Optimal Inclusion Probabilities Under Multi-purpose Sampling
Computes the population vector of optimal inclusion probabilities under the Holmbergs's Approach
PikHol(n, sigma, e, Pi)
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. |
Assuming that all of the characteristic of interest are equally important, the Holmberg's sampling design yields the following inclusion probabilities
π_{(opt)k}=\frac{n^*√{a_{qk}}}{∑_{k\in U}√{a_{qk}}}
where
n^*≥q \frac{(∑_{k\in U}√{a_{qk}})^2}{(1+c)Q+∑_{k\in U}a_{qk}}
and
a_{qk}= ∑_{q=1}^Q \frac{σ^2_{qk}}{∑_{k\in U}≤ft( \frac{1}{π_{qk}}-1\right)σ^2_{qk}}
Note that σ^2_{qk} is a size measure associated with the k-th element in the q-th characteristic of interest.
The function returns a vector of inclusion probabilities.
Hugo Andres Gutierrez Rojas hagutierrezro@gmail.com
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
####################### #### 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)
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