The responses of different genotypes to an environmental gradient are often nonlinear and nonparallel. Current tests for differential genotypic responses are based largely on linear regression models (stability analysis) or on evaluations of all quadruples for crossover interactions (COIs) from a two‐way genotype × environment (G × E) table if the environments are unquantified. The objective of this study was to develop a new statistical analysis for comparing nonlinear genotypic response curves over an environmental gradient. We first conducted an investigation to find the points where the two nonparallel curves intersected. If the intersection points lie within the attainable environmental range, the two nonparallel curves involve COI; if the points lie at the boundaries or outside the attainable range, the nonparallel curves do not involve COI. We then developed statistical tests for comparing a full and a reduced model describing the two nonparallel curves. The tests were used to analyze a wheat (Triticum aestivum L.) germination test (WGT) data under the reciprocal of a linear function and a barley (Hordeum vulgare L.) cultivar trial (BCT) data under Cauchy function. The WGT analysis shows that at least one pair of cultivars involve COI over the temperature gradient, which went undetected by a previous test based on all possible quadruples. The BCT analysis revealed that 56% of 780 possible pairs of 40 genotypes differ significantly from each other, providing more insights into the patterns of complex G × E interactions. Our analysis is therefore a viable alternative to the existing procedures.

中文翻译：

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测试非线性基因型×环境相互作用

不同基因型对环境梯度的响应通常是非线性且不平行的。当前对差异基因型反应的测试主要基于线性回归模型（稳定性分析），或者如果环境未经量化，则基于双向基因型×环境（G×E）表评估所有四倍的交叉相互作用（COI）。这项研究的目的是开发一种新的统计分析，用于比较环境梯度上的非线性基因型反应曲线。我们首先进行调查，找出两条不平行的曲线相交的点。如果相交点在可达到的环境范围内，则两条不平行的曲线涉及COI；如果这些点位于边界处或可达到范围之外，则非平行曲线不涉及COI。然后，我们开发了统计测试，用于比较描述两条非平行曲线的完整模型和简化模型。该测试用于分析小麦（线性函数的倒数下的普通小麦萌发测试（WGT ）数据，柯西函数下的大麦（Hordeum vulgare L.）品种试验（BCT）数据。WGT分析表明，在温度梯度上至少有一对品种涉及COI，以前的测试基于所有可能的四倍数未能检测到。BCT分析显示，在780个可能的40个基因型对中，有56％彼此显着不同，从而提供了对复杂G×E相互作用模式的更多见解。因此，我们的分析是对现有程序的可行替代方案。