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Confidence interval for a share with a design effect

Source: R/share_intervals.R
share_interval.Rd

Computes Wilson, Jeffreys or logit intervals for the share x / n using the effective sample size n / deff, so that the extra variance due to clustering (for example of postings by source) widens the interval.

Usage

share_interval(
  x,
  n,
  method = c("wilson", "jeffreys", "logit"),
  deff = 1,
  df = Inf,
  conf = 0.95
)

Arguments

x

Numeric vector of counts (postings with the code).

n

Numeric vector of denominators, same length as x or length 1.

method

Interval method: "wilson" (default), "jeffreys" or "logit".

deff

Design effect, length 1 or length(x). Must be positive; NA gives NA bounds.

df

Degrees of freedom of the t quantile, length 1 or length(x). Inf (default) uses the normal quantile. Must be positive; NA gives NA bounds. Not used by "jeffreys".

conf

Confidence level, default 0.95.

Value

A data.table with one row per element of x and columns quota (x / n), lo, hi, n_eff (n / deff) and metodo.

Details

With effective counts n_eff = n / deff, x_eff = x / deff, share p = x / n and quantile z = qt(1 - (1 - conf) / 2, df) (equal to the normal quantile when df = Inf):

  • Wilson: (p + z^2 / (2 n_eff)) / (1 + z^2 / n_eff) +/- z / (1 + z^2 / n_eff) * sqrt(p (1 - p) / n_eff + z^2 / (4 n_eff^2)).

  • Jeffreys: quantiles (1 - conf) / 2 and 1 - (1 - conf) / 2 of Beta(x_eff + 0.5, n_eff - x_eff + 0.5); the lower bound is 0 when x = 0 and the upper bound is 1 when x = n (Brown, Cai and DasGupta, 2001).

  • Logit: plogis(logit(p) +/- z * sqrt(deff / (n p (1 - p)))); when x = 0 or x = n the Haldane correction replaces p with (x + 0.5) / (n + 1) and n with n + 1.

The design effect is the ratio between the variance of the estimator under the actual design and its binomial variance (Kish, 1965); see cluster_deff() for its estimation with sources as clusters and smooth_deff() for its stabilisation along a code hierarchy.

References

Kish, L. (1965). Survey Sampling. Wiley.

Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16(2), 101-133.

Examples

share_interval(x = c(0, 12, 300), n = 1000)
#>    quota           lo          hi n_eff metodo
#>    <num>        <num>       <num> <num> <char>
#> 1: 0.000 2.168404e-19 0.003826758  1000 wilson
#> 2: 0.012 6.877648e-03 0.020857268  1000 wilson
#> 3: 0.300 2.724068e-01 0.329123861  1000 wilson
share_interval(x = c(0, 12, 300), n = 1000, deff = 4, df = 20)
#>    quota          lo         hi n_eff metodo
#>    <num>       <num>      <num> <num> <char>
#> 1: 0.000 0.000000000 0.01710722   250 wilson
#> 2: 0.012 0.003840213 0.03685644   250 wilson
#> 3: 0.300 0.243386262 0.36345663   250 wilson
share_interval(x = 12, n = 1000, method = "logit", deff = 2)
#>    quota          lo         hi n_eff metodo
#>    <num>       <num>      <num> <num> <char>
#> 1: 0.012 0.005400909 0.02644777   500  logit