The misuse of the nonlinear field normalization method: Nonlinear field normalization citation counts at the paper level should not be added or averaged

Nonlinear field normalization citation counts at the paper level obtained using nonlinear field normalization methods should not be added or averaged. Unfortunately, there are many cases adding or averaging the nonlinear normalized citation counts of individual papers that can be found in the academic literature, indicating that nonlinear field normalization methods have long been misused in academia. In this paper, we performed the following three research works. First, we analyzed why the nonlinear normalized citation counts of individual papers should not be added or averaged from the perspective of theoretical analysis in mathematics: we provided mathematical proofs for the crucial steps of the analysis. Second, we systematically classified the existing main field normalization methods into linear and nonlinear field normalization methods. Third, we used real citation data to explore the error effects caused by adding or averaging the nonlinear normalized citation counts on practical research evaluation results. The above three research works provide a theoretical basis for the proper use of field normalization methods in the future. Furthermore, because our mathematical proof is applicable to all nonlinear data in the entire real number domain, our research works are also meaningful for the whole field of data and information science.