Hierarchical Bayes Ratio Model for a Sub-Proportion of a Parent Indicator

Estimates an indicator that is a subset of a parent indicator (for instance extreme poverty within poverty, severe stunting within stunting) as the product of the parent estimate and a modelled sub-proportion: \(\theta_d = s_d P_d\), where \(P_d\) is a given (e.g. officially released) estimate of the parent indicator and \(s_d\) is the share of the parent population that has the sub-characteristic. The sub-proportion follows a binomial area-level model (hb_area with family = "binomial") whose trials are the (effective) numbers of sampled parent cases. Since \(0 \le s_d \le 1\), the estimates never exceed the parent (\(\theta_d \le P_d\)), and an optional benchmark on the logit scale of \(s_d\) keeps that property.

Usage

hb_ratio(
  formula,
  data,
  trials,
  parent,
  parent_se = NULL,
  domain = NULL,
  benchmark_target = NULL,
  benchmark_weight = NULL,
  benchmark_group = NULL,
  n_draws = 2000,
  seed = 1L,
  print_result = TRUE,
  ...
)

Arguments

formula: Formula for the direct sub-proportion, s ~ x1 + x2: the response is the direct estimate of \(s_d\) (for instance the direct ratio of the sub-indicator to the parent), or the number of parent cases with the sub-characteristic.
data: Data frame with one row per area.
trials: Vector, column name, or formula: (effective) numbers of sampled parent cases, e.g. from effective_trials with the parent cases as n.
parent: Vector, column name, or formula: the parent estimate \(P_d\) (non-negative).
parent_se: Optional vector, column name, or formula: standard errors of parent. With them, sd, mse, rse and the intervals include the uncertainty of the parent (treated as independent of \(s_d\)); without them they are conditional on parent.
domain: Optional column name or formula of area identifiers.
benchmark_target: Optional benchmark target for the weighted mean of \(\theta_d\) (scalar, or named by benchmark_group).
benchmark_weight: Vector, column name, or formula of area weights for the benchmark (e.g. population sizes); needed with benchmark_target.
benchmark_group: Optional vector, column name, or formula: groups with their own target.
n_draws: Number of posterior draws (default 2000).
seed: Integer seed of the posterior draws and of the parent draws (default 1); NULL uses the session's random number stream.
print_result: Logical. If TRUE (default), prints a summary.
...: Further arguments to hb_area (e.g. spatial, W, engine = inla_engine(...)); family is "binomial".

Value

An object of class c("fastsae_hb_ratio", "fastsae_hb", "fastsae") with

df_hb: One row per area: domain, y (direct sub-proportion times parent), hb (posterior mean of \(\theta_d\)), sd, mse, rse, ci_lower, ci_upper, rse_cond (RSE given the parent), ratio, ratio_sd (posterior mean and SD of the possibly benchmarked \(s_d\)), ratio_direct, parent, parent_se and trials.
draws: Matrix of posterior draws of \(\theta_d\) (areas x draws) behind df_hb (for instance for add_reliability_flags(draws = )).
ratio_fit: The hb_area fit of the sub-proportion (use it for diagnose() of the model).
benchmark: NULL, or a list with the targets, groups and the weighted means of the benchmarked estimates.

and estcoef, random_effect_var, goodness, convergence, inla_mode_status, formula, model and call.

Details

The benchmark solves, for each posterior draw, for one shift \(a_g\) per group on the logit scale such that \(\sum_{d \in g} w_d P_d\, \mathrm{logit}^{-1}(\mathrm{logit}(s_d) + a_g) / \sum_{d \in g} w_d = T_g\), so the benchmarked \(s_d\) stay in (0, 1) and the estimates below their parent. The parent enters at its given value; the parent draws for the total uncertainty are \(\max(P_d + \mathrm{se}_d z, 0)\), \(z \sim N(0, 1)\).

Modelling the sub-proportion borrows strength for the part of the indicator that the parent does not already explain, and it keeps the two indicators coherent. In an application to extreme poverty within poverty for 514 Indonesian districts, the ratio model gave a relative standard error of at most 50% in 124 districts against 77 for a binomial HB model of the extreme poverty rate itself, and no district estimate above its poverty rate (against 7).

Only the INLA engine is supported.

Examples

# \donttest{
if (requireNamespace("INLA", quietly = TRUE)) {
  set.seed(1)
  D <- 60
  d <- data.frame(area = 1:D, x = rnorm(D), pop = rpois(D, 5e4))
  d$p0 <- plogis(-2 + 0.3 * rnorm(D))                 # released parent rates
  d$n_poor <- rpois(D, 40)                             # sampled parent cases
  s_true <- plogis(-1.5 + 0.5 * d$x + rnorm(D, 0, 0.3))
  d$s_dir <- rbinom(D, d$n_poor, s_true) / d$n_poor
  fit <- hb_ratio(s_dir ~ x,
    data = d, trials = "n_poor", parent = "p0", domain = "area",
    benchmark_target = 0.025, benchmark_weight = "pop", print_result = FALSE
  )
  head(fit$df_hb)
  all(fit$df_hb$hb <= d$p0)
}
#> ℹ Response `y` appears to be proportions; using `y * trials` as the number of cases (not rounded).
#> [1] TRUE
# }