Precision of a ratio of two totals at a given sample size

Evaluate the standard error, margin of error, and coefficient of variation a planned design achieves for a ratio of two population totals.

Usage

prec_ratio(r = NULL, ...)

# Default S3 method
prec_ratio(
  r = NULL,
  n = NULL,
  ...,
  cv_num = NULL,
  cv_den = NULL,
  component_cor = NULL,
  alpha = 0.05,
  N = Inf,
  deff = 1,
  resp_rate = 1,
  df = NULL,
  plan = NULL
)

# S3 method for class 'svyplan_n'
prec_ratio(r, ...)

Arguments

r: For the default method: the anticipated ratio, mean(y) / mean(x). May be negative, but not zero. For svyplan_n objects: a sample size result from n_ratio().
...: Additional arguments passed to methods. Unused arguments are rejected.
n: Sample size to evaluate, gross of nonresponse. The responding sample is n * resp_rate.
cv_num: Coefficient of variation of the numerator variable, strictly positive.
cv_den: Coefficient of variation of the denominator variable, strictly positive.
component_cor: Correlation between numerator and denominator across units, in [-1, 1].
alpha: Significance level, default 0.05.
N: Population size. Inf (default) means no finite population correction.
deff: Design effect of the ratio estimator (> 0). See n_ratio() for what this is the design effect of.
resp_rate: Expected response rate, in (0, 1]. Default 1.
df: Degrees of freedom of the variance estimator, switching the interval quantile from normal to t. NULL (default) applies none.
plan: Optional svyplan() object providing design defaults.

Value

A svyplan_prec object with type = "ratio" and method = "linearization":

se: Standard error of the estimated ratio, on the ratio scale.
moe: Half-width of the confidence interval, q * se.
cv: Relative standard error, se / abs(r).
rmoe: Margin of error relative to r, moe / abs(r).
params: The inputs, plus unit_relvar and n.

Details

At a gross sample n, with n_net = n * resp_rate and n_eff = n_net / deff,

$$SE(\hat{R}) = |R| \sqrt{L_R (1 - n_{net}/N) / n_{eff}}$$

where $L_R$ is the unit relative variance defined in n_ratio(). The interval is the symmetric first-order one and is not a Fieller interval. The method omits the ratio estimator's bias and assumes a denominator safely away from zero.

Why there is no solve-for-level mode

prec_mean() accepts a target cv and solves for the smallest detectable mu, because the standard error of a mean does not involve the mean. That inverse does not exist here. The relative standard error of a ratio,

$$CV(\hat{R}) = \sqrt{\mathrm{deff} \cdot L_R (1/n_{net} - 1/N)}$$

does not involve the magnitude of R at all, so either every non-zero ratio with those component moments meets a cv target or none does.

A moe target is algebraically invertible, giving $|R| = \mathrm{moe} / (q \sqrt{L_R \cdot fpc / n_{eff}})$, and is still not offered. It recovers only the magnitude, so it cannot return the estimand that was asked for, and the question it answers, the largest ratio whose absolute margin of error stays inside a bound at a fixed sample size, is not one survey planning asks.

Examples

# Precision of an issued sample of 1200
prec_ratio(r = 420, n = 1200, cv_num = 1.20, cv_den = 0.45,
           component_cor = 0.65)
#> Sampling precision for ratio (linearization)
#> n = 1200
#> se = 11.7581, moe = 23.0455, cv = 0.0280, rmoe = 0.0549

# With a design effect and nonresponse
prec_ratio(r = 420, n = 1200, cv_num = 1.20, cv_den = 0.45,
           component_cor = 0.65, deff = 1.3, resp_rate = 0.85)
#> Sampling precision for ratio (linearization)
#> n = 1200 (net: 1020)
#> se = 14.5412, moe = 28.5002, cv = 0.0346, rmoe = 0.0679

# Round trip: a size and the precision it buys agree exactly
size <- n_ratio(r = 2, cv_num = 1.1, cv_den = 0.6, component_cor = 0.7,
                cv = 0.05)
prec_ratio(size)$cv
#> [1] 0.05