In many clinical studies, the Lin's concordance correlation coefficient (CCC) is a common tool to assess the agreement of a continuous response measured by two raters or methods. However, the need of measures of agreement may arise for more complex situations, such as when the responses are measured on more than one occasion by each rater or method. In this work, we propose a new CCC in the presence of repeated measurements, called the multivariate concordance correlation coefficient. We constructed the multivariate CCC based on a matrix that possesses the properties needed to characterize the level of agreement between two p x 1 vectors of random variables. For ease of interpretation, we transformed this matrix to a scalar whose value is scaled to range between -1 and 1 by using three distinct functions, namely trace, highest eigenvalue, and determinant. It can be shown that the multivariate CCC reduces to Lin's CCC when p = 1. For inference, we proposed an asymptotically unbiased estimator based on U-statistics and derived its asymptotic distribution for each form of the function. The proposed estimators are proven to be asymptotically normal and their performances are evaluated via simulation studies. To obtain a confidence interval or a test statistic, we considered a sample moment estimator of the asymptotic variance and the Z-transformation to improve the normal approximation and bound the confidence limits. The simulation studies confirmed that overall in terms of accuracy, precision, and the coverage probabilities, the estimator of the multivariate CCC based on the determinant function works relatively well in general cases even with small samples. However, for a skewed underlying distribution with moderate or weaker correlation between the two variables, the trace multivariate CCC is slightly more robust. Finally, We used real data from an Asthma Clinical Research Network(ACRN) study and the Penn State Young Women's Health Study for demonstration.
