混合模型这个统计怪兽是从什么时候开始，在生态学领域大行其道的呢？大概是2010年之后，随着Trends in Ecology and Evolution发表了这篇Generalized linear mixed models: a practical guide for ecology and evolution经典论文的发表，点燃了混合模型的星星之火(Bolker et al. 2009)。 随着nlme，尤其是lme4, glmmTMB, brms等混合模型强力工具的不断涌现，混合模型的应用更是呈爆发式增长，尤其是在生态学领域，更是几乎达到了无混合、不文章的恐怖地步。以生态学第一专业期刊Ecology letters为例，2023年第12期（也是目前能看到的最新一期），一共发表了12篇论文，其中1篇为View point, 1篇为纯理论研究，另外10篇用到了统计模型的论文中，有8篇都使用了混合模型。应用广度恐怖如斯，实在堪称当今第一应用统计模型。

“How should ecologists and evolutionary biologists analyze nonnormal data that involve random effects? Nonnormal data such as counts or proportions often defy classical statistical procedures. Generalized linear mixed models (GLMMs) provide a more flexible approach for analyzing nonnormal data when random effects are present. The explosion of research on GLMMs in the last decade has generated considerable uncertainty for practitioners in ecology and evolution. Despite the availability of accurate techniques for estimating GLMM parameters in simple cases, complex GLMMs are challenging to fit and statistical inference such as hypothesis testing remains difficult. We review the use (and misuse) of GLMMs in ecology and evolution, discuss estimation and inference and summarize ‘best-practice’ data analysis procedures for scientists facing this challenge”.

另外，值得一提的是，这篇论文的第一作者Benjamin M. Bolker（下图）并非等闲之辈，这位大师正是lme4这一混合模型圣经包的核心作者之一(Bates et al. 2015)。

图片来源：https://ms.mcmaster.ca/~bolker/

2015年Bolker在journal of statistical software发表的lme4手册性论文Fitting Linear Mixed-Effects Models Using lme4目前的引用量更是高达令人瞠目结舌71000次, 足见混合模型在科研360行中应用之火爆(Bates et al. 2015)。另据赖江山老师2019年针对生态领域6万篇论文的分析，lme4的使用频率排名高居第一，另一款混合模型拟合工具nlme包则排名第三(Lai et al. 2019)。可见无混合、不生态，并未言过其实。值得注意的是，在生物学、心理学乃至神经生物学领域，混合模型的大规模应用似乎才刚刚开始，以至于在这些领域的学者们，纷纷以“Beyond t test and ANOVA: applications of mixed-effects models for more rigorous statistical analysis in neuroscience research” (Yu et al. 2021), ”Catching Up on Multilevel Modeling” (Hoffman and Walters 2022), “embracing mixed modelling to address biological complexity” (Bertram et al. 2022)等略显夸张的标题呼吁混合模型的使用。 可见虽然生态学是否是一门硬科学至今仍为学界所诟病，但我们对统计模型的应用，似乎是大幅领先其他学科的（虽然生态学的“数据过烂”可能是重要原因）。

参考文献：

Bates, D., M. M£¿chler, B. Bolker, and S. Walker. 2015. Fitting Linear Mixed-Effects Models using lme4. Journal of Statistical Software 67:1-48.

Bertram, M. G., J. M. Martin, E. S. McCallum, L. A. Alton, J. A. Brand, B. W. Brooks, D. Cerveny, J. Fick, A. T. Ford, G. Hellström, M. Michelangeli, S. Nakagawa, G. Polverino, M. Saaristo, A. Sih, H. Tan, C. R. Tyler, B. B. M. Wong, and T. Brodin. 2022. Frontiers in quantifying wildlife behavioural responses to chemical pollution. Biological review 97:1346-1364.

Bolker, B. M., M. E. Brooks, C. J. Clark, S. W. Geange, J. R. Poulsen, M. H. H. Stevens, and J.-S. S. White. 2009. Generalized linear mixed models: a practical guide for ecology and evolution. Trends in Ecology & Evolution 24:127-135.

Hoffman, L., and R. W. Walters. 2022. Catching Up on Multilevel Modeling.  73:659-689.

Lai, J., C. J. Lortie, R. A. Muenchen, J. Yang, and K. Ma. 2019. Evaluating the popularity of R in ecology. Ecosphere 10:e02567.

Yu, Z., M. Guindani, S. F. Grieco, L. Chen, T. C. Holmes, and X. Xu. 2021. Beyond t test and ANOVA: applications of mixed-effects models for more rigorous statistical analysis in neuroscience research. Neuron.

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