## 1. Introduction

In many scientific studies, one of the research objectives is to assess agreement of observations made by two raters or methods. For example, in the area of medical diagnostic testing, the main research interest is to compare the results of a new technique with the gold standard practice. When the observations are measured on a continuous scale, the concordance correlation coefficient (CCC), introduced by Lin (1989), is one of the most popular measures of agreement. The CCC evaluates the agreement between two readings from the same sample by measuring how far each paired data point deviates from the 45 ° line through the origin, called the concordance line. The value of CCC ranges between −1 and 1 with the equality to 1 for perfect positive agreement, 0 for no agreement, and −1 for perfect negative agreement. Unlike the traditional approaches, for example the Pearson correlation coefficient, the paired *t*-test, and the least squares test, which sometimes fail to detect departure from the concordance line or falsely reject strong agreement, the CCC can fully assess the desired reproducibility characteristics.

However, in many fields of science, especially medical sciences, the need of measure of agreement between the two raters or methods often arises when the data are obtained at several occasions. For example, in a longitudinal asthma clinical trial, one of the research goals was to study the amount of agreement between plasma cortisol AUC (area under the curve) measured every hour and every two hours at several visits. In this situation, we need a repeated measure CCC that can quantify the overall agreement between two random vectors of repeated measurements.

For paired or unpaired repeated measurements study design, Chinchilli et al. (1996) developed a weighted CCC based on a random coefficient model that allows the within-subject variances to change across subjects. For each subject, the CCC was constructed as an average of q CCC's of the least squares random vectors, whose variance–covariance matrices were of dimension *q*×*q*. Then, the global CCC was defined as a weighted average of the coefficients using a weight function based on the amount of variation within each subject.

King, Chinchilli, and Carrasco (2007) proposed another version of the CCC in the presence of repeated measurements. They characterized the amount of agreement between two *p*× 1 random vectors, **X** and **Y**, by , where **D** is a *p*×*p* nonnegative definite matrix of weights among the different repeated measurements. Then, the repeated measure CCC was defined as

Carrasco, King, and Chinchilli (2009) developed a CCC for longitudinal repeated measurements through the appropriate specification of the intraclass correlation coefficient from a variance components linear mixed model. The authors showed that this CCC is equivalent to the repeated measure CCC proposed by King et al. (2007) when the weight matrix **D** is the identity matrix.

In this work, we introduce a new repeated measure CCC that not only can be proven to possess the properties needed to measure the amount of overall agreement between two *p*× 1 vectors of random variables but also has more intuitive appeal than the former methods. In Section 2, we first construct a matrix that characterizes the overall agreement between the two vectors. Then, to ease the problem of interpretation, we transform this matrix to a scalar based on a matrix norm and scale its value to range between −1 and 1. We call our repeated measure CCC the *matrix-based concordance correlation coefficient* (MCCC). To estimate the MCCC, we consider an estimator based on U-statistics. For inference, we derive the asymptotic distribution of the proposed estimator. To obtain a confidence interval or a test statistic, we consider a consistent estimator of the asymptotic variance and the *Z*-transformation to improve the normal approximation and bound the confidence limits. In Section 3, a Monte Carlo simulation is performed to assess the properties of the estimator of the MCCC based on finite samples. Finally, in Section 4, some real examples are used to demonstrate the application of the MCCC.