.k <- length(Mmodel.coef)
Mmodel.coef.sim <- rmvnorm(sims, mean = Mmodel.coef, sigma = Mmodel.var.cov)
beta2.sim <- Mmodel.coef.sim[, T.out]
sigma.2 <- summary(model.m)$sigma
sig2.shape <- model.m$df/2
sig2.invscale <- (model.m$df/2) * sigma.2^2
sigma.2.sim <- sqrt(1 / rgamma(sims, shape = sig2.shape, scale = 1/sig2.invscale))
## Step 1-2: Simulate Y model parameters
Ymodel.coef <- model.y$coef
if(robustSE){
Ymodel.var.cov <- vcovHC(model.y)
} else if(!is.null(cluster)) {
dta <- merge(y.data, as.data.frame(cluster), sort=FALSE, by="row.names")
fm <- update(model.y, data=dta)
YModel.var.cov <- getvcov(dta, fm, dta[,ncol(dta)])
} else {
Ymodel.var.cov <- vcov(model.y)
}
y.k <- length(Ymodel.coef)
Ymodel.coef.sim <- rmvnorm(sims, mean = Ymodel.coef, sigma = Ymodel.var.cov)
colnames(Ymodel.coef.sim) <- names(Ymodel.coef)
gamma.tilde <- Ymodel.coef.sim[, M.out]
# Step 2: Compute ACME via the procedure in IKT
## Step 2-1: Estimate Error Correlation from inconsistent estimate of Y on M
rho12.sim <- (sigma.2.sim * gamma.tilde) / (1 + sqrt(sigma.2.sim^2 * gamma.tilde^2))
## Step 2-2: Calculate alpha_1, beta_1 and xi_1
YTmodel.coef.sim <- ( Ymodel.coef.sim[, !colnames(Ymodel.coef.sim) %in% M.out] *
sqrt(1-rho12.sim^2) %x% t(rep(1,y.k-1)) +
Mmodel.coef.sim * (rho12.sim/sigma.2.sim) %x% t(rep(1, y.k-1)) )
## Step 2-3: Calculate Gamma
## Data matrices for the Y model less M
y.mat.1 <- y.mat.0 <- model.matrix(model.y)[, !colnames(model.matrix(model.y)) %in% M.out]
y.mat.1[,T.out] <- 1
y.mat.0[,T.out] <- 0
## Initialize objects before the rho loop
d0 <- d1 <- z0 <- z1 <-
upper.d0 <- upper.d1 <- lower.d0 <- lower.d1 <-
upper.z0 <- upper.z1 <- lower.z0 <- lower.z1 <-
ind.d0 <- ind.d1 <- ind.z0 <- ind.z1 <-
tau <- nu0 <- nu1 <-
upper.tau <- upper.nu0 <- upper.nu1 <-
lower.tau <- lower.nu0 <- lower.nu1 <- rep(NA, length(rho))
d0.sim <- d1.sim <- z0.sim <- z1.sim <-
sigma.1star.sim <- beta3.sim <-
tau.sim <- nu0.sim <- nu1.sim <- rep(NA, sims)
## START OF RHO LOOP
for(i in 1:length(rho)){
gamma.sim <- (-rho[i] + rho12.sim * sqrt((1 - rho[i]^2)/(1 - rho12.sim^2)))/sigma.2.sim
for(k in 1:sims){
sigma.1star.sim[k] <- sqrt(gamma.sim[k]^2 *
sigma.2.sim[k]^2 + 2 * gamma.sim[k] * rho[i] * sigma.2.sim[k] + 1)
YTcoef.k <- YTmodel.coef.sim[k,]
if("indirect" %in% etype.vec){
d0.sim[k] <- weighted.mean( pnorm(y.mat.0 %*% YTcoef.k +
gamma.sim[k] * beta2.sim[k] / sigma.1star.sim[k]) -
pnorm(y.mat.0 %*% YTcoef.k), w = weights )
d1.sim[k] <- weighted.mean( pnorm(y.mat.1 %*% YTcoef.k) -
pnorm(y.mat.1 %*% YTcoef.k -
gamma.sim[k] * beta2.sim[k] / sigma.1star.sim[k]),
w = weights )
tau.sim[k] <- weighted.mean( pnorm(y.mat.1 %*% YTcoef.k) -
pnorm(y.mat.0 %*% YTcoef.k), w = weights )
nu0.sim[k] <- d0.sim[k]/tau.sim[k]
nu1.sim[k] <- d1.sim[k]/tau.sim[k]
}
if("direct" %in% etype.vec){
beta3.sim[k] <- sigma.1star.sim[k] * YTcoef.k[T.out] -
beta2.sim[k] * gamma.sim[k]
z0.sim[k] <- weighted.mean( pnorm(y.mat.0 %*% YTcoef.k +
beta3.sim[k] / sigma.1star.sim[k]) -
pnorm(y.mat.0 %*% YTcoef.k), w = weights )
z1.sim[k] <- weighted.mean( pnorm(y.mat.1 %*% YTcoef.k) -
pnorm(y.mat.1 %*% YTcoef.k -
beta3.sim[k] / sigma.1star.sim[k]), w = weights )
}
}
## Step 2-4: Compute Outputs
if("indirect" %in% etype.vec){
d0[i] <- mean(d0.sim) # ACME(t=0)
d1[i] <- mean(d1.sim) # ACME(t=1)
upper.d0[i] <- quantile(d0.sim, high)
upper.d1[i] <- quantile(d1.sim, high)
lower.d0[i] <- quantile(d0.sim, low)
lower.d1[i] <- quantile(d1.sim, low)
tau[i] <- mean(tau.sim) # ATE
nu0[i] <- mean(nu0.sim) # Proportion Mediated (t=0)
nu1[i] <- mean(nu1.sim) # Proportion Mediated (t=0)
upper.tau[i] <- quantile(tau.sim, high)
upper.nu0[i] <- quantile(nu0.sim, high)
upper.nu1[i] <- quantile(nu1.sim, high)
lower.tau[i] <- quantile(tau.sim, low)
lower.nu0[i] <- quantile(nu0.sim, low)
lower.nu1[i] <- quantile(nu1.sim, low)
ind.d0[i] <- as.numeric(lower.d0[i] < 0 & upper.d0[i] > 0)
ind.d1[i] <- as.numeric(lower.d1[i] < 0 & upper.d1[i] > 0)
}
if("direct" %in% etype.vec){
z0[i] <- mean(z0.sim) # ACME(t=0)
z1[i] <- mean(z1.sim) # ACME(t=1)
upper.z0[i] <- quantile(z0.sim, high)
upper.z1[i] <- quantile(z1.sim, high)
lower.z0[i] <- quantile(z0.sim, low)
lower.z1[i] <- quantile(z1.sim, low)
ind.z0[i] <- as.numeric(lower.z0[i] < 0 & upper.z0[i] > 0)
ind.z1[i] <- as.numeric(lower.z1[i] < 0 & upper.z1[i] > 0)
}
## END OF RHO LOOP
}
# Save R2 tilde values
r.sq.m <- summary(model.m)$r.squared
y.mat <- model.matrix(model.y)
fitted <- y.mat %*% Ymodel.coef
var.ystar <- var(fitted)
r.sq.y <- var.ystar/(1 + var.ystar)
R2tilde.prod <- rho^2 * (1 - r.sq.m) * (1 - r.sq.y)
## END OF CASE 3: Binary Outcome + Continuous Mediator
} else {
stop("mediate object fitted with non-supported model combinations")
}
#########################################################
## Compute Output
#########################################################
# Calculate rho at which ACME=0
if(INT || type=="bo"){
if("indirect" %in% etype.vec){
ii <- which(abs(d0 - 0) == min(abs(d0 - 0)))
kk <- which(abs(d1 - 0) == min(abs(d1 - 0)))
err.cr.1.d <- rho[ii]
err.cr.2.d <- rho[kk]
err.cr.d <- c(err.cr.1.d, err.cr.2.d)
}
if("direct" %in% etype.vec){
ii.z <- which(abs(z0 - 0) == min(abs(z0 - 0)))
kk.z <- which(abs(z1 - 0) == min(abs(z1 - 0)))
err.cr.1.z <- rho[ii.z]
err.cr.2.z <- rho[kk.z]
err.cr.z <- c(err.cr.1.z, err.cr.2.z)
}
} else {
if("indirect" %in% etype.vec){
ii <- which(abs(d0 - 0) == min(abs(d0 - 0)))
err.cr.d <- rho[ii]
}
if("direct" %in% etype.vec){
ii.z <- which(abs(z0 - 0) == min(abs(z0 - 0)))
err.cr.z <- rho[ii.z]
}
}
if("indirect" %in% etype.vec){
R2star.d.thresh <- err.cr.d^2
R2tilde.d.thresh <- err.cr.d^2 * (1 - r.sq.m) * (1 - r.sq.y)
}
if("direct" %in% etype.vec){
R2star.z.thresh <- err.cr.z^2
R2tilde.z.thresh <- err.cr.z^2 * (1 - r.sq.m) * (1 - r.sq.y)
}
out <- list(
d0=d0, d1=d1,
upper.d0=upper.d0, lower.d0=lower.d0,
upper.d1=upper.d1, lower.d1=lower.d1,
z0=z0, z1=z1,
upper.z0=upper.z0, lower.z0=lower.z0,
upper.z1=upper.z1, lower.z1=lower.z1,
tau=tau, upper.tau=upper.tau, lower.tau=lower.tau,
nu0=nu0, nu1=nu1, upper.nu0=upper.nu0, upper.nu1=upper.nu1,
lower.nu0=lower.nu0, lower.nu1=lower.nu1,
rho = rho, rho.by=rho.by, sims=sims,
err.cr.d=err.cr.d, err.cr.z=err.cr.z,
ind.d0=ind.d0, ind.d1=ind.d1, ind.z0=ind.z0, ind.z1=ind.z1,
R2star.prod=R2star.prod, R2tilde.prod=R2tilde.prod,
R2star.d.thresh=R2star.d.thresh, R2tilde.d.thresh=R2tilde.d.thresh,
R2star.z.thresh=R2star.z.thresh, R2tilde.z.thresh=R2tilde.z.thresh,
r.square.y=r.sq.y, r.square.m=r.sq.m,
INT=INT, conf.level = x$conf.level, effect.type=effect.type, type=type,
robustSE = robustSE, cluster = cluster)
class(out) <- "medsens"
out
}
#' Summarizing Results from Sensitivity Analysis for Causal Mediation Effects
#'
#' Functions to report results from the sensitivity analysis for causal
#' mediation effects via \code{\link{medsens}} in a tabular form.
#'
#' @aliases summary.medsens print.summary.medsens
#'
#' @param object output from medsens function.
#' @param x output from summary.medsens function.
#' @param ... additional arguments affecting the summary produced.
#'
#' @author Dustin Tingley, Harvard University,
#' \email{dtingley@@gov.harvard.edu}; Teppei Yamamoto, Massachusetts Institute
#' of Technology, \email{teppei@@mit.edu}; Jaquilyn Waddell-Boie, Princeton
#' University, \email{jwaddell@@princeton.edu}; Luke Keele, Penn State
#' University, \email{ljk20@@psu.edu}; Kosuke Imai, Princeton University,
#' \email{kimai@@princeton.edu}.
#'
#' @seealso \code{\link{medsens}}, \code{\link{summary}}.
#'
#' @references Tingley, D., Yamamoto, T., Hirose, K., Imai, K. and Keele, L.
#' (2014). "mediation: R package for Causal Mediation Analysis", Journal of
#' Statistical Software, Vol. 59, No. 5, pp. 1-38.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2011). Unpacking the
#' Black Box of Causality: Learning about Causal Mechanisms from Experimental
#' and Observational Studies, American Political Science Review, Vol. 105, No.
#' 4 (November), pp. 765-789.
#'
#' Imai, K., Keele, L. and Tingley, D. (2010) A General Approach to Causal
#' Mediation Analysis, Psychological Methods, Vol. 15, No. 4 (December), pp.
#' 309-334.
#'
#' Imai, K., Keele, L. and Yamamoto, T. (2010) Identification, Inference, and
#' Sensitivity Analysis for Causal Mediation Effects, Statistical Science,
#' Vol. 25, No. 1 (February), pp. 51-71.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2009) "Causal Mediation
#' Analysis Using R" in Advances in Social Science Research Using R, ed. H. D.
#' Vinod New York: Springer.
#'
#' @export
summary.medsens <- function(object, ...)
structure(object, class = c("summary.medsens", class(object)))
#' @rdname summary.medsens
#' @export
print.summary.medsens <- function(x, ...){
etype.vec <- x$effect.type
if(etype.vec == "both"){
etype.vec <- c("indirect","direct")
}
if(x$INT || x$type=="bo"){
if("indirect" %in% etype.vec){
tab.d0 <- cbind(round(x$rho,4), round(x$d0,4), round(x$lower.d0,4),
round(x$upper.d0, 4), x$ind.d0, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.d0)==1){
tab.d0 <- as.matrix(tab.d0[x$ind.d0==1, -5])
tab.d0 <- t(tab.d0)
} else {
tab.d0 <- tab.d0[x$ind.d0==1, -5]
}
colnames(tab.d0) <- c("Rho", "ACME(control)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.d0) <- NULL
tab.d1 <- cbind(round(x$rho,4), round(x$d1,4), round(x$lower.d1,4),
round(x$upper.d1, 4), x$ind.d1, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.d1)==1){
tab.d1 <- as.matrix(tab.d1[x$ind.d1==1, -5])
tab.d1 <- t(tab.d1)
} else {
tab.d1 <- tab.d1[x$ind.d1==1, -5]
}
colnames(tab.d1) <- c("Rho", "ACME(treated)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.d1) <- NULL
cat("\nMediation Sensitivity Analysis: Average Mediation Effect\n")
if(sum(x$ind.d0) > 0){
cat("\nSensitivity Region: ACME for Control Group\n\n")
print(tab.d0)
}
cat("\nRho at which ACME for Control Group = 0:", round(x$err.cr.d[1], 4))
cat("\nR^2_M*R^2_Y* at which ACME for Control Group = 0:", round(x$R2star.d.thresh[1], 4))
cat("\nR^2_M~R^2_Y~ at which ACME for Control Group = 0:", round(x$R2tilde.d.thresh[1], 4), "\n\n")
if(sum(x$ind.d1) > 0){
cat("\nSensitivity Region: ACME for Treatment Group\n\n")
print(tab.d1)
}
cat("\nRho at which ACME for Treatment Group = 0:", round(x$err.cr.d[2], 4))
cat("\nR^2_M*R^2_Y* at which ACME for Treatment Group = 0:", round(x$R2star.d.thresh[2], 4))
cat("\nR^2_M~R^2_Y~ at which ACME for Treatment Group = 0:", round(x$R2tilde.d.thresh[2], 4), "\n\n")
invisible(x)
}
if("direct" %in% etype.vec){
tab.z0 <- cbind(round(x$rho,4), round(x$z0,4), round(x$lower.z0,4),
round(x$upper.z0, 4), x$ind.z0, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.z0)==1){
tab.z0 <- as.matrix(tab.z0[x$ind.z0==1, -5])
tab.z0 <- t(tab.z0)
} else {
tab.z0 <- tab.z0[x$ind.z0==1, -5]
}
colnames(tab.z0) <- c("Rho", "ADE(control)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.z0) <- NULL
tab.z1 <- cbind(round(x$rho,4), round(x$z1,4), round(x$lower.z1,4),
round(x$upper.z1, 4), x$ind.z1, round(x$R2star.prod,4), round(x$R2tilde.prod, 4))
if(sum(x$ind.z1)==1){
tab.z1 <- as.matrix(tab.z1[x$ind.z1==1, -5])
tab.z1 <- t(tab.z1)
} else {
tab.z1 <- tab.z1[x$ind.z1==1, -5]
}
colnames(tab.z1) <- c("Rho", "ADE(treated)", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.z1) <- NULL
cat("\nMediation Sensitivity Analysis: Average Direct Effect\n")
if(sum(x$ind.z0) > 0){
cat("\nSensitivity Region: ADE for Control Group\n\n")
print(tab.z0)
}
cat("\nRho at which ADE for Control Group = 0:", round(x$err.cr.z[1], 4))
cat("\nR^2_M*R^2_Y* at which ADE for Control Group = 0:", round(x$R2star.z.thresh[1], 4))
cat("\nR^2_M~R^2_Y~ at which ADE for Control Group = 0:", round(x$R2tilde.z.thresh[1], 4), "\n\n")
if(sum(x$ind.z1) > 0){
cat("\nSensitivity Region: ADE for Treatment Group\n\n")
print(tab.z1)
}
cat("\nRho at which ADE for Treatment Group = 0:", round(x$err.cr.z[2], 4))
cat("\nR^2_M*R^2_Y* at which ADE for Treatment Group = 0:", round(x$R2star.z.thresh[2], 4))
cat("\nR^2_M~R^2_Y~ at which ADE for Treatment Group = 0:", round(x$R2tilde.z.thresh[2], 4), "\n\n")
invisible(x)
}
} else {
if("indirect" %in% etype.vec){
tab <- cbind(round(x$rho,4), round(x$d0, 4), round(x$lower.d0, 4),
round(x$upper.d0, 4), x$ind.d0, round(x$R2star.prod,4), round(x$R2tilde.prod,4))
if(sum(x$ind.d0)==1){
tab <- as.matrix(tab[x$ind.d0==1, -5])
tab <- t(tab)
} else {
tab <- tab[x$ind.d0==1, -5]
}
colnames(tab) <- c("Rho", "ACME", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab) <- NULL
cat("\nMediation Sensitivity Analysis for Average Causal Mediation Effect\n")
if(sum(x$ind.d0) > 0){
cat("\nSensitivity Region\n\n")
print(tab)
}
cat("\nRho at which ACME = 0:", round(x$err.cr.d, 4))
cat("\nR^2_M*R^2_Y* at which ACME = 0:", round(x$R2star.d.thresh, 4))
cat("\nR^2_M~R^2_Y~ at which ACME = 0:", round(x$R2tilde.d.thresh, 4), "\n\n")
invisible(x)
}
if("direct" %in% etype.vec){
tab.z <- cbind(round(x$rho,4), round(x$z0, 4), round(x$lower.z0, 4),
round(x$upper.z0, 4), x$ind.z0, round(x$R2star.prod,4), round(x$R2tilde.prod,4))
if(sum(x$ind.z0)==1){
tab.z <- as.matrix(tab.z[x$ind.z0==1, -5])
tab.z <- t(tab.z)
} else {
tab.z <- tab.z[x$ind.z0==1, -5]
}
colnames(tab.z) <- c("Rho", "ADE", paste(100*x$conf.level, "% CI Lower", sep=""),
paste(100*x$conf.level, "% CI Upper", sep=""), "R^2_M*R^2_Y*", "R^2_M~R^2_Y~")
rownames(tab.z) <- NULL
cat("\nMediation Sensitivity Analysis for Average Direct Effect\n")
if(sum(x$ind.z0) > 0){
cat("\nSensitivity Region\n\n")
print(tab.z)
}
cat("\nRho at which ADE = 0:", round(x$err.cr.z, 4))
cat("\nR^2_M*R^2_Y* at which ADE = 0:", round(x$R2star.z.thresh, 4))
cat("\nR^2_M~R^2_Y~ at which ADE = 0:", round(x$R2tilde.z.thresh, 4), "\n\n")
invisible(x)
}
}
}
#' Plotting Results from Sensitivity Analysis for Causal Mediation Effects
#'
#' This function is used to plot results from the 'medsens' function. Causal
#' average mediation effects (as well as average direct effects and proportions
#' mediated for selected models) can be plotted against two alternative
#' sensitivity parameters.
#'
#' @details The sensitivity analysis for causal mediation effects can be
#' conducted in terms of two alternative sensitivity parameters, which both
#' quantify the degree of violation of the sequential ignorability assumption.
#' The "rho" parameter represents the correlation between the two error terms
#' of the (latent) linear models for the mediator and outcome variables. A
#' large value of rho indicates the existence of important common unobserved
#' predictors for both the mediator and outcome and therefore a high degree of
#' sequential ignorability violation, while a value close to zero implies
#' there is no such confounders.
#'
#' The resulting "rho" figures plot the estimated true values of ACME (or ADE,
#' proportion mediated) against rho, along with the confidence intervals. When
#' rho is zero, sequantial ignorability holds, so the estimated value at that
#' point will be equal to the estimate returned by the \code{\link{mediate}}.
#' The confidence level is determined by the 'conf.level' value of the
#' original \code{\link{mediate}} object.
#'
#' The "R2" parameters represent the proportions of the mediator and outcome
#' variances that are explained by an unobserved pre-treatment confounder,
#' thereby indicating the importance of such a confounder in each model. When
#' 'r.type' is "residual", the R2 parameters represent the proportions of the
#' residual variances of the mediator and outcome models that become explained
#' by the inclusion of the hypothetical pre-treatment confounder. These are
#' denoted as "R square stars" in Imai, Keele and Yamamoto (2010) and can also
#' be specified as "star" or using a numeric value 1 in \code{medsens.plot}.
#' When 'r.type' is "total", the R2s represent the total mediator and outcome
#' variances the unobserved confounder would explain. This option can also be
#' specified using "tilde" or a numeric value 2.
#'
#' For both types of the "R2" parameters, 'sign.prod' indicates the
#' hypothesized direction in which the unobserved confounder affects the
#' mediator and outcome. (The name derives from the fact that this direction
#' is mathematically represented by the sign of the product of two regression
#' coefficients.) If "positive" (or a numeric value 1) is given, the
#' confounder is assumed to affect the mediator and outcome in the same
#' direction. If "negative" (or a numeric value -1), the effect is assumed to
#' be in opposite directions.
#'
#' The resulting contours in the "R2" plots represent the values of the ACME
#' (or ADE) for different combinations of the mediator R2 and outcome R2
#' values. When both values are zero (the lower-left corner of the plot), the
#' unobserved pre-treatment confounder has no effect on either mediator or
#' outcome and therefore sequantial ignorability is satisfied.
#'
#' @param x 'medsens' object, typically output from \code{medsens}.
#' @param sens.par a character string indicating the sensitivity parameter to be
#' used. Default plots effects as functions of "rho". See Details.
#' @param r.type type of the R square parameter to be used in "R2" plots. If
#' "residual", effects are plotted against the proportions of the residual
#' variances that are explained by the unobserved confounder. If "total", the
#' proportions of the total variances are used as sensitivity parameters. Only
#' relevant if 'sens.par' is "R2".
#' @param sign.prod a value indicating the direction of hypothesized confounding
#' in the sensitivity analysis. If "positive", the confounder is assumed to
#' affect the mediator and outcome variable in the same direction; if
#' "negative" the effects are assumed to be in opposite directions. Only
#' relevant if sens.par is set to "R2".
#' @param pr.plot a logical value. If 'TRUE', the "proportions mediated" will be
#' plotted instead of the average causal mediation effects or direct effects.
#' Currently only available if the object 'medsens' is based on the linear
#' mediator and binary probit outcome models. Default is 'FALSE'.
#' @param smooth.effect a logical value indicating whether the estimated
#' mediation effects are smoothed via \code{\link{lowess}} before being
#' plotted. Default is 'FALSE'.
#' @param smooth.ci a logical value indicating whether the confidence bands are
#' smoothed via \code{\link{lowess}} before being plotted. Default is 'FALSE'.
#' @param ask a logical value. If 'TRUE', the user is asked for input before a
#' new figure is plotted. Default is to ask only if the number of plots on
#' current screen is fewer than necessary.
#' @param levels vector of levels at which to draw contour lines. Only relevant
#' if 'sens.par' is set to "R2". If 'NULL', default values in
#' \code{\link{contour.default}} are used.
#' @param xlab label for the x axis. Default labels are used if 'NULL'.
#' @param ylab label for the y axis. Default labels are used if 'NULL'.
#' @param xlim limits of the x axis. If 'NULL' default values are used.
#' @param ylim limits of the y axis. If 'NULL' default values are used.
#' @param main main title for the plot. If 'NULL', default titles are used.
#' @param lwd width of the lines used in graphs.
#' @param ... additional arguments to be passed to plotting functions.
#'
#' @section Warning: The 'smooth.effect' and 'smooth.ci' options should be used
#' with caution since the smoothing could affect substantive implications of
#' the graphical analysis in a significant way.
#'
#' @author Dustin Tingley, Harvard University,
#' \email{dtingley@@gov.harvard.edu}; Teppei Yamamoto, Massachusetts Institute
#' of Technology, \email{teppei@@mit.edu}; Jaquilyn Waddell-Boie, Princeton
#' University, \email{jwaddell@@princeton.edu}; Luke Keele, Penn State
#' University, \email{ljk20@@psu.edu}; Kosuke Imai, Princeton University,
#' \email{kimai@@princeton.edu}.
#'
#' @seealso \code{\link{medsens}}, \code{\link{plot}}, \code{\link{contour}}.
#'
#' @references Tingley, D., Yamamoto, T., Hirose, K., Imai, K. and Keele, L.
#' (2014). "mediation: R package for Causal Mediation Analysis", Journal of
#' Statistical Software, Vol. 59, No. 5, pp. 1-38.
#'
#' Imai, K., Keele, L. and Tingley, D. (2010) A General Approach to Causal
#' Mediation Analysis, Psychological Methods, Vol. 15, No. 4 (December), pp.
#' 309-334.
#'
#' Imai, K., Keele, L. and Yamamoto, T. (2010) Identification, Inference, and
#' Sensitivity Analysis for Causal Mediation Effects, Statistical Science,
#' Vol. 25, No. 1 (February), pp. 51-71.
#'
#' Imai, K., Keele, L., Tingley, D. and Yamamoto, T. (2009) "Causal Mediation
#' Analysis Using R" in Advances in Social Science Research Using R, ed. H. D.
#' Vinod New York: Springer.
#' @export
plot.medsens <- function(x, sens.par = c("rho", "R2"),
r.type = c("residual", "total"), sign.prod = c("positive", "negative"),
pr.plot = FALSE, smooth.effect = FALSE, smooth.ci = FALSE,
ask = prod(par("mfcol")) < nplots, levels = NULL,
xlab = NULL, ylab = NULL, xlim = NULL, ylim = NULL,
main = NULL, lwd = par("lwd"), ...){
# Compatibility with previous version
if(r.type[1L] == 1 | r.type[1L] == "star") r.type <- "residual"
if(r.type[1L] == 2 | r.type[1L] == "tilde") r.type <- "total"
if(sign.prod[1L] == 1) sign.prod <- "positive"
if(sign.prod[1L] == -1) sign.prod <- "negative"
# Match arguments
sens.par <- match.arg(sens.par)
r.type <- match.arg(r.type)
sign.prod <- match.arg(sign.prod)
# Effect types
etype.vec <- x$effect.type
if(x$effect.type == "both"){
etype.vec <- c("indirect","direct")
}
# Determine number of plots necessary
nplots <- ifelse(pr.plot, 2,
ifelse(x$INT || x$type == "bo", 2, 1) * ifelse(x$effect.type == "both", 2, 1))
if (ask) {
oask <- devAskNewPage(TRUE)
on.exit(devAskNewPage(oask))
}
##################################################
## A. Plots for proportions mediated
##################################################
if(pr.plot){
if(sens.par != "rho"){
stop("proportion mediated can only be plotted against rho")
}
if(x$type != "bo"){
stop("proportion mediated is only implemented for binary outcome")
}
out <- as.list(rep(NA, 2))
out[[1]] <- list(nu = x$nu0, lo = x$lower.nu0, up = x$upper.nu0)
out[[2]] <- list(nu = x$nu1, lo = x$lower.nu1, up = x$upper.nu1)
r <- x$rho
for(i in 1:2){
o <- out[[i]]
if(is.null(ylim)){
yl <- c(min(o$nu), max(o$nu))
} else yl <- ylim
if(is.null(main))
ma <- bquote(paste("Proportion Mediated"[.(i-1)], (rho)))
else ma <- main
if(smooth.effect)
nu <- lowess(r, o$nu)$y
else nu <- o$nu
if(smooth.ci){
lower <- lowess(r, o$lo)$y
upper <- lowess(r, o$up)$y
} else {
lower <- o$lo
upper <- o$up
}
plot.default(r, nu, type = "n", xlab = "", ylab = "",
main = ma, xlim = xlim, ylim = yl, ...)
polygon(x = c(r, rev(r)), y = c(lower, rev(upper)),
border = FALSE, col = 8, lty = 2, lwd = lwd)
lines(r, nu, lty = 1, lwd = lwd)
abline(h = 0)
abline(v = 0)
if(is.null(xlab))
title(xlab = expression(paste("Sensitivity Parameter: ", rho)),
line = 2.5, cex.lab = .9)
else title(xlab = xlab)
if(is.null(ylab))
title(ylab = bquote(paste("Proportion Mediated: ", bar(nu), (.(i-1)))),
line = 2.5, cex.lab = .9)
else title(ylab = ylab)
}
##################################################
## B. Plots for ACME/ADE wrt rho
##################################################
} else if(sens.par == "rho"){
out <- as.list(NA, 4)
out[[1]] <- list(eff = x$d0, lo = x$lower.d0, up = x$upper.d0,
labs = c("ACME", "Average Mediation Effect", 0))
out[[2]] <- list(eff = x$d1, lo = x$lower.d1, up = x$upper.d1,
labs = c("ACME", "Average Mediation Effect", 1))
out[[3]] <- list(eff = x$z0, lo = x$lower.z0, up = x$upper.z0,
labs = c("ADE", "Average Direct Effect", 0))
out[[4]] <- list(eff = x$z1, lo = x$lower.z1, up = x$upper.z1,
labs = c("ADE", "Average Direct Effect", 1))
r <- x$rho
rby <- x$rho.by
wh <- x$INT || x$type == "bo"
out[[1]]$show <- "indirect" %in% etype.vec
out[[2]]$show <- wh && "indirect" %in% etype.vec
out[[3]]$show <- "direct" %in% etype.vec
out[[4]]$show <- wh && "direct" %in% etype.vec
for(i in 1:length(out)){
o <- out[[i]]
if(!o$show) next
if(is.null(ylim)){
yl <- c(min(o$eff), max(o$eff))
} else yl <- ylim
if(is.null(main)){
if(wh){
ma <- bquote(paste(.(o$labs[1])[.(o$labs[3])], "(", rho, ")"))
} else {
ma <- bquote(paste(.(o$labs[1]), "(", rho, ")"))
}
} else ma <- main
if(smooth.effect)
eff <- lowess(r, o$eff)$y
else eff <- o$eff
if(smooth.ci){
lower <- lowess(r, o$lo)$y
upper <- lowess(r, o$up)$y
} else {
lower <- o$lo
upper <- o$up
}
plot.default(r, eff, type = "n", xlab = "", ylab = "",
main = ma, xlim = xlim, ylim = yl, ...)
polygon(x = c(r, rev(r)), y = c(lower, rev(upper)),
border = FALSE, col = 8, lty = 2, lwd = lwd)
lines(r, eff, lty = 1, lwd = lwd)
abline(h = 0)
abline(v = 0)
abline(h = weighted.mean(c(eff[floor(1/rby)], eff[ceiling(1/rby)]),
c(1 - 1/rby + floor(1/rby), 1/rby - floor(1/rby))),
lty = 2, lwd = lwd)
if(is.null(xlab))
title(xlab = expression(paste("Sensitivity Parameter: ", rho)),
line = 2.5, cex.lab = .9)
else title(xlab = xlab)
if(is.null(ylab)){
if(wh){
title(ylab = bquote(.(o$labs[2])[.(o$labs[3])]),
line = 2.5, cex.lab = .9)
} else {
title(ylab = o$labs[2], line = 2.5, cex.lab = .9)
}
} else title(ylab = ylab)
}
##################################################
## C. Plots for ACME/ADE wrt R2
##################################################
} else {
R2Mstar <- seq(0, 1 - x$rho.by, 0.01)
R2Ystar <- seq(0, 1 - x$rho.by, 0.01)
R2Mtilde <- (1 - x$r.square.m) * seq(0, 1 - x$rho.by, 0.01)
R2Ytilde <- (1 - x$r.square.y) * seq(0, 1 - x$rho.by, 0.01)
if(r.type == "residual"){
R2M <- R2Mstar; R2Y <- R2Ystar
} else {
R2M <- R2Mtilde; R2Y <- R2Ytilde
}
dlength <- length(seq(0, (1 - x$rho.by)^2, 0.0001))
R2prod.mat <- outer(R2Mstar, R2Ystar)
Rprod.mat <- round(sqrt(R2prod.mat), digits=4)
if(is.null(xlim))
xlim <- c(0,1)
if(is.null(ylim))
ylim <- c(0,1)
out <- as.list(NA, 4)
out[[1]] <- list(eff = x$d0, labs = c("ACME", "Average Mediation Effect", 0))
out[[2]] <- list(eff = x$d1, labs = c("ACME", "Average Mediation Effect", 1))
out[[3]] <- list(eff = x$z0, labs = c("ADE", "Average Direct Effect", 0))
out[[4]] <- list(eff = x$z1, labs = c("ADE", "Average Direct Effect", 1))
wh <- x$INT || x$type == "bo"
out[[1]]$show <- "indirect" %in% etype.vec
out[[2]]$show <- wh && "indirect" %in% etype.vec
out[[3]]$show <- "direct" %in% etype.vec
out[[4]]$show <- wh && "direct" %in% etype.vec
for(i in 1:length(out)){
o <- out[[i]]
if(!o$show) next
if(is.null(levels)){
levels <- pretty(quantile(o$eff, probs=c(0.1,0.9)), 10)
}
if(sign.prod == "positive"){
eff <- approx(o$eff[((length(o$eff)+1)/2):length(o$eff)], n=dlength)$y
sgnlab <- "1"
} else {
eff <- rev(approx(o$eff[1:((length(o$eff)+1)/2)], n=dlength)$y)
sgnlab <- "-1"
}
effmat <- matrix(eff[Rprod.mat/0.0001+1], nrow=length(R2M))
ma <- switch(wh + 1, bquote(paste(.(o$labs[1]))), # wh == F
bquote(paste(.(o$labs[1])[.(o$labs[3])]))) # wh == T
if(is.null(main) & r.type == "residual")
ma <- bquote(paste(.(ma), "(", R[M]^{2},"*,", R[Y]^2,"*), sgn",
(lambda[2]*lambda[3])==.(sgnlab)))
else if(is.null(main) & r.type == "total")
ma <- bquote(paste(.(ma), "(", tilde(R)[M]^{2}, "," , tilde(R)[Y]^2, "), sgn",
(lambda[2]*lambda[3])==.(sgnlab)))
else ma <- main
contour(R2M, R2Y, effmat, levels = levels, main = ma,
xlab = xlab, ylab = ylab, ylim = ylim, xlim = xlim, lwd = lwd, ...)
contour(R2M, R2Y, effmat, levels = 0, lwd = lwd + 1, add = TRUE, drawlabels = FALSE)
if(is.null(xlab) & r.type == "residual")
title(xlab=expression(paste(R[M]^{2},"*")), line=2.5, cex.lab=.9)
else if(is.null(xlab) & r.type == "total")
title(xlab=expression(paste(tilde(R)[M]^{2})), line=2.5, cex.lab=.9)
if(is.null(ylab) & r.type == "residual")
title(ylab=expression(paste(R[Y]^2,"*")), line=2.5, cex.lab=.9)
else if(is.null(ylab) & r.type == "total")
title(ylab=expression(paste(tilde(R)[Y]^2)), line=2.5, cex.lab=.9)
axis(2, at = seq(0, 1, by = .1))
axis(1, at = seq(0, 1, by = .1))
}
}
}
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