Sample size for a mean

Compute the required sample size for estimating a population mean with a specified margin of error or coefficient of variation.

Usage

n_mean(var = NULL, ...)

# Default S3 method
n_mean(
  var = NULL,
  ...,
  sd = NULL,
  mu = NULL,
  moe = NULL,
  cv = NULL,
  rmoe = NULL,
  alpha = 0.05,
  N = Inf,
  deff = 1,
  resp_rate = 1,
  df = NULL,
  plan = NULL
)

# S3 method for class 'svyplan_prec'
n_mean(var, ..., moe = NULL, cv = NULL, rmoe = NULL)

Arguments

var: For the default method: population variance $S^2$. Estimate from a pilot study, a previous survey, or published data for a similar population. When uncertain, use a conservative (larger) estimate to avoid under-sizing. For svyplan_prec objects: a precision result from prec_mean().
...: Additional arguments passed to methods. Unused arguments are rejected.
sd: Population standard deviation, an alternative spelling of var. Supply exactly one of var or sd. Stratum frames and published survey reports usually quote standard deviations.
mu: Population mean. Required when cv or rmoe is specified, both being defined against the mean.
moe: Desired margin of error, the half-width of the confidence interval, in the same units as the variable. For example, if measuring income in dollars, moe = 50 means the 95 percent CI should be no wider than +/- $50. Specify exactly one of moe, cv, or rmoe.
cv: Target coefficient of variation (relative standard error). For example, cv = 0.05 means the standard error should be at most 5 percent of the estimate. Use cv when you want precision to scale with the estimate (common in economic surveys). Use moe when you want a fixed absolute precision. Requires mu. Specify exactly one of moe, cv, or rmoe.
rmoe: Target margin of error relative to mu, so rmoe = 0.05 asks for a 95 percent interval whose half-width is 5 percent of the mean. It is moe / abs(mu), taken on the magnitude so that a negative mean gives a positive margin of error, and it requires mu. Specify exactly one of moe, cv, or rmoe.
alpha: Significance level, default 0.05.
N: Population size. Inf (default) means no finite population correction. Setting a finite N reduces the required sample size when the sampling fraction is non-negligible (rule of thumb: matters when n/N > 5 percent).
deff: Design effect multiplier (> 0). See n_prop() for guidance on estimating DEFF. 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 required sample size is inflated by 1 / resp_rate.
df: Degrees of freedom of the variance estimator the planned design will have, typically sampled PSUs minus strata, and available from design_df(). It switches the interval quantile from normal to t. NULL (default) applies no adjustment. See n_prop() for the full account.
plan: Optional svyplan() object providing design defaults.

Value

A svyplan_n object with type = "mean":

n: Required sample size, continuous and gross. It already carries deff and the 1 / resp_rate inflation, so it counts the units to release, not the completed interviews. $n and as.double() keep the unrounded value, which is what makes the round trip through prec_mean() exact; print() and as.integer() round it up to the whole units you would field. Take the field figure from as.integer() rather than from $n.
se, moe, cv, rmoe: Precision the design achieves at that n, the same values prec_mean() reports for the same inputs. moe is qnorm(1 - alpha / 2) * se, and the interval is symmetric about the mean. cv and rmoe are NA unless mu was supplied, both needing a mean to be relative to.
params: The validated inputs (var, alpha, N, deff, resp_rate, mu when given, and whichever of moe, cv, or rmoe was the target). Dispersion is always stored as var, including when you supplied sd. predict(), confint() and the prec_mean() round trip read the design back from here.

Details

Two modes:

MOE mode: n = deff * z^2 * var / (moe^2 + deff * z^2 * var / N). An rmoe target enters here as moe = rmoe * abs(mu).
CV mode: Computes CVpop = sqrt(var) / abs(mu), then n = deff * CVpop^2 / (cv^2 + deff * CVpop^2 / N).

deff appears in the denominator as well as the numerator: it inflates the variance the finite population correction is then applied to, rather than scaling a size already corrected. The two coincide only at infinite N. For var = 100, moe = 2, N = 100 and deff = 2, inflating afterwards would give 97.98 against the correct 65.76.

Finite population correction

Setting N to a finite value reduces the required sample size when the sampling fraction (n/N) is non-negligible (rule of thumb: matters when n/N > 5 percent). Unlike n_prop(), no N/(N-1) adjustment is needed because var is already defined on N-1 degrees of freedom. See n_prop() for a fuller explanation of FPC.

All methods use the normal (z) quantile. This is standard for survey sampling where the sample size is large enough for the CLT to apply.

Sample size for a total

A separate n_total() function is not needed because the sample size for a population total $\hat{Y} = N \bar{y}$ is identical to the sample size for the mean. The two are related by a factor of $N$:

CV mode: $CV(\hat{Y}) = CV(\bar{y})$, so the required sample size is the same. Use n_mean(var, mu, cv) directly.
MOE mode: $MOE(\hat{Y}) = N \times MOE(\bar{y})$, so divide the target MOE for the total by $N$: n_mean(var, moe = moe_total / N, N = N).

See Examples below.

References

Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley.

Valliant, R., Dever, J. A., and Kreuter, F. (2018). Practical Tools for Designing and Weighting Survey Samples (2nd ed.). Springer.

Examples

# MOE mode
n_mean(var = 100, moe = 2)
#> Sample size for mean
#> n = 97 (var = 100.00, moe = 2.000, deff = 1)

# CV mode
n_mean(var = 100, mu = 50, cv = 0.05)
#> Sample size for mean
#> n = 16 (var = 100.00, cv = 0.050, deff = 1)

# Relative MOE mode: interval half-width 5 percent of the mean
n_mean(var = 100, mu = 50, rmoe = 0.05)
#> Sample size for mean
#> n = 62 (var = 100.00, rmoe = 0.050, deff = 1)

# With FPC, design effect, and response rate
n_mean(var = 100, moe = 2, N = 5000, deff = 1.5, resp_rate = 0.8)
#> Sample size for mean
#> n = 176 gross (net: 141) (var = 100.00, moe = 2.000, deff = 1.50, resp_rate = 0.80)

## Sample size for a total
# Target: estimate total income (N = 10000) with MOE of 500000
n_mean(var = 2500, moe = 500000 / 10000, N = 10000)
#> Sample size for mean
#> n = 4 (var = 2500.00, moe = 50.000, deff = 1)

# CV mode: identical for means and totals
n_mean(var = 2500, mu = 300, cv = 0.05, N = 10000)
#> Sample size for mean
#> n = 12 (var = 2500.00, cv = 0.050, deff = 1)