optimStrat (version 2.4)

expvar: Expected variance

Description

Compute the expected variance of five sampling strategies.

Usage

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

Value

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.

Arguments

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.

Details

The expected variance of a sample of size n is computed for five sampling strategies (\(\pi\)ps--reg, STSI--reg, STSI--HT, \(\pi\)ps--pos and STSI--pos).

The strategies are defined assuming that the underlying superpopulation model is of the form $$Y_{k}=\delta_{0}+\delta_{1}x_{k}^{\delta_{2}}+\epsilon_{k}$$ with \(E\epsilon_{k}=0\), \(V\epsilon_{k}=\delta_{3}^{2}x_{k}^{2\delta_{4}}\) and \(Cov(\epsilon_{k} , \epsilon_{l}) = 0\). But the true generating model is of the form $$Y_{k}=\beta_{0}+\beta_{1}x_{k}^{\beta_{2}}+\epsilon_{k}$$ with \(E\epsilon_{k}=0\), \(V\epsilon_{k} = \beta_{3}^{2}x_{k}^{2\beta_{4}}\) and \(Cov(\epsilon_{k},\epsilon_{l})=0\).

The parameters \(\beta_2\) and \(\beta_4\) are given by b. The parameters \(\delta_2\) and \(\delta_4\) 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 \(\pi\)ps--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 \(\pi\)ps--reg is calculated. If 2 in st, the expected variance of STSI--reg is calculated, and so on.

References

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.

See Also

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

Examples

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)