Sampling precision for a proportion

Compute the sampling error (SE, margin of error, CV) for estimating a population proportion given a sample size. This is the inverse of n_prop().

Usage

prec_prop(p, ...)

# Default S3 method
prec_prop(
  p = NULL,
  n,
  ...,
  cv = NULL,
  rmoe = NULL,
  alpha = 0.05,
  N = Inf,
  deff = 1,
  resp_rate = 1,
  method = c("wald", "wilson", "logodds", "beta"),
  df = NULL,
  plan = NULL
)

# S3 method for class 'svyplan_n'
prec_prop(p, ...)

Arguments

p: For the default method: expected proportion, in (0, 1). Leave it NULL and give cv or rmoe to solve for it instead. For svyplan_n objects: a sample size result from n_prop().
...: Additional arguments passed to methods. Unused arguments are rejected.
n: Sample size, measured as gross units drawn and bounded by a finite N.
cv: Target relative standard error, supplied instead of p to solve for the smallest proportion the design measures that precisely. Specify exactly one of p, cv, or rmoe.
rmoe: Target margin of error relative to the proportion, moe / p, supplied instead of p to solve for the smallest proportion the design can report at that relative half-width. Unlike the cv solve this one reads the interval rather than the variance and so depends on method. Specify exactly one of p, cv, or rmoe.
alpha: Significance level, default 0.05.
N: Population size. Inf (default) means no finite population correction.
deff: Design effect multiplier (> 0). Values < 1 are valid for efficient designs (e.g., stratified sampling with Neyman allocation).
resp_rate: Expected response rate, in (0, 1]. Default 1 (no adjustment). The effective sample size is n * resp_rate.
method: One of "wald" (default), "wilson", "logodds", or "beta" (Korn-Graubard). See n_prop() for how to choose.
df: Degrees of freedom of the variance estimator, typically sampled PSUs minus strata, and available from design_df(). It switches the interval quantile from normal to t under every method. NULL (default) applies no adjustment. See n_prop().
plan: Optional svyplan() object providing design defaults.

Value

A svyplan_prec object with type = "proportion":

se: Sampling standard error of the planned estimate, computed on the net sample n * resp_rate. It describes the estimator, so it is the same under all four methods and moves only with p, n, deff, N, and resp_rate.
moe: Half the length of the interval the chosen method builds at that standard error. Only under "wald" is this qnorm(1 - alpha / 2) * se; the other three close at a different half-width, which is the whole of what distinguishes them. None of the three is symmetric about p, so p - moe and p + moe are not the endpoints; use confint() for those.
cv: Relative standard error, se / p, and therefore also the same under all four methods.
rmoe: Margin of error relative to the proportion, moe / p. It reads the interval, so like moe it is method-specific.
method: The interval method used.
expected_cases: Positive cases the design expects to yield, n * resp_rate * p, the same quantity n_prop() reports and the one its min_cases constraint acts on.
solved: "p" when the proportion was solved for, and absent otherwise. The solved value is in params$p, whichever of cv or rmoe asked for it.
params: The validated inputs (p, n, alpha, N, deff, resp_rate, df, and rmoe when the proportion was solved from one). predict(), confint() and the n_prop() round trip read the design back from here.

Nothing here is rounded: n is taken as given, so passing a continuous n back from n_prop() reproduces its se, moe, cv and rmoe exactly.

Details

Computes the standard error for the given sample size and design parameters, then derives the margin of error and coefficient of variation. The variance equation is se^2 = deff * p * (1 - p) * fpc(n_net) / n_net, where n_net = n * resp_rate is the expected number of responding units: deff multiplies the SRS variance at the realized sample size, and the finite population correction uses the actual sampling fraction n_net / N. Equivalently, se^2 = p * (1 - p) / n_eff with the effective sample size n_eff = n_net / (deff * fpc(n_net)), the size at which an infinite-population simple random sample would carry the same variance.

The FPC uses the Cochran (1977, Ch. 3) form: the finite-population correction for a Bernoulli proportion is (N - n_net) / (N - 1), not the simpler 1 - n_net / N used for means. All four methods read the same variance through n_eff, so all four respond to deff and N and all four return zero at a census. They differ only in the interval built around that variance; see n_prop() for how to choose.

That division of labour is worth stating plainly, because it decides what each returned quantity answers. se and cv describe the estimator and are identical under all four methods; changing method on the same design does not move them. moe and rmoe describe the interval and are the only place the methods differ. Reading se back out of a half-width as moe / q would make the reported standard error of the estimate change when nothing but the interval construction had.

The four precision quantities

The package reports the same precision four ways, along two axes. One axis is what is being described, the estimator's sampling variance or the interval drawn around it; the other is whether it is stated in the estimand's units or as a fraction of it.

	Absolute	Relative
Sampling	`se`	`cv`
Interval	`moe`	`rmoe`

The sampling row is free of alpha and of method. The interval row depends on both: alpha fixes the confidence level and method fixes the construction, and rmoe = moe / p inherits each. rmoe = q * cv holds under "wald" alone, where q is the interval quantile the design uses, normal by default and t on df degrees of freedom when df is supplied; under the other three the interval does not close at q * se, and reading a relative margin of error off the CV that way overstates the precision by 4 to 10 percent at p = 0.02, n = 900, deff = 2.

n_prop() takes any of moe, rmoe, and cv as the target, cv under "wald" alone for the reason above, and all four quantities are reported back by both functions.

$moe is half the length of that interval, which equals q * se under "wald" alone. Only the Wald interval is symmetric about p, so for the other three p - moe and p + moe are not the limits: use confint() for those. The "beta" method makes this most visible, since its interval is deliberately asymmetric for a rare outcome.

Supplying cv in place of p solves the same equation in the remaining direction. n_prop() fixes the proportion and the precision and returns the sample size. This function fixes the proportion and the sample size and returns the precision. Giving n and cv returns the proportion. That third reading answers which estimates a fielded design can carry: because se(p) / p falls as p rises, the solution is a floor, and every larger proportion meets the target. It is the planning form of a publication threshold. expected_cases is reported alongside it, so the precision floor and the min_cases floor n_prop() applies can be read together.

The solution is closed form, p = 1 / (1 + n_eff * cv^2), and because cv reads the sampling variance rather than an interval, it is the same under all four methods: method changes the interval this design would report, not the smallest proportion it can measure to a given relative standard error. Every positive cv has a root in (0, 1), so no target is unattainable. A census is an error, having no sampling variance for any proportion to be measured against.

Giving n and rmoe asks the same question of the interval instead of the variance, and the answer is method-specific: the smallest proportion the design can report at a relative margin of error of rmoe under the chosen construction. There is no closed form, and two properties the cv solve does not share bound it. The Wilson and Korn-Graubard half-widths do not vanish as p approaches 1, so their relative margin of error has a positive floor and a target below it is refused with the floor named. The back-transformed log-odds half-width turns upward once the logit spread outgrows logit(p), near p = 0.999 at n = 1500 but as low as p = 0.97 at n = 30, so under "logodds" the target is met on an interval rather than on every larger proportion, and the reported root is its lower end.

When called on a svyplan_n object, parameters are extracted from the stored result. Any argument of the default method (e.g. method, deff, N) can be overridden through .... Unknown argument names are an error. Passing a different method evaluates the stored sample size under that formula. The round-trip will not be exact because n was determined under the original method.

Examples

# Precision with n = 400
prec_prop(p = 0.3, n = 400)
#> Sampling precision for proportion (wald)
#> n = 400
#> se = 0.0229, moe = 0.0449, cv = 0.0764, rmoe = 0.1497
#> expected cases = 120.0

# With design effect and response rate
prec_prop(p = 0.3, n = 400, deff = 1.5, resp_rate = 0.8)
#> Sampling precision for proportion (wald)
#> n = 400 (net: 320)
#> se = 0.0314, moe = 0.0615, cv = 0.1046, rmoe = 0.2050
#> expected cases = 96.0

# Korn-Graubard interval for a rare outcome in a clustered design
rare <- prec_prop(p = 0.02, n = 900, deff = 2, method = "beta", df = 25)
confint(rare)
#>        2.5 %     97.5 %
#>  0.008766989 0.03876296

# Smallest proportion 1500 units can report at a 10 percent CV
floor <- prec_prop(n = 1500, cv = 0.10, N = 2e6)
floor$params$p
#> [1] 0.06245608

# The same question of the interval: a 20 percent relative margin of
# error under the Korn-Graubard construction
prec_prop(n = 1500, rmoe = 0.20, method = "beta", N = 2e6)$params$p
#> [1] 0.06327848

# How that floor moves with the sample size
predict(floor, expand.grid(n = c(500, 1500, 5000)))
#>      n          p          se         moe  cv      rmoe
#> 1  500 0.16663201 0.016663201 0.032659274 0.1 0.1959964
#> 2 1500 0.06245608 0.006245608 0.012241167 0.1 0.1959964
#> 3 5000 0.01955979 0.001955979 0.003833649 0.1 0.1959964