A Modified Group Method of Data Handling (MGMDH) is a component of a synthetic method of constructing multivariate polynomial regression given by a redundant representation. The MGMDH is used to construct multivariate linear regression given by a redundant representation in the case when a decomposition method, which is also a component of the synthetic method, allowed to estimate with a given accuracy the values of unknown coefficients for nonlinear terms of a multivariate polynomial regression. As statistical studies have shown, the MGMDH efficiently finds the correct structure of a multivariate linear regression when the realizations of a random factor in the tests are an order of magnitude smaller than the modules of the corresponding values of the regression to be determined. Only in this case, the use of the regularity criterion in the MGMDH almost always allows finding the correct structure of a multivariate linear regression given by a redundant representation. In this paper, the MGMDH is adapted for the case when during the tests the modules of the random factor implementations and the values of the regression to be determined take values of the same order, which significantly increases the efficiency of using the MGMDH for constructing multivariate linear regressions given by redundant representations. It is obvious that the adapted MGMDH for multivariate linear regressions given by redundant representations presented in this work is easily transformed using the standardization operation for the general problem of constructing multivariate regression given by a redundant representation in the case when the unknown coefficients are linear in the regression structure.