el.cor.test: EL and EEL test for a correlation coefficient
In corrfuns: Correlation Coefficient Related Functions
EL and EEL test for a correlation coefficient R Documentation
EL and EEL test for a correlation coefficient
Description
EL and EEL test for a correlation coefficient.
Usage
el.cor.test(y, x, rho, tol = 1e-07)
el.cor.test(y, x, rho, tol = 1e-07)
Arguments
y
A numerical vector.
x
A numerical vector.
rho
The hypothesized value of the true partial correlation.
tol
The tolerance vlaue to terminate the Newton-Raphson algorithm.
Details
The empirical likelihood (EL) or the exponential empirical likelihood (EEL) test is performed for the Pearson correlation coefficient. At first we standardise the data so that the sample correlation equal the inner product between the two variables, \hat{r}=\sum_{i=1}^nx_iy_i, where n is the sample size.
The EL works by minimizing the quantity \sum_{i=1}^n\log{nw_i} subject to the constraints \sum_{i=1}^nw_i(x_iy_i-\rho)=0, \sum_{i=1}^nw_i=1 and \rho is the hypothesised correlation coefficient, under the H_0. After some algebra the form of the weights w_i becomes
w_i=\frac{1}{n}\frac{1}{1+\lambda(x_iy_i-\rho)},
where \lambda is the Lagrange multiplier of the first (zero sum) constraint. Thus, the zero sum constraint becomes \sum_{i=1}^n\frac{x_iy_i-\rho}{1 + \lambda(x_iy_i-\rho)}=0 and this equality is solved with respect to \lambda via the Newton-Raphson algortihm. The derivative of this function is -\sum_{i=1}^n\frac{(x_iy_i-\rho)^2}{\left[1 + \lambda(x_iy_i-\rho)\right]^2}=0.
The EL works by minimizing the quantity \sum_{i=1}^nw_i\log{nw_i} subject to the same constraints as before, \sum_{i=1}^nw_i(x_iy_i-\rho)=0 or \sum_{i=1}^nw_i(x_iy_i)=\rho, \sum_{i=1}^nw_i=1. After some algebra the form of the weights w_i becomes
w_i=\frac{e^{\lambda x_iy_i}}{\sum_{j=1}^ne^{\lambda x_jy_j}},
where, again, \lambda is the Lagrange multiplier of the first (zero sum) constraint. Thus, the zero sum constraint becomes \frac{\sum_{i=1}^nx_iy_ie^{\lambda x_iy_i}}{\sum_{j=1}^ne^{\lambda x_jy_j}}-\rho=0 and this equality is solved with respect to \lambda via the Newton-Raphson algortihm. The derivative of this function is
\frac{\sum_{i=1}^n(x_iy_i)^2e^{\lambda x_iy_i} * \sum_{i=1}^ne^{\lambda x_iy_i} - \left(\sum_{i=1}^nx_iy_ie^{\lambda x_iy_i}\right)^2}{\left(\sum_{j=1}^ne^{\lambda x_jy_j}\right)^2}.
Value
A list including:
iters
The number of iterations required by the Newton-Raphson. If no convergence occured this is NULL.
info
A vector with three values, the value of \lambda, the test statistic and its associated asymptotic p-value.
If no convergence occured, the value of the \lambda is NA, the value of test statistic is 10^5
and the p-value is 0. No convergence can be interpreted as rejection of the hypothesis test.
p
The probabilities of the EL or of the EEL. If no covnergence occured this is NULL.
Author(s)
Michail Tsagris
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Efron B. (1981) Nonparametric standard errors and confidence intervals. Canadian Journal of
Statistics, 9(2): 139–158.
Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.
See Also
perm.elcortest, correl, permcor
Examples
el.cor.test( iris[, 1], iris[, 2], 0 )$info
eel.cor.test( iris[, 1], iris[, 2], 0 )$info
corrfuns documentation built on April 3, 2025, 7:27 p.m.
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| EL and EEL test for a correlation coefficient | R Documentation |
EL and EEL test for a correlation coefficient
Description
EL and EEL test for a correlation coefficient.
Usage
el.cor.test(y, x, rho, tol = 1e-07)
el.cor.test(y, x, rho, tol = 1e-07)
Arguments
y |
A numerical vector. |
x |
A numerical vector. |
rho |
The hypothesized value of the true partial correlation. |
tol |
The tolerance vlaue to terminate the Newton-Raphson algorithm. |
Details
The empirical likelihood (EL) or the exponential empirical likelihood (EEL) test is performed for the Pearson correlation coefficient. At first we standardise the data so that the sample correlation equal the inner product between the two variables, \hat{r}=\sum_{i=1}^nx_iy_i, where n is the sample size.
The EL works by minimizing the quantity \sum_{i=1}^n\log{nw_i} subject to the constraints \sum_{i=1}^nw_i(x_iy_i-\rho)=0, \sum_{i=1}^nw_i=1 and \rho is the hypothesised correlation coefficient, under the H_0. After some algebra the form of the weights w_i becomes
w_i=\frac{1}{n}\frac{1}{1+\lambda(x_iy_i-\rho)},
where \lambda is the Lagrange multiplier of the first (zero sum) constraint. Thus, the zero sum constraint becomes \sum_{i=1}^n\frac{x_iy_i-\rho}{1 + \lambda(x_iy_i-\rho)}=0 and this equality is solved with respect to \lambda via the Newton-Raphson algortihm. The derivative of this function is -\sum_{i=1}^n\frac{(x_iy_i-\rho)^2}{\left[1 + \lambda(x_iy_i-\rho)\right]^2}=0.
The EL works by minimizing the quantity \sum_{i=1}^nw_i\log{nw_i} subject to the same constraints as before, \sum_{i=1}^nw_i(x_iy_i-\rho)=0 or \sum_{i=1}^nw_i(x_iy_i)=\rho, \sum_{i=1}^nw_i=1. After some algebra the form of the weights w_i becomes
w_i=\frac{e^{\lambda x_iy_i}}{\sum_{j=1}^ne^{\lambda x_jy_j}},
where, again, \lambda is the Lagrange multiplier of the first (zero sum) constraint. Thus, the zero sum constraint becomes \frac{\sum_{i=1}^nx_iy_ie^{\lambda x_iy_i}}{\sum_{j=1}^ne^{\lambda x_jy_j}}-\rho=0 and this equality is solved with respect to \lambda via the Newton-Raphson algortihm. The derivative of this function is
\frac{\sum_{i=1}^n(x_iy_i)^2e^{\lambda x_iy_i} * \sum_{i=1}^ne^{\lambda x_iy_i} - \left(\sum_{i=1}^nx_iy_ie^{\lambda x_iy_i}\right)^2}{\left(\sum_{j=1}^ne^{\lambda x_jy_j}\right)^2}.
Value
A list including:
iters |
The number of iterations required by the Newton-Raphson. If no convergence occured this is NULL. |
info |
A vector with three values, the value of |
p |
The probabilities of the EL or of the EEL. If no covnergence occured this is NULL. |
Author(s)
Michail Tsagris
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Efron B. (1981) Nonparametric standard errors and confidence intervals. Canadian Journal of Statistics, 9(2): 139–158.
Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.
See Also
perm.elcortest, correl, permcor
Examples
el.cor.test( iris[, 1], iris[, 2], 0 )$info
eel.cor.test( iris[, 1], iris[, 2], 0 )$info
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