The investigation of the structure of genotype-by-environment interaction is an important topic in multi-environment trials, in which a series of tests are undertaken across multiple environmental conditions. This study proposes a generalisation of joint regression analysis for situations when the response (e.g. yield) is non-linear across environments and can be expressed as a second (or higher) order polynomial or another non-linear function. We propose a selection technique based on the modification of two tests after determining the common form regression function for all genotypes: (i) a test for parallelism of regression curves; and (ii) a test of coincidence for those regressions. When the parallelism hypothesis is ruled out, subgroups of genotypes with parallel (or coincident) responses should be found. The Scheffe multiple comparison approach for regression coefficients in second-order polynomials allows for the classification of genotypes into two categories: one with upward-facing concavity (i.e. potential yield growth), and the other with downward-facing concavity (i.e. the yield approaches saturation). With an example of yield from a non-orthogonal series of experiments with winter rye, theoretical conclusions for genotype comparison and genotype selection are demonstrated (Secale cereale L.). To demonstrate that our meteorology is entirely relevant to incomplete data sets, we randomly erased 10% of that data, which are common in multi-environment trials. The hypothesis of parallelism of regression curves was rejected, which is natural in multi-environment trials with interaction between genotype and environment. The main difference in the two subgroups of genotypes where the responses are parallel is that one group had upward-facing concavity (i.e. potential yield growth) and the other had downward-facing concavity (i.e. the yield approaches saturation), which can help breeders in their genotype selection. The approach proposed in this paper is general and applicable to any series of experiments conducted in multi-environment trials or simply to the case of two-way classified data.