Est.Corr.NHT {samplingVarEst}

Estimator of a correlation coefficient using the Narain-Horvitz-Thompson point estimator

Description

Estimates a population correlation coefficient of two variables using the Narain (1951); Horvitz-Thompson (1952) point estimator.

Usage

Est.Corr.NHT(VecY.s, VecX.s, VecPk.s, N)

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 that 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 that 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.

N

the population size. It must be an integer or a double-precision scalar with zero-valued fractional part.

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 (implemented by the current function) is given by:

\hat{C} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})(x_k - \hat{\bar{x}}_{NHT})}{\sqrt{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{NHT})^2}\sqrt{\sum_{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} \sum_{k\in U} y_k,

\hat{\bar{y}}_{NHT} = \frac{1}{N}\sum_{k\in s} w_k y_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

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.

See Also

Est.Corr.Hajek
VE.Jk.Tukey.Corr.NHT

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
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)