PPS design quantities for a frame of units

y and size describe the frame – every unit that could be drawn – and n is the number of draws. That is the only reading under which pi_i = n x_i / sum_k x_k is a probability: the denominator is the population size total, not a sample total. A unit with pi_i > 1 must be taken with certainty, so it is an error here rather than a silently truncated probability.

Usage

Ppsamp(y, size, n)

Arguments

y

Values for the frame units.

size

Positive size measure, same length as y.

n

Number of draws, at least 1.

Value

List with pi, p, estimate, total, hh_variance, hh_se, srs_variance, deff, X, N, n.

Details

The Hansen-Hurwitz design variance is available in closed form from the frame and is exactly zero when y is proportional to size – the case PPS is designed for, and the anchor used here.

Formula: p_i = x_i / X; pi_i = n p_i; Var(Yhat_HH) = (1/n) sum_i p_i (y_i / p_i - Y)^2.

References

Hansen, M. H. & Hurwitz, W. N. (1943). On the theory of sampling from finite populations. Annals of Mathematical Statistics 14(4):333-362. doi:10.1214/aoms/1177731356 .

Examples

Ppsamp(y = c(1, 2, 3, 4, 5, 6, 7, 8), size = c(1, 2, 3, 4, 5, 6, 7, 8), n = 3L)
#> $pi
#> [1] 0.08333333 0.16666667 0.25000000 0.33333333 0.41666667 0.50000000 0.58333333
#> [8] 0.66666667
#> 
#> $p
#> [1] 0.02777778 0.05555556 0.08333333 0.11111111 0.13888889 0.16666667 0.19444444
#> [8] 0.22222222
#> 
#> $estimate
#> [1] 36
#> 
#> $total
#> [1] 36
#> 
#> $hh_variance
#> [1] 0
#> 
#> $hh_se
#> [1] 0
#> 
#> $srs_variance
#> [1] 112
#> 
#> $deff
#> [1] 0
#> 
#> $X
#> [1] 36
#> 
#> $N
#> [1] 8
#> 
#> $n
#> [1] 3
#> 
#> $method
#> [1] "PPS selection probabilities and Hansen-Hurwitz design variance"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"