#' The function for hierarchical imputation of categorical variables.
#'
#' The function is called by the wrapper and relies on \code{MCMCglmm}.\cr
#' While in the single level function (\code{imp_cat_single}) we used regression trees
#' to impute data, here we run a multilevel multinomial model.
#' The basic idea is that for each category of the target variable (expect the reference category)
#' an own formula is set up, saying for example that the chances to end up in category
#' j increase with increasing X5. So there is an own regression coefficient \eqn{beta_{5,j}} present.
#' In a multilevel setting, this regression coefficient \eqn{beta_{5,j}} might be different for
#' different clusters: for cluster 27 it would be \eqn{beta_{5,j,27} = beta_{5,j} + u_{5,27}}.
#' This also leads to own random effect covariance matrices for each category.
#' All those random effect variance parameters can be collected
#' in one (quite large) covariance matrix where (for example)
#' not only the random intercepts variance and random slopes variance and their covariance
#' is present. Instead, there is even a covariance between the random slopes in category s
#' and the random intercepts in category p. Beside the difficulties in interpretation,
#' these covariances have shown to be numerically instable so they are set to be 0.
#' @param y_imp A Vector with the variable to impute.
#' @param X_imp A data.frame with the fixed effects variables.
#' @param Z_imp A data.frame with the random effects variables.
#' @param clID A vector with the cluster ID.
#' @param nitt An integer defining number of MCMC iterations (see MCMCglmm).
#' @param thin An integer defining the thinning interval (see MCMCglmm).
#' @param burnin An integer defining the percentage of draws from the gibbs sampler
#' that should be discarded as burn in (see MCMCglmm).
#' @return A n x 1 data.frame with the original and imputed values.
imp_cat_multi <- function(y_imp,
X_imp,
Z_imp,
clID,
nitt = 1e5,
thin = 1e3,
burnin = 1e4){
if(min(table(y_imp)) < 2) {
stop("Too few observations per category in a categorical target variable.")
}
if(!is.factor(y_imp)){
warning("We suggest to make all your categorical variables to be a factor.")
y_imp <- as.factor(y_imp)
}
# ----------------------------- preparing the X and Z data ------------------
# remove excessive variables
X_imp <- remove_excessives(X_imp)
# standardise the covariates in X (which are numeric and no intercept)
X_imp_stand <- stand(X_imp)
# -- standardise the covariates in Z (which are numeric and no intercept)
Z_imp_stand <- stand(Z_imp)
#the missing indactor indicates, which values of y are missing.
missind <- is.na(y_imp)
n <- nrow(X_imp_stand)
number_clusters <- length(table(clID))
#starting model
ph <- sample_imp(y_imp)[, 1]
tmp_0_all <- data.frame(target = ph)
xnames_0 <- paste("X", 1:ncol(X_imp_stand), sep = "")
tmp_0_all[xnames_0] <- X_imp_stand
tmp_0_sub <- tmp_0_all[!missind, , drop = FALSE]
reg_1_all <- nnet::multinom(target ~ 0 + ., data = tmp_0_all, trace = FALSE)
reg_1_sub <- nnet::multinom(target ~ 0 + ., data = tmp_0_sub, trace = FALSE)
X_model_matrix_1_all <- stats::model.matrix(reg_1_all)
xnames_1 <- paste("X", 1:ncol(X_model_matrix_1_all), sep = "")
znames <- paste("Z", 1:ncol(Z_imp_stand), sep = "")
#remove unneeded variables
unneeded <- apply(stats::coefficients(reg_1_sub), 2, function(x) any(is.na(x)))
xnames_2 <- xnames_1[!unneeded]
tmp_2_all <- data.frame(target = y_imp)
tmp_2_all[, xnames_2] <- X_model_matrix_1_all[, !unneeded, drop = FALSE]
# note: the [] part is only relevant if ncol(X_model_matrix) == 1
# (see http://stackoverflow.com/questions/40872034/what-happens-when-a-data-frame-gets-new-columns)
tmp_2_all[, znames] <- Z_imp_stand[, 1:ncol(Z_imp_stand)]
tmp_2_all[, "ClID"] <- clID
tmp_2_sub <- tmp_2_all[!missind, , drop = FALSE]
# -------------- calling the gibbs sampler to get imputation parameters----
fixformula <- stats::formula(paste("target~ - 1 + ",
paste(xnames_2, ":trait", sep = "", collapse = "+"), sep = ""))
randformula <- stats::formula(paste("~us( - 1 + ", paste(znames, ":trait", sep = "", collapse = "+"),
"):ClID", sep = ""))
J <- length(table(y_imp)) #number of categories
number_fix_parameters <- ncol(X_model_matrix_1_all) * (J-1)
# Get the number of random effects variables
number_random_effects <- length(znames)
number_random_parameters <- number_random_effects * (J - 1)
#Fix residual variance R at 1
# cf. http://stats.stackexchange.com/questions/32994/what-are-r-structure-g-structure-in-a-glmm
J_matrix <- array(1, dim = c(J, J) - 1) # matrix of ones
I_matrix <- diag(J - 1) #identiy matrix
IJ <- (I_matrix + J_matrix)/J # see Hadfields Course notes p 97
prior <- list(R = list(V = IJ, fix = TRUE),
G = list(G1 = list(V = diag(number_random_parameters), nu = 2)))
MCMCglmm_draws <- MCMCglmm::MCMCglmm(fixed = fixformula,
random = randformula,
rcov = ~us(trait):units,
data = tmp_2_sub,
family = "categorical",
verbose = FALSE, pr = TRUE,
prior = prior,
saveX = TRUE, saveZ = TRUE,
nitt = nitt,
thin = thin,
burnin = burnin)
pointdraws <- MCMCglmm_draws$Sol
xdraws <- pointdraws[, 1:number_fix_parameters, drop = FALSE]
zdraws <- pointdraws[, number_fix_parameters +
1:(number_random_parameters * number_clusters), drop = FALSE]
number_of_draws <- nrow(pointdraws)
select.record <- sample(1:number_of_draws, 1, replace = TRUE)
# -------------------- drawing samples with the parameters from the gibbs sampler --------
#now generate new P(Y = A|x * beta) = x*beta/(1+ sum(exp(x*beta))) etc.
#set up random intercepts and slopes
y_ret <- data.frame(matrix(nrow = n, ncol = 1))
###start imputation
exp_beta <- array(dim = c(n, J - 1))
#For each individual (in the rows) a coefficient for the J - 1 categories of the target variable
#will be saved (in the columns).
fix_eff_sample <- matrix(xdraws[select.record, ], byrow = TRUE, ncol = J - 1)
rownames(fix_eff_sample) <- paste("covariate", 1:ncol(X_model_matrix_1_all))
colnames(fix_eff_sample) <- paste("effect for category", 1:(J-1))
#xdraws has the following form:
#c(x1 effect for category 1, x1 effect for cat 2, ..., x1 effect for last cat,
# x2 effect for cat 1, ..., x2 effect for last cat,
#... last covariates effect for cat 1, ..., last covariates effect for last cat)
rand_eff_sample <- matrix(zdraws[select.record, ], nrow = number_clusters)
rownames(rand_eff_sample) <- paste("cluster", 1:number_clusters)
colnames(rand_eff_sample) <- paste(paste("effect of Z", 1:number_random_effects,
" on category ", sep = ""),
rep(1:(J-1), each = number_random_effects), sep = "")
# zdraws has the following form:
# effect of Z1 on category 1 in cluster 1,
# effect of Z1 on category 1 in cluster 2,
# effect of Z1 on category 1 in cluster 3,
# ...
# effect of Z1 on category 2 in cluster 1,
# effect of Z1 on category 2 in cluster 2,
# effect of Z1 on category 2 in cluster 3,
# ...
# effect of Z2 on category 1 in cluster 1,
# effect of Z2 on category 1 in cluster 2,
# effect of Z2 on category 1 in cluster 3,
# ...
# effect of Z2 on category 2 in cluster 1,
# effect of Z2 on category 2 in cluster 2,
# effect of Z2 on category 2 in cluster 3,
# ...
# effect of last covariate on last category (without the reference category) in last cluster.
#rand_eff_sample has then the form
#row_i is the:
#effect of Z1 on category 1 in cluster_i,
#effect of Z1 on category 2 in cluster_i,
#effect of Z2 on category 1 in cluster_i,
#effect of Z2 on category 2 in cluster_i.
for(k in 1:ncol(exp_beta)){
# make for each cluster a matrix with the random effect coefficients
rand_betas <- rand_eff_sample[, k, drop = FALSE]
exp_beta[, k] <-
as.matrix(exp(as.matrix(tmp_2_all[, xnames_2, drop = FALSE]) %*%
fix_eff_sample[, k, drop = FALSE] +#fix effects
rowSums(tmp_2_all[, znames, drop = FALSE] * rand_betas[clID, , drop = FALSE])))#random effects
#explanation for the fixed effects part:
#MCMCglmm_draws$Sol is ordered in the following way: beta_1 for category 1, beta_1 for category_2
#... beta 1 for category_k, beta_2 for category_1, beta_2 for category_2 ...
#so we have to skip some values as we proceed by category and not beta.
}
y_temp <- array(dim = n)
for(i in 1:n){
# ensure that the reference category is in the correct position
mytmp_prob <- exp_beta[i, ]/(1 + sum(exp_beta[i, ]))
my_prob <- c(1 - sum(mytmp_prob), mytmp_prob) #the first category is the reference category in MCMCglmm
y_temp[i] <- sample(levels(y_imp), size = 1, replace = TRUE, prob = my_prob)
}
y_ret <- data.frame(y_imp = as.factor(ifelse(is.na(y_imp), y_temp, as.character(y_imp))))
# --------- returning the imputed data --------------
return(y_ret)
}
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