Est.Corr.Hajek {samplingVarEst} | R Documentation |
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)
Arguments
VecY.s |
vector of the variable of interest Y; its length is equal to |
VecX.s |
vector of the variable of interest X; its length is equal to |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to |
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
.
Value
The function returns a value for the correlation coefficient point estimator.
Author(s)
Emilio Lopez Escobar.
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])