Every k-th unit: all k systematic samples and the exact variance

A systematic sample has only k possible outcomes, so its design variance is exact rather than estimated. deff is its ratio to the simple-random-sampling variance.

Usage

Sysamp(y, k)

Arguments

y

The WHOLE population, in the order it would be sampled; length(y) must be an exact multiple of k.

k

Sampling interval.

Value

List with means, population_mean, variance, se, srs_variance, deff, rho, N, n, k.

Details

Formula: V(ybar_sy) = (1/k) sum_{i=1}^{k} (ybar_i - Ybar)^2; deff = V(ybar_sy) / [(1 - f) S^2 / n]

References

Cochran (1977), Sampling Techniques, 3rd edition, Chapter 8. Chapter 8 was NOT in the scanned excerpt available to this batch, so the standard published form is used; rho is reported by inverting V(ybar_sy) = (S^2/n)[1 + (n - 1) rho].

Examples

Sysamp(y = matrix(c(1, 2, 3, 4, 5, 6), nrow = 2), k = 3L)
#> $means
#> [1] 2.5 3.5 4.5
#> 
#> $population_mean
#> [1] 3.5
#> 
#> $variance
#> [1] 0.6666667
#> 
#> $se
#> [1] 0.8164966
#> 
#> $srs_variance
#> [1] 1.166667
#> 
#> $deff
#> [1] 0.5714286
#> 
#> $rho
#> [1] -0.6190476
#> 
#> $N
#> [1] 6
#> 
#> $n
#> [1] 2
#> 
#> $k
#> [1] 3
#> 
#> $method
#> [1] "Systematic sampling, exact design variance"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"