#' The function to impute interval data variables
#'
#' This functions imputes interval data variables. Those are variables,
#' that consists of a lower and upper (numeric) boundary. Technically
#' those boundaries are contained in a string, separated by a semi colon.
#' E.g. if a person reports there income to be something between 3000 and 4000 dollars,
#' its value in the interval covariate would be \code{"3000;4000"}.
#' Left (resp. right) censored data can be denoted by \code{"-Inf;x"} (resp. \code{"x;Inf"}),
#' with \code{x} being the (numeric) observed value.
#' @param y_imp A Vector from the class \code{interval} with the variable to impute.
#' @param X_imp A data.frame with the fixed effects variables.
#' @param pvalue A numeric between 0 and 1 denoting the threshold of p-values a variable in the imputation
#' model should not exceed. If they do, they are excluded from the imputation model.
#' @param rounding_degrees A numeric vector with the presumed rounding degrees.
#' @return A n x 1 data.frame with the original and imputed values.
#' Note that this function won't return \code{interval} data as its purpose is to
#' "break" the interval answers into precise answers.
imp_interval <- function(y_imp,
X_imp,
pvalue = 0.2,
rounding_degrees = c(1, 10, 100, 1000)){
# ----------------------------- preparing the X data ------------------
# remove excessive variables
X_imp <- cleanup(X_imp)
# standardize X
X <- stand(X_imp, rounding_degrees = rounding_degrees)
# has to be numeric, so it must only consists of precise observations
decomposed <- decompose_interval(interval = y_imp)
# classify the data into the three types of observations:
# 1. precise data (like 3010 or 3017 - in interval notation "3010;3010", "3017;3017")
# 2. imprecise data (like "3000;3600")
# 3. missing data (NA - in interval notation "-Inf;Inf")
#get the indicator of the missing values
indicator_precise <- !is.na(decomposed[, "precise"])
indicator_imprecise <- !is.na(decomposed[, "lower_imprecise"])
indicator_missing <- is.infinite(decomposed[, "lower_general"]) &
is.infinite(decomposed[, "upper_general"])
if(any(decomposed[, "lower_general"] > decomposed[, "upper_general"], na.rm = TRUE)){
stop("in your interval covariate, some values in the lower bound exceed the upper bound.")
}
y_precise_template <- sample_imp(center.interval(y_imp, inf2NA = TRUE))[, 1]
n <- length(y_precise_template)
y_mean <- mean(y_precise_template)
y_sd <- stats::sd(y_precise_template)
#it can happen that in the template only identical values are present,
#leading to 0 variance.
if(y_sd <= 0) y_sd <- 1
#add 1 to avoid an exactly zero intercept when modeling y_stand ~ x_stand
# If both, X and Y, are standardized, the intercept
#will be exactly 0 and thus not significantly different from 0.
#So in order to avoid this variable to be removed later in the code, we add +1.
decomposed_stand <- (decomposed - y_mean)/y_sd + 1
y_precise_template_stand <- (y_precise_template - y_mean)/y_sd + 1
low_sample <- decomposed_stand[, "lower_general"]
up_sample <- decomposed_stand[, "upper_general"]
y_precise_template <- msm::rtnorm(n = n, lower = low_sample,
upper = up_sample,
mean = y_precise_template_stand,
sd = 1)
ph <- y_precise_template
tmp_0_all <- data.frame(target = ph, X)
xnames_1 <- colnames(X)
tmp_formula <- paste("target~ 0 + ", paste(xnames_1, collapse = "+"), sep = "")
reg_1_all <- stats::lm(stats::formula(tmp_formula), data = tmp_0_all)
X_model_matrix_1_all <- stats::model.matrix(reg_1_all)
xnames_1 <- paste("X", 1:ncol(X_model_matrix_1_all), sep = "")
colnames(X_model_matrix_1_all) <- xnames_1
tmp_0_all <- data.frame(target = ph)
tmp_0_all[, xnames_1] <- X_model_matrix_1_all
#From this initial model matrix X_model_matrix_1_all
#now step by step irrelavant variables are removed.
# Principally those models are based on precise observations only.
# But in some data situation, there might be no presice observations, only intervals;
# then all precise and interval data has to be used.
# Precise data can be use directly, from imprecise data, a draw from within their bounds is used
if(sum(indicator_precise) < 30){
use_indicator <- indicator_precise | indicator_imprecise
}else{
use_indicator <- indicator_precise
}
X_model_matrix_1_sub <- X_model_matrix_1_all[use_indicator, , drop = FALSE]
# The first step of the reduction is to remove variables having a non-measurable effect
# (e.g. due to colinearity) on y.
# tmp_1 shall include the covariates (like X_model_matrix) and additionally the target variable
ph_sub <- ph[use_indicator]
tmp_1_sub <- data.frame(target = ph_sub)
xnames_1 <- colnames(X_model_matrix_1_sub)
tmp_1_sub[, xnames_1] <- X_model_matrix_1_sub
tmp_formula <- paste("target~ 0 + ", paste(xnames_1, collapse = "+"), sep = "")
reg_1_sub <- stats::lm(stats::formula(tmp_formula) , data = tmp_1_sub)
#remove unneeded variables
X_model_matrix_1_sub <- X_model_matrix_1_sub[, !is.na(stats::coefficients(reg_1_sub)),
drop = FALSE]
# Remove insignificant variables from the imputation model
check <- TRUE
while(check){
tmp_1_sub <- data.frame(target = ph_sub)
xnames_1 <- colnames(X_model_matrix_1_sub)
tmp_1_sub[, xnames_1] <- X_model_matrix_1_sub
tmp_formula <- paste("target~ 0 + ", paste(xnames_1, collapse = "+"), sep = "")
reg_1_sub <- stats::lm(stats::formula(tmp_formula), data = tmp_1_sub)
pvalues <- summary(reg_1_sub)$coefficients[, 4]
insignificant_variables <- which(pvalues > pvalue)
most_insignificant <- insignificant_variables[which.max(pvalues[insignificant_variables])]
if(length(most_insignificant) == 0){
check <- FALSE
}else{
#.. drop the insignificant variable from the model.matrix, but only if at least 1 variable remains
tmp_MM <- stats::model.matrix(reg_1_sub)[, -most_insignificant, drop = FALSE]
if(ncol(tmp_MM) == 0){
check <- FALSE
}else{
X_model_matrix_1_sub <- tmp_MM
}
}
}
tmp_2_all <- tmp_0_all[, colnames(tmp_1_sub), drop = FALSE]
#####maximum likelihood estimation using starting values
####estimation of the parameters
lmstart2 <- stats::lm(target ~ 0 + ., data = tmp_1_sub)
betastart2 <- as.vector(lmstart2$coef)
sigmastart2 <- stats::sigma(lmstart2)
#####maximum likelihood estimation using the starting values
starting_values <- c(betastart2, sigmastart2)
###exclude obs below (above) the 0.5% (99.5%) income quantile before maximizing
###the likelihood. Reason: Some extrem outliers cause problems during the
###maximization
quants <- stats::quantile(y_precise_template, c(0.005, 0.995), na.rm = TRUE)
# in X and y_in_negloglik only those observations that are no outliers shall be included.
# Observations with a missing Y are to be included as well even if they could be an outlier.
# Therefore w
keep <- (y_precise_template >= quants[1] & y_precise_template <= quants[2]) |
is.na(y_precise_template)
m2 <- stats::optim(par = starting_values, negloglik2_intervalsonly,
X = as.matrix(tmp_2_all[, xnames_1, drop = FALSE])[keep, , drop = FALSE],
lower_bounds = decomposed_stand[, "lower_imprecise"][keep],
upper_bounds = decomposed_stand[, "upper_imprecise"][keep],
method = "BFGS",#alternative: "Nelder-Mead"
control = list(maxit = 10000), hessian = TRUE)
par_ml2 <- m2$par
hess <- m2$hessian
# link about nearest covariance matrix:
# http://quant.stackexchange.com/questions/2074/what-is-the-best-way-to-fix-a-covariance-matrix-that-is-not-positive-semi-defi
# nearPD(hess)$mat
# isSymmetric(Sigma_ml2)
Sigma_ml2 <- tryCatch(
{
solve(hess)
},
error = function(cond) {
cat("Hessian matrix couldn't be inverted (in the imputation function for interval variables).
Still, you should get a result, but which needs special attention.\n")
tmp <- matrix(0, nrow = length(par_ml2), ncol = length(par_ml2))
diag(tmp) <- abs(par_ml2)/100
return(tmp)
},
warning = function(cond) {
cat("There seems to be a problem with the Hessian matrix in the imputation of the rounded continuous variable\n")
cat("Here is the original warning message:\n")
cat(as.character(cond))
return(solve(hess))
},
finally = {
}
)
# make sure, that the main diagonal elements are non-zero
###set starting values equal to the observed income
###rounded income will be replaced by imputations later
imp_tmp <- y_precise_template
####draw new parameters (because it is a Bayesian imputation)
#a negative draw for the variance parameter has to be rejected
Sigma_ml3 <- as.matrix(Matrix::nearPD(Sigma_ml2)$mat)
invalid <- TRUE
counter <- 0
while(invalid & counter < 1000){
counter <- counter + 1
pars <- mvtnorm::rmvnorm(1, mean = par_ml2, sigma = Sigma_ml3)
invalid <- pars[length(pars)] <= 0
}
#first eq on page 63 in Drechsler, Kiesl, Speidel (2015)
# derive imputation model parameters from previously drawn parameters
#The covariance matrix from equation (3)
beta_hat <- as.matrix(pars[1:(length(pars) - 1)], ncol = 1)
Sigma <- pars[length(pars)]^2
mymean <- as.matrix(tmp_2_all[, xnames_1, drop = FALSE]) %*% beta_hat
###################################
#BEGIN IMPUTING INTERVALL-DATA AND COMPLETELY MISSING DATA
#for this purpose we have to replace the lower and upper bounds
# of those observations with an NA in y_imp by -Inf and Inf
expanded_lower <- decomposed_stand[, "lower_general"]
expanded_upper <- decomposed_stand[, "upper_general"]
#draw values from the truncated normal distributions
# the bounds are straight forward for the interval data.
# for the missing data, the bounds are -Inf and +Inf,
# which is equivalent to draw from a unbounded normal distribution.
# for precise observations, the bounds are here set to be NA,
# resulting in NA draws for those observations.
# The imputation for precise but rounded data follows in the next section.
# precise and not rounded data need no impuation at all.
tnorm_draws <- msm::rtnorm(n = n,
lower = expanded_lower,
upper = expanded_upper,
mean = mymean,
sd = sqrt(Sigma))
#undo the standardization
y_ret <- (tnorm_draws - 1) * y_sd + y_mean
return(data.frame(y_ret = y_ret))
}
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