expvar: Expected variance
In optimStrat: Choosing the Sample Strategy

Description Usage Arguments Details Value References See Also Examples

Compute the expected variance of five sampling strategies.

1 2	expvar(b, d, x, n, H, Rxy, stratum1 = NULL, stratum2 = NULL, st = 1:5, short = FALSE)

`b`	a numeric vector of length two giving the true shapes of the trend and spread terms.
`d`	a numeric vector of length two giving the assumed shapes of the trend and spread terms.
`x`	a positive numeric vector giving the values of the auxiliary variable.
`n`	a positive integer indicating the desired sample size.
`H`	a positive integer giving the desired number of strata/poststrata. Ignored if `stratum1` and `stratum2` are given.
`Rxy`	a number giving the correlation between the auxiliary variable and the study variable.
`stratum1`	a list giving stratum and sample sizes per stratum (see ‘Details’).
`stratum2`	a list giving stratum and sample sizes per stratum (see ‘Details’).
`st`	a numeric vector indicating the strategies for which the expected variance is to be calculated (see ‘Details’).
`short`	logical. If `FALSE` (the default) a vector of length five is returned. If `TRUE` only the strategies given by `st` are returned.

The expected variance of a sample of size n is computed for five sampling strategies (πps–reg, STSI–reg, STSI–HT, πps–pos and STSI–pos).

The strategies are defined assuming that the underlying superpopulation model is of the form

Y_k = δ_0 + δ_1 x_k^δ_2 + ε_k

with Eε_k = 0, Vε_k = δ_3^2 x_k^2δ_4 and Cov(ε_k , ε_l) = 0. But the true generating model is of the form

Y_k = β_0 + β_1 x_k^β_2 + ε_k

with Eε_k = 0, Vε_k = β_3^2 x_k^2β_4 and Cov(ε_k , ε_l) = 0.

The parameters β_2 and β_4 are given by b. The parameters δ_2 and δ_4 are given by d.

stratum1 and stratum2 are lists with two components (each with length length(x)): stratum indicates the stratum to which each element belongs and nh indicates the sample sizes to be selected in each stratum. They can be created via optiallo. stratum1 gives the stratification for STSI–HT and the poststrata for πps–pos and STSI–pos; whereas stratum2 gives the stratification for STSI–reg and STSI–pos. If NULL, optiallo is used for defining H strata/poststrata.

st indicates which variances to be calculated. If 1 in st, the expected variance of πps–reg is calculated. If 2 in st, the expected variance of STSI–reg is calculated, and so on.

If short=FALSE a vector of length five is returned giving the expected variance of the strategies given in st. NA is returned for those strategies not given in st. If short=TRUE, the NAs are omitted.

Bueno, E. (2018). A Comparison of Stratified Simple Random Sampling and Probability Proportional-to-size Sampling. Research Report, Department of Statistics, Stockholm University 2018:6. http://gauss.stat.su.se/rr/RR2018_6.pdf.

optiallo for how to stratify an auxiliary variable and allocate the sample size; desvar for calculating the variance of the five strategies.

x<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
expvar(b=c(1,1),d=c(1,1),x,n=500,H=6,Rxy=0.9)
expvar(b=c(1,1),d=c(1,1),x,n=500,H=6,Rxy=0.9,st=1:3)
expvar(b=c(1,1),d=c(1,1),x,n=500,H=6,Rxy=0.9,st=1:3,short=TRUE)

st1<- optiallo(n=500,x,H=6)
post1<- optiallo(n=500,x^1.5,H=10)
expvar(b=c(1,1),d=c(1,1),x,n=500,H=6,Rxy=0.9,
   stratum1=post1,stratum2=st1)