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#' @name MLCDW
#' @aliases MLCDW
#' @title Multinomial logistic calibration estimator under dual frame approach with auxiliary information from the whole population
#'
#' @description Produces estimates for class totals and proportions using multinomial logistic regression from survey data obtained
#' from a dual frame sampling design using a model calibrated dual frame approach with auxiliary information from the whole population. Confidence intervals are also computed, if required.
#'
#' @usage MLCDW (ysA, ysB, pik_A, pik_B, domains_A, domains_B, xsA, xsB, x, ind_sam, N_A,
#' N_B, N_ab = NULL, met = "linear", conf_level = NULL)
#' @param ysA A data frame containing information about one or more factors, each one of dimension \eqn{n_A}, collected from \eqn{s_A}.
#' @param ysB A data frame containing information about one or more factors, each one of dimension \eqn{n_B}, collected from \eqn{s_B}.
#' @param pik_A A numeric vector of length \eqn{n_A} containing first order inclusion probabilities for units included in \eqn{s_A}.
#' @param pik_B A numeric vector of length \eqn{n_B} containing first order inclusion probabilities for units included in \eqn{s_B}.
#' @param domains_A A character vector of size \eqn{n_A} indicating the domain each unit from \eqn{s_A} belongs to. Possible values are "a" and "ab".
#' @param domains_B A character vector of size \eqn{n_B} indicating the domain each unit from \eqn{s_B} belongs to. Possible values are "b" and "ba".
#' @param xsA A numeric vector of length \eqn{n_A} or a numeric matrix or data frame of dimensions \eqn{n_A} x \eqn{m}, with \eqn{m} the number of auxiliary variables, containing auxiliary information in frame A for units included in \eqn{s_A}.
#' @param xsB A numeric vector of length \eqn{n_B} or a numeric matrix or data frame of dimensions \eqn{n_B} x \eqn{m}, with \eqn{m} the number of auxiliary variables, containing auxiliary information in frame B for units included in \eqn{s_B}.
#' @param x A numeric vector or length \eqn{N} or a numeric matrix or data frame of dimensions \eqn{N} x \eqn{m}, with \eqn{m} the number of auxiliary variables, containing auxiliary information for every unit in the population.
#' @param ind_sam A numeric vector of length \eqn{n = n_A + n_B} containing the identificators of units of the population (from 1 to \eqn{N}) that belongs to \eqn{s_A} or \eqn{s_B}
#' @param N_A A numeric value indicating the size of frame A
#' @param N_B A numeric value indicating the size of frame B
#' @param N_ab (Optional) A numeric value indicating the size of the overlap domain
#' @param met (Optional) A character vector indicating the distance that must be used in calibration process. Possible values are "linear", "raking" and "logit". Default is "linear".
#' @param conf_level (Optional) A numeric value indicating the confidence level for the confidence intervals, if desired.
#' @details Multinomial logistic calibration estimator in dual frame using auxiliary information from the whole population for a proportion is given by
#' \deqn{\hat{P}_{MLCi}^{DW} = \frac{1}{N} \left(\sum_{k \in s_A \cup s_B} w_k^{\circ} z_{ki}\right), \hspace{0.3cm} i = 1,...,m}
#' with \eqn{m} the number of categories of the response variable, \eqn{z_i} the indicator variable for the i-th category of the response variable,
#' and \eqn{w^{\circ}} calibration weights which are calculated having into account a different set of constraints, depending on the case. For instance, if \eqn{N_A, N_B} and \eqn{N_{ab}} are known, calibration constraints are
#' \deqn{\sum_{k \in s_a}w_k^{\circ} = N_a, \sum_{k \in s_{ab}}w_k^{\circ} = \eta N_{ab}, \sum_{k \in s_{ba}}w_k^{\circ} = (1 - \eta) N_{ab}, \sum_{k \in s_{b}}w_k^{\circ} = N_{b}} and \deqn{\sum_{k \in s_A \cup s_B}w_k^\circ p_{ki}^{\circ} = \sum_{k \in U} p_{ki}^\circ}
#' with \eqn{\eta \in (0,1)} and \deqn{p_{ki}^{\circ} = \frac{exp(x_k^{'}\beta_i^{\circ})}{\sum_{r=1}^m exp(x_k^{'}\beta_r^{\circ})},}
#' being \eqn{\beta_i^\circ} the maximum likelihood parameters of the multinomial logistic model considering weights \eqn{d_k^{\circ} =\left\{\begin{array}{lcc}
#' d_k^A & \textrm{if } k \in a\\
#' \eta d_k^A & \textrm{if } k \in ab\\
#' (1 - \eta) d_k^B & \textrm{if } k \in ba \\
#' d_k^B & \textrm{if } k \in b
#' \end{array}
#' \right.}.
#' @return \code{MLCDW} returns an object of class "MultEstimatorDF" which is a list with, at least, the following components:
#' \item{Call}{the matched call.}
#' \item{Est}{class frequencies and proportions estimations for main variable(s).}
#' @references Molina, D., Rueda, M., Arcos, A. and Ranalli, M. G. (2015)
#' \emph{Multinomial logistic estimation in dual frame surveys}
#' Statistics and Operations Research Transactions (SORT). To be printed.
#' @seealso \code{\link{JackMLCDW}}
#' @examples
#' data(DatMA)
#' data(DatMB)
#' data(DatPopM)
#'
#' IndSample <- c(DatMA$Id_Pop, DatMB$Id_Pop)
#' N_FrameA <- nrow(DatPopM[DatPopM$Domain == "a" | DatPopM$Domain == "ab",])
#' N_FrameB <- nrow(DatPopM[DatPopM$Domain == "b" | DatPopM$Domain == "ab",])
#' N_Domainab <- nrow(DatPopM[DatPopM$Domain == "ab",])
#
#' #Let calculate proportions of categories of variable Prog using MLCDW estimator
#' #using Read as auxiliary variable
#' MLCDW(DatMA$Prog, DatMB$Prog, DatMA$ProbA, DatMB$ProbB, DatMA$Domain, DatMB$Domain,
#' DatMA$Read, DatMB$Read, DatPopM$Read, IndSample, N_FrameA, N_FrameB)
#'
#' #Now, let suppose that the overlap domian size is known
#' MLCDW(DatMA$Prog, DatMB$Prog, DatMA$ProbA, DatMB$ProbB, DatMA$Domain, DatMB$Domain,
#' DatMA$Read, DatMB$Read, DatPopM$Read, IndSample, N_FrameA, N_FrameB, N_Domainab)
#'
#' #Let obtain 95% confidence intervals together with the estimations
#' MLCDW(DatMA$Prog, DatMB$Prog, DatMA$ProbA, DatMB$ProbB, DatMA$Domain, DatMB$Domain,
#' DatMA$Read, DatMB$Read, DatPopM$Read, IndSample, N_FrameA, N_FrameB, N_Domainab,
#' conf_level = 0.95)
#' @export
MLCDW = function (ysA, ysB, pik_A, pik_B, domains_A, domains_B, xsA, xsB, x, ind_sam, N_A, N_B, N_ab = NULL, met = "linear", conf_level = NULL){
ysA <- as.data.frame(ysA)
ysB <- as.data.frame(ysB)
xsA <- as.matrix(xsA)
xsB <- as.matrix(xsB)
x <- as.matrix(x)
if (any(is.na(ysA)))
stop("There are missing values in sample from frame A.")
if (any(is.na(ysB)))
stop("There are missing values in sample from frame B.")
if (any(is.na(pik_A)))
stop("There are missing values in pikl from frame A.")
if (any(is.na(pik_B)))
stop("There are missing values in pikl from frame B.")
if (any(is.na(domains_A)))
stop("There are missing values in domains from frame A.")
if (any(is.na(domains_B)))
stop("There are missing values in domains from frame B.")
if (nrow(ysA) != length(pik_A) | nrow(ysA) != length(domains_A) | length(domains_A) != length(pik_A))
stop("Arguments from frame A have different sizes.")
if (nrow(ysB) != length(pik_B) | nrow(ysB) != length(domains_B) | length(domains_B) != length(pik_B))
stop("Arguments from frame B have different sizes.")
if (ncol(ysA) != ncol(ysB))
stop("Number of variables does not match.")
if (length(which(domains_A == "a")) + length(which(domains_A == "ab")) != length(domains_A))
stop("Domains from frame A are not correct.")
if (length(which(domains_B == "b")) + length(which(domains_B == "ba")) != length(domains_B))
stop("Domains from frame B are not correct.")
if (is.null (N_ab) == "FALSE" & (is.null (N_A) == "TRUE" | is.null (N_B) == "TRUE"))
stop("A value for N_ab has been provided, but values for N_A or N_B are missing. This is not a possible option.")
cl <- match.call()
estimations <- list()
interv <- list()
c <- ncol(ysA)
xs <- rbind(xsA, xsB)
N <- nrow(x)
R <- ncol(x)
n_A <- nrow(ysA)
n_B <- nrow(ysB)
n <- n_A + n_B
ones_ab_A <- Domains (rep (1, n_A), domains_A, "ab")
ones_ab_B <- Domains (rep (1, n_B), domains_B, "ba")
Vhat_Nhat_ab_A <- varest(ones_ab_A, pik = pik_A)
Vhat_Nhat_ab_B <- varest(ones_ab_B, pik = pik_B)
eta_0 <- Vhat_Nhat_ab_B / (Vhat_Nhat_ab_A + Vhat_Nhat_ab_B)
domains <- factor(c(as.character(domains_A), as.character(domains_B)))
delta_a <- Domains (rep (1, n), domains, "a")
delta_ab <- Domains (rep (1, n), domains, "ab")
delta_b <- Domains (rep (1, n), domains, "b")
delta_ba <- Domains (rep (1, n), domains, "ba")
pik <- c(pik_A, pik_B)
dd <- 1/pik
d <- dd * delta_a + dd * eta_0 * delta_ab + dd * (1-eta_0) * delta_ba + dd * delta_b
for (k in 1:c){
ys <- factor(c(as.character(ysA[,k]),as.character(ysB[,k])))
lev <- levels(ys)
m <- length(lev)
mat <- matrix (NA, 2, m)
rownames(mat) <- c("Class Tot.", "Prop.")
colnames(mat) <- lev
z <- disjunctive(ys)
mod <- multinom(formula = ys ~ 0 + xs, weights = d, trace = FALSE)
beta_tilde <- rbind(rep(0, R), summary(mod)$coefficients)
denom <- rowSums(exp(x %*% t(beta_tilde)))
p <- exp (x %*% t(beta_tilde)) / denom
if (is.null(N_ab)){
Xs <- cbind(delta_a + delta_ab + delta_ba, delta_b + delta_ab + delta_ba, p[ind_sam,])
total <- c(N_A, N_B, colSums(p))
}
else {
Xs <- cbind(delta_a, delta_ab, delta_ba, delta_b, p[ind_sam,])
total <- c(N_A - N_ab, eta_0 * N_ab, (1 - eta_0) * N_ab, N_B - N_ab, colSums(p))
}
g <- calib (Xs, d, total, method = met)
mat[1,] <- colSums (g * d * z)
mat[2,] <- 1/N * mat[1,]
estimations[[k]] <- mat
if (!is.null(conf_level)){
alpha <- ginv(t(Xs) %*% diag(d) %*% Xs) %*% t(Xs) %*% diag(d) %*% z
e <- z - Xs %*% alpha
e <- e * d
Vhat_AMLCDW <- apply(e, 2, var)
Vhat_PMLCDW <- 1/N^2 * Vhat_AMLCDW
interval <- matrix (NA, 6, m)
rownames(interval) <- c("Class Tot.", "Lower Bound", "Upper Bound", "Prop.", "Lower Bound", "Upper Bound")
colnames(interval) <- lev
interval[1,] <- mat[1,]
interval[2,] <- mat[1,] + qnorm(1 - (1 - conf_level) / 2) * sqrt(Vhat_AMLCDW)
interval[3,] <- mat[1,] - qnorm(1 - (1 - conf_level) / 2) * sqrt(Vhat_AMLCDW)
interval[4,] <- mat[2,]
interval[5,] <- mat[2,] + qnorm(1 - (1 - conf_level) / 2) * sqrt(Vhat_PMLCDW)
interval[6,] <- mat[2,] - qnorm(1 - (1 - conf_level) / 2) * sqrt(Vhat_PMLCDW)
interv[[k]] <- interval
}
}
results = list(Call = cl, Est = estimations, ConfInt = interv)
class(results) = "EstimatorMDF"
attr(results, "attributesMDF") = conf_level
return(results)
}
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