expgreg: Expected variance of the general regression estimator
In optimStrat: Choosing the Sample Strategy

Description Usage Arguments Details Value References See Also Examples

View source: R/expgreg.R

Compute the expected design variance of the general regression estimator of the total of a study variable under different sampling designs.

1 2	expgreg(x, b11, b12, b21, b22, d12, Rfy, n, design = NULL, stratum = NULL, x_des = NULL, inc.p = NULL, ...)

`x`	design matrix with the variables to be used into the GREG estimator.
`b11`	a numeric vector of length equal to the number of variables in `x` giving the coefficients of the trend term in the true superpopulation model (see ‘Details’).
`b12`	a numeric vector of length equal to the number of variables in `x` giving the exponents of the trend term in the true superpopulation model (see ‘Details’).
`b21`	a numeric vector of length equal to the number of variables in `x` giving the coefficients of the spread term in the true superpopulation model (see ‘Details’).
`b22`	a numeric vector of length equal to the number of variables in `x` giving the exponents of the spread term in the true superpopulation model (see ‘Details’).
`d12`	a numeric vector of length equal to the number of variables in `x` giving the exponents of the trend term in the assumed superpopulation model (see ‘Details’).
`Rfy`	a number giving the square root of the coefficient of determination between the auxiliary variables and the study varible.
`n`	either a positive number indicating the (expected) sample size (when `design` is one of 'srs', 'poi', 'pips' or `NULL`) or a numeric vector indicating the sample size of the strata to which each element belongs (when `design` is 'stsi') (see ‘Examples’).
`design`	a character string giving the sampling design. It must be one of 'srs' (simple random sampling without replacement), 'poi' (Poisson sampling), 'stsi' (stratified simple random sampling), 'pips' (Pareto πps sampling) or `NULL` (see ‘Details’).
`stratum`	a vector indicating the stratum to which every unit belongs. Only used if `design` is 'stsi'.
`x_des`	a positive numeric vector giving the values of the auxiliary variable that is used for defining the inclusion probabilities. Only used if `design` is 'poi' or 'pips'.
`inc.p`	a matrix giving the first and second order inclusion probabilities. Only used if `design` is `NULL`.
`...`	other arguments passed to `lm` (see ‘Details’).

The expected variance of the general regression estimator under different sampling designs is computed.

It is assumed that the underlying superpopulation model is of the form

Y_k = f(x_k|δ_1) + ε_k

with Eε_k = 0, Vε_k = σ_0^2 g(x_k|δ_2)^2 and Cov(ε_k , ε_l) = 0.

But the true generating model is in fact of the form

Y_k = f(x_k|β_1) + ε_k

with Eε_k = 0, Vε_k = σ^2 g(x_k|β_2)^2 and Cov(ε_k , ε_l) = 0.

Where

f(x_k|δ_1) = Σ_[j=1]^J δ_[1,j] x_[j,k]^δ_[1,J+j],

f(g_k|δ_2) = Σ_[j=1]^J δ_[2,j] x_[j,k]^δ_[2,J+j],

f(x_k|β_1) = Σ_[j=1]^J β_[1,j] x_[j,k]^β_[1,J+j],

f(g_k|β_2) = Σ_[j=1]^J β_[2,j] x_[j,k]^β_[2,J+j].

the coefficients β_[1,j] (j = 1,...,J) are given by b11;
the exponents β_[1,j] (j=J+1,...,2J) are given by b12;
the coefficients β_[2,j] (j = 1,...,J) are given by b21;
the exponents β_[2,j] (j = J+1,...,2J) are given by b22;
the exponents δ_[1,j] (j = J+1,...,2J) are given by d12.

The expected variance of the GREG estimator is approximated by

E(V(t_hat)) = V(t*_hat) + σ*^2 Σ_[k=1]^N (1/π_k - 1)g(x_k|β_2)^2

where

V(t*_hat) = Σ_[k=1]^N Σ_[l=1]^N π_kl (z_k*z_l)/(π_k*π_l) - (Σ_[k=1]^N z_k)^2

and

σ*^2 = S^2_f/(g^2)_bar*(1/R_fy^2 - 1),

z_k = (x_k^β - x_k^δ*A)*β**_1,

S^2_f = Σ_[k=1]^N (f(x_k|β_1) - f_bar)^2 / N,

(g^2)_bar = Σ_[k=1]^N g(x_k|β_2)^2 / N,

x_k^β = (x_[1k]^(β_[1,J+1]),…,x_[Jk]^(β_[1,2J])),

x_k^δ = (x_[1k]^(δ_[1,J+1]),…,x_[Jk]^(δ_[1,2J])),

β**_1 = (β_[1,1],…,β_[1,J])',

A = (Σ_[k=1]^N w_k*x_k^δ'*x_k^δ)^-1 Σ_[k=1]^N w_k*x_k^δ'*x_k^β.

N is the population size and π_k and π_kl are, respectively, the first and second order inclusion probabilities. w_k is a weight associated to each element and it represents the inverse of the conditional variance (up to a scalar) of the underlying superpopulation model (see ‘Examples’).

If design=NULL, the matrix of inclusion probabilities is obtained proportional to the matrix p.inc. If design is other than NULL, the formula for the variance is simplified in such a way that the inclusion probabilities matrix is no longer necessary. In particular:

if design='srs', only the sample size n is required;
if design='stsi', both the stratum ID stratum and the sample size per stratum n, are required;
if design is either 'pips' or 'poi', the inclusion probabilities are obtained proportional to the values of x_des, corrected if necessary.

A numeric value giving the expected variance of the general regression estimator for the desired design under the working and true models.

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.

expvar for the simultaneous calculation of the expected variance of five sampling strategies under a superpopulation model; vargreg for the variance of the GREG estimator; desvar for the simultaneous calculation of the variance of six sampling strategies; optimApp for an interactive application of expgreg.

x1<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
x2<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
x3<- 1 + sort( rgamma(5000, shape=4/9, scale=108) )
x<- cbind(x1,x2,x3)
expgreg(x,b11=c(1,-1,0),b12=c(1,1,0),b21=c(0,0,1),b22=c(0,0,0.5),
        d12=c(1,1,0),Rfy=0.8,n=150,"pips",x_des=x3)
expgreg(x,b11=c(1,-1,0),b12=c(1,1,0),b21=c(0,0,1),b22=c(0,0,0.5),
        d12=c(1,1,0),Rfy=0.8,n=150,"pips",x_des=x2)
expgreg(x,b11=c(1,-1,0),b12=c(1,1,0),b21=c(0,0,1),b22=c(0,0,0.5),
        d12=c(1,1,0),Rfy=0.8,n=150,"pips",x_des=x2,weights=1/x1)

st1<- optiallo(n=150,x=x3,H=6)
expgreg(x,b11=c(1,-1,0),b12=c(1,1,0),b21=c(0,0,1),b22=c(0,0,0.5),
        d12=c(1,1,0),Rfy=0.8,n=st1$nh,"stsi",stratum=st1$stratum)
expgreg(x,b11=c(1,-1,0),b12=c(1,1,0),b21=c(0,0,1),b22=c(0,0,0.5),
        d12=c(1,0,1),Rfy=0.8,n=st1$nh,"stsi",stratum=st1$stratum)
expgreg(x,b11=c(1,-1,0),b12=c(1,1,0),b21=c(0,0,1),b22=c(0,0,0.5),
        d12=c(1,0,1),Rfy=0.8,n=st1$nh,"stsi",stratum=st1$stratum,weights=1/x1)