Canonical Concordance Correlation Analysis

Abstract

A multivariate technique named Canonical Concordance Correlation Analysis (CCCA) is introduced. In contrast to the classical Canonical Correlation Analysis (CCA) which is based on maximization of the Pearson’s correlation coefficient between the linear combinations of two sets of variables, the CCCA maximizes the Lin’s concordance correlation coefficient which accounts not just for the maximum correlation but also for the closeness of the aggregates’ mean values and the closeness of their variances. While the CCA employs the centered data with excluded means of the variables, the CCCA can be understood as a more comprehensive characteristic of similarity, or agreement between two data sets measured simultaneously by the distance of their mean values and the distance of their variances, together with the maximum possible correlation between the aggregates of the variables in the sets. The CCCA is expressed as a generalized eigenproblem which reduces to the regular CCA if the means of the aggregates are equal, but for the different means it yields a different from CCA solution. The properties and applications of this type of multivariate analysis are described. The CCCA approach can be useful for solving various applied statistical problems when closeness of the aggregated means and variances, together with the maximum canonical correlations are needed for a general agreement between two data sets.

1. Introduction

Multivariate statistical analysis for two sets of variables is usually performed by the Canonical Correlation Analysis (CCA), which is a tool for finding parameters of their aggregation into the total scores with maximum correlation. The CCA was proposed by H. Hotelling [1] for studying correlations between two aggregated variables, and has been extended in multiple works, including generalizations to nonlinear analysis and several data sets [2,3,4,5]. Its applications are known in various fields, from management and information systems to machine learning and biometrics [6,7,8,9,10,11,12,13]. The classical CCA is based on the maximization of the Pearson’s correlation between the aggregates of two sets of variables.

In place of the Pearson’s correlation, a related measure of agreement can be used—namely, Lin’s concordance correlation coefficient [14] which accounts not only for the correlation but also for the closeness of the variables’ means values. Various features of Lin’s coefficient are discussed in [15], including measuring agreement among raters, instruments, predictors, and intra-class correlations (ibid., with the references within). Results obtained by Lin’s coefficient define the best linear predictor with the restriction that its variance equals the variance of the dependent variable itself [15,16], that also corresponds to the so-called diagonal, or geometric mean regression [17].

The current paper introduces an alternative to the CCA multivariate technique of the Canonical Concordance Correlation Analysis (CCCA) which maximizes the Lin’s concordance correlation coefficient between the aggregates of two data sets. It simultaneously takes into account not just the correlation but also the closeness of the aggregates’ means values and the closeness of their variances. The regular CCA employs the centered data with excluded means of the variables, but the CCCA uses a more comprehensive characteristic of closeness, similarity, or agreement between data sets. It incorporates the maximum possible correlation between the aggregates of the variables in the data sets with optimization by the minimum distance between their mean values and the minimum distance of their variances.

The CCCA can be expressed as a generalized eigenproblem for the matrices of variances and covariance for data sets, in a special form of block-matrices and outer-product vectors of means for the combined vector of loadings for variables in both data sets. The CCCA reduces to the regular CCA if the means of the aggregates are equal, otherwise the CCCA solution differs from the corresponding CCA solution. Particularly, it is shown that in contrast to the regular CCA with only positive correlation values, the CCCA can produce also the negative concordance correlations with the optimal features, that extends the properties of the canonical correlations. The properties and applications of this type of multivariate analysis for the sample data are considered in this work.

The paper is organized as follows. Regular CCA and its modification to CCCA are described in Section 2 and Section 3, numerical example and comparisons are given in Section 4, and Section 5 summarizes the findings. Appendix A contains additional mathematical results.

2. Pair Correlation and Canonical Correlations

Consider two sampled data sets presented as matrices X and Y of the orders N × n and N × m, where N is the number of observations, and n and m are the numbers of variables x_j (j = 1, 2, …, n) and y_k (k = 1, 2, …, m), or columns of X and Y, respectively. The mean values (denoted by bars) of these variables $\overline{x_j}$ and $\overline{y_k}$ can be assembled as the elements of the column vectors $\overline{x}$ and $\overline{y}$of the n-th and m-th order, respectively. Let the matrices X and Y contain the centered data, so the centered second moments (denoted by C), or sample variances and covariances of these variables, can be presented in the matrix form as follows:

$$var\left(x\right)=C_{xx}=\frac{1}{N}{X}^{\prime }X, var\left(y\right)=C_{yy}=\frac{1}{N}{Y}^{\prime }Y, cov\left(x,y\right)=C_{xy}=\frac{1}{N}{X}^{\prime }Y,$$

(1)

where the prime indicates transposition. The C_xx and C_yy are square matrices of the n-th and m-th order, C_xy and its transposition C_yx (which equals $\frac{1}{N}{Y}^{\prime }X$) are rectangular matrices of the n by m, and of m by n order, respectively.

Let us briefly describe some main formulae of the canonical correlation analysis, or CCA. The scores of each data set aggregation variables are:

$$v=Xa, w=Yb,$$

(2)

where v and w are vector-columns of the N-th order, and a and b are the vector-columns of parameters of the order n and m, respectively. Pearson’s correlation between two variables (2) can be expressed via their covariance and variances as

$$\rho =\frac{cov\left(v,w\right)}{\sqrt{var\left(v\right)var\left(w\right)}}.$$

(3)

The canonical correlation of two data sets, or the pair correlation of their scores v and w, measures the connection between these sets. With definitions (1) and (2), the expression (3) equals:

$$\rho =\frac{\frac{1}{N}{v}^{\prime }w}{\sqrt{\left(\frac{1}{N}{v}^{\prime }v\right)\left(\frac{1}{N}{w}^{\prime }w\right)}}=\frac{a^{\prime }\left(\frac{1}{N}{X}^{\prime }Y\right)b}{\sqrt{\left(a^{\prime }(\frac{1}{N}{X}^{\prime }X\right)a)\left(b^{\prime }\left(\frac{1}{N}{Y}^{\prime }Y\right)b\right)}}=\frac{a^{\prime }{C}_{xy}b}{\sqrt{\left(a^{\prime }{C}_{xx}a\right)\left(b^{\prime }{C}_{yy}b\right)}}.$$

(4)

CCA calculates the vectors a and b via maximizing the correlation (4), which can be achieved using the conditional objective

$$\rho =a^{\prime }{C}_{xy}b-\frac{\lambda }{2}(a^{\prime }{C}_{xx}a-1)-\frac{\mu }{2}\left(b^{\prime }{C}_{yy}b-1\right),$$

(5)

where $\lambda$ and $\mu$ are Lagrange multipliers. Differentiating $\rho$ (5) with respect to the vectors a and b and equating to zero yields the system of equations:

$$C_{xy}b=\lambda C_{xx}a,C_{yx}a=\mu C_{yy}b.$$

(6)

Multiplying row vectors $a^{\prime }$ and $b^{\prime }$ by the first and second Equation (6), respectively, and using conditions (5) yields the equality of the Lagrange multipliers one to the other and to the Pearson’s coefficient of correlation,

$$\lambda =\mu =a^{\prime }{C}_{xy}b=\rho .$$

(7)

It means that two normalizing conditions used in (5) can be reduced to one combined condition for the sum of both quadratic forms of the variances from (4). Some other normalizing conditions are described in Appendix A.

Solving the second Equation (6) for the vector b and substituting it into the first equation, and also solving the first Equation (6) for the vector a and substituting it into the second one, leads to the expressions:

$$C_{xx}^{-1}{C}_{xy}{C}_{yy}^{-1}{C}_{yx}a={\lambda }^{2}a,C_{yy}^{-1}{C}_{yx}{C}_{xx}^{-1}{C}_{xy}b={\lambda }^{2}b.$$

(8)

The eigenproblems (8) present the classical CCA solution as the eigenvalues for the squared canonical correlations ${\lambda }^2={\rho }^2$ and the corresponding pairs of eigenvectors a and b. The eigenproblems (8) yield all canonical correlations and the corresponding pairs of the eigenvectors a and b for the first, second, third, etc., canonical variables, which number equals the minimum rank of the data matrices X and Y, that corresponds to the solutions with nonzero ${\lambda }^2$. It is sufficient to solve only one of the eigenproblems in (8) of the lesser order, then the dual vectors can be obtained from one of the linear systems (6). For example, if there are ten x variables and twenty y variables, we can solve the first eigenproblem (8) of the matrix of the 10-th order, then for each ${\lambda }_j$ and $a_j$ (j = 1, 2, …, 10) to solve the second linear system (6) for the dual vectors $b_j={\lambda }_j^{-1}{C}_{yy}^{-1}{C}_{yx}{a}_j$.

Equations (6) and (7) can be presented via one generalized eigenproblem for the extended block-matrices:

$$\left(\begin{array}{cc}{0}_{nn}& C_{xy}\\ C_{yx}& 0_{mm}\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\lambda \left(\begin{array}{cc}{C}_{xx}& 0_{nm}\\ 0_{mn}& C_{yy}\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right),$$

(9)

where 0_nn is the zero matrix of the nth order, and similar notations are used for the orders of the other zero matrices. In place of two eigenproblems (8), here is one generalized eigenproblem (9) for the square matrices of the n + m order. From the combined eigenvector in (9) we can extract two sub-vectors a and b which after normalizing would coincide with the eigenvectors obtained from the two separate problems (8).

It is important to note the following. Suppose the minimum rank of the matrices in (8) equals n, then the first eigenproblem in (8) yields n positive eigenvalues ${\lambda }^2$, while the second eigenproblem in (8), among its m eigenvalues, has the same n positive eigenvalues ${\lambda }^2$ plus m-n eigenvalues equal zero. The generalized eigenproblems (9), among its eigenvalues $\lambda$, has 2n values of opposite signs in pairs $\lambda =\pm \sqrt{{\lambda }^2}$ (they correspond to the singular values in the SVD, or the singular value decomposition), plus m-n zero eigenvalues. The eigenvectors corresponding to these paired eigenvalues in the eigenproblem (9) are the same by one part, a or b of the total vector, but of the opposite sign for the other part, b or a, respectively. It produces pairs of the canonical correlations (4) of the opposite values.

3. Concordance Correlation Coefficient and Canonical Concordance Correlation Analysis

Consider another measure of the relation between two variables—Lin’s concordance correlation coefficient ${\rho }_c$which can be expressed as follows:

$${\rho }_c=\frac{2\rho \sqrt{var\left(v\right)var\left(w\right)}}{var\left(v\right)+var\left(w\right)+{\left[E\left(v\right)-E\left(w\right)\right]}^2}=\frac{2cov\left(v,w\right)}{var\left(v\right)+var\left(w\right)+{\left[E\left(v\right)-E\left(w\right)\right]}^2},$$

(10)

where $\rho$ is Pearson’s correlation between two variables (2), and E(v) − E(w) is the difference of the expectations at the population level, or the sample means of these variables at the sample level. Using (10) in place of (3), we can repeat the derivation (4)–(9) for the proposed Canonical Concordance Correlation Analysis, or CCCA. The expression (4) for the measure of agreement (10) of the scores (2) is as follows:

$$\begin{array}{c}{\rho }_c=\frac{2\left(\frac{1}{N}{v}^{\prime }w\right)}{\left(\frac{1}{N}{v}^{\prime }v\right)+\left(\frac{1}{N}{w}^{\prime }w\right)+[a^{\prime }\overline{x}-b^{\prime }\overline{y}{]}^2}=\frac{2a^{\prime }\left(\frac{1}{N}{X}^{\prime }Y\right)b}{a^{\prime }\left(\frac{1}{N}{X}^{\prime }X\right)a+b^{\prime }\left(\frac{1}{N}{Y}^{\prime }Y\right)b+[a^{\prime }\overline{x}-b^{\prime }\overline{y}{]}^2}=\\ \frac{2a^{\prime }{C}_{xy}b}{a^{\prime }{C}_{xx}a+b^{\prime }{C}_{yy}b+[a^{\prime }\overline{x}-b^{\prime }\overline{y}{]}^2},\end{array}$$

(11)

where $\overline{x}$ and $\overline{y}$ are column vectors of the sample means, and matrices of sample second moments are the same as in relations (1).

CCCA calculates the vectors a and b via maximizing the concordance correlation coefficient (11), which is a Rayleigh quotient of two bilinear forms and can be optimized similarly to (5) by the conditional objective of the numerator subject to the unit value of the denominator:

$${\rho }_c=2a^{\prime }{C}_{xy}b-\lambda (a^{\prime }{C}_{xx}a+b^{\prime }{C}_{yy}b+{\left[a^{\prime }\overline{x}-b^{\prime }\overline{y}\right]}^2-1).$$

(12)

In contrast to the regular canonical correlation (4) containing the product of two variances and two normalizing conditions (although, due to (7) reduceable to one combined condition with the sum of variances), the additive items in the denominator (11) can be restricted by just one condition used in (12). It is possible because we can vary parameters of the aggregating vectors a and b for achieving this aim.

Partial derivatives of the objective (12) by the vectors a and b equal to zero define the system of equations for finding these vectors:

$$\{\begin{array}{c}\frac{\partial {\rho }_c}{\partial a^{\prime }}=2C_{xy}b-2\lambda (C_{xx}a+\left(a^{\prime }\overline{x}-b^{\prime }\overline{y}\right)\overline{x})=0\\ \frac{\partial {\rho }_c}{\partial b^{\prime }}=2C_{yx}a-2\lambda (C_{yy}b-\left(a^{\prime }\overline{x}-b^{\prime }\overline{y}\right)\overline{y})=0\end{array}.$$

(13)

This system can be rearranged as follows:

$$\{\begin{array}{c}{C}_{xy}b=\lambda (C_{xx}a+\left(\overline{x}{\overline{x}}^{\prime }\right)a-\left(\overline{x}{\overline{y}}^{\prime }\right)b)\\ C_{yx}a=\lambda (C_{yy}b+\left(\overline{y}{\overline{y}}^{\prime }\right)b-\left(\overline{y}{\overline{x}}^{\prime }\right)a)\end{array},$$

(14)

where $\left(\overline{x}{\overline{x}}^{\prime }\right)$is the outer-product of the x-means vector by itself, so it is the square matrix of the n-th order. The $\left(\overline{x}{\overline{y}}^{\prime }\right)$ is the outer-product of the x-means by y-means vectors, that it is the rectangular matrix of the n by m order, and similarly the other outer-products in (14) are constructed.

Solving one equation from (14) for one of the vectors and substituting it into the other equation generates quadratic in $\lambda$ eigenproblems difficult for practical solving. However, it is possible to present the system (14) as one generalized eigenproblem for the extended block-matrices:

$$\left(\begin{array}{cc}{0}_{nn}& C_{xy}\\ C_{yx}& 0_{mm}\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right)=\lambda \left(\begin{array}{cc}{C}_{xx}+\overline{x}{\overline{x}}^{\prime }& -\overline{x}{\overline{y}}^{\prime }\\ -\overline{y}{\overline{x}}^{\prime }& C_{yy}+\overline{y}{\overline{y}}^{\prime }\end{array}\right)\left(\begin{array}{c}a\\ b\end{array}\right),$$

(15)

Similar to the CCA solution (9), the CCCA solution can be given for the one generalized eigenproblem (15) for the square block-matrices of the n + m order. Separating the obtained combined eigenvectors in (15) to two sub-vectors a and b and normalizing each of them yields the eigenvectors for the Canonical Concordance Correlation Analysis.

The matrix at the left-hand side is the same in both problems (9) and (15). If all variables have zero mean value then the right-hand side matrix (15) reduces to the right-hand side matrix in (9), so the CCCA reduces to the classical CCA solution. Similar to the discussed above feature of the generalized eigenproblems (9), the eigenproblem (15) has n positive and n negative eigenvalues and the rest m-n values equal zero. Due to the structure of the corresponding eigenvectors, the values of concordance canonical coefficients (11) would appear in pairs of the opposite signs, although, in contrast to the model (9), not equal by the absolute value.

Multiplying the row vectors (a′, b′) of the obtained eigenvectors from the left-hand side on the relation (15) and taking into account the normalizing condition in (12), yields the equality

$$2a^{\prime }{C}_{xy}b=\lambda (a^{\prime }{C}_{xx}a+b^{\prime }{C}_{yy}b+{\left[a^{\prime }\overline{x}-b^{\prime }\overline{y}\right]}^2)=\lambda ={\rho }_c.$$

(16)

Therefore, the eigenvalues $\lambda$ in (15) define the spectrum of the optimal concordance correlation coefficients ${\rho }_c$ in the CCCA approach.

The generalized matrix problem can be reduced to the regular eigenproblem if to solve first the eigenproblem of the matrix from the right-hand side (15), then to find the square root of this matrix, and using it the generalized problem can be transformed to the regular eigenproblem of a symmetric matrix The eigenproblem solution of a matrix with the added outer-product of a vector, as in (15), is related to the eigenproblem for the matrix itself [18,19,20]. Most modern software packages solve an eigenproblem of a non-symmetric matrix, so a simple way to reduce the generalized problem (15) to a regular eigenproblem is to invert the matrix from the right-hand side and multiply it by the matrix from the left-hand side (15), that is presented in the analytical form in Appendix A.

In contrast to the Lin’s pairwise concordance correlation coefficient, which is not a scale invariant measure because it depends on the units of the variables, the CCCA is a scale invariant measure because parameters of the vectors a and b transform all the variables units to the same comparable units of the combined variables (2).

Let us consider the difference between the pair correlation and the Lin’s concordance correlation coefficient ${\rho }_c$ (10):

$$\rho -{\rho }_c=\rho \left(1-\frac{2\sqrt{var\left(v\right)var\left(w\right)}}{var\left(v\right)+var\left(w\right)+{\left[E\left(v\right)-E\left(w\right)\right]}^2}\right)=\rho \left(\frac{{\left(\sqrt{var\left(v\right)}-\sqrt{var\left(w\right)}\right)}^2+{\left[E\left(v\right)-E\left(w\right)\right]}^2}{var\left(v\right)+var\left(w\right)+{\left[E\left(v\right)-E\left(w\right)\right]}^2}\right).$$

(17)

The expression in parentheses at the right-hand side (17) is always non-negative. Then, for $\rho \ge 0$ the left-hand side is $\rho -{\rho }_c\ge 0$, so $\rho \ge {\rho }_c$, thus, the pair correlation is bigger than the concordance coefficient. In the opposite case, for $\rho <0$, the $\rho -{\rho }_c<0$, so $\rho <{\rho }_c.$ Therefore, the Pearson pair correlation is always bigger than the Lin’s concordance coefficient by the absolute value:

$$\left|\rho \right|\ge \left|{\rho }_c\right|.$$

(18)

This relation holds for the canonical correlations as well. Only for the situation of equal means and equal variances, the concordance correlation (10) can reach the pair correlation (2), ${\rho }_c=\rho$.

4. Numerical Example

For a numerical illustration, let us consider a dataset LifeCycleSavings by 50 countries used in [8] and available in the R statistical package for an example on the canonical correlation analysis performed by the function cancor. In this demographical financial problem, the first matrix X contains two variables: x₁—percentage of population under 15 years old (pop15); and x₂—percentage of population over 75 years old (pop 75). The second matrix Y consists of three variables: y₁—the saving ratio (sr) presented by the aggregate personal saving divided by the disposable income; y₂—the real per-capita disposable income (dpi); and y₃—the percentage rate of change in per-capita disposable income (ddpi). These characteristics had been averaged over the decade to remove the business cycle or other short-term fluctuations as described in [21].

Table 1. The dataset LifeCycleSavings, the means and standard deviations.

Characteristics	x1	x2	y1	y2	y3
mean	35.09	2.29	9.67	1106.76	3.76
std	9.15	1.29	4.48	990.87	2.87

shows the mean values and standard deviation (std) for these data.

The matrices of variances and covariance (1) for this data are:

$$C_{xx}=\left(\begin{array}{cc}82.079& -10.517\\ -10.517& 1.633\end{array}\right), C_{yy}=\left(\begin{array}{ccc}19.673& 958.717& 3.841\\ 958.717& 962,184.7& -360.849\\ 3.841& -360.849& 8.071\end{array}\right),$$

$$C_{xy}=\left(\begin{array}{ccc}-18.305& 6720.091& -1.231\\ 1.794& 986.429& 0.0919\end{array}\right).$$

With these matrices the CCA (9) solution was obtained. The same matrices and the mean values from Table 1 yield the CCCA (15) solution. Each combined vector (a, b) is normalized so that the total of its squared elements equals one. The rank of the matrices equals the minimum size r = min (n, m) = min (2, 3) = 2, so there are two main canonical pairs of vectors for CCA and for CCCA.

Table 2. CCA solution.

Variables	CCA by the Eigenproblem (9)
	(a1, b1)	(a2, b2)	(a3, b3)	(a4, b4)
x1	0.1808	−0.1366	−0.1808	0.1366
x2	−0.9655	−0.9815	0.9655	0.9815
y1	−0.1681	0.1259	−0.1681	0.1259
y2	−0.0026	−0.0003	−0.0026	−0.0003
y3	−0.0828	−0.0463	−0.0828	−0.0463
Canonical correlation coefficient, $\rho$	0.8248	0.3653	−0.8248	−0.3653
Added concordance correlation coefficient, ${\rho }_c$	0.1358	0.0034	−0.8014	−0.0051

presents the CCA results obtained by the combined eigenproblem (9).

For the CCA solution (9), there are two positive canonical correlations, ${\rho }_1=0.8248$ and ${\rho }_2=0.3653$, and two negative ones of the same absolute value, ${\rho }_3=-0.8248$ and ${\rho }_4=-0.3653$, shown in the first two and the last two columns of Table 2, respectively. The first two solutions coincide (up to normalization) with the solutions which can be obtained by the classical CCA eigenproblems (8). We can see by Table 2 that the CCA vectors a₃ = −a₁, and a₄ = −a₂, that explains the opposite signs of the CCA values of correlations (3) ${\rho }_3=-{\rho }_1$ and ${\rho }_4=-{\rho }_2$ of the aggregated scores (2) with one vector a of an opposite sign.

With each paired vectors a and b, we can additionally estimate the corresponding values of the concordance correlation (11)—those are shown in the last row of Table 2: ${\rho }_{c1}=0.1358$ and ${\rho }_{c2}=0.0034$, and also ${\rho }_{c3}=-0.8014$ and ${\rho }_{c4}=-0.0051$. These values correspond to the inequality (18). Three of these ${\rho }_c$are close to zero, but one of them$, {\rho }_{c3}$, is big by the absolute value. Thus, for the generalized solution (9), it is possible to find such a concordance correlation solution which is close to the classical CCA (i.e., −0.8014 vs. −0.8248). This specific solution with the negative correlation identifies the vectors a₃ and b₃ with which we reach the best agreement between two data sets measured by the Lin’s concordance correlation, so with the high pair correlation and close mean values of the aggregate variables.

Table 3. CCCA solution.

Variables	CCCA by the Eigenproblem (15)
	(a1, b1)	(a2, b2)	(a3, b3)	(a4, b4)
x1	0.0084	−0.0505	−0.2131	−0.0969
x2	−0.9980	−0.3061	0.9546	−0.8014
y1	−0.0419	0.3663	−0.1891	−0.5870
y2	−0.0013	−0.0007	−0.0028	0.0013
y3	−0.0456	−0.8773	−0.0875	−0.0616
Added correlation coefficient, $\rho$	0.8082	0.1767	−0.8247	−0.3237
Canonical concordance correlation coefficient, ${\rho }_c$	0.8080	0.0167	−0.8247	−0.0944

is organized similar to Table 2 and presents the CCCA results obtained by the combined eigenproblem (15).

Table 3 contains four pairs of the dual vectors a and b in its columns, with two positive and two negative concordance correlations ${\rho }_c$, all different by the absolute value, as shown in the last row. For each pair of the vectors a and b, the additional regular pair correlations $\rho$ (4) of the aggregate variables were calculated—they are presented in the second from the bottom row. Again, the inequality (18) is satisfied for each solution. The first solution (a₁, b₁) yields a high canonical concordance correlation ${\rho }_{c1}=0.8080$ which is very close to the regular pair correlation of the scores ${\rho }_1=0.8082$. The third solution is even better by the equal absolute value of the correlations (actually, with more precision the values are ${\rho }_{c3}=-0.824679$ and ${\rho }_3=-0.824681$, so they differ only by 0.000002). Therefore, the solution (a_3, b₃) with the negative relation of the scores identifies the vectors for the best agreement between two data sets measured by the Lin’s concordance correlation and by the regular correlation, with the close mean values of the aggregate variables, so the good agreement of the data in two sets.

Comparison between the CCA and CCCA vectors in Table 2 and Table 3 shows the following. The square root of sum of squares (SRSS) for difference of solutions given in the columns of these table equals, respectively, 0.2193, 1.1009, 0.0403, and 1.9343, so the best similarity occurs for the third dual vectors (a_3, b₃). The pair correlation of the vector solutions in each column of Table 2 and Table 3 are, respectively, 0.9693, 0.2618, 0.9997, and −0.8204, so the best correspondence by the absolute value between the CCA and CCCA solutions is reached for the third pair of vectors. They can be used both as the best CCA solution and the best CCCA solution. It is important to note that this optimal solution corresponds to the negative canonical correlation in Table 2 which can be obtained via the combined eigenproblem (9), while the classical CCA eigenproblems (8) produce only positive eigenvalues. Concerning the eigenvalues, the CCA in Table 2 yields the maximum possible canonical correlations, but the additionally estimated concordance correlations are very low, with exception of the third solution. The eigenvalues of the CCCA in Table 3 are all higher than the concordance coefficients in the previous table, and the related to them estimations of the correlations are high, especially for the first and third solutions.

5. Summary

The paper introduced a multivariate statistical method of the Canonical Concordance Correlation Analysis, or CCCA, and compares it with a related Canonical Correlation Analysis, or CCA. The classical CCA is based on the maximization of the Pearson’s correlation between the aggregates of two sets of variables. In place of the Pearson correlation, a related measure of the Lin’s concordance correlation coefficient which accounts both for the correlation and the closeness of the variables’ mean values was adopted. Both the CCA and CCCA methods are presented in a unified framework of the generalized eigenproblems for the combined vectors of loading for two sets of variables. The CCCA eigenproblem can be reduced to the CCA eigenproblem if the means of the aggregates become equal.

The properties and applications of this type of multivariate analysis for the sample data are considered. It is shown that the combined generalized eigenproblem of CCA method is preferable in finding all positive and negative eigenvalues defining the canonical correlations of both signs. The closeness of solutions produced by the CCA and CCCA can serve as indication on the optimal vectors of loading which yield the high correlations of the aggregate scores together with the close mean values of the aggregates, so the best agreement between two data sets. Applying CCA, we get the best possible canonical correlations but not necessarily a high level of agreement measured by the concordance correlations. However, applying the CCCA, we reach the maximum possible agreement given by the canonical concordance correlations, together with the high level of the correlations among the aggregated variables of two data sets.

The proposed new type of modern multivariate analysis can serve for various practical research projects operating with two data sets, particularly, for data fusion [22] and for other problems [15] which require finding a maximum agreement, similarity, or closeness between variables. Future studies on canonical concordance correlation analysis can extend this method to the robust solutions, evaluations for three and more data sets, for information presented in multidimensional arrays or matrices with three and more entries, and for more adequate operating on complex data of the modern science and applications.