Common Error 5: Nonlinear Monotonic Transformations on X, M, and Y Without Rational Justification

To meet parametric assumptions (e.g., normality), researchers sometimes apply transformations (e.g., log, square root) to variables X, M, or Y. In our review, 13.50% of studies did so. Arbitrary transformations pose three main risks: they often lack theoretical justification and can distort true relationships (Busemeyer & Jones, 1983); they may yield unreliable MMR results; and they can inflate Type I error or obscure genuine interaction effects. A robust approach is needed: comprehensive diagnostic tests (e.g., Q-Q plots) should precede any transformation. When transformations are used, both the mathematical operation and its substantive rationale must be documented. For non-normal data, alternatives like bootstrapping (Russell & Dean, 2000), robust regression, or permutation-based methods are recommended.

常见问题五：缺乏合理依据地对变量进行非线性单调变换

为了满足多变量正态性、同方差性等参数检验假设，研究中常对变量（X, M, Y）进行诸如取对数、平方根、反正弦等非线性单调变换。我们的综述发现，有13.50%的调节研究进行了此类操作。然而，缺乏理论依据的随意变换会带来三重主要风险：首先，它可能扭曲变量间真实的底层关系（Busemeyer & Jones, 1983）；其次，特别是对结果变量（Y）的变换，可能导致不可靠的多元调节回归结果；最后，变换可能人为地夸大Ⅰ类错误率（误报效应），或掩盖真正的交互效应。

问题实质：

变量变换改变了数据的原始尺度和分布形态。若无坚实的理论支持，这种变换就像是在“修饰”数据以迎合统计假设，其结果可能是一个基于扭曲事实的“统计幻象”，而非反映真实的理论关系。

解决建议：

Ÿ  诊断先行，谨慎变换：在进行任何变换前，必须对参数假设进行全面的诊断检验（如Q-Q图、异方差性检验）。不能仅因分布非正态就机械变换。

Ÿ  提供明确的理论与数学依据：如果变换确有必要（例如，经济学中常基于理论对变量取对数），研究者必须在文中明确记录所采用的数学变换及其实质性的理论理由。

Ÿ  优先考虑稳健的替代方法：

• 对于估计ΔR²的置信区间，可采用Bootstrap方法，它不依赖于严格的分布假设（Russell & Dean, 2000）。

• 对于严重非正态的数据，可考虑使用稳健回归技术或基于置换的检验方法，这些非参数或半参数方法通常比传统的参数方法更可靠。

Reference

• Busemeyer, J. R., & Jones, L. E. (1983). Analysis of      multiplicative combination rules when the causal variables are measured      with error. Psychological Bulletin, 93(3), 549-562.

• Carte, T. A., & Russell, C. J. (2003). In pursuit of      moderation: Nine common errors and their solutions. MIS Quarterly,      27(3), 479-501.

• Dawson, J. F. (2014). Moderation in management research: What,      why, when, and how. Journal of Business and Psychology, 29(1),      1-19.

• Russell, C. J., & Dean, M. A. (2000). To log or not to log:      Bootstrap as an alternative to the parametric estimation of moderation      effects in the presence of skewed dependent variables. Organizational      Research Methods, 3(2), 166-185.

• Xu, Y., & Shiau, W. L. (2026). Moderation analysis in      business and management research: Common issues, solutions, and guidelines      for future research. International Journal of Information      Management, 86, 102995.