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#' Pairs Cluster Bootstrapped p-Values For mlogit
#'
#' This software estimates p-values using pairs cluster bootstrapped t-statistics for multinomial logit models (Cameron, Gelbach, and Miller 2008). The data set is repeatedly re-sampled by cluster, a model is estimated, and inference is based on the sampling distribution of the pivotal (t) statistic.
#'
#' @param mod A model estimated using \code{mlogit}.
#' @param dat The data set used to estimate \code{mod}.
#' @param cluster A formula of the clustering variable.
#' @param ci.level What confidence level should CIs reflect?
#' @param boot.reps The number of bootstrap samples to draw.
#' @param cluster.se Use clustered standard errors (= TRUE) or ordinary SEs (= FALSE) for bootstrap replicates.
#' @param report Should a table of results be printed to the console?
#' @param prog.bar Show a progress bar of the bootstrap (= TRUE) or not (= FALSE).
#' @param output.replicates Should the cluster bootstrap coefficient replicates be output (= TRUE) or not (= FALSE)?
#' @param seed Random number seed for replicability (default is NULL).
#'
#' @return A list with the elements
#' \item{p.values}{A matrix of the estimated p-values.}
#' \item{ci}{A matrix of confidence intervals.}
#' @author Justin Esarey
#' @note Code to estimate GLM clustered standard errors by Mahmood Arai: http://thetarzan.wordpress.com/2011/06/11/clustered-standard-errors-in-r/, although modified slightly to work for \code{mlogit} models. Cluster SE degrees of freedom correction = (M/(M-1)) with M = the number of clusters.
#' @examples
#' \dontrun{
#'
#' #######################################
#' # example one: train ticket selection
#' #######################################
#' require(mlogit)
#' data("Train", package="mlogit")
#' Train$choiceid <- 1:nrow(Train)
#'
#' Tr <- dfidx(Train, shape = "wide", varying = 4:11, sep = "_",
#' choice = "choice", idx = list(c("choiceid", "id")),
#' idnames = c(NA, "alt"))
#' Tr$price <- Tr$price/100 * 2.20371
#' Tr$time <- Tr$time/60
#'
#' ml.Train <- mlogit(choice ~ price + time + change + comfort | -1, Tr)
#'
#' # compute pairs cluster bootstrapped p-values
#' # note: few reps to speed up example
#' cluster.bs.tr <- cluster.bs.mlogit(ml.Train, Tr, ~ id, boot.reps=100)
#'
#'
#' ##################################################################
#' # example two: predict type of heating system installed in house
#' ##################################################################
#' require(mlogit)
#' data("Heating", package = "mlogit")
#' H <- Heating
#' H$region <- as.numeric(H$region)
#' H.ml <- dfidx(H, shape="wide", choice="depvar", varying=c(3:12),
#' idx = list(c("idcase", "region")))
#' m <- mlogit(depvar~ic+oc, H.ml)
#'
#' # compute pairs cluster bootstrapped p-values
#' cluster.bs.h <- cluster.bs.mlogit(m, H.ml, ~ region, boot.reps=1000)
#'
#' }
#' @rdname cluster.bs.mlogit
#' @import stats
#' @importFrom dfidx dfidx idx idx_name
#' @importFrom utils write.table
#' @importFrom utils setTxtProgressBar
#' @importFrom utils txtProgressBar
#' @importFrom lmtest coeftest
#' @importFrom sandwich estfun
#' @importFrom sandwich sandwich
#' @importFrom mlogit mlogit hmftest mFormula is.mFormula mlogit.optim cov.mlogit cor.mlogit rpar scoretest med rg stdev qrpar prpar drpar
#' @references Esarey, Justin, and Andrew Menger. 2017. "Practical and Effective Approaches to Dealing with Clustered Data." \emph{Political Science Research and Methods} forthcoming: 1-35. <URL:http://jee3.web.rice.edu/cluster-paper.pdf>.
#' @references Cameron, A. Colin, Jonah B. Gelbach, and Douglas L. Miller. 2008. "Bootstrap-Based Improvements for Inference with Clustered Errors." \emph{The Review of Economics and Statistics} 90(3): 414-427. <DOI:10.1162/rest.90.3.414>.
#' @export
cluster.bs.mlogit<-function(mod, dat, cluster, ci.level = 0.95, boot.reps = 1000, cluster.se = TRUE,
report = TRUE, prog.bar = TRUE, output.replicates = FALSE,
seed = NULL){
# compensate for bizarre R formula updating bug
# thanks to Jason Thorpe for reporting!
form.old <- update(mod$formula, 1 ~ 1 )
while(form.old != mod$formula){
form.old <- mod$formula
invisible(mod <- update(mod, new = .~.))
}
if(is.null(seed)==F){ # if user supplies a seed, set it
tryCatch(set.seed(seed),
error = function(e){return("seed must be a valid integer")},
warning = function(w){return(NA)})
}
form <- mod$formula # what is the formula of this model?
variables <- all.vars(form) # what variables are in this model?
used.idx <- which(rownames(dat) %in% rownames(mod$mod)) # what observations are used?
dat <- dat[used.idx,] # keep only active observations
ind.variables <- names(coefficients(mod)) # what independent variables are in this model?
"%w/o%" <- function(x, y) x[!x %in% y] # create a without function (see ?match)
dv <- variables %w/o% all.vars(update(form, 1 ~ .)) # what is the dependent variable?
# obtain the clustering variable
clust.name <- all.vars(cluster) # name of the cluster variable
dat.rs <- as.data.frame(subset(idx(dat), select = clust.name )) # select cluster variable from data set
dat.rs$id.zz <- idx(dat, n=1) # choice index
dat.rs$ti.zz <- idx(dat, n=2) # alternative index
clust <- reshape(dat.rs, timevar="ti.zz", # reshape long to wide, store as clust
idvar=c("id.zz", clust.name), direction="wide")[[clust.name]]
if(sum(is.na(clust)>0)){stop("missing cluster indices")} # check for missing cluster indices
G <- length(unique(clust)) # how many clusters are in this model?
# load in a function to create clustered standard errors for mlogit models
# initial code by Mahmood Arai: http://thetarzan.wordpress.com/2011/06/11/clustered-standard-errors-in-r/
# slightly modified for mlogit models by Justin Esarey on 3/3/2015
cl.mlogit <- function(fm, cluster){
# fm: a fitted mlogit model
# cluster: a data vector with the cluster
# identity of each observation in fm
#require(sandwich, quietly = TRUE)
#require(lmtest, quietly = TRUE)
M <- length(unique(cluster))
N <- length(cluster)
K <- length(coefficients(fm))
dfc <- (M/(M-1))
uj <- apply(estfun(fm),2, function(x) tapply(x, cluster, sum));
vcovCL <- dfc*sandwich(fm, meat.=crossprod(uj)/N)
coeftest(fm, vcovCL)
}
if(cluster.se == T){
se.clust <- cl.mlogit(mod, clust)[ind.variables,2] # retrieve the clustered SEs
beta.mod <- coefficients(mod)[ind.variables] # retrieve the estimated coefficients
w <- beta.mod / se.clust # calculate the t-test statistic
}else{
se.beta <- summary(mod)$CoefTable[ind.variables,2] # retrieve the vanilla SEs
beta.mod <- coefficients(mod)[ind.variables] # retrieve the estimated coefficients
w <- beta.mod / se.beta # calculate the t-test statistic
}
# keep track of the beta bootstrap replicates for possible output
rep.store <- matrix(data=NA, nrow=boot.reps, ncol=length(beta.mod))
colnames(rep.store) <- ind.variables
w.store <- matrix(data=NA, nrow=boot.reps, ncol=length(ind.variables)) # store bootstrapped test statistics
if(prog.bar==TRUE){pb <- txtProgressBar(min = 0, max = boot.reps, initial = 0, style = 3)}
for(i in 1:boot.reps){
if(prog.bar==TRUE){setTxtProgressBar(pb, value=i)}
boot.sel <- sample(1:G, size=G, replace=T) # randomly select clusters
# pick the observations corresponding to the randomly selected clusters
boot.ind <- c() # where the selected obs will be stored
boot.clust <- c() # create + store a new cluster index for the bootstrap data
for(k in 1:G){
obs.sel <- which(dat[[clust.name]] == # which observations are in the sampled cluster?
unlist(unique(clust))[boot.sel[k]])
boot.ind <- c(boot.ind, obs.sel) # append the selected obs index to existing index
boot.clust <- c(boot.clust, rep(k, length(obs.sel))) # store the new bootstrap cluster index
}
boot.dat <- dat[boot.ind,] # create the bootstrapped data
eval(parse(text=paste("boot.dat$idx[[\"", clust.name, # add boot-specific cluster variable
"\"]]=boot.clust", sep="")))
alt.num <- length(unique(idx(dat, n=2))) # how many alternatives are there?
ch.num <- round( dim(boot.dat)[1] / alt.num ) # how many choices are there? (rounding for imprecision)
eval(parse(text=paste("boot.dat$idx[[\"",idx_name(dat, n=1), # create new choice index
"\"]]=rep(1:ch.num, each=alt.num)", sep="")))
# eval(parse(text=paste("boot.dat$idx[[\"", idx_name(dat, n=2), # create new alternative index
# "\"]]=value=rep(1:alt.num, ch.num)", sep="")))
rownames(boot.dat) <- NULL # purge old (duplicated) row names
boot.mod.call <- mod$call # get original model call
boot.mod.call[[3]] <- quote(boot.dat) # modify call for bootstrap data set
boot.mod <- suppressWarnings(tryCatch(eval(boot.mod.call), # estimate model on bootstrap dataset
error = function(e){return(NULL)}))
fail <- is.null(boot.mod) # determine whether the mlogit process created an error
# obtain the bootstrap clustering variable
boot.dat.rs <- as.data.frame(
subset(idx(boot.dat), select = clust.name )) # select cluster variable from BS data set
boot.dat.rs$id.zz <- idx(boot.dat, n=1) # choice index
boot.dat.rs$ti.zz <- idx(boot.dat, n=2) # alternative index
boot.clust.n <- reshape(boot.dat.rs, timevar="ti.zz", # reshape long to wide, store as clust
idvar=c("id.zz", clust.name),
direction="wide")[[clust.name]]
if(fail==0){ # proceed if the mlogit model was not in error
if(cluster.se == T){
se.boot <- tryCatch(cl.mlogit(boot.mod, boot.clust.n)[ind.variables,2],
error = function(e){return(NA)},
warning = function(w){return(NA)}) # retrieve the bootstrap clustered SE
beta.boot <- tryCatch(coefficients(boot.mod)[ind.variables],
error = function(e){return(NA)},
warning = function(w){return(NA)}) # store the bootstrap beta coefficient
w.store[i,] <- (beta.boot-beta.mod) / se.boot # store the bootstrap test statistic
rep.store[i,] <- beta.boot # store the bootstrap beta for output
}else{
se.boot <- tryCatch(summary(boot.mod)$CoefTable[ind.variables,2],
error = function(e){return(NA)},
warning = function(w){return(NA)}) # retrieve the bootstrap vanilla SE
beta.boot <- tryCatch(coefficients(boot.mod)[ind.variables],
error = function(e){return(NA)},
warning = function(w){return(NA)}) # retrieve the bootstrap beta coefficient
w.store[i,] <- (beta.boot-beta.mod) / se.boot # calculate the t-test statistic
rep.store[i,] <- beta.boot # store the bootstrap beta for output
}
}else{
w.store[i,] <- NA # if model didn't converge, store NA as a result
rep.store[i,] <- NA
}
}
if(prog.bar==TRUE){close(pb)}
num.fail <- length(attr(na.omit(w.store), "na.action")) # count the number of times something went wrong
w.store <- na.omit(w.store) # drop the erroneous bootstrap replicates
comp.fun<-function(vec2, vec1){as.numeric(vec1>vec2)} # a simple function comparing v1 to v2
p.store.s <- t(apply(X = abs(w.store), FUN=comp.fun, MARGIN = 1, vec1 = abs(w))) # compare the BS test stats to orig. result
p.store <- 1 - ( colSums(p.store.s) / dim(w.store)[1] ) # calculate the cluster bootstrap p-value
# compute critical t-statistics for CIs
crit.t <- apply(X=abs(w.store), MARGIN=2, FUN=quantile, probs=ci.level )
if(cluster.se == TRUE){
ci.lo <- beta.mod - crit.t*se.clust
ci.hi <- beta.mod + crit.t*se.clust
}else{
ci.lo <- beta.mod - crit.t*se.beta
ci.hi <- beta.mod + crit.t*se.beta
}
print.ci <- cbind(ind.variables, ci.lo, ci.hi)
print.ci <- rbind(c("variable name", "CI lower", "CI higher"), print.ci)
out.ci <- cbind(ci.lo, ci.hi)
rownames(out.ci) <- ind.variables
colnames(out.ci) <- c("CI lower", "CI higher")
out <- matrix(p.store, ncol=1)
colnames(out) <- c("clustered bootstrap p-value")
rownames(out) <- ind.variables
out.p <- cbind(ind.variables, out)
out.p <- rbind(c("variable name", "clustered bootstrap p-value"), out.p)
printmat <- function(m){
write.table(format(m, justify="right"), row.names=F, col.names=F, quote=F, sep = " ")
}
if(report==T){
if(num.fail!=0){
cat("\n", "\n", "\n", "****", "Warning: ", num.fail, " out of ", boot.reps, "bootstrap replicate models failed to estimate.", "****", "\n")
}
cat("\n", "Cluster Bootstrap p-values: ", "\n", "\n")
printmat(out.p)
cat("\n", "Confidence Intervals (derived from bootstrapped t-statistics): ", "\n", "\n")
printmat(print.ci)
}
out.list<-list()
out.list[["p.values"]]<-out
out.list[["ci"]] <- out.ci
if(output.replicates == TRUE){out.list[["replicates"]] <- rep.store}
return(invisible(out.list))
}
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