Est.Corr.NHT | R Documentation |
Estimates a population correlation coefficient of two variables using the Narain (1951); Horvitz-Thompson (1952) point estimator.
Est.Corr.NHT(VecY.s, VecX.s, VecPk.s, N)
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 |
N |
the population size. It must be an integer or a double-precision scalar with zero-valued fractional part. |
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 (implemented by the current function) is given by:
\hat{C} = \frac{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})(x_k - \hat{\bar{x}}_{NHT})}{√{∑_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})^2}√{∑_{k\in s} w_k (x_k - \hat{\bar{x}}_{NHT})^2}}
where \hat{\bar{y}}_{NHT} is the Narain (1951); Horvitz-Thompson (1952) estimator for the population mean \bar{y} = N^{-1} ∑_{k\in U} y_k,
\hat{\bar{y}}_{NHT} = \frac{1}{N}∑_{k\in s} w_k y_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.
Horvitz, D. G. and Thompson, D. J. (1952) A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663–685.
Narain, R. D. (1951) On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169–175.
Est.Corr.Hajek
VE.Jk.Tukey.Corr.NHT
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 N <- dim(oaxaca)[1] #Defines the population size 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.NHT(y1[s==1], x[s==1], pik.U[s==1], N) #Computes the correlation coefficient estimator for y2 and x Est.Corr.NHT(y2[s==1], x[s==1], pik.U[s==1], N)
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