PikHol {TeachingSampling}

Optimal Inclusion Probabilities Under Multi-purpose Sampling

Description

Computes the population vector of optimal inclusion probabilities under the Holmbergs's Approach

Usage

PikHol(n, sigma, e, Pi)

Arguments

n

Vector of optimal sample sizes for each of the characteristics of interest.

sigma

A matrix containing the size measures for each characteristics of interest.

e

Maximum allowed error under the ANOREL approach.

Pi

Matrix of first order inclusion probabilities. By default, this probabilites are proportional to each sigma.

Details

Assuming that all of the characteristic of interest are equally important, the Holmberg's sampling design yields the following inclusion probabilities

\pi_{(opt)k}=\frac{n^*\sqrt{a_{qk}}}{\sum_{k\in U}\sqrt{a_{qk}}}

where

n^*\geq \frac{(\sum_{k\in U}\sqrt{a_{qk}})^2}{(1+c)Q+\sum_{k\in U}a_{qk}}

and

a_{qk}= \sum_{q=1}^Q \frac{\sigma^2_{qk}}{\sum_{k\in U}\left( \frac{1}{\pi_{qk}}-1\right)\sigma^2_{qk}}

Note that \sigma^2_{qk} is a size measure associated with the k-th element in the q-th characteristic of interest.

Value

The function returns a vector of inclusion probabilities.

Author(s)

Hugo Andres Gutierrez Rojas [e-mail]

References

Holmberg, A. (2002), On the Choice of Sampling Design under GREG Estimation in Multiparameter Surveys. RD Department, Statistics Sweden.
Sarndal, C-E. and Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling. Springer.
Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

Examples

#######################
#### First example ####
#######################

# Uses the Lucy data to draw an otpimal sample
# in a multipurpose survey context
data(Lucy)
attach(Lucy)
# Different sample sizes for two characteristics of interest: Employees and Taxes
N <- dim(Lucy)[1]
n <- c(350,400)
# The size measure is the same for both characteristics of interest,
# but the relationship in between is different
sigy1 <- sqrt(Income^(1))
sigy2 <- sqrt(Income^(2))
# The matrix containign the size measures for each characteristics of interest
sigma<-cbind(sigy1,sigy2)
# The vector of optimal inclusion probabilities under the Holmberg's approach
Piks<-PikHol(n,sigma,0.03)
# The optimal sample size is given by the sum of piks
n=round(sum(Piks))
# Performing the S.piPS function in order to select the optimal sample of size n
res<-S.piPS(n,Piks)
sam <- res[,1]
# The information about the units in the sample is stored in an object called data
data <- Lucy[sam,]
attach(data)
names(data)
# Pik.s is the vector of inclusion probability of every single unit
# in the selected sample
Pik.s <- res[,2]
# The variables of interest are: Income, Employees and Taxes
# This information is stored in a data frame called estima
estima <- data.frame(Income, Employees, Taxes)
E.piPS(estima,Pik.s)

########################
#### Second example ####
########################

# We can define our own first inclusion probabilities
data(Lucy)
attach(Lucy)

N <- dim(Lucy)[1]
n <- c(350,400)

sigy1 <- sqrt(Income^(1))
sigy2 <- sqrt(Income^(2))
sigma<-cbind(sigy1,sigy2)
pikas <- cbind(rep(400/N, N), rep(400/N, N))

Piks<-PikHol(n,sigma,0.03, pikas)

n=round(sum(Piks))
n

res<-S.piPS(n,Piks)
sam <- res[,1]

data <- Lucy[sam,]
attach(data)
names(data)

Pik.s <- res[,2]
estima <- data.frame(Income, Employees, Taxes)
E.piPS(estima,Pik.s)