Package 'lancor'

Lancaster correlation

Description

Computes the Lancaster correlation coefficient.

Usage

lcor(x, y = NULL, type = c("rank", "linear"))

Arguments

x	a numeric vector, or a matrix or data frame with two columns.
y	NULL (default) or a vector with same length as x.
type	a character string indicating which lancaster correlation is to be computed. One of "rank" (default), or "linear": can be abbreviated.

Details

Let $F_X$ and $F_Y$ be the distribution functions of $X$ and $Y$, and define

$X^* = \Phi^{-1}(F_X(X)), \quad Y^* = \Phi^{-1}(F_Y(Y)),$

where $\Phi^{-1}$ is the standard normal quantile function. Furthermore for $X$ and $Y$ with finite fourth moment, let

$\tilde{X} = (X - \mathbb{E}(X)) / \operatorname{sd}(X), \quad \tilde{Y} = (Y - \mathbb{E}(Y)) / \operatorname{sd}(Y).$

Then

$\rho_L(X,Y) = \max\{|\operatorname{Cor}_{\text{Pearson}}(X^*,Y^*)|,\; | \operatorname{Cor}_{\text{Pearson}}((X^*)^2,(Y^*)^2)|\}$

and

$\rho_{L,1}(X,Y) = \max\{|\operatorname{Cor}_{\text{Pearson}}(X,Y)|,\; | \operatorname{Cor}_{\text{Pearson}}((\tilde{X})^2,(\tilde{Y})^2)|\}$

are called the Lancaster correlation coefficient and the linear Lancaster correlation coefficient, respectively. Two estimation methods are supported:

• Linear estimator for $\bold{\rho_{L,1}}$ (type = "linear"): Consider $\rho_{L1} = \operatorname{Cor}_{\text{Pearson}}(X,Y)$ and $\rho_{L2} = \operatorname{Cor}_{\text{Pearson}}((\tilde{X})^2,(\tilde{Y})^2)$. Let $\hat\rho_{L1}$ be the sample Pearson correlation and $\hat\rho_{L2}$ the empirical correlation of the squares of the empirically standardized observations, and set $\hat\rho_{L,1} = \max\{\,|\hat\rho_{L1}|,\;|\hat\rho_{L2}|\,\}$.

• Rank-based estimator for $\bold{\rho_{L}}$ (type = "rank"): Consider $\rho_{R1} = \operatorname{Cor}_{\text{Pearson}}(X^*,Y^*)$ and $\rho_{R2} = \operatorname{Cor}_{\text{Pearson}}((X^*)^2,(Y^*)^2)$. Let $Q_i$ and $R_i$ be the ranks of $X_i$ and $Y_i$, within $X_1,...,X_n$ or $Y_1,...,Y_n$ respectively. Define

  $\hat\rho_{R1} = \frac{1}{n\,s_a^2}\sum_{j=1}^n a(Q_j)\,a(R_j),$

  $\hat\rho_{R2} = \frac{1}{n\,s_b^2}\sum_{j=1}^n \bigl(b(Q_j)-\bar b\bigr)\,\bigl(b(R_j)-\bar b\bigr),$

  where the scores are, for $j=1,...,n$,

  $a(j) = \Phi^{-1}\!\Bigl(\frac{j}{n+1}\Bigr), \quad b(j)=a(j)^2,$

  $\bar b=\frac{1}{n}\sum_{j=1}^n b(j), \quad s_a^2 = \frac{1}{n}\sum_{j=1}^n\bigl(a(j)-\bar a\bigr)^2, \quad s_b^2 = \frac{1}{n}\sum_{j=1}^n\bigl(b(j)-\bar b\bigr)^2.$

  Finally, the rank‐based Lancaster correlation is

  $\hat\rho_{L} = \max\bigl\{\,|\hat\rho_{R1}|, |\hat\rho_{R2}|\bigr\}.$

Value

the sample Lancaster correlation.

Author(s)

Hajo Holzmann, Bernhard Klar

References

Holzmann, Klar (2024). "Lancester correlation - a new dependence measure linked to maximum correlation". doi:10.1111/sjos.12733

See Also

lcor.comp, lcor.ci, lcor.test

Examples

Sigma <- matrix(c(1,0.1,0.1,1), ncol=2)
R <- chol(Sigma)
n <- 1000
x <- matrix(rnorm(n*2), n)
lcor(x, type = "rank")
lcor(x, type = "linear")

x <- matrix(rnorm(n*2), n)
nu <- 2
y <- x / sqrt(rchisq(n, nu)/nu)
cor(y[,1], y[,2], method = "spearman")
lcor(y, type = "rank")

---

Confidence intervals for the Lancaster correlation coefficient

Description

Computes confidence intervals for the Lancaster correlation coefficient. Lancaster correlation is a bivariate measures of dependence.

Usage

lcor.ci(
  x,
  y = NULL,
  conf.level = 0.95,
  type = c("rank", "linear"),
  con = TRUE,
  R = 1000,
  method = c("plugin", "boot", "pretest")
)

Arguments

x	a numeric vector, or a matrix or data frame with two columns.
y	NULL (default) or a vector with same length as x.
conf.level	confidence level of the interval.
type	a character string indicating which lancaster correlation is to be computed. One of "rank" (default), or "linear": can be abbreviated.
con	logical; if TRUE (default), conservative asymptotic confidence intervals are computed.
R	number of bootstrap replications.
method	a character string indicating how the asymptotic covariance matrix is computed if type ="linear". One of "plugin" (default), "boot" or "symmetric": can be abbreviated.

Details

Computes asymptotic and bootstrap-based confidence intervals for the (linear) Lancaster correlation coefficient $\rho_L$ ($\rho_{L,1}$). For more details see lcor.

Asymptotic confidence intervals are derived under two cases (analogously for $\rho_{L}$; see Holzmann and Klar (2024)):

Case 1: If $|\rho_{L1}|\neq|\rho_{L2}|$, the $1-\alpha$ asymptotic interval is

$\left[ \max\{\hat\rho_{L,1} - z_{1-\alpha/2}\,s/\sqrt{n}, 0\},\ \min\{\hat\rho_{L,1} + z_{1-\alpha/2}\,s/\sqrt{n}, 1\} \right],$

where $z_{1-\alpha/2}$ is the standard normal quantile and $s$ is an estimator of the corresponding standard deviation.

Case 2: If $|\rho_{L1}|=|\rho_{L2}|=a>0$, let $\tau$ denote the correlation between the two components and let $q_{1-\alpha/2}$ be the $1-\alpha/2$ quantile of the asymptotic distribution of $\sqrt{n}(\hat\rho_{L,1} - a)$. A conservative asymptotic interval is

$\left[ \max\{\hat\rho_{L,1} - q_{1-\alpha/2}/\sqrt{n}, 0\},\ \min\{\hat\rho_{L,1} + z_{1-\alpha/2}\,s/\sqrt{n}, 1\} \right].$

Additionally, bootstrap-based intervals can be obtained by resampling and estimating the covariance matrix of the rank or linear correlation components.

Value

a vector containing the lower and upper limits of the confidence interval.

Author(s)

Hajo Holzmann, Bernhard Klar

References

Holzmann, Klar (2024). "Lancester correlation - a new dependence measure linked to maximum correlation". doi:10.1111/sjos.12733

See Also

lcor, lcor.comp, lcor.test

Examples

n <- 1000
x <- matrix(rnorm(n*2), n)
nu <- 2
y <- x / sqrt(rchisq(n, nu)/nu) # multivariate t
lcor(y, type = "rank")
lcor.ci(y, type = "rank")

---

Lancaster correlation and its components

Description

Computes the Lancaster correlation coefficient and its components.

Usage

lcor.comp(x, y = NULL, type = c("rank", "linear"), plot = FALSE)

Arguments

x	a numeric vector, or a matrix or data frame with two columns.
y	NULL (default) or a vector with same length as x.
type	a character string indicating which lancaster correlation is to be computed. One of "rank" (default), or "linear": can be abbreviated.
plot	logical; if TRUE, scatterplots of the transformed x and y values and of their squares are drawn.

Details

For more details see lcor.

Value

a vector containing the two components rho1 and rho2 and the sample Lancaster correlation.

Author(s)

Hajo Holzmann, Bernhard Klar

References

Holzmann, Klar (2024). "Lancester correlation - a new dependence measure linked to maximum correlation". doi:10.1111/sjos.12733

See Also

lcor, lcor.comp, lcor.test

Examples

Sigma <- matrix(c(1,0.1,0.1,1), ncol=2)
R <- chol(Sigma)
n <- 1000
x <- matrix(rnorm(n*2), n)
nu <- 8
y <- x / sqrt(rchisq(n, nu)/nu) #multivariate t
cor(y[,1], y[,2])
lcor.comp(y, type = "linear")

x <- matrix(rnorm(n*2), n)
nu <- 2
y <- x / sqrt(rchisq(n, nu)/nu) #multivariate t
cor(y[,1], y[,2], method = "spearman")
lcor.comp(y, type = "rank", plot = TRUE)

---

Lancaster correlation test

Description

Lancaster correlation test of bivariate independence. Lancaster correlation is a bivariate measures of dependence.

Usage

lcor.test(
  x,
  y = NULL,
  type = c("rank", "linear"),
  nperm = 999,
  method = c("permutation", "asymptotic", "symmetric")
)

Arguments

x	a numeric vector, or a matrix or data frame with two columns.
y	NULL (default) or a vector with same length as x
type	a character string indicating which lancaster correlation is to be computed. One of "rank" (default), or "linear": can be abbreviated.
nperm	number of permutations.
method	a character string indicating how the p-value is computed if type ="linear". One of "permutation" (default), "asymptotic" or "symmetric": can be abbreviated.

Details

For more details on the testing procedure see $Remark \, 2$ in Holzmann, Klar (2024).

Value

A list containing the following components:

lcor	the value of the test statistic
pval	the p-value of the test

Author(s)

Hajo Holzmann, Bernhard Klar

References

Holzmann, Klar (2024). "Lancester correlation - a new dependence measure linked to maximum correlation". doi:10.1111/sjos.12733

See Also

lcor, lcor.comp, lcor.ci and for for performing an ACE permutation test of independence see acepack (https://cran.r-project.org/package=acepack).

Examples

n <- 200
x <- matrix(rnorm(n*2), n)
nu <- 2
y <- x / sqrt(rchisq(n, nu)/nu)
cor.test(y[,1], y[,2], method = "spearman")
lcor.test(y, type = "rank")

---

Covariance matrix of components of Lancaster correlation coefficient

Description

Estimate of covariance matrix of the two components of Lancaster correlation. Lancaster correlation is a bivariate measures of dependence.

Usage

Sigma.est(xx)

Arguments

xx	a matrix or data frame with two columns.

Details

For more details see the Appendix in Holzmann, Klar (2024).

Value

the estimated covariance matrix.

Author(s)

Hajo Holzmann, Bernhard Klar

References

Holzmann, Klar (2024). "Lancester correlation - a new dependence measure linked to maximum correlation". doi:10.1111/sjos.12733

See Also

lcor.ci

Examples

Sigma <- matrix(c(1,0.1,0.1,1), ncol=2)
R <- chol(Sigma)
n <- 1000
x <- matrix(rnorm(n*2), n)
nu <- 8
y <- x / sqrt(rchisq(n, nu)/nu) #multivariate t
Sigma.est(y)

---