samplingVarEst (version 1.4)

Est.Corr.Hajek: Estimator of a correlation coefficient using the Hajek point estimator

Description

Estimates a population correlation coefficient of two variables using the Hajek (1971) point estimator.

Usage

Est.Corr.Hajek(VecY.s, VecX.s, VecPk.s)

Value

The function returns a value for the correlation coefficient point estimator.

Arguments

VecY.s

vector of the variable of interest Y; its length is equal to \(n\), the 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 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 first-order inclusion probabilities; its length is equal to \(n\), the sample size. Values in VecPk.s must be greater than zero and less than or equal to one. There must not be missing values.

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) (implemented by the current function), 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 \(s\).

References

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.

See Also

Est.Corr.NHT
VE.Jk.Tukey.Corr.Hajek
VE.Jk.CBS.HT.Corr.Hajek
VE.Jk.CBS.SYG.Corr.Hajek
VE.Jk.B.Corr.Hajek
VE.Jk.EB.SW2.Corr.Hajek

Examples

data(oaxaca)                                #Loads the Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00) #Reconstructs the 1st order incl. probs.
s     <- oaxaca$sHOMES00                    #Defines the sample to be used
y1    <- oaxaca$POP10                       #Defines the variable of interest y1
y2    <- oaxaca$POPMAL10                    #Defines the variable of interest y2
x     <- oaxaca$HOMES10                     #Defines the variable of interest x
#Computes the correlation coefficient estimator for y1 and x
Est.Corr.Hajek(y1[s==1], x[s==1], pik.U[s==1])
#Computes the correlation coefficient estimator for y2 and x
Est.Corr.Hajek(y2[s==1], x[s==1], pik.U[s==1])