VE.Jk.EB.SW2.Corr.Hajek: The self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the estimator of a correlation coefficient using the Hajek point estimator

Description

Computes the self-weighted two-stage sampling Escobar-Berger (2013) jackknife variance estimator for the estimator of a correlation coefficient of two variables using the Hajek (1971) point estimator.

Usage

VE.Jk.EB.SW2.Corr.Hajek(VecY.s, VecX.s, VecPk.s, nII, VecPi.s,
                         VecCluLab.s, VecCluSize.s)

Value

The function returns a value for the estimated variance.

Arguments

VecY.s: vector of the variable of interest Y; its length is equal to $n$, the total sample size. Its length has to be the same as the length of VecPk.s and VecX.s. There must not be missing values.
VecX.s: vector of the variable of interest X; its length is equal to $n$, the total sample size. Its length has to be the same as the length of VecPk.s and VecY.s. There must not be missing values.
VecPk.s: vector of the elements' first-order inclusion probabilities; its length is equal to $n$, the total sample size. Values in VecPk.s must be greater than zero and less than or equal to one. There must not be missing values.
nII: the second stage sample size, i.e. the fixed number of ultimate sampling units that were selected within each cluster. Its size must be less than or equal to the minimum cluster size in the sample.
VecPi.s: vector of the clusters' first-order inclusion probabilities; its length is equal to $n$, the total sample size. Hence values are expected to be repeated in the utilised sample dataset. Values in VecPi.s must be greater than zero and less than or equal to one. There must not be missing values.
VecCluLab.s: vector of the clusters' labels for the elements; its length is equal to $n$, the total sample size. The labels must be integer numbers.
VecCluSize.s: vector of the clusters' sizes; its length is equal to $n$, the total sample size. Hence values are expected to be repeated in the utilised sample dataset. None of the sizes must be smaller than nII.

Author

Emilio Lopez Escobar.

Details

For the population correlation coefficient of two variables $y$ and $x$: $$C = \frac{\sum_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{\sqrt{\sum_{k\in U} (y_k - \bar{y})^2}\sqrt{\sum_{k\in U} (x_k - \bar{x})^2}}$$ the point estimator of $C$, assuming that $N$ is unknown (see Sarndal et al., 1992, Sec. 5.9), is: $$\hat{C}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sqrt{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})^2}\sqrt{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}}$$ where $\hat{\bar{y}}_{Hajek}$ is the Hajek (1971) point estimator of the population mean $\bar{y} = N^{-1} \sum_{k\in U} y_k$, $$\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}$$ and $w_k=1/\pi_k$ with $\pi_k$ denoting the inclusion probability of the $k$-th element in the sample $s$. If $s$ is a self-weighted two-stage sample, the variance of $\hat{C}_{Hajek}$ can be estimated by the Escobar-Berger (2013) jackknife variance estimator (implemented by the current function): $$\hat{V}(\hat{C}_{Hajek}) = v_{clu} + v_{obs}$$ $$v_{clu} = \sum_{i\in s} (1-\pi_{Ii}^{*}) \varsigma_{(Ii)}^{2} - \frac{1}{\hat{d}}\left(\sum_{i\in s} (1-\pi_{Ii}) \varsigma_{(Ii)}\right)^{2}$$ $$v_{obs} = \sum_{k\in s} \phi_k \varepsilon_{(k)}^{2}$$ where $\hat{d}={\sum}_{i\in s}{(1-\pi_{Ii})}$, $\phi_k = I\{k\in s_{i}\}\pi_{Ii}^{*}(M_{i}-n_{II})/(M_{i}-1)$, $\pi_{Ii}^{*} = \pi_{Ii}n_{II}(M_{i}-1)/(n_{II}-1)M_{i}$, with $s_{i}$ denoting the sample elements from the $i$-th cluster, $I\{k\in s_{i}\}$ is an indicator that takes the value $1$ if the $k$-th observation is within the $i$-th cluster and $0$ otherwise, $\pi_{Ii}$ is the inclusion probability of the $i$-th cluster in the sample $s$, $M_{i}$ is the size of the $i$-th cluster, $n_{II}$ is the sample size within each cluster, $n_{I}$ is the number of sampled clusters, and where $$\varsigma_{(Ii)}=\frac{n_{I}-1}{n_{I}} (\hat{C}_{Hajek}-\hat{C}_{Hajek(Ii)})$$ $$\varepsilon_{(k)}=\frac{n-1}{n} (\hat{C}_{Hajek}-\hat{C}_{Hajek(k)})$$ where $\hat{C}_{Hajek(Ii)}$ and $\hat{C}_{Hajek(k)}$ have the same functional form as $\hat{C}_{Hajek}$ but omitting the $i$-th cluster and the $k$-th element, respectively, from the sample $s$. Note that this variance estimator utilises implicitly the Hajek (1964) approximations that are designed for large-entropy sampling designs, large samples and large populations, i.e. care should be taken with highly-stratified samples, e.g. Berger (2005).

References

Berger, Y. G. (2005) Variance estimation with highly stratified sampling designs with unequal probabilities. Australian & New Zealand Journal of Statistics, 47, 365--373.

Escobar, E. L. and Berger, Y. G. (2013) A jackknife variance estimator for self-weighted two-stage samples. Statistica Sinica, 23, 595--613.

Hajek, J. (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. The Annals of Mathematical Statistics, 35, 4, 1491--1523.

Hajek, J. (1971) Comment on An essay on the logical foundations of survey sampling by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), p. 236. Holt, Rinehart and Winston.

Sarndal, C.-E. and Swensson, B. and Wretman, J. (1992) Model Assisted Survey Sampling. Springer-Verlag, Inc.

Examples

Run this code

data(oaxaca)                          #Loads the Oaxaca municipalities dataset
s         <- oaxaca$sSW_10_3          #Defines the sample to be used
SampData  <- oaxaca[s==1, ]           #Defines the sample dataset
nII       <- 3                        #Defines the 2nd stage fixed sample size
CluLab.s  <- SampData$IDDISTRI        #Defines the clusters' labels
CluSize.s <- SampData$SIZEDIST        #Defines the clusters' sizes
piIi.s    <- (10 * CluSize.s / 570)   #Reconstructs clusters' 1st order incl. probs.
pik.s     <- piIi.s * (nII/CluSize.s) #Reconstructs elements' 1st order incl. probs.
y1.s      <- SampData$POP10           #Defines the variable y1
y2.s      <- SampData$POPMAL10        #Defines the variable y2
x.s       <- SampData$HOMES10         #Defines the variable x
#Computes the var. est. of the corr. coeff. point estimator using y1
VE.Jk.EB.SW2.Corr.Hajek(y1.s, x.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s)
#Computes the var. est. of the corr. coeff. point estimator using y2
VE.Jk.EB.SW2.Corr.Hajek(y2.s, x.s, pik.s, nII, piIi.s, CluLab.s, CluSize.s)

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