Open Access Published by De Gruyter Open Access October 20, 2023

# Test of bivariate independence based on angular probability integral transform with emphasis on circular-circular and circular-linear data

• Juan José Fernández-Durán and María Mercedes Gregorio-Domínguez
From the journal Dependence Modeling

## Abstract

The probability integral transform of a continuous random variable X with distribution function F X is a uniformly distributed random variable U = F X ( X ) . We define the angular probability integral transform (APIT) as θ U = 2 π U = 2 π F X ( X ) , which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum modulus 2 π of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation modulus 2 π , and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the APITs of two random variables, X 1 and X 2 , and test for the circular uniformity of their sum (difference) modulus 2 π , this is equivalent to test of independence of the original variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the family, to evaluate the power of the proposed independence test by generating samples from NNTS alternative distributions that could be at a closer proximity with respect to the circular uniform null distribution.

MSC 2010: 62H11; 62H05

## 1 Introduction

Testing for independence is one of the most important tasks in statistics, for example, when constructing the joint distribution of a set of random variables or considering the conditional dependence of one variable in terms of other variables as in regression models. According to Herwatz and Maxand [23], one can consider the following tests of independence: bivariate (pairwise), mutual, and groupwise independence tests. Hereinafter, we only consider absolutely continuous random variables. A random variable with a density function with support on an interval of the real line is a linear random variable, and one with support on the unit circle is a circular random variable. The distribution function of a circular random is a periodic function, and then, it has an arbitrary starting direction. In our case, we considered that all the circular random variables are defined on the interval ( 0 , 2 π ] with common starting direction equal to zero and are angles in radians measured all counterclockwise or clockwise. In this sense, if Θ is a circular random variable with starting direction zero and measured in radians counterclockwise,

F Θ ( θ ) = P ( 0 < Θ θ ) for 0 θ 2 π

and

F Θ ( θ + 2 k π ) = F Θ ( θ ) ,

for k an integer number. The density function for an absolutely continuous circular random variable satisfies being a periodic function such that f Θ ( θ ) 0 , f Θ ( θ + 2 k π ) = f Θ ( θ ) , and 0 2 π f Θ ( θ ) d θ = 1 . For two random variables, X 1 and X 2 , bivariate (pairwise) independence tests have null hypothesis, H 0 : F X 1 , X 2 ( x 1 , x 2 ) = F X 1 ( x 1 ) F X 2 ( x 2 ) , where F X 1 , X 2 ( x 1 , x 2 ) = P { X 1 x 1 , X 2 x 2 } is the bivariate joint distribution function and F X 1 ( x 1 ) = P { X 1 x 1 } and F X 2 ( x 2 ) = P { X 2 x 2 } are the corresponding marginal distribution functions. A mutual independence test for a set of random variables, X 1 , X 2 , , X d considers the null hypothesis H 0 : F X 1 , X 2 , , X d ( x 1 , x 2 , , x d ) = k = 1 d F X k ( x k ) , where F X 1 , X 2 , , X d ( x 1 , x 2 , , x d ) is the joint distribution and F X k ( x k ) for k = 1 , 2 , , d are the marginal univariate distribution functions. Finally, groupwise independence tests consider the independence between disjoint subsets of random variables. If the parametric functional form of F X 1 , X 2 , , X d and F X k for k = 1 , 2 , , d are specified, a likelihood ratio test of independence from a sample of random vectors X ̲ i = ( X i 1 , X i 2 , , X i d ) of size n ( i = 1 , 2 , , n ), can be constructed. The most commonly used test for independence is the chi-squared test of independence for contingency tables, which is not adequate when dealing with absolutely continuous random variables. Alternatively to the likelihood ratio independence test, other tests for independence were developed by considering nonparametric (distribution free) methods, rank tests [24,31], and measures of dependence (association) derived from the empirical copula process [6,1921,43,44]. The empirical copula, C n , for a vector of d absolutely continuous linear random variables, X ̲ = ( X 1 , X 2 , , X d ) , and a sample of size n is defined as follows:

(1) C n ( u 1 , u 2 , , u d ) = 1 n j = 1 n I ( F ˆ 1 ( X i 1 ) u 1 , F ˆ 2 ( X i 2 ) u 2 , , F ˆ d ( X i d ) u 2 ) ,

where I ( ) is an indicator function, which is equal to one if the condition in its argument is satisfied and zero otherwise, and F ˆ 1 , F ˆ 2 , , F ˆ d are the empirical distribution functions of the random variables X 1 , X 2 , , X d . A test of independence based on the distance between the empirical copula and independence copula for absolutely continuous linear random variables is implemented in the R package c o p u l a [25,32].

Among the nonparametric tests of independence, there is a family of tests based on some functional of the empirical independence process [5], which is defined as the distance between the empirical joint distribution function and product of the empirical univariate distribution functions. Historically, the most used functionals have been the Cramér-von Mises and Kolmogorov-Smirnov functionals [3,6,7]. For example, Hoeffding [24] considered the Cramér-von Mises functional to generate a rank test of independence between two random variables. Modern rank tests of independence have been developed by Kallenberg and Ledwina [29]. Kernel-based methods have also been used to estimate the empirical independence process, as in the study by Pfister et al. [40]. Mardia and Kent [34] used the general Rao score test to generate independence tests. Csörgö [5] developed independence tests based on the multivariate empirical characteristic function, and Einmahl and McKeague [8] developed the tests based on the empirical likelihood. The measures of dependence derived from entropy were defined by Joe [27] and from mutual information by Berrett and Samworth [2].

When constructing tests of independence, one can take advantage of the characteristics of the multivariate joint distribution. For example, for the multivariate normal distribution, one can test for independence by testing for an identity correlation matrix. Some of these pairwise tests are the Pearson’s [37] product moment correlation coefficient test, Kendall’s [30] rank correlation coefficient test, and Spearman’s [46] rank correlation coefficient test. Of course, for a pair of Gaussian random variables, rejecting null correlation implies rejecting pairwise independence, but applying pairwise independence (correlation) tests is not adequate to test for mutual independence for a set with more than two Gaussian random variables. The Wilks test [49] is an optimal test of independence for multivariate Gaussian populations and for the case of a bivariate groupwise independence test for the vectors X ̲ D 1 and X ̲ D 2 with X ̲ = X ̲ D 1 D 2 = ( X ̲ D 1 , X ̲ D 2 ) considers the following test statistic for a sample of size n ,

(2) W = Σ ˆ D 1 D 2 Σ ˆ D 1 Σ ˆ D 2 ,

where Σ ˆ D 1 D 2 = j = 1 n ( x ̲ j x ̲ ¯ ) ( x ̲ j x ̲ ¯ ) is the estimated covariance matrix of the complete vector of observations x ̲ = x ̲ D 1 D 2 , which is partitioned into Σ ˆ D 1 and Σ ˆ D 2 with Σ ˆ D 1 = j = 1 n ( x ̲ D 1 , j x ̲ ¯ D 1 ) ( x ̲ D 1 , j x ̲ ¯ D 1 ) and Σ ˆ D 2 = j = 1 n ( x ̲ D 2 , j x ̲ ¯ D 2 ) ( x ̲ D 2 , j x ̲ ¯ D 2 ) . The statistic W then measures the extent of the distance between the determinant of Σ ˆ D 1 D 2 and the product of the determinants of Σ ˆ D 2 and Σ ˆ D 2 . The equality relationship is satisfied in the multivariate Gaussian population under the null hypothesis of independence between X ̲ D 1 and X ̲ D 2 . Asymptotically and under regularity conditions, n ln ( W ) follows a chi-squared distribution with D 1 D 2 degrees of freedom, where D 1 and D 2 are the cardinalities of sets D 1 and D 2 , respectively.

For the circular-circular (angular-angular) and circular-linear (angular-linear) cases, in which the objective is to test for bivariate independence between two circular random variables and one circular and one linear random variable, respectively, independence tests were developed by considering the specification of measures of dependence and studying their (asymptotic) distributions. By applying Kendall’s tau and Spearman’s rho general measures of dependence based on the concept of concordance, or the construction of distribution-free correlation coefficients based on ranks to a pair of circular random variables or a circular and a linear random variables, tests of independence were developed by Fisher and Lee [1416] and reviewed by Fisher [17] and Mardia and Jupp [35].

The objective of this study is to develop a test of bivariate (pairwise) independence for two random variables by considering the angular probability transform of each variables, which correspond to circular uniform distributions on ( 0 , 2 π ], and an additional result of the theory of circular statistics ([17,26,35,38,47]). The test was evaluated using flexible nonnegative trigonometric sums (NNTS) distributions [10,11]. Although the proposed test of independence is a general one, it is especially suitable to test for the independence in the circular-circular and circular-linear cases. Thus, a measure of dependence was developed.

This article is divided into six sections, including the introduction. In Section 2, Johnson and Wehrly’s [28] model is presented as a motivation for performing the test of bivariate independence for two random variables, and here, the theory of NNTS circular distributions is included. Section 3 presents the proposed bivariate independence test, a measure of dependence, and its application to simulated data to study the power of the test. The Section 4 includes a simulation study to evaluate the power of the proposed test in the linear-linear, circular-linear, and circular-circular cases. The Section 5 describes the application of the proposed independence test to real datasets. Finally, conclusions are presented in Section 6.

## 2 Bivariate Johnson and Wehrly model and NNTS family of circular densities

Sklar [45] theorem specifies that the joint cumulative distribution function of two continuous random variables, X 1 and X 2 , with corresponding cumulative distribution functions F X 1 ( x 1 ) and F X 2 ( x 2 ) , can be expressed as follows:

(3) F X 1 , X 2 ( x 1 , x 2 ) = C ( F X 1 ( x 1 ) , F X 2 ( x 2 ) ) ,

where C ( u , v ) is the copula (distribution) function that corresponds to a bivariate joint distribution of two identically distributed uniform random variables on the interval [ 0 , 1 ] , U = F X 1 ( X 1 ) and V = F X 2 ( X 2 ) . A linear-linear copula function must be an increasing function satisfying that C ( u , 1 ) = u , C ( 1 , v ) = v , C ( 0 , v ) = C ( u , 0 ) = 0 [36]. By differentiation of equation (3), one obtains the joint density function of X 1 and X 2 , f X 1 , X 2 ( x 1 , x 2 ) , as follows:

(4) f X 1 , X 2 ( x 1 , x 2 ) = c ( F X 1 ( x 1 ) , F X 2 ( x 2 ) ) f X 1 ( x 1 ) f X 2 ( x 2 ) ,

where f X 1 ( x 1 ) and f X 2 ( x 2 ) are the marginal density functions of X 1 and X 2 , respectively. The function c ( u , v ) is the copula density function defined as c ( u , v ) = 2 C ( u , v ) u v . Johnson and Wehrly [28] and Wehrly and Johnson [48] proposed a large family of joint density functions for circular-circular ( Θ 1 and Θ 2 ) and circular-linear ( Θ and T ) random vectors in the following way:

(5) f Θ 1 , Θ 2 ( θ 1 , θ 2 ) = 2 π g ( ( 2 π ( F Θ 1 ( θ 1 ) ± F Θ 2 ( θ 2 ) ) ) ( mod 2 π ) ) f Θ 1 ( θ 1 ) f Θ 2 ( θ 2 )

and

(6) f Θ , T ( θ , t ) = 2 π g ( ( 2 π ( F Θ ( θ ) ± F T ( t ) ) ) ( mod 2 π ) ) f Θ ( θ ) f T ( t ) .

Fernández-Durán [9] identified the structure of the Johnson and Wehrly’s model in terms of the theory of copula functions through Sklar’s [45] theorem [36] satisfying

(7) c ( u , v ) = 2 π g ( ( 2 π ( u ± v ) ) ( mod 2 π ) ) .

The function g must be the density function of an angular (circular) random variable on the interval ( 0 , 2 π ] . For the case of circular-circular and circular-linear bivariate copulas, the function c has also to be periodic in their circular arguments to satisfy the periodicity of the density function of a circular random variable, and this is the reason that in the Johnson and Wehrly model the joining function g is the density of a circular random variable.

Johnson and Wehrly derived bivariate circular–circular and circular–linear models by considering conditional arguments. When function g corresponds to a uniform circular density on the circle, g ( θ ) = 1 2 π , the joint density of the Johnson and Wehrly model corresponds to the independence case in which the joint density of the circular-circular (circular-linear) model is the product of the marginal univariate densities.

This property of the Johnson and Wehrly model motivated our independence test by approximating the circular density function g with a density function from NNTS family of densities [10,11], which includes uniform circular density as a particular case. In addition, by using the NNTS family of circular densities, it is possible to generate joining densities g that are in closer proximity to the circular uniform density as desired; this is explained below. Pewsey and Kato [39] developed a goodness-of-fit test for the circular-circular model of Wehrly and Johnson [48] in equation (5). Their test considers the independence between each circular random variable, Θ 1 and Θ 2 , and the argument of the joining function g , Ω = 2 π { F Θ 2 ( θ 2 ) ± F Θ 1 ( θ 1 ) } ( mod 2 π ) . In a similar way to this article, their independence test is implemented by testing for the toroidal uniformity of the two bivariate random vectors, ( 2 π F Θ 1 ( θ 1 ) , 2 π F Ω ( ω ) ) and ( 2 π F Θ 2 ( θ 2 ) , 2 π F Ω ( ω ) ) .

The circular density function based on NNTS for a circular (angular) random variable Θ ( 0 , 2 π ] [10] is defined as follows:

(8) f Θ ( θ ; M , c ̲ ) = 1 2 π k = 0 M c k e i k θ 2 = 1 2 π k = 0 M l = 0 M c k c ¯ l e i ( k l ) θ ,

where i = 1 , c k are complex numbers c k = c r k + i c c k for k = 0 , , M and c ¯ k = c r k i c c k is the conjugate of c k . To have a valid density function, f Θ ( θ ; M , c ̲ ) , which integrates to one,

(9) k = 0 M c k 2 = 1 ,

where c k 2 = c r k 2 + c c k 2 . The parameter space of the vector of parameters, c ̲ = ( c 0 , c 1 , , c M ) , is a subset of the surface of a hypersphere since c ̲ and c ̲ gives the same NNTS density and the conjugate of c ̲ written in reverse order also gives the same NNTS density as c ̲ . Then, for identifiability of the parameter vector c ̲ and given equation (9), the following constraints are imposed: c c 0 = 0 and c r 0 0 , i.e., c 0 is a nonnegative real number and c 0 2 c M 2 . There is a total of 2 M free parameters c , and M is an additional parameter that determines the total number of terms in the sum defining the density and it is related to the maximum number of modes that the density can have. Note that an NNTS model with M = M 1 is nested on NNTS models with M = M 2 such that M 2 > M 1 . The circular uniform density on ( 0 , 2 π ] corresponds to an NNTS density with M = 0 , f Θ ( θ ; M = 0 , c ̲ ) = 1 2 π , and it is nested in all NNTS models with M > 0 or, for an NNTS model with M > 0 , and as c 0 approaches the value of 1, the NNTS density converges to the circular uniform density or, equivalently, for an NNTS density with c ̲ = ( 1 , 0 , 0 , , 0 ) , that is, with c 0 = 1 and the other elements of c ̲ equal to zero corresponds to the circular uniform density. It is this property that will be used to evaluate the power for the proposed independence test by generating samples from NNTS alternative densities with values of c 0 as close to one as desired. Fernández-Durán and Gregorio-Domínguez [12] developed an efficient algorithm based on optimization on manifolds to obtain the maximum likelihood estimates of the c parameters. This algorithm is included with other routines for the analysis of circular data based on NNTS models in the free R [42] package CircNNTSR [13].

## 3 Proposed test for bivariate independence

For absolutely continuous independent and identically distributed (i.i.d.) circular uniform random variables, U 1 , U 2 , , U d U ( 0 , 2 π ] , consider k = 1 d U k ( mod 2 π ) U ( 0 , 2 π ] , that is, the sum modulus 2 π of i.i.d. circular uniform random variables is also circular and uniformly distributed. This is a consequence of the fact that the characteristic function of a circular uniform random variable, ψ ( t ) = E ( e i t Θ ) , is equal to one when t = 0 and equal to zero elsewhere ( t 0 ). Then, if the sum modulus 2 π of circular uniform random variables is not circular uniformly distributed, this implies that the circular uniform random variables are not independent [35, p. 35]. The proposed test of independence is based on this result by considering the angular probability integral transform (APIT) of arbitrary (linear or circular) absolutely continuous random variables. Let X 1 , X 2 , , X d be d arbitrary absolutely continuous random variables. The angular probability transform of X k is defined as the angular (circular) random variable APIT ( X k ) = 2 π F X k ( X k ) , which is uniformly distributed on the unit circle. By considering the null hypothesis of mutual joint independence, APIT ( X 1 ) , APIT ( X 2 ) , , APIT ( X d ) are i.i.d. U ( 0 , 2 π ] , then k = 1 d ± APIT ( X k ) ( mod 2 π ) U ( 0 , 2 π ] is also circular and uniformly distributed. The proposed test for bivariate independence is based on testing for the circular uniformity of ( APIT ( X 1 ) + APIT ( X 2 ) ) ( mod 2 π ) ( ( APIT ( X 1 ) APIT ( X 2 ) ) ( mod 2 π ) ) for absolutely continuous (circular or linear) random variables X 1 and X 2 . Testing for bivariate independence is equivalent to testing for uniformity of the sum (difference) modulus 2 π of the APITs. To test for circular uniformity of the sum (difference) modulus 2 π of the angular integral transforms, we considered the tests of Rayleigh and Pycke [41]. The Rayleigh test considers an alternative unimodal circular density and has a test statistic for a sample θ 1 , θ 2 , , θ n , which is defined as follows [35]:

(10) T RT = 2 n R ¯ 2 ,

where R ¯ is the sample mean resultant length. The statistic T RT asymptotically has a chi-squared distribution with two degrees of freedom. The p-values of the Rayleigh test used in this article correspond to the approximation given by Fisher [17, p. 70] that includes a correction for small sample sizes. The Pycke test considers an alternative multimodal density, and its test statistic for a sample θ 1 , θ 2 , , θ n is defined as follows:

(11) T P T = 1 n i = 1 n j = 1 n 2 ( cos ( θ i θ j ) 0.5 ) 1.5 ( 2 0.5 cos ( θ i θ j ) ) .

The critical values of the Pycke test are obtained via simulation.

The steps of the proposed independence test for two absolutely continuous random variables, X 1 and X 2 , are as follows. First, the empirical distributions F ˆ X 1 and F ˆ X 2 were calculated from the observed values of each random variable. Second, the APITs were calculated by multiplying the pseudo-observations by 2 π ( 2 π F ˆ X 1 and 2 π F ˆ X 2 ). Third, the sum (difference) modulus 2 π of the two APITs was calculated. Finally, the Rayleigh or Pycke test was applied to the vector of the observed values of the sum (difference) modulus 2 π of the APITs. In the case of a positive association between the random variables, the proposed independence test must be applied to the difference modulus 2 π of the APITs, and in the case of a negative association, it must be applied to the sum modulus 2 π of the APITs. The case in which there is no prior indication regarding whether the association is positive or negative, one can calculate the two test statistics, one for the sum modulus 2 π and other for the difference modulus 2 π of the APITs, and modify the minimum of the two p -values in accordance to some multiple testing correction such as Bonferroni procedure [4,22]. In practice, the user can plot the histogram of the sum (difference) modulus 2 π of APITs and decide on the use of the Rayleigh or Pycke test in terms of the observed number of modes. In case of doubt, the Pycke test should be preferred. Other omnibus tests for circular uniformity can be considered as those of Ajne et al. [35].

Derived from the fitting of an NNTS model with M = 1 to the sum (difference) modulus 2 π of APITs, the following measure of dependence can be defined as follows:

(12) λ ˆ c 0 = M + 1 M ( 1 c ˆ 0 2 ) = 2 ( 1 c ˆ 0 2 ) ,

where the correction term M + 1 M = 2 comes from the fact that the NNTS density, in the case M = 1 , with the highest concentration around zero has a parameter vector c ̲ in which the squared norm of each of its components is equal to 1 M + 1 = 1 2 . This implies that 1 2 c 0 2 1 and λ ˆ c 0 , for M = 1 , takes values in the interval [ 0 , 1 ] with values close to zero, implying low dependence (independence) and values closer to one, further implying high dependence between the considered random variables. For independent random variables, the theoretical value of λ ˆ c 0 is equal to zero. The measure of dependence λ ˆ c 0 is particularly useful in the circular-circular and circular-linear cases.

## 4 Simulation study

In this section, we present a simulation study to compare the power of the proposed test with the Wilks test and a test of independence based on the empirical copula. We simulated the data from different multivariate distributions using known parameters that define the dependence structure and known marginal densities. We considered the sample sizes of 20, 50, 100, and 200. For a given significance level α (10, 5, and 1%), the powers of the tests were obtained by generating 100 samples of the specified sample size from the bivariate density, by calculating the values of the test statistics for each of the 100 samples, and determining the number of times that the test statistics considered a value that rejected the null hypothesis of independence at the given value of α . Thus, the reported powers of the tests considered values in the range of 0–100 and can be interpreted in terms of percentages. The Rayleigh test of circular uniformity was performed using the circular R package [1]. The Pycke test of circular uniformity was performed using the CircMLE R package [18,33]. The R package c o p u l a was used to calculate the empirical copula test, and the measure of dependence λ ˆ c 0 was obtained by fitting an NNTS model with M = 1 using the R package CircNNTSR [13].

### 4.1 Circular-linear models

Table 1 includes the powers of the proposed ART and APT, and the WT and ECT when simulating samples from the circular-linear model of Johnson and Wehrly with the circular marginal density being an NNTS density with M = 3 , which is plotted in the first plot of Figure 1; three different linear marginals (exponential, Gaussian, and Cauchy); and a joining circular density that corresponds to an NNTS density with M = 3 with five different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999) to account for different degrees of association between the circular and linear random variables, as depicted in the last plot of Figure 1, which includes the plots of the angular joining functions for the five different values of parameter c 0 . The case c 0 = 0.9999 corresponds to an almost circular uniform density and to the null hypothesis of independence (refer the last plot of Figure 1). In general, the proposed ART and APT demonstrated significantly larger powers when compared to the ECT, which was only similar for large sample sizes and low values of c 0 (0.7, 0.8, and 0.9), further representing highly dependent circular and linear random variables. The power of the WT was considerably lower when compared to that of the ART, APT, and ECT.

Table 1

NNTS angular joining density circular-linear power study

Marginals SS c 0 λ ˆ c 0 = 2 ( 1 c ˆ 0 2 ) α = 10 % α = 5 % α = 1 %
RT PT WT ECT RT PT WT ECT RT PT WT ECT
Θ NNTS ( M = 3 ) 20 0.7 0.61 91 89 29 43 89 88 22 32 68 64 8 3
X Exp ( 1 ) 20 0.8 0.66 88 85 26 48 83 79 22 39 67 64 6 5
20 0.9 0.62 86 85 39 47 83 80 21 35 58 61 8 11
20 0.99 0.22 34 35 18 11 28 26 10 9 10 7 3 2
20 0.9999 0.15 13 10 12 10 3 5 7 7 1 1 1 2
50 0.7 0.46 100 100 33 79 100 99 20 57 97 97 11 28
50 0.8 0.51 100 100 33 86 100 100 18 61 99 98 6 29
50 0.9 0.5 100 100 43 89 100 100 34 72 99 98 9 48
50 0.99 0.15 71 81 21 40 64 70 13 20 40 41 4 3
50 0.9999 0.04 12 10 7 10 6 6 3 4 1 2 0 2
100 0.7 0.45 100 100 51 100 100 100 39 100 100 100 19 89
100 0.8 0.49 100 100 47 100 100 100 30 99 100 100 16 94
100 0.9 0.47 100 100 72 100 100 100 56 100 100 100 26 90
100 0.99 0.12 90 95 31 55 82 92 22 41 66 80 7 18
100 0.9999 0.02 10 7 11 9 5 4 4 4 1 0 1 2
200 0.7 0.44 100 100 71 100 100 100 64 100 100 100 42 100
200 0.8 0.49 100 100 65 100 100 100 58 100 100 100 31 100
200 0.9 0.45 100 100 86 100 100 100 76 100 100 100 56 100
200 0.99 0.11 100 100 39 97 100 100 31 79 99 100 13 55
200 0.9999 0.01 11 14 10 12 9 7 3 8 3 1 0 0
Θ NNTS ( M = 3 ) 20 0.7 0.62 89 86 13 36 79 80 7 25 55 52 2 4
X N ( 0 , 1 ) 20 0.8 0.63 89 85 17 39 78 76 8 23 53 52 6 6
20 0.9 0.64 94 88 20 48 86 78 8 37 62 57 1 9
20 0.99 0.32 38 30 17 23 28 19 11 19 11 10 1 8
20 0.9999 0.16 13 18 18 14 7 8 10 12 1 2 3 3
50 0.7 0.56 100 100 19 92 100 100 8 77 99 99 1 34
50 0.8 0.59 100 100 12 89 100 100 7 78 99 99 2 33
50 0.9 0.49 100 100 29 85 100 100 18 64 98 99 7 30
50 0.99 0.13 59 63 21 27 52 55 13 11 30 33 5 5
50 0.9999 0.04 14 12 8 10 5 6 4 5 2 2 1 1
100 0.7 0.43 100 100 12 100 100 100 7 100 100 100 2 95
100 0.8 0.5 100 100 14 100 100 100 10 100 100 100 0 97
100 0.9 0.48 100 100 44 100 100 100 29 98 100 100 13 93
100 0.99 0.13 91 97 21 64 86 91 14 36 66 76 4 23
100 0.9999 0.02 10 14 7 13 5 7 4 3 1 1 1 2
200 0.7 0.44 100 100 18 100 100 100 9 100 100 100 4 100
200 0.8 0.48 100 100 16 100 100 100 10 100 100 100 3 100
200 0.9 0.46 100 100 64 100 100 100 56 100 100 100 31 100
200 0.99 0.11 100 100 30 92 100 100 15 81 97 100 8 50
200 0.9999 0.01 11 7 9 4 6 3 8 1 0 0 3 0
Θ NNTS ( M = 3 ) 20 0.7 0.52 81 75 17 28 73 66 13 14 46 46 6 6
X Cauchy ( 0 , 1 ) 20 0.8 0.58 82 84 15 40 78 72 9 22 53 50 1 5
20 0.9 0.6 81 79 22 34 70 71 14 22 56 52 3 9
20 0.99 0.26 34 37 12 21 22 25 8 9 7 10 1 4
20 0.9999 0.15 12 16 12 15 6 10 10 6 2 6 3 0
50 0.7 0.49 100 100 26 84 100 100 17 68 99 99 1 35
50 0.8 0.5 100 100 15 89 100 100 8 80 99 98 0 35
50 0.9 0.48 100 100 13 88 100 100 6 67 99 99 2 32
50 0.99 0.13 62 69 13 28 52 56 8 19 26 26 2 5
50 0.9999 0.05 12 12 15 9 4 5 9 3 1 2 3 1
100 0.7 0.45 100 100 13 100 100 100 10 100 100 100 4 89
100 0.8 0.49 100 100 14 100 100 100 10 99 100 100 3 89
100 0.9 0.47 100 100 12 100 100 100 7 97 100 100 2 83
100 0.99 0.12 97 94 12 52 88 94 10 34 74 84 3 15
100 0.9999 0.02 9 14 7 7 4 5 6 5 0 1 2 0
200 0.7 0.43 100 100 17 100 100 100 11 100 100 100 0 100
200 0.8 0.48 100 100 17 100 100 100 13 100 100 100 1 100
200 0.9 0.46 100 100 15 100 100 100 7 100 100 100 1 100
200 0.99 0.1 100 100 13 92 99 100 5 84 97 100 3 56
200 0.9999 0.01 14 13 10 7 7 5 8 2 2 1 1 1

The powers of the proposed test implemented using the Rayleigh (ART) and Pycke (APT) circular uniformity tests, the Wilks test (WT) and the empirical copula test (ECT) are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a Johnson and Wehrly circular-linear density function constructed from an NNTS angular joining density with M = 3 , an NNTS marginal density function with M = 3 (Figure 1) and, three different linear marginals (exponential, Gaussian, and Cauchy). The NNTS angular joining density is defined with five different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case with c 0 = 1 corresponds to the null independence model.

Figure 1

Circular-circular copula model: The first two plots show the marginal NNTS circular densities ( M 1 = 3 and M 2 = 2 ) and the last third plot show the angular (circular) joining density for different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case c 0 = 1 corresponds to the circular uniform density (null independence model).

### 4.2 Circular-circular models

For Johnson and Wehrly’s circular-circular model, we used the same angular joining density and one of the marginal circular densities as that used in the circular-linear model. Figure 1 depicts the plots of the marginal circular densities that correspond to NNTS densities with M = 3 and M = 2 , and angular joining densities that correspond to NNTS densities with M = 3 for five different values of the parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999) for the circular-circular model of Johnson and Wehrly. Similar to the results in the circular-linear model, the ART and APT demonstrated larger power values when compared to the WT and ECT tests. Given the multimodality of all the circular densities involved, of the two proposed independence tests, APT exhibited a larger power when compared to ART, which was similar only for the largest sample size of 200. Moreover, for highly dependent circular random variables ( c 0 = 0.7 , 0.8, or 0.9) and large sample sizes of 100 or 200, the power of the ECT test was similar to those of the ART and APT. The average values of the dependence measure λ ˆ c 0 listed in the fourth column of Tables 1 and 2 assumed similar values when c 0 = 0.7 , 0.8, and 0.9, but these values were smaller when c 0 = 0.99 and 0.9999, further reflecting the fact that values of c 0 near one are associated with random variables with a very weak association.

Table 2

NNTS angular joining density circular-circular power study

Marginals SS c 0 λ ˆ c 0 = 2 ( 1 c ˆ 0 2 ) α = 10 % α = 5 % α = 1 %
RT PT WT ECT RT PT WT ECT RT PT WT ECT
Θ 1 NNTS ( M 1 = 3 ) 20 0.7 0.63 87 80 16 38 78 73 11 29 59 61 1 11
Θ 2 NNTS ( M 2 = 2 ) 20 0.8 0.6 84 81 13 41 72 74 9 28 57 57 3 10
20 0.9 0.59 81 79 20 36 76 67 15 28 46 49 5 7
20 0.99 0.25 34 33 19 15 26 23 13 12 10 8 7 4
20 0.9999 0.13 11 13 12 10 7 7 7 10 1 0 1 1
50 0.7 0.45 100 100 16 88 99 99 7 63 99 98 1 41
50 0.8 0.5 100 100 15 91 100 100 5 70 98 98 0 38
50 0.9 0.49 100 100 28 92 100 100 14 70 100 100 6 34
50 0.99 0.14 70 73 25 36 55 63 16 17 36 36 7 6
50 0.9999 0.05 17 14 9 12 6 6 4 3 0 1 3 1
100 0.7 0.42 100 100 13 100 100 100 9 100 100 100 3 92
100 0.8 0.47 100 100 17 100 100 100 9 100 100 100 2 93
100 0.9 0.46 100 100 38 100 100 100 28 100 100 100 10 96
100 0.99 0.12 90 95 32 60 83 92 23 41 64 85 10 15
100 0.9999 0.02 9 14 17 8 4 7 7 1 3 1 1 0
200 0.7 0.43 100 100 25 100 100 100 19 100 100 100 6 100
200 0.8 0.48 100 100 18 100 100 100 12 100 100 100 6 100
200 0.9 0.45 100 100 51 100 100 100 45 100 100 100 21 100
200 0.99 0.1 100 100 42 86 100 100 31 80 99 99 15 40
200 0.9999 0.01 8 10 12 9 4 6 6 4 0 0 2 1

The powers of the proposed test implemented using the ART and APT circular uniformity tests, the WT, and the empirical copula test (ECT) are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a Johnson and Wehrly circular-circular density function constructed from an NNTS angular joining density with M = 3 and NNTS marginal density functions with M 1 = 3 and M 2 = 2 . The NNTS angular joining density is defined with five different values of the dependence parameter c 0 (0.7, 0.8, 0.9, 0.99, and 0.9999). The case with c 0 = 1 corresponds to the null independence model. The plots of the NNTS angular joining density and NNTS circular marginal densities are shown in Figure 1.

### 4.3 Linear-linear models

Tables 3 and 4 list the powers of different tests while simulating samples from a bivariate distribution in which both variables are linear. In the first case, a Gaussian copula was used, and in the second case, a Frank copula was used.

Table 3

Gaussian copula linear-linear power study

Marginals SS ρ λ ˆ c 0 = 2 ( 1 c ˆ 0 2 ) α = 10 % α = 5 % α = 1 %
ART APT WT ECT ART APT WT ECT ART APT WT ECT
X 1 Exp ( 1 ) 20 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
X 2 Exp ( 2 ) 20 0.75 0.57 75 69 98 97 68 60 92 92 50 45 83 75
20 0.5 0.25 38 36 64 60 24 25 51 47 9 7 40 24
20 0.25 0.15 17 20 33 29 10 8 24 15 3 4 9 3
20 0 0.14 10 14 12 10 7 9 8 5 0 0 2 3
50 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
50 0.75 0.48 99 98 100 100 99 96 100 100 91 89 100 100
50 0.5 0.14 57 51 93 94 46 38 87 90 23 21 71 85
50 0.25 0.06 18 17 52 42 10 7 42 28 2 1 14 15
50 0 0.05 7 6 11 8 1 3 6 4 1 0 4 0
100 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
100 0.75 0.43 100 100 100 100 100 100 100 100 99 99 100 100
100 0.5 0.11 79 76 97 99 72 61 96 99 50 43 94 98
100 0.25 0.03 32 30 71 69 20 15 58 55 4 5 37 36
100 0 0.02 8 12 13 10 4 3 10 5 1 1 4 1
200 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
200 0.75 0.4 100 100 100 100 100 100 100 100 100 100 100 100
200 0.5 0.08 99 98 100 100 97 97 100 100 88 82 100 100
200 0.25 0.02 33 27 90 94 21 24 86 90 10 7 63 74
200 0 0.01 10 13 13 6 6 7 7 1 1 1 1 0
X 1 N ( 0 , 1 ) 20 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
X 2 N ( 0 , 1 ) 20 0.75 0.6 81 75 99 97 70 64 99 93 50 37 94 72
20 0.5 0.26 27 21 73 66 19 13 65 54 7 4 42 25
20 0.25 0.14 13 10 38 21 4 7 25 12 1 2 11 2
20 0 0.15 13 8 15 7 4 6 8 5 1 1 2 2
50 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
50 0.75 0.49 99 99 100 100 98 97 100 100 96 92 100 100
50 0.5 0.12 59 51 99 95 40 40 98 92 26 22 92 75
50 0.25 0.05 19 15 58 45 8 11 47 32 1 1 28 13
50 0 0.04 5 8 13 10 1 5 6 5 1 1 1 3
100 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
100 0.75 0.44 100 100 100 100 100 100 100 100 100 100 100 100
100 0.5 0.1 81 75 100 100 69 66 100 100 47 44 99 100
100 0.25 0.03 23 18 80 71 12 11 75 56 3 4 57 33
100 0 0.02 4 6 9 5 1 5 4 1 0 0 2 0
200 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
200 0.75 0.4 100 100 100 100 100 100 100 100 100 100 100 100
200 0.5 0.09 98 97 100 100 96 92 100 100 88 83 100 100
200 0.25 0.02 41 28 98 93 22 19 93 90 9 8 83 79
200 0 0.01 11 11 10 11 8 5 5 5 0 0 0 2
X 1 Cauchy ( 0 , 1 ) 20 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
X 2 Cauchy ( 0 , 1 ) 20 0.75 0.57 75 69 85 97 68 60 77 92 50 45 60 80
20 0.5 0.25 38 36 42 60 24 25 35 47 9 7 23 26
20 0.25 0.15 17 20 18 30 10 8 15 14 3 4 12 6
20 0 0.14 10 14 16 10 7 9 13 5 0 0 7 4
50 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
50 0.75 0.48 99 98 93 100 99 96 90 100 91 89 81 100
50 0.5 0.14 57 51 55 94 46 38 43 90 23 21 31 85
50 0.25 0.06 18 17 18 42 10 7 15 28 2 1 9 15
50 0 0.05 7 6 8 8 1 3 7 4 1 0 5 0
100 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
100 0.75 0.43 100 100 96 100 100 100 95 100 99 99 88 100
100 0.5 0.11 79 76 63 99 72 61 53 99 50 43 40 98
100 0.25 0.03 32 30 21 69 20 15 16 55 4 5 11 36
100 0 0.02 8 12 10 10 4 3 6 5 1 1 2 1
200 0.99 1 100 100 100 100 100 100 100 100 100 100 100 100
200 0.75 0.4 100 100 96 100 100 100 95 100 100 100 89 100
200 0.5 0.08 99 98 67 100 97 97 62 100 88 82 44 100
200 0.25 0.02 33 27 19 94 21 24 13 90 10 7 10 74
200 0 0.01 10 13 5 6 6 7 5 1 1 1 2 0

The powers of the proposed test implemented using the ART and APT circular uniformity tests, the WT and the ECT tests are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a linear-linear density function constructed from a Gaussian copula and three different marginals (exponential, Gaussian, and Cauchy). The Gaussian copula is defined with an equicorrelated correlation matrix with five different common correlation values of 0, 0.25, 0.5, 0.75, and 0.99. The case with common correlation equal to zero corresponds to the null independence model.

Table 4

Frank copula linear-linear power study

Marginals SS φ λ ˆ c 0 = 2 ( 1 c ˆ 0 2 ) α = 10 % α = 5 % α = 1 %
ART APT WT ECT ART APT WT ECT ART APT WT ECT
X 1 Exp ( 1 ) 20 50 1 100 100 100 100 100 100 100 100 100 100 100 100
X 2 Exp ( 2 ) 20 15 0.95 100 100 100 100 100 99 100 100 99 97 98 100
20 10 0.83 97 95 98 100 96 90 97 99 88 78 92 98
20 5 0.4 55 48 72 89 46 41 62 77 31 19 38 59
20 0 0.14 14 10 4 6 4 2 1 3 1 0 0 0
50 50 1 100 100 100 100 100 100 100 100 100 100 100 100
50 15 0.99 100 100 100 100 100 100 100 100 100 100 100 100
50 10 0.9 100 100 100 100 100 100 100 100 100 100 100 100
50 5 0.34 97 89 95 100 94 84 94 99 81 67 83 99
50 0 0.05 14 12 3 13 6 7 2 7 1 2 0 1
100 50 1 100 100 100 100 100 100 100 100 100 100 100 100
100 15 1 100 100 100 100 100 100 100 100 100 100 100 100
100 10 0.89 100 100 100 100 100 100 100 100 100 100 100 100
100 5 0.31 100 100 100 100 100 100 100 100 100 97 100 100
100 0 0.02 9 9 3 9 4 4 2 4 1 0 2 0
200 50 1 100 100 100 100 100 100 100 100 100 100 100 100
200 15 1 100 100 100 100 100 100 100 100 100 100 100 100
200 10 0.88 100 100 100 100 100 100 100 100 100 100 100 100
200 5 0.29 100 100 100 100 100 100 100 100 100 100 100 100
200 0 0.01 8 10 5 8 4 5 2 6 1 1 0 2
X 1 N ( 0 , 1 ) 20 50 1 100 100 100 100 100 100 100 100 100 100 100 100
X 2 N ( 0 , 1 ) 20 15 0.95 100 100 100 100 100 99 100 100 99 97 100 100
20 10 0.83 97 95 100 100 96 90 99 99 88 78 99 97
20 5 0.4 55 48 94 90 46 41 88 72 31 19 69 54
20 0 0.14 14 10 10 6 4 2 5 3 1 0 1 0
50 50 1 100 100 100 100 100 100 100 100 100 100 100 100
50 15 0.99 100 100 100 100 100 100 100 100 100 100 100 100
50 10 0.9 100 100 100 100 100 100 100 100 100 100 100 100
50 5 0.34 97 89 99 100 94 84 99 99 81 67 99 99
50 0 0.05 14 12 13 13 6 7 6 7 1 2 0 1
100 50 1 100 100 100 100 100 100 100 100 100 100 100 100
100 15 1 100 100 100 100 100 100 100 100 100 100 100 100
100 10 0.89 100 100 100 100 100 100 100 100 100 100 100 100
100 5 0.31 100 100 100 100 100 100 100 100 100 97 100 100
100 0 0.02 9 9 5 9 4 4 3 4 1 0 0 0
200 50 1 100 100 100 100 100 100 100 100 100 100 100 100
200 15 1 100 100 100 100 100 100 100 100 100 100 100 100
200 10 0.88 100 100 100 100 100 100 100 100 100 100 100 100
200 5 0.29 100 100 100 100 100 100 100 100 100 100 100 100
200 0 0.01 8 10 9 8 4 5 6 6 1 1 1 2
X 1 Cauchy ( 0 , 1 ) 20 50 1 100 100 95 100 100 100 92 100 100 100 91 100
X 2 Cauchy ( 0 , 1 ) 20 15 0.95 100 100 81 100 100 99 73 100 99 97 62 100
20 10 0.83 97 95 71 100 96 90 64 99 88 78 47 97
20 5 0.4 55 48 42 90 46 41 33 72 31 19 19 54
20 0 0.14 14 10 12 6 4 2 7 3 1 0 2 0
50 50 1 100 100 98 100 100 100 95 100 100 100 93 100
50 15 0.99 100 100 81 100 100 100 72 100 100 100 62 100
50 10 0.9 100 100 66 100 100 100 63 100 100 100 52 100
50 5 0.34 97 89 43 100 94 84 38 99 81 67 25 99
50 0 0.05 14 12 12 13 6 7 9 7 1 2 4 1
100 50 1 100 100 92 100 100 100 89 100 100 100 81 100
100 15 1 100 100 70 100 100 100 65 100 100 100 49 100
100 10 0.89 100 100 64 100 100 100 54 100 100 100 35 100
100 5 0.31 100 100 32 100 100 100 25 100 100 97 20 100
100 0 0.02 9 9 7 9 4 4 5 4 1 0 5 0
200 50 1 100 100 86 100 100 100 82 100 100 100 77 100
200 15 1 100 100 62 100 100 100 55 100 100 100 43 100
200 10 0.88 100 100 49 100 100 100 42 100 100 100 35 100
200 5 0.29 100 100 30 100 100 100 26 100 100 100 14 100
200 0 0.01 8 10 5 9 4 5 4 7 1 1 2 2

The powers of the proposed test implemented using the ART and APT circular uniformity tests, the WT and the ECT are compared when simulating 100 times samples of sizes 20, 50, 100, and 200 from a linear-linear density function constructed from a Frank copula and three different marginals (exponential, Gaussian, and Cauchy). The Frank copula is defined with five different values of the dependence parameter φ (0, 5, 10, 15, and 50). The limit case with φ = 0 corresponds to the null independence model.

#### 4.3.1 Bivariate Gaussian copula

The bivariate Gaussian (normal) copula correspond to a multivariate distribution, which is defined as follows:

(13) C ( u 1 , u 2 ) = Φ Γ ( Φ 1 ( u 1 ) , Φ 1 ( u 2 ) ) ,

where Φ Γ is the multivariate normal distribution with a zero mean vector and correlation matrix Γ and Φ ( ) is the univariate standard normal distribution function. By using an identity matrix as a correlation matrix, Γ = I , the independence copula, C ( u 1 , u 2 ) = u 1 u 2 , is obtained.

Table 3 compares the powers of the proposed independence test when using a ART and APT circular uniformity tests with respect to those of the WT and ECT independence tests when using simulated samples from a bivariate linear-linear distribution with a Gaussian copula and three different cases of marginal distributions following Herwatz and Maxand [23]: exponential, Gaussian, and Cauchy. For the Gaussian copula, we considered five different values of the correlation coefficient ρ (0, 0.25, 0.5, 0.75, and 0.99). When the correlation coefficient is equal to zero, it corresponds to the null independence hypothesis, and the reported powers correspond to the sizes of the tests expected to be similar to the significance levels α (10, 5, and 1%). In general terms, both WT and ECT have higher powers than the proposed ART and APT. However, for large sample sizes and high values of the correlation coefficient (0.75 and 0.99), the ART and APT have powers similar to those of the WT and ECT. As expected, for the exponential and Cauchy marginals, the power of the WT reduced when compared to the Gaussian marginal case for which it was designed; that is, the power of the WT deteriorates for marginals that are not Gaussian. In the case of Gaussian marginals, the WT for large sample sizes has high power, and in some cases, its power is larger than the ECT power. In terms of the sizes of all the tests, it appears that all tests have approximately the correct sizes, given that a total of 100 samples were used. The fourth column of Table 3 includes the averages of the values of the dependence measure λ ˆ c 0 which, as expected, increase as the value of the correlation coefficient ρ increases.

#### 4.3.2 Bivariate Frank copula

The Frank bivariate copula is defined as follows:

(14) C ( u , v ) = 1 φ ln 1 + ( e φ u 1 ) ( e φ v 1 ) e φ 1 ,

where u , v ( 0 , 1 ] . Parameter φ assumes values in the interval ( 0 , ) and in the limit φ = 0 , the Frank copula corresponds to the independence copula. In Table 4, the Gaussian copula of the previous study is replaced by the Frank copula using five different values of the dependence parameter φ (0, 5, 10, 15, and 50). For the case of independence obtained in the limit when φ = 0 , the powers are also obtained when the limit is considered as φ = 0 . In general, in Table 4, we obtained the same conclusions as those obtained in Table 3, further indicating that the characteristics of the proposed tests: ART and APT are more suitable for the circular-circular and circular-linear cases than to the linear-linear case.

## 5 Application to real circular-circular and circular-linear data

### 5.1 Test of bivariate independence

#### 5.1.1 Circular-linear real examples

Figure 2 depicts the scatterplots of the considered real examples. We applied the proposed independence test to the circular-linear data on wind direction (circular variable) and ozone concentration (linear variable) originally analyzed by Johnson and Wehrly [28], and later included them as dataset B.18 in Fisher [17]. A total of 19 measurements were taken at a weather station in Milwaukee at 6 o’clock in the morning every fourth day starting on April 18 and ending on June 29, 1975. The scatterplot of this data is included in the top left plot of Figure 2, which presents the values of the wind direction and ozone concentration, further indicating a possible positive association between the circular and linear variables and considering the periodicity of the circular random variable. By applying the Pycke and Rayleigh circular uniformity tests to the difference modulus 2 π of the angular probability transforms of the circular and linear variables, we obtained p-values of 0.0133 and 0.0077, respectively, thus rejecting the null hypothesis of independence (uniformity) at a 5% significance level for the wind direction and ozone concentration. The value of the dependence measure λ ˆ c 0 was calculated to be 0.4383. Fisher [17] reached the same conclusion by considering an expected sine-wave functional form for the conditional expected value of ozone concentration given the wind direction.

Figure 2

Scatterplots of the circular-linear (upper plots) and circular-circular (lower plots) real datasets. The upper left scatterplot shows the wind direction (relative to north) and ozone concentration (ppb) datapoints of the dataset of Johnson and Wehrly (1977). The upper right plot corresponds to the small blue periwinkles dataset on travelled distance (cm) and direction analyzed by Fisher (1993). The bottom left plot corresponds to the Johnson and Wehrly (1977) dataset on pairs of wind directions (relative to north) at 6:00 am and 12:00 noon in a weather monitoring station. At last, the bottom right includes the pairs of dihedral angles in segments alanine-alanine-alanine of proteins originally analyzed by Fernández-Durán (2007).

A second example analyzed by Fisher [17] is a dataset on the directions and distances travelled by 31 small blue periwinkles after undergoing transplantation from their normal place of living. The top-right plot depicted in Figure 2 includes the scatterplot for this dataset, further indicating a possible negative association between the direction and travelled distance. When applying the uniformity test to the sum modulus 2 π of the angular probability transforms of the direction and distance, a p -value of 0.0087 for the Pycke test and a p -value of 0.0096 for the Rayleigh test were obtained, which rejected independence in accordance with the results obtained by Fisher [17] when fitting a circular-linear regression model with von Mises errors with non-constant dispersion. The dependence measure λ ˆ c 0 was calculated to be 0.2668.

#### 5.1.2 Circular-circular real examples

The first example in the circular-circular test of independence corresponds to pairs of wind directions measured at a weather monitoring station at Milwaukee. The measurements were taken at 6:00 and 12:00 o’clock for 21 consecutive days and were originally included in Johnson and Wehrly [28]. The bottom-left plot depicted in Figure 2 includes a scatterplot of the pairs of wind directions, which indicates a possible positive association between the two angles. Fisher [17] listed this dataset as the B.21 dataset, and the main conclusion of Fisher [17] was that there exists a strong positive association between the wind directions when applying a hypothesis test based on a circular-circular correlation coefficient. When applying the proposed methodology to the difference modulus 2 π of the angular probability transforms, a p -value of 0.0148 for the Pycke test and a p-value of 0.0075 for the Rayleigh test were obtained, which rejected the null hypothesis of independence between the two angles at a 5% significance level. The dependence measure λ ˆ c 0 was calculated to be one.

The second example corresponds to 233 pairs ( ϕ , ψ ) of dihedral angles in segments alanine-alanine-alanine of proteins that were originally analyzed by Fernández-Durán [11] using bivariate NNTS models demonstrating the dependence between the two angles. The scatterplot of these two angles is included in the bottom-right plot of Figure 2, and the type of association between the two angles, whether it is positive or negative is not evident. When applying the independence test to the difference modulus 2 π of the angular probability transforms of the two angles, we obtained p -values of 0.0014 and 0.0064 for the Rayleigh and Pycke circular uniformity tests, respectively, thus clearly rejecting the null hypothesis of independence in favor of a positive association.

## 6 Conclusion

By using the result that the sum modulus 2 π of independent circular uniform random variables is circular and uniformly distributed, a general test of independence based on the angular integral probability transform was developed. We demonstrated its use, particularly when at least one of the variables is an angle, that is a circular random variable, further implying that testing for independence in the circular case could be equivalent to testing for circular uniformity. From the simulation study presented in this article, it is clear that the proposed test is particularly useful for bivariate cases of circular-circular and circular-linear pairs of random variables with more power than the Wilks and empirical copula independence tests. We reached this conclusion by simulating samples from NNTS densities, in which the degree of closeness to the circular uniform distribution was under control. Although the proposed independence test can be applied to the linear-linear case, its power is smaller than that of the commonly used independence tests converging to the power of the common independence tests when the sample size increases. The proposed independence test can be put in practice and it demonstrated superior performance when at least one circular random variable is among the random variables used to test for independence. In addition, a new measure of dependence was introduced based on the fitting of an NNTS density by considering M = 1 to the sum (difference) modulus 2 π of the APITs.

## Acknowledgments

We express our sincere gratitude to the Asociación Mexicana de Cultura, A.C. for their support.

1. Funding information: No funding was received to assist with the preparation of this manuscript.

2. Conflict of interest: The authors have no conflict of interest to declare that are relevant to the content of this article.

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