# Simulations of Wilson vs. Wald CI Intervals

In “Wilson Confidence Intervals for Binomial Proportions With Multiple Imputation for Missing Data” (A. Lott & J. Reiter, 2018), the authors run simulation studies comparing coverage of MI-Wilson and MI-Wald confidence intervals, among a few other slight variations of the two. This is good motivation for using the phat versions of the mi_wilson and mi_wald functions. While we don’t implement the simulations here, we lay out a foundation and demonstrate one use of the mi_wald_phat and mi_wilson_phat functions.

We first load the MI-Wilson library as follows:

library(MIWilson)

We then create a simple master dataset with binary values and induce MCAR missingness; this is carried out by the create_missing_data function. With the incomplete master dataset, we create multiple imputations using Bayesian principles (see paper for details), using the create_imps function.

#creating missing data
create_missing_data <- function(n, p, m, MIA_perc) {

complete = incomplete = rbinom(n, 1, p)

#setting up number of missing values, dataset with missing values
blanks = floor(MIA_perc * n)
idcs = 1:length(complete)
incomplete[sample(idcs, blanks)] = NA

return(incomplete)

}

#creating multiple imputations
create_imps <- function(n, m, incomplete) {

count_one = table(incomplete)
count_zero = table(incomplete)

imputations = matrix(nrow = n, ncol = m)
for (i in 1:m) {
p_star = rbeta(1, count_one + 1, count_zero + 1)
incomp_idx = which(is.na(incomplete))

curr_imp = incomplete
curr_imp[incomp_idx] = rbinom(length(incomp_idx), 1, p_star)

imputations[,i] = curr_imp
}

return(imputations)

}

To demonstrate, we create a master dataset with a true binomial proportion of $$p=0.5$$ and induce MCAR missingness for 30% of the dataset. We then produce $$m=10$$ imputations and use them to create MI-Wilson and MI-Wald confidence intervals for $$p$$.

n = 100
p = 0.7
m = 10
MIA_perc = 0.3

incomplete = create_missing_data(n, p, m, MIA_perc)
imputations = create_imps(n, m, incomplete)

phats = colSums(imputations)/nrow(imputations)
mi_wald_phat(phats = phats, n = nrow(imputations))
#>  "Qbar:  0.712"
#>  "Tm:  0.00287002222222222"
#>  "dof:  108.597364325479"
#>  0.6231227 0.8008773
mi_wilson_phat(phats = phats, n =nrow(imputations))
#>  "Qbar:  0.712"
#>  "Rm:  0.404257863891879"
#>  "dof:  108.597364325479"
#>  0.6164037 0.7918188