PracTools (version 1.8)

nEdge: Compute the total sample size for a stratified, simple random sample based on an Edgeworth approximation

Description

Compute the total stratified, simple random sample size for various allocations that is large enough to insure adequate coverage of a normal approximation confidence interval (CI) for a population mean.

Usage

nEdge(ci.lev, side, epsilon = 0.005, dat, pop.sw = TRUE, wts = NULL, hcol=NULL, ycol,
      alloc = NULL, Costh = NULL, nh.lowerbd = NULL
)

Value

If there are no strata, the value is a list with components:

CI type

one-sided lower, one-sided upper, or two-sided

epsilon

tolerance on CI coverage

Total sample size

numeric sample size

allocation

type of allocation to strata or NULL if no strata are used

g1

skewness of y

g2

excess kurtosis of y, i.e., 4th central moment divided by the square of the variance minus 3

q1

1st polynomial in Edgeworth expansion of the t-statistic

q2

2nd polynomial in Edgeworth expansion of the t-statistic

If the sample design has strata, the value is a list with the first four items above followed by

Stratum values

Data frame with columns for stratum, number of population units in each stratum, number of sample units allocated to each stratum (nh), value of ch which depends on the value of alloc (see Qing & Valliant 2026), stratum skewness g1h, stratum excess kurtosis g2h

If alloc = "totcost" or "totvar", the total cost of the allocation is returned and Stratum values contains a column for Costh.

Arguments

ci.lev

desired confidence level for a 1- or 2-sided normal approximation confidence interval based on an estimated mean; must be in the interval (0,1)

side

either "two.sided", "one.sided.upper" or "one.sided.lower" for type of confidence interval

epsilon

tolerance on coverage probability; the sample should be large enough that CI coverage is within \(\pm\) epsilon of ci.lev; must be in the interval (0,1)

dat

either a population or sample data frame

pop.sw

TRUE if dat is for a full population; FALSE if dat is for a sample

wts

vector of weights if dat is a sample; if dat is for a population, wts = NULL

hcol

column of dat that contains the stratum ID; strata can be character or numeric

ycol

column of dat that contains the analysis variable; must be numeric

alloc

allocation to the strata; must be one of prop, equal, neyman, totcost, totvar, or NULL

Costh

vector of costs per unit in each stratum; these exclude fixed costs that do not vary with the sample size

nh.lowerbd

Scalar value of the minimum value of the sample size allowed for any stratum

Author

Richard Valliant, Siyu Qing

Details

nEdge computes the total sample size needed in either a stratified or unstratified simple random sample so that the coverage probability of a confidence interval is within a specified tolerance (epsilon) of a nominal confidence level (ci.lev). The calculation assumes that there is a single estimated mean of the variable ycol that is of key importance in a sample. Confidence intervals for the finite population mean are usually computed using the normal approximation whose accuracy depends on the underlying structure of the analytic variable and the total sample size.

In some applications, assuring that CIs have near nominal coverage is critical. For example, for some items on business tax returns the US Internal Revenue Service allows sample estimates to be used but sets precision standards based on the lower (or upper) limit of a 1-sided CI. In some cases the computed sample size from nEdge may be too small to meet other goals of a sample (such as a precision requirement) and may need to be adjusted.

Using a three-term Edgeworth approximation to the distribution of the estimated overall mean in Qing & Valliant (2026), which extends results in Qing & Valliant (2025), nEdge computes the total sample size needed so that a CI will have coverage equal to the nominal value in ci.lev plus or minus the tolerance epsilon. The remainder term in the approximation used in nEdge is \(O(n^{-3/2})\). In previous versions of PracTools, the remainder term in the approximation used in nEdge was \(O(n^{-1})\), i.e. this version of nEdge provides a more accurate sample size approximation. See Qing & Valliant (2026) for comparisons between the two levels of approximation.

This was also the level of approximation used in the function nEdgeSRS in previous versions of Practools. nEdgeSRS has been deleted from the package because nEdge also can be used for simple random sampling. The calculation in nEdge assumes that the sampling fraction in each stratum is negligible. The total sample size returned by nEdge is based on the overall Edgeworth criterion; the resulting stratum sample sizes may not be large enough so that the normal approximation is adequate for each stratum estimator.

When dat is a sample, the weights (wts) used in the estimator of the mean are assumed to be scaled for estimating population totals. They can be inverse selection probabilities, i.e. ones used in the \(\pi\)-estimator, or weights that have been adjusted to account for nonresponse or coverage errors.

The value of nh.lowerbd is usually set to be 2 or more to permit variances to be estimated.

References

Qing, S. and Valliant, R. (2025). Extending Cochran's Sample Size Rule to Stratified Simple Random Sampling with Applications to Audit Sampling. Journal of Official Statistics, 41(1), 309-328.

Qing, S. and Valliant, R. (2026). Minimum Sample Size Rule for STSRS: A Three-Term Edgeworth Expansion for the Studentized Sample Mean, accepted by Journal of Survey Statistics and Methodology.

U.S. Internal Revenue Service (2011). 26 CFR 601.105: Examination of returns and claims for refund, credit or abatement: determination of correct tax liability. Washington DC. https://www.irs.gov/pub/irs-drop/rp-11-42.pdf

See Also

nCont, nLogOdds, nProp, nPropMoe, nWilson

Examples

require(PracTools)
set.seed(1289129963)
pop <- HMT(N=10000, H=5)
    # run for full population, 2-sided CI, Neyman allocation
nEdge(ci.lev = 0.95, side = "two.sided", epsilon = 0.005, dat = pop,
      pop.sw = TRUE, wts = NULL, hcol = "strat", ycol = "y", alloc = "neyman",
      Costh = NULL)
nEdge(ci.lev = 0.95, side = "two.sided", epsilon = 0.005,dat = pop,
      pop.sw = TRUE, wts = NULL, hcol = "strat", ycol = "y", alloc = "totcost",
      Costh = c(5, 10, 15, 20, 25)
)
      
    # run for a stratified sample, 1-sided lower CI, equal allocation
require(sampling)
sam <- strata(data=pop, stratanames="strat", size=c(30, 40, 50, 60, 70), method=c("srswor"),
              description=TRUE)
samdat <- pop[sam$ID_unit,]
w = 1/sam$Prob
nEdge(ci.lev = 0.99, side = "one.sided.lower", epsilon = 0.01, dat = samdat,
      pop.sw = FALSE, wts = w, hcol = "strat", ycol = "y", alloc = "equal",
      Costh = NULL, nh.lowerbd = 2)
    
    # simple radom sample size      
nEdge(ci.lev = 0.95, side = "two.sided", epsilon = 0.005, dat = pop,
      pop.sw = TRUE, wts = NULL, hcol = NULL, ycol = "y")