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R/nw_est.R
In causaldrf: Estimating Causal Dose Response Functions
##' The Nadaraya-Watson modified estimator
##'
##' This is a kernel based regression method that uses a kernel as a weighting
##' function to estimate the ADRF. The normal kernel is weighted by the inverse of the estimated
##' GPS. See Flores et al. (2012) for more details.
##'
##'
##' @param Y is the the name of the outcome variable contained in \code{data}.
##' @param treat is the name of the treatment variable contained in
##' \code{data}.
##' @param treat_formula an object of class "formula" (or one that can be
##' coerced to that class) that regresses \code{treat} on a linear combination
##' of \code{X}: a symbolic description of the model to be fitted.
##' @param data is a dataframe containing \code{Y} and \code{treat} and
##' \code{X}.
##' @param grid_val contains the treatment values to be evaluated.
##' @param bandw is the bandwidth. Default is 1.
##' @param treat_mod a description of the error distribution to be used in the
##' model for treatment. Options include: \code{"Normal"} for normal model,
##' \code{"LogNormal"} for lognormal model, \code{"Sqrt"} for square-root transformation
##' to a normal treatment, \code{"Poisson"} for Poisson model,
##' \code{"NegBinom"} for negative binomial model, \code{"Gamma"} for gamma
##' model.
##' @param link_function is either "log", "inverse", or "identity" for the
##' "Gamma" \code{treat_mod}.
##' @param ... additional arguments to be passed to the treatment regression function.
##'
##' @details
##'
##' This method is a version of the Nadarya-Watson estimator
##' Nadaraya (1964) which is a local constant regression but
##' weighted by the inverse of the estimated GPS.
##'
##'
##' @return \code{nw_est} returns an object of class "causaldrf",
##' a list that contains the following components:
##' \item{param}{parameter estimates for a nw fit.}
##' \item{t_mod}{the result of the treatment model fit.}
##' \item{call}{the matched call.}
##'
##' @seealso \code{\link{nw_est}}, \code{\link{iw_est}}, \code{\link{hi_est}}, \code{\link{gam_est}},
##' \code{\link{add_spl_est}},
##' \code{\link{bart_est}}, etc. for other estimates.
##'
##' \code{\link{t_mod}}, \code{\link{overlap_fun}} to prepare the \code{data}
##' for use in the different estimates.
##'
##' @references Schafer, J.L., Galagate, D.L. (2015). Causal inference with a
##' continuous treatment and outcome: alternative estimators for parametric
##' dose-response models. \emph{Manuscript in preparation}.
##'
##' Flores, Carlos A., et al. "Estimating the effects of length of exposure to
##' instruction in a training program: the case of job corps."
##' \emph{Review of Economics and Statistics} \bold{94.1} (2012): 153-171.
##'
##' Nadaraya, Elizbar A. "On estimating regression."
##' \emph{Theory of Probability and Its Applications} \bold{9.1} (1964): 141--142.
##'
##' @examples
##'
##' ## Example from Schafer (2015).
##'
##' example_data <- sim_data
##'
##' nw_list <- nw_est(Y = Y,
##' treat = T,
##' treat_formula = T ~ B.1 + B.2 + B.3 + B.4 + B.5 + B.6 + B.7 + B.8,
##' data = example_data,
##' grid_val = seq(8, 16, by = 1),
##' bandw = bw.SJ(example_data$T),
##' treat_mod = "Normal")
##'
##' sample_index <- sample(1:1000, 100)
##'
##' plot(example_data$T[sample_index],
##' example_data$Y[sample_index],
##' xlab = "T",
##' ylab = "Y",
##' main = "nw estimate")
##'
##' lines(seq(8, 16, by = 1),
##' nw_list$param,
##' lty = 2,
##' lwd = 2,
##' col = "blue")
##'
##' legend('bottomright',
##' "nw estimate",
##' lty=2,
##' lwd = 2,
##' col = "blue",
##' bty='Y',
##' cex=1)
##'
##' rm(example_data, nw_list, sample_index)
##'
##'
##' @usage
##'
##' nw_est(Y,
##' treat,
##' treat_formula,
##' data,
##' grid_val,
##' bandw,
##' treat_mod,
##' link_function,
##' ...)
##'
##' @export
##'
nw_est <- function (Y,
treat,
treat_formula,
data,
grid_val,
bandw,
treat_mod,
link_function,
...)
{
# Y is the name of the Y variable
# treat is the name of the treatment variable
# treat_formula is the formula for the treatment model
# data will contain all the data: X, treat, and Y
# grid_val is the set of grid points on T
# bandw is the bandwidth
# treat_mod is th treatment model to fit
# link_function is the link function used, if needed
# The outcome is a list of 2 objects:
# (1) estimated values of ADRF at gridvalues
# (2) result of treatment model
#save input
tempcall <- match.call()
#some basic input checks
if (!("Y" %in% names(tempcall))) stop("No Y variable specified")
if (!("treat" %in% names(tempcall))) stop("No treat variable specified")
if (!("treat_formula" %in% names(tempcall))) stop("No treat_formula model specified")
if (!("data" %in% names(tempcall))) stop("No data specified")
if (!("grid_val" %in% names(tempcall))) stop("No grid_val specified")
if (!("bandw" %in% names(tempcall))) stop("No bandw specified")
if (!("treat_mod" %in% names(tempcall)) | ("treat_mod" %in% names(tempcall) & !(tempcall$treat_mod %in% c("NegBinom", "Poisson", "Gamma", "LogNormal", "Sqrt", "Normal")))) stop("No valid family specified (\"NegBinom\", \"Poisson\", \"Gamma\", \"Log\", \"Sqrt\", \"Normal\")")
if (tempcall$treat_mod == "Gamma") {if(!(tempcall$link_function %in% c("log", "inverse"))) stop("No valid link function specified for family = Gamma (\"log\", \"inverse\")")}
if (tempcall$treat_mod == "binomial") {if(!(tempcall$link_function %in% c("logit", "probit", "cauchit", "log", "cloglog"))) stop("No valid link function specified for family = binomial (\"logit\", \"probit\", \"cauchit\", \"log\", \"cloglog\")")}
if (tempcall$treat_mod == "ordinal" ) {if(!(tempcall$link_function %in% c("logit", "probit", "cauchit", "cloglog"))) stop("No valid link function specified for family = ordinal (\"logit\", \"probit\", \"cauchit\", \"cloglog\")")}
#make new dataframe for newly computed variables, to prevent variable name conflicts
tempdat <- data.frame(
Y = data[,as.character(tempcall$Y)],
treat = data[,as.character(tempcall$treat)]
)
# make a formula for the treatment model
# formula_t <- eval(parse(text = paste(deparse(tempcall$treat, width.cutoff = 500), deparse(tempcall$treat_formula, width.cutoff = 500), sep = "")))
formula_t <- tempcall$treat_formula
if (treat_mod == "NegBinom") {
samp_dat <- data
result <- MASS::glm.nb(formula = formula_t, link = log, data = samp_dat, ...)
cond_mean <- result$fitted.values
cond_var <- cond_mean + cond_mean^2/result$theta
prob_nb_est <- (cond_var - cond_mean)/cond_var
gps_fun_NB <- function(tt) {
dnbinom(x = tt, size = result$theta, mu = result$fitted.values,
log = FALSE)
}
gps_fun <- gps_fun_NB
}
else if (treat_mod == "Poisson") {
samp_dat <- data
result <- glm(formula = formula_t, family = "poisson", data = samp_dat, ...)
cond_mean <- result$fitted.values
samp_dat$gps_vals <- dpois(x = tempdat$treat, lambda = cond_mean)
gps_fun_Pois <- function(t) {
dpois(t, lambda = cond_mean)
}
gps_fun <- gps_fun_Pois
}
else if (treat_mod == "Gamma") {
samp_dat <- data
result <- glm(formula = formula_t, family = Gamma(link = link_function),
data = samp_dat, ...)
est_treat <- result$fitted
shape_gamma <- as.numeric(MASS::gamma.shape(result)[1])
theta_given_X <- result$fitted.values/shape_gamma
theta_treat_X <- tempdat$treat/shape_gamma
gps_fun_Gamma <- function(t) {
dgamma(t, shape = shape_gamma, scale = theta_given_X)
}
gps_fun <- gps_fun_Gamma
}
else if (treat_mod == "LogNormal") {
samp_dat <- data
samp_dat[, as.character(tempcall$treat)] <- log(samp_dat[, as.character(tempcall$treat)])
result <- lm(formula = formula_t, data = samp_dat, ...)
est_log_treat <- result$fitted
sigma_est <- summary(result)$sigma
gps_fun_Log <- function(tt) {
dnorm(log(tt), mean = est_log_treat, sd = sigma_est)
}
gps_fun <- gps_fun_Log
}
else if (treat_mod == "Sqrt") {
samp_dat <- data
samp_dat[, as.character(tempcall$treat)] <- sqrt(samp_dat[, as.character(tempcall$treat)])
result <- lm(formula = formula_t, data = samp_dat, ...)
est_sqrt_treat <- result$fitted
sigma_est <- summary(result)$sigma
gps_fun_sqrt <- function(tt) {
dnorm(sqrt(tt), mean = est_sqrt_treat, sd = sigma_est)
}
gps_fun <- gps_fun_sqrt
}
else if (treat_mod == "Normal") {
samp_dat <- data
result <- lm(formula = formula_t, data = samp_dat, ...)
gps_fun_Normal <- function(tt) {
dnorm(tt, mean = result$fitted, sd = summary(result)$sigma)
}
gps_fun <- gps_fun_Normal
}
else {
stop("No valid treat_mod specified. Please try again)")
}
if (treat_mod == "LogNormal") {
K_h <- function(t, h) {
dnorm((log(tempdat$treat) - log(t))/h, 0, 1)/h
}
K_h_tild <- function(t, h) {
K_h(t, h)/gps_fun(t)
}
K_est <- numeric(length(grid_val))
for (i in 1:length(grid_val)) {
K_est[i] <- sum(tempdat$Y * K_h_tild(grid_val[i], bandw))/sum(K_h_tild(grid_val[i],
bandw))
}
}
else if (treat_mod == "Sqrt") {
K_h <- function(t, h) {
dnorm((sqrt(tempdat$treat) - sqrt(t))/h, 0, 1)/h
}
K_h_tild <- function(t, h) {
K_h(t, h)/gps_fun(t)
}
K_est <- numeric(length(grid_val))
for (i in 1:length(grid_val)) {
K_est[i] <- sum(tempdat$Y * K_h_tild(grid_val[i], bandw))/sum(K_h_tild(grid_val[i],
bandw))
}
}
else {
K_h <- function(t, h) {
dnorm((tempdat$treat - t)/h, 0, 1)/h
}
K_h_tild <- function(t, h) {
K_h(t, h)/gps_fun(t)
}
K_est <- numeric(length(grid_val))
for (i in 1:length(grid_val)) {
K_est[i] <- sum(tempdat$Y * K_h_tild(grid_val[i], bandw))/sum(K_h_tild(grid_val[i],
bandw))
}
}
z_object <- list(param = K_est,
t_mod = result,
call = tempcall)
class(z_object) <- "causaldrf"
z_object
}
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##' The Nadaraya-Watson modified estimator
##'
##' This is a kernel based regression method that uses a kernel as a weighting
##' function to estimate the ADRF. The normal kernel is weighted by the inverse of the estimated
##' GPS. See Flores et al. (2012) for more details.
##'
##'
##' @param Y is the the name of the outcome variable contained in \code{data}.
##' @param treat is the name of the treatment variable contained in
##' \code{data}.
##' @param treat_formula an object of class "formula" (or one that can be
##' coerced to that class) that regresses \code{treat} on a linear combination
##' of \code{X}: a symbolic description of the model to be fitted.
##' @param data is a dataframe containing \code{Y} and \code{treat} and
##' \code{X}.
##' @param grid_val contains the treatment values to be evaluated.
##' @param bandw is the bandwidth. Default is 1.
##' @param treat_mod a description of the error distribution to be used in the
##' model for treatment. Options include: \code{"Normal"} for normal model,
##' \code{"LogNormal"} for lognormal model, \code{"Sqrt"} for square-root transformation
##' to a normal treatment, \code{"Poisson"} for Poisson model,
##' \code{"NegBinom"} for negative binomial model, \code{"Gamma"} for gamma
##' model.
##' @param link_function is either "log", "inverse", or "identity" for the
##' "Gamma" \code{treat_mod}.
##' @param ... additional arguments to be passed to the treatment regression function.
##'
##' @details
##'
##' This method is a version of the Nadarya-Watson estimator
##' Nadaraya (1964) which is a local constant regression but
##' weighted by the inverse of the estimated GPS.
##'
##'
##' @return \code{nw_est} returns an object of class "causaldrf",
##' a list that contains the following components:
##' \item{param}{parameter estimates for a nw fit.}
##' \item{t_mod}{the result of the treatment model fit.}
##' \item{call}{the matched call.}
##'
##' @seealso \code{\link{nw_est}}, \code{\link{iw_est}}, \code{\link{hi_est}}, \code{\link{gam_est}},
##' \code{\link{add_spl_est}},
##' \code{\link{bart_est}}, etc. for other estimates.
##'
##' \code{\link{t_mod}}, \code{\link{overlap_fun}} to prepare the \code{data}
##' for use in the different estimates.
##'
##' @references Schafer, J.L., Galagate, D.L. (2015). Causal inference with a
##' continuous treatment and outcome: alternative estimators for parametric
##' dose-response models. \emph{Manuscript in preparation}.
##'
##' Flores, Carlos A., et al. "Estimating the effects of length of exposure to
##' instruction in a training program: the case of job corps."
##' \emph{Review of Economics and Statistics} \bold{94.1} (2012): 153-171.
##'
##' Nadaraya, Elizbar A. "On estimating regression."
##' \emph{Theory of Probability and Its Applications} \bold{9.1} (1964): 141--142.
##'
##' @examples
##'
##' ## Example from Schafer (2015).
##'
##' example_data <- sim_data
##'
##' nw_list <- nw_est(Y = Y,
##' treat = T,
##' treat_formula = T ~ B.1 + B.2 + B.3 + B.4 + B.5 + B.6 + B.7 + B.8,
##' data = example_data,
##' grid_val = seq(8, 16, by = 1),
##' bandw = bw.SJ(example_data$T),
##' treat_mod = "Normal")
##'
##' sample_index <- sample(1:1000, 100)
##'
##' plot(example_data$T[sample_index],
##' example_data$Y[sample_index],
##' xlab = "T",
##' ylab = "Y",
##' main = "nw estimate")
##'
##' lines(seq(8, 16, by = 1),
##' nw_list$param,
##' lty = 2,
##' lwd = 2,
##' col = "blue")
##'
##' legend('bottomright',
##' "nw estimate",
##' lty=2,
##' lwd = 2,
##' col = "blue",
##' bty='Y',
##' cex=1)
##'
##' rm(example_data, nw_list, sample_index)
##'
##'
##' @usage
##'
##' nw_est(Y,
##' treat,
##' treat_formula,
##' data,
##' grid_val,
##' bandw,
##' treat_mod,
##' link_function,
##' ...)
##'
##' @export
##'
nw_est <- function (Y,
treat,
treat_formula,
data,
grid_val,
bandw,
treat_mod,
link_function,
...)
{
# Y is the name of the Y variable
# treat is the name of the treatment variable
# treat_formula is the formula for the treatment model
# data will contain all the data: X, treat, and Y
# grid_val is the set of grid points on T
# bandw is the bandwidth
# treat_mod is th treatment model to fit
# link_function is the link function used, if needed
# The outcome is a list of 2 objects:
# (1) estimated values of ADRF at gridvalues
# (2) result of treatment model
#save input
tempcall <- match.call()
#some basic input checks
if (!("Y" %in% names(tempcall))) stop("No Y variable specified")
if (!("treat" %in% names(tempcall))) stop("No treat variable specified")
if (!("treat_formula" %in% names(tempcall))) stop("No treat_formula model specified")
if (!("data" %in% names(tempcall))) stop("No data specified")
if (!("grid_val" %in% names(tempcall))) stop("No grid_val specified")
if (!("bandw" %in% names(tempcall))) stop("No bandw specified")
if (!("treat_mod" %in% names(tempcall)) | ("treat_mod" %in% names(tempcall) & !(tempcall$treat_mod %in% c("NegBinom", "Poisson", "Gamma", "LogNormal", "Sqrt", "Normal")))) stop("No valid family specified (\"NegBinom\", \"Poisson\", \"Gamma\", \"Log\", \"Sqrt\", \"Normal\")")
if (tempcall$treat_mod == "Gamma") {if(!(tempcall$link_function %in% c("log", "inverse"))) stop("No valid link function specified for family = Gamma (\"log\", \"inverse\")")}
if (tempcall$treat_mod == "binomial") {if(!(tempcall$link_function %in% c("logit", "probit", "cauchit", "log", "cloglog"))) stop("No valid link function specified for family = binomial (\"logit\", \"probit\", \"cauchit\", \"log\", \"cloglog\")")}
if (tempcall$treat_mod == "ordinal" ) {if(!(tempcall$link_function %in% c("logit", "probit", "cauchit", "cloglog"))) stop("No valid link function specified for family = ordinal (\"logit\", \"probit\", \"cauchit\", \"cloglog\")")}
#make new dataframe for newly computed variables, to prevent variable name conflicts
tempdat <- data.frame(
Y = data[,as.character(tempcall$Y)],
treat = data[,as.character(tempcall$treat)]
)
# make a formula for the treatment model
# formula_t <- eval(parse(text = paste(deparse(tempcall$treat, width.cutoff = 500), deparse(tempcall$treat_formula, width.cutoff = 500), sep = "")))
formula_t <- tempcall$treat_formula
if (treat_mod == "NegBinom") {
samp_dat <- data
result <- MASS::glm.nb(formula = formula_t, link = log, data = samp_dat, ...)
cond_mean <- result$fitted.values
cond_var <- cond_mean + cond_mean^2/result$theta
prob_nb_est <- (cond_var - cond_mean)/cond_var
gps_fun_NB <- function(tt) {
dnbinom(x = tt, size = result$theta, mu = result$fitted.values,
log = FALSE)
}
gps_fun <- gps_fun_NB
}
else if (treat_mod == "Poisson") {
samp_dat <- data
result <- glm(formula = formula_t, family = "poisson", data = samp_dat, ...)
cond_mean <- result$fitted.values
samp_dat$gps_vals <- dpois(x = tempdat$treat, lambda = cond_mean)
gps_fun_Pois <- function(t) {
dpois(t, lambda = cond_mean)
}
gps_fun <- gps_fun_Pois
}
else if (treat_mod == "Gamma") {
samp_dat <- data
result <- glm(formula = formula_t, family = Gamma(link = link_function),
data = samp_dat, ...)
est_treat <- result$fitted
shape_gamma <- as.numeric(MASS::gamma.shape(result)[1])
theta_given_X <- result$fitted.values/shape_gamma
theta_treat_X <- tempdat$treat/shape_gamma
gps_fun_Gamma <- function(t) {
dgamma(t, shape = shape_gamma, scale = theta_given_X)
}
gps_fun <- gps_fun_Gamma
}
else if (treat_mod == "LogNormal") {
samp_dat <- data
samp_dat[, as.character(tempcall$treat)] <- log(samp_dat[, as.character(tempcall$treat)])
result <- lm(formula = formula_t, data = samp_dat, ...)
est_log_treat <- result$fitted
sigma_est <- summary(result)$sigma
gps_fun_Log <- function(tt) {
dnorm(log(tt), mean = est_log_treat, sd = sigma_est)
}
gps_fun <- gps_fun_Log
}
else if (treat_mod == "Sqrt") {
samp_dat <- data
samp_dat[, as.character(tempcall$treat)] <- sqrt(samp_dat[, as.character(tempcall$treat)])
result <- lm(formula = formula_t, data = samp_dat, ...)
est_sqrt_treat <- result$fitted
sigma_est <- summary(result)$sigma
gps_fun_sqrt <- function(tt) {
dnorm(sqrt(tt), mean = est_sqrt_treat, sd = sigma_est)
}
gps_fun <- gps_fun_sqrt
}
else if (treat_mod == "Normal") {
samp_dat <- data
result <- lm(formula = formula_t, data = samp_dat, ...)
gps_fun_Normal <- function(tt) {
dnorm(tt, mean = result$fitted, sd = summary(result)$sigma)
}
gps_fun <- gps_fun_Normal
}
else {
stop("No valid treat_mod specified. Please try again)")
}
if (treat_mod == "LogNormal") {
K_h <- function(t, h) {
dnorm((log(tempdat$treat) - log(t))/h, 0, 1)/h
}
K_h_tild <- function(t, h) {
K_h(t, h)/gps_fun(t)
}
K_est <- numeric(length(grid_val))
for (i in 1:length(grid_val)) {
K_est[i] <- sum(tempdat$Y * K_h_tild(grid_val[i], bandw))/sum(K_h_tild(grid_val[i],
bandw))
}
}
else if (treat_mod == "Sqrt") {
K_h <- function(t, h) {
dnorm((sqrt(tempdat$treat) - sqrt(t))/h, 0, 1)/h
}
K_h_tild <- function(t, h) {
K_h(t, h)/gps_fun(t)
}
K_est <- numeric(length(grid_val))
for (i in 1:length(grid_val)) {
K_est[i] <- sum(tempdat$Y * K_h_tild(grid_val[i], bandw))/sum(K_h_tild(grid_val[i],
bandw))
}
}
else {
K_h <- function(t, h) {
dnorm((tempdat$treat - t)/h, 0, 1)/h
}
K_h_tild <- function(t, h) {
K_h(t, h)/gps_fun(t)
}
K_est <- numeric(length(grid_val))
for (i in 1:length(grid_val)) {
K_est[i] <- sum(tempdat$Y * K_h_tild(grid_val[i], bandw))/sum(K_h_tild(grid_val[i],
bandw))
}
}
z_object <- list(param = K_est,
t_mod = result,
call = tempcall)
class(z_object) <- "causaldrf"
z_object
}
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