Effective sample sizes \(n_d / \mathrm{deff}_d\) for binomial area-level models
(trials of hb_area and hb_twofold), from the direct
proportions, their design variances and the sample sizes. The design effect of an area is
undefined when its direct proportion is 0 or 1 (no design variance); such areas take a
design effect pooled over their group, or a supplied fallback.
Usage
effective_trials(
y,
vardir,
n,
group = NULL,
data = NULL,
method = c("pooled", "area"),
deff_fallback = NULL,
deff_min = 1
)Arguments
- y
Vector, column name, or one-sided formula: direct proportions in \([0, 1]\).
- vardir
Vector, column name, or one-sided formula: design variances of
y.- n
Vector, column name, or one-sided formula: sample sizes (number of sampled units in the denominator of
y).- group
Optional vector, column name, or one-sided formula: groups (e.g. provinces) within which design effects are pooled.
- data
Optional data frame containing the variables.
- method
"pooled"(default): every area takes the design effect of its group, \(\sum_d v_d / \sum_d \{y_d (1 - y_d) / n_d\}\) over the areas of the group with \(0 < y_d < 1\) (the ratio of the summed design variances to the summed simple random sampling variances)."area": each area takes its own design effect \(v_d / \{y_d (1 - y_d) / n_d\}\) where it is defined, and the pooled one otherwise.- deff_fallback
Optional vector, column name, or one-sided formula: design effects for the areas whose own design effect is undefined, used before the pooled ones with
method = "area"(for instance the design effect of a related indicator of the same survey, which is defined whereyis 0).- deff_min
Lower bound of the design effects (default 1, no gain over simple random sampling).
Value
A data frame with one row per area and columns n, deff (the design
effect used, at least deff_min), deff_source ("area",
"fallback", "group" or "overall") and trials
(\(n_d / \mathrm{deff}_d\)). Areas with missing y or n get NA.
Details
Single-area design effects are noisy and undefined at \(y_d \in \{0, 1\}\), which is
common for rare indicators. In a simulation of a two-stage cluster sample (12 provinces x 6
districts), effective sample sizes with design effects pooled per province gave 95% interval
coverage of 0.925 (districts) and 0.937 (provinces), close to 0.929 and 0.946 with the true
design effects, against 0.911 and 0.897 when the design was ignored
(benchmarks/validation/binomial_neff.R in the source repository). A group without any
area with \(0 < y_d < 1\) takes the design effect pooled over all areas.
References
Kish, L. (1965). Survey Sampling. John Wiley & Sons.
Chen, C., Wakefield, J., and Lumley, T. (2014). The use of sampling weights in Bayesian hierarchical models for small area estimation. Spatial and Spatio-temporal Epidemiology, 11, 33–43.
See also
hb_area (family = "binomial"), gvf_smooth.
Examples
set.seed(1)
d <- data.frame(
prov = rep(1:3, each = 5), n = rpois(15, 80),
y = c(0, rbeta(14, 2, 30))
)
d$vardir <- 1.6 * d$y * (1 - d$y) / d$n
effective_trials(~y, vardir = ~vardir, n = ~n, group = ~prov, data = d)
#> n deff deff_source trials
#> 1 74 1.6 group 46.250
#> 2 91 1.6 group 56.875
#> 3 91 1.6 group 56.875
#> 4 83 1.6 group 51.875
#> 5 66 1.6 group 41.250
#> 6 84 1.6 group 52.500
#> 7 86 1.6 group 53.750
#> 8 85 1.6 group 53.125
#> 9 77 1.6 group 48.125
#> 10 72 1.6 group 45.000
#> 11 74 1.6 group 46.250
#> 12 77 1.6 group 48.125
#> 13 79 1.6 group 49.375
#> 14 79 1.6 group 49.375
#> 15 88 1.6 group 55.000
# own design effects where defined, pooled ones for y = 0
effective_trials("y", "vardir", "n", group = "prov", data = d, method = "area")
#> n deff deff_source trials
#> 1 74 1.6 group 46.250
#> 2 91 1.6 area 56.875
#> 3 91 1.6 area 56.875
#> 4 83 1.6 area 51.875
#> 5 66 1.6 area 41.250
#> 6 84 1.6 area 52.500
#> 7 86 1.6 area 53.750
#> 8 85 1.6 area 53.125
#> 9 77 1.6 area 48.125
#> 10 72 1.6 area 45.000
#> 11 74 1.6 area 46.250
#> 12 77 1.6 area 48.125
#> 13 79 1.6 area 49.375
#> 14 79 1.6 area 49.375
#> 15 88 1.6 area 55.000