1. Statistical approaches for testing hypotheses of heterogeneity in fitness functions are needed to accommodate studies of phenotypic selection with repeated sampling across study units, populations or years. In this study, we tested directly for among‐population variation in complex fitness functions and demonstrate a new approach for locating the region of the trait distribution where variation in fitness and traits is greatest.
2. We modelled heterogeneity in fitness functions among populations by treating regression coefficients of fitness on traits as random variates. We applied random regression using two model specifications, (i) spline‐based curve and (ii) stepwise, to a 2‐year study of selection among 16 populations of the gall wasp, Belonocnema treatae. Log‐likelihood ratio tests of variance components and 10‐fold cross‐validation were used to assess the evidence that selection varied among populations.
3. Ten‐fold cross‐validation prediction error sums of squares (PSS) indicated that spline‐based fitness functions were population specific and that the strength of evidence for heterogeneity in selection differed between years. Hypothesis testing of variance components from both models was consistent with the PSS results. Both the stepwise model and the local prediction error estimates of spline‐based fitness functions identified the region(s) of the phenotype distribution harbouring the greatest heterogeneity among populations.
4. The adopted framework advances our understanding of phenotypic selection in natural populations by extending the analysis of spline‐based fitness functions to testing for heterogeneity among study units and isolating the regions of the phenotypic distribution where this variation is most pronounced.

中文翻译：

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使用随机回归对适应度函数之间的异质性进行建模。

1. 需要用统计方法来检验适应度函数异质性的假设，以适应表型选择的研究，并在研究单位、人群或年份之间重复采样。在这项研究中，我们直接测试了复杂适应度函数中的人群间变异，并展示了一种定位适应度和性状变异最大的性状分布区域的新方法。
2. 我们通过将性状的适应度回归系数视为随机变量来对人群适应度函数的异质性进行建模。我们使用两种模型规范（（i）基于样条的曲线和（ii）逐步）将随机回归应用于一项为期 2 年的瘿蜂Belonocnema treatae种群选择研究。方差分量的对数似然比检验和 10 倍交叉验证用于评估人群之间选择不同的证据。
3. 十倍交叉验证预测误差平方和（PSS）表明，基于样条的适应度函数是特定于人群的，并且选择异质性的证据强度因年份而异。两个模型的方差分量的假设检验与 PSS 结果一致。逐步模型和基于样条的适应度函数的局部预测误差估计都确定了人群中具有最大异质性的表型分布区域。
4. 采用的框架通过将基于样条的适应度函数的分析扩展到测试研究单位之间的异质性并隔离这种变异最明显的表型分布区域，增进了我们对自然群体表型选择的理解。