Help for package lancor

Type:

Package

Title:

Statistical Inference via Lancaster Correlation

Version:

0.1.3

Maintainer:

Bernhard Klar <bernhard.klar@kit.edu>

Description:

Implementation of the methods described in Holzmann, Klar (2024) <doi:10.1111/sjos.12733>. Lancaster correlation is a correlation coefficient which equals the absolute value of the Pearson correlation for the bivariate normal distribution, and is equal to or slightly less than the maximum correlation coefficient for a variety of bivariate distributions. Rank and moment-based estimators and corresponding confidence intervals are implemented, as well as independence tests based on these statistics.

Imports:

arrangements, boot, graphics, sn, stats

License:

GPL-2

Encoding:

UTF-8

RoxygenNote:

7.3.2

Suggests:

testthat (≥ 3.0.0)

Config/testthat/edition:

NeedsCompilation:

Packaged:

2025-08-22 07:21:55 UTC; Klar

Author:

Bernhard Klar

[aut, cre], Hajo Holzmann [aut], Lucas Iglesias [ctb]

Repository:

CRAN

Date/Publication:

2025-08-22 07:50:10 UTC

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

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)

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

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

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

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

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

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Lancaster correlation

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Confidence intervals for the Lancaster correlation coefficient

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Lancaster correlation and its components

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Lancaster correlation test

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples