We propose a multilevel main effects matrix factor model (MMEFM) to extract meaningful row and column effects from groups of matrix data. As a generalization of a multilevel matrix factor model, we provide rigorous definitions of MMEFM and identifications, together with iterative algorithms for the estimators of all global and local components in the main effects and common component, including global/local main effects and core ranks, with rates of convergence spelt out. An important feature of the model is that all main effects can be nonstationary and still be consistently estimated at each time point. At the same time, our estimated global common factors are alignment independent, and hence are truly global, and can be presented and given interpretations more easily. We propose a Bootstrap procedure for detecting the existence and identifying the groups of data that are driven by the global common factors. The power of the test, together with the accuracy of various estimators, are demonstrated in extensive simulation settings. Estimation performance is also compared to other state-of-the-art methods for matrix time series by simulations and a set of U.S. equity data. All numerical works are done using the R package MMEFM, which is available at CRAN.

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