Est.Corr.Hajek | R Documentation |
Estimates a population correlation coefficient of two variables using the Hajek (1971) point estimator.
Est.Corr.Hajek(VecY.s, VecX.s, VecPk.s)
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 that of |
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 that of |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to n, the sample size. Values in |
For the population correlation coefficient of two variables y and x:
C = \frac{∑_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{√{∑_{k\in U} (y_k - \bar{y})^2}√{∑_{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{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{√{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})^2}√{∑_{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} ∑_{k\in U} y_k,
\hat{\bar{y}}_{Hajek} = \frac{∑_{k\in s} w_k y_k}{∑_{k\in s} w_k}
and w_k=1/π_k with π_k denoting the inclusion probability of the k-th element in the sample s.
The function returns a value for the correlation coefficient point estimator.
Emilio Lopez Escobar.
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.
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
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])
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