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#' Between-group standard deviation (BGSD)
#'
#' Between-Group Standard Deviation (BGSD) is an absolute measure of inequality
#' that considers all population subgroups. Subgroups are weighted according
#' to their population share.
#'
#' BGSD is calculated as the square root of the weighted average of squared
#' differences between the subgroup estimates and the setting average.
#' Squared differences are weighted by each subgroup’s population share.
#' For more information on this inequality measure see Schlotheuber,
#' A., & Hosseinpoor, A. R. (2022) below.
#'
#' 95% confidence intervals are calculated using a methodology of simulated
#' estimates. The dataset is simulated a large number of times (e.g., 100)
#' and BGSD is calculated for each of the simulated samples. The 95%
#' confidence intervals are based on the 2.5th and 97.5th percentiles of the
#' BGSD results.
#'
#' **Interpretation:** BGSD has only positive values, with larger values
#' indicating higher levels of inequality. BGSD is zero if there is no
#' inequality. It has the same unit as the health indicator.
#'
#' **Type of summary measure:** Complex; absolute; weighted
#'
#' **Applicability:** Non-ordered; more than two subgroups
#'
#' @param pop The number of people within each subgroup.
#' Population size must be available for all subgroups.
#' @param est The subgroup estimate. Estimates must be
#' available for all subgroups.
#' @param se The standard error of the subgroup estimate.
#' If this is missing, 95% confidence intervals of BGSD cannot be calculated.
#' @param scaleval The scale of the indicator. For example, the
#' scale of an indicator measured as a percentage is 100. The
#' scale of an indicator measured as a rate per 1000 population is 1000.
#' @param sim The number of simulations to estimate 95% confidence intervals
#' @param seed The random number generator (RNG) state for the 95% confidence
#' interval simulation
#' @param ... Further arguments passed to or from other methods.
#' @examples
#' # example code
#' data(NonorderedSample)
#' head(NonorderedSample)
#' with(NonorderedSample,
#' bgsd(pop = population,
#' est = estimate,
#' se = se,
#' scaleval = indicator_scale
#' )
#' )
#' @references Schlotheuber, A., & Hosseinpoor, A. R. (2022).
#' Summary measures of health inequality: A review of existing
#' measures and their application. International Journal of
#' Environmental Research and Public Health, 19 (6), 3697.
#' @return The estimated BGSD value, corresponding estimated standard error,
#' and confidence interval as a `data.frame`.
#' @export
#'
bgsd <- function(pop,
est,
se = NULL,
scaleval,
sim = NULL,
seed = 123456,
...) {
# Variable checks
## Stop
if(anyNA(est) & sum(is.na(est))/length(est) > .15){
stop('Estimates are missing in more than 15% of subgroups')
}
if(anyNA(est)){
pop <- pop[!is.na(est)]
if(!is.null(se)) se <- se[!is.na(est)]
if(!is.null(scaleval)) scaleval <- scaleval[!is.na(est)]
est <- est[!is.na(est)]
}
if(anyNA(pop)){
stop('Population is missing in some subgroups')
}
if(!is.numeric(pop)){
stop('Population needs to be numeric')
}
if(all(pop == 0)){
stop('The population is of size 0 in all cells')
}
## Warning
if(any(is.na(se)) | is.null(se))
warning("Standard errors are missing in all or some subgroups, confidence
intervals will not be computed.")
# Calculate summary measure
popsh <- pop/sum(pop)
weighted.mean <- sum(popsh * est)
bgsd <- sqrt(sum(popsh * (est - weighted.mean)^2))
# Calculate 95% confidence intervals
boot.lcl <- NA
boot.ucl <- NA
bgsd_sim <- c()
if(!any(is.na(se)) & !is.null(se)) {
if(is.null(sim)){
sim <- 100
}
input_data <- data.frame(est,
se,
scaleval)
set.seed(seed)
for (j in 1:sim) {
# Simulate each estimate in the dataset
simulated_data <- input_data %>%
rowwise() %>%
mutate(simulation =
{result <- if (scaleval != 100) {
repeat {
result <- rnorm(1, mean = est, sd = se)
if (result > 0) break
}
result
} else {
repeat {
result <- rnorm(1, mean = est, sd = se)
if (result >= 0 & result <= 100) break
}
result
}
}) %>%
ungroup()
# Calculate weighted mean
if(!is.null(pop)){
mean_sim <- sum(popsh * simulated_data$simulation)
}
# Calculate summary measure using simulated estimates
bgsd_sim[j] <- with(
simulated_data, sqrt(sum(popsh * (simulation - mean_sim)^2)))
}
boot.lcl <- quantile(bgsd_sim, probs = c(0.025), na.rm = TRUE)
boot.ucl <- quantile(bgsd_sim, probs = c(0.975), na.rm = TRUE)
}
# Return data frame
return(data.frame(measure = "bgsd",
estimate = bgsd,
lowerci = boot.lcl,
upperci = boot.ucl)
)
}
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