Abstract: We investigate multivariate calibrations from a modern perspective with a focus on incorporating auxiliary variables and handling complex data dependencies with random effects. By introducing auxiliary variables, the roles of the variables in multivariate calibration problems are no longer restricted to being either response or explanatory, which offers much flexibility and adaptability to a broader range of practical problems. Our analysis reveals that a new shrinkage approach, that connects the conventional generalized least squares and the inverse regression approaches, offers much improved performance. To accommodate complex dependence in contemporary studies, we develop a computationally efficient expectation-maximization algorithm for solving multivariate calibration problems with random effects. The shrinkage approach shows promising performance in numerical simulations and an empirical study.

Key words and phrases: Inverse regression, linear mixed-effect models, multivariate calibration, multivariate response variables, shrinkage estimation.