Design-based estimate of a total with its variance

Horvitz-Thompson estimate of the population total of y from a select_units() sample, with the variance estimator that matches the design: the local-mean estimator of Grafström and Schelin (2014) for spatially balanced samples (it needs no joint inclusion probabilities), the variance among replicate estimates for replicated systematic samples, and the simple-random-sampling formula for equal-probability simple random samples. A systematic sample drawn as a single replicate has no unbiased design-based variance estimator; the local-mean estimator is used as the customary approximation. The simple-random-sampling variance of the same sample is always reported as a reference.

Usage

design_variance(sample, y, neighbours = 3L)

Arguments

sample

A fieldopt_sample.

y

Column of the study variable (observed on the sampled units; other rows may be NA).

neighbours

Number of nearest sampled units in the neighbourhood of the local-mean estimator.

Value

A one-row tibble: total, variance, se, cv, variance_srs, n, N, variance_method and df (degrees of freedom of the variance estimate: -1 with r replicates, the number of sampled units minus the number of strata otherwise).

Details

The local-mean estimator averages over each sampled unit and its neighbours nearest sampled units (ties included), as BalancedSampling::vsb() does with its default k = 3. In the package's experiments (three frames, Gaussian fields at four spatial ranges) it is close to unbiased when the variable has no spatial structure and conservative when it is smooth; the replicate estimator is unbiased but has only -1 degrees of freedom, so intervals should use the t quantile with df (reported) rather than the normal one.

References

Grafström, A. and Schelin, L. (2014). How to select representative samples. Scandinavian Journal of Statistics, 41(2), 277–290.

Examples

set.seed(2)
frame <- data.frame(x = runif(80), y = runif(80))
frame$crop <- 10 + 20 * frame$x + rnorm(80)
s <- select_units(frame, n = 20)
design_variance(s, y = "crop")
#> # A tibble: 1 × 9
#>   total variance    se     cv variance_srs     n     N variance_method    df
#>   <dbl>    <dbl> <dbl>  <dbl>        <dbl> <int> <int> <chr>           <int>
#> 1 1620.    1442.  38.0 0.0234        9260.    20    80 local-mean         19
r <- select_units(frame, n = 20, method = "systematic", replicates = 4)
design_variance(r, y = "crop")
#> # A tibble: 1 × 9
#>   total variance    se     cv variance_srs     n     N variance_method    df
#>   <dbl>    <dbl> <dbl>  <dbl>        <dbl> <int> <int> <chr>           <int>
#> 1 1596.    1952.  44.2 0.0277       10160.    20    80 replicates          3